a+ls mathematics curriculum planning manual · pre-algebra ia 43 1 8 pre-algebra ib 37 1 8 algebra...

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Mathematics

Curriculum Planning Manual

3LIT3321

Table of Contents

i

Mathematics Curriculum Planning Manual Teachers’ Guides and Scope & Sequences

Mathematics Overview ................................................ 1 Mathematics I–VII ..................................................... 5

Mathematics I ....................................................... 10 Mathematics II ..................................................... 17 Mathematics III .................................................... 23 Mathematics IV ..................................................... 30 Mathematics V ...................................................... 36 Mathematics VI ..................................................... 42 Mathematics VII .................................................... 47

Math Foundations IA ................................................ 52 Math Foundations IB ................................................ 60 Math Foundations IIA ............................................... 71 Math Foundations IIB ............................................... 81 Mathematics for Grade Levels 8-12 ............................. 91 Pre-Algebra IA ......................................................... 95 Pre-Algebra IB ....................................................... 108 Algebra IA ............................................................ 121 Algebra IB ............................................................ 138 Algebra I: A Function Approach Part 1 ....................... 153 Algebra I: A Function Approach Part 2 ....................... 166 Algebra II: Part 1 and 2 .......................................... 178

Algebra II: Part 1 ................................................ 179 Algebra II: Part 2 ................................................ 185

Geometry IA ......................................................... 189 Geometry IB ......................................................... 205 Trigonometry ........................................................ 221

Technical Requirements

For detailed workstation specifications, please visit the American Education Corporation’s web site:

www.amered.com/awl_requirements_wba.php

Additionally, some courses require Adobe® Acrobat Reader®, Adobe Flash®, and/or Adobe Shockwave® plug-ins for your browsers. These are available for free from http://www.adobe.com. The required software version numbers are listed on our website (see above).

For those using our Web-based A+LS™ (WBA+) product, the initial WBA+ screen offers links to download the necessary Adobe Acrobat and Flash files.

For detailed instructions for configuring your browsers to work with A+LS (such as Active X), please see Troubleshooting in the following document on our website:

A Teachers Guide to Web-based A+LS

To access Encyclopaedia Britannica® and other Internet based resources found in some courses, a connection to the Internet is required. To view multimedia from these sites a fast connection is recommended.

http://www.amered.com/docs/A+LS/ALS_35_Web_based_ALS_Teachers.pdf

Table of Contents

ii

Calculus I and II .................................................... 225 Limits ................................................................ 226 Definite Integrals ................................................ 229

iii

Copyright © 2013 K12 Inc. All rights reserved. K12 is a registered trademark of K12 Inc. The K12 logo and other marks referenced herein are trademarks of K12 Inc. and its subsidiaries, and other marks are owned by third parties.

Adobe, Acrobat Reader, and Flash are registered trademarks of the Adobe Systems Incorporated.

Encyclopædia Britannica is a registered trademark of Encyclopædia Britannica, Inc.

MetaMetrics, Lexile, and Quantile are registered trademarks of MetaMetrics, Inc.

A+nyWhere Learning Systems Teacher’s Guide

1

Mathematics Overview Grade Levels 1-12 A+LS™ Mathematics are comprehensive, completely integrated courses for grades 1–12. The Mathematics courses develop strategies to solve real-world problems and are designed to move the student beyond the level of basic knowledge into critical thinking and learning activities.

• Mathematics is a collection of one- and two- semester

long courses.

• All lessons in the math titles contain a study guide, a practice test, and a mastery test. Many lessons have an essay or other constructed response.

• Some Mathematics lessons are certified by MetaMetrics® with Lexile® scores.

• Some courses are enriched by National Library of

Virtual Manipulatives (NLVM) applets or Encyclopædia Britannica Online School Edition (EB) workspaces that contain learning materials. Learning materials may contain articles, games, images, maps, and/or videos.

A+nyWhere Learning System Mathematics Teacher’s Guide Grade Levels 1–12

2

• The content in these titles is designed to meet and exceed the requirements of the National Council of Teachers in

Mathematics. The standards for school mathematics describe a comprehensive set of goals for mathematics instruction.

• Students will understand meanings of operations and how they relate to one another.

• Students will understand numbers, ways of representing numbers, relationships among numbers, and number systems.

• Students will compute fluently and make reasonable estimates. In addition, students will build new mathematical knowledge through problem solving.

Third-Party Content in A+LS Lessons

The National Library of Virtual Manipulatives is a library of uniquely interactive, web-based virtual manipulatives or concept tutorials for mathematics instruction supported by the National Science Foundation.

Each applet provides interactive math lessons and activities.

The launch icon for NLVM objects, , is located on the study guide pages of the A+LS lesson.


3

The Encyclopædia Britannica® Online School Edition has teacher resources and student learning materials. The materials include a wide range of interactive lessons, research projects, animations, and worksheets that support many A+LS lessons.

Each workspace may contain an article, diagram, study guide, video, or interactive media.

The launch icon for EB objects, , is located at the top of the A+LS screen in the study guide section.


4

The Mathematics courses each contain a variety of lessons and differ in length, grade level, and available features. Listed at right are the courses found within the entire curriculum planning manual.

Course Name

Number of

Lessons

Length of Course in Semesters

Grade Levels

Lexile Measure NLVM

Mathematics I 49 2 1 500L Yes

Mathematics II 42 2 2 570L Yes

Mathematics III 50 2 3 660L Yes

Mathematics IV 36 2 4 690L Yes

Mathematics V 37 2 5 730L Yes

Mathematics VI 30 2 6 860L Yes

Mathematics VII 30 2 7 790L Yes

Pre-Algebra IA 43 1 8

Pre-Algebra IB 37 1 8

Algebra IA 63 1 9–10

Algebra IB 54 1 9–10

Algebra I: A Functional Approach Part 1 37 1 9–10 Yes

Algebra I: A Functional Approach Part 2 37 1 9–10 Yes

Algebra II: Part 1 44 1 10–12 Yes

Algebra II: Part 2 33 1 10–12 Yes

Geometry IA 45 1 9–10

Geometry IB 46 1 9–10

Trigonometry 29 1 11–12

Calculus I 30 1 12

Calculus II 26 1 12


5

Mathematics I–VII Grade Levels 1–7

A+LS Mathematics I–VII introduces students to the following concepts, skills, and more:

• numbers and counting

• addition and subtraction story problems

• multiplication

• writing decimals to the tenths and hundredths positions

• writing, adding and subtracting decimals

• uses of line and circle graphs

• circles, perimeter, circumference, pyramids, and probability

• capacity and weight units, temperature, lines and rays

• positive and negative integers

• calculating perimeter, area, volume, using a number line

• graphing ordered pairs on a coordinate axis

• positive and negative rational numbers

• translating word phrases into algebraic expressions with integers, slope, binomials, determinants, and Cramer’s rule

A+nyWhere Learning System Mathematics Teacher’s Guide Grade Levels 1-7

6

A+LS Mathematics are comprehensive, completely integrated courses for grades 1–7. The Mathematics courses develop strategies to solve real-world problems and are designed to move the student beyond the level of basic knowledge into critical thinking and learning activities.

• Mathematics I-VII is presented as a collection of year-

long courses.

• All lessons in the seven titles contain a study guide, a practice test, and mastery test. Many lessons have an essay or other constructed response.

• These Mathematics lessons are certified by MetaMetrics with Lexile scores.

• Some courses are enriched by National Library of

Virtual Manipulatives (NLVM) applets or Encyclopædia Britannica Online School Edition (EB) workspaces that contain learning materials. Learning materials may include articles, games, images, maps, and/or videos.


7











8

The Encyclopædia Britannica Online School Edition has teacher resources and student learning materials. The materials include a wide range of interactive lessons, research projects, animations, and worksheets that support many A+LS lessons.




9

The Mathematics courses I–VII each contains a variety of lessons and differs in length, grade level, and available features. Additional features for these courses are found in the table below.

Course Name

Number of Lessons


Grade Levels

Lexile Measure NLVM

Mathematics I 49 2 1 Yes Yes

Mathematics II 42 2 2 Yes Yes

Mathematics III 50 2 3 Yes Yes

Mathematics IV 36 2 4 Yes Yes

Mathematics V 37 2 5 Yes Yes

Mathematics VI 30 2 6 Yes Yes

Mathematics VII 30 2 7 Yes Yes For information on Pre-Algebra and Algebra mid-level courses see the curriculum planning manual for Mathematics Grade Levels 9–12.

A+nyWhere Learning System Teacher’s Guide

10

Mathematics I Grade Level 1

Mathematics I is an introduction to mathematical concepts. The lessons cover numbers and counting; ordering numbers; ordinal numbers; addition readiness; vertical addition; subtraction readiness; number sense; vertical subtraction; fact families; word problems; addition sentences; subtraction sentences; identifying the operation needed to solve a problem; reading and creating graphs; identifying and counting coins; measuring length, weight, and temperature; telling time; three-dimensional figures; symmetry; and fractions. The Mathematics I course includes other topics such as:

• adding with number lines

• mathematical patterns

• counting by tens

• number sets

A+nyWhere Learning System Mathematics I Scope & Sequence Grade Level 1

Quantile Lesson Title Lesson Content Measure Activities

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1 Using Numbers to 5 Identify the numbers 1 through 5. Match numbers with the amounts they stand for. 50Q

2 Using Numbers to 10

Identify the numbers 6 through 10. Match numbers with the amounts they stand for. EM Essay: Counting

3 Number Sets Learn to compare numbers and amounts by using the symbols for greater than, less than, and equals. 50Q Study: EB Learning

Material

4 Numbers and

Counting Patterns to 100

Use the 100's chart and counting patterns to count to 100. Find missing numbers. EM

Study: EB Learning Material

NLVM Activity Essay: Counting

5 Ordering Numbers Learn the concepts of greater than and less than. Practice putting numbers in correct counting order. 50Q


Essay: Counting

6 Ordinal Numbers to Fifth

Learn about the ordinal numbers that match the numbers 1 through 5. EM

Study: Drag and Drop Activity

Essay: Logic

7 Ordinal Numbers to Tenth

Review the ordinal numbers that match the numbers 1 through 5. Learn about the ordinal numbers that match the numbers 6 through 10.

EM Study: EB Learning Material

8 More or Less Compare two sets of objects by counting and using more or less. 50Q Study: NLVM Activity

9 Addition Readiness Learn how to add by putting numbers together to make a new, larger number. EM


Essay: Writing Equations



12

10 Adding With Number Lines Learn how to use number lines to add. EM Study: NLVM Activity


11 Vertical Addition Learn how to add numbers vertically. EM Essay: Writing Equations

12 Subtraction Readiness

Learn how to subtract by taking a part away from a whole to create an new, smaller number. EM Study: EB Learning

Material

13 Number Sense Sort, group, and count objects. Learn to compare and order numbers. 50Q Essay: Writing Equations

14 Subtracting With Number Lines Learn how to use number lines to subtract. EM Study: NLVM Activity


15 Vertical Subtraction Learn how to subtract numbers vertically. EM Essay: Writing Equations

16 Fact Families Recognize fact families and find missing numbers in addition and subtraction problems. EM

Study: EB Learning Material NLVM Activity


17 Counting On Use the “counting on” strategy to add. EM Study: EB Learning

Material NLVM Activity

18 Addition and Subtraction

Patterns Find out how to use patterns to add and subtract. 40Q Essay: Problem Solving

19 Addition Sentences Write addition sentences using given information and answer questions about how many. EM Study: NLVM Activity



13

20 Adding Three Numbers Learn how to work addition problems using three numbers. EM


EB Learning Material NLVM Activities (2)

21 Continuing Patterns Discover how to continue a pattern by figuring out what comes next. EM Study: NLVM Activity

22 Subtraction Sentences Create subtraction sentences from existing data. EM Study: NLVM Activity

Essay: Problem Solving

23 Missing Numbers Learn how to find missing numbers in addition and subtraction problems. EM Essay: Writing Equations

24 Word Problems Learn the steps needed to solve word problems. EM Study: NLVM Activity Essay: Problem Solving

25 Counting By Tens Learn how to count in groups of tens. 90Q


Essay: Counting

26 Meaning of Addition and Subtraction

Recognize clue words for addition and subtraction word problems. Identify the operation needed to solve problems. EM Essay: Writing Equations

27 Understanding Numbers

Learn how to identify the value of each digit in a number. Recognize and write numbers up to 99 in standard and expanded form. 10Q


NLVM Activity Essay: Writing Equations



14

28 Before, After, and Between Numbers

Identify and write numbers that come before, after, or in between given numbers. 50Q


Essay: Counting

29 Skip Counting Learn how to skip count by twos, fives, and tens. EM Study: EB Learning

Material Essay: Counting

30 Reading Graphs Use graphs to represent data and solve problems. 390Q Study: NLVM Activity Essay: Writing Equations

31 Creating Graphs Learn how to represent data by completing graphs. 70Q Study: NLVM Activity Essay: Written Essay

32 Introduction to Money Identify and count pennies and nickels. 70Q Essay: Problem Solving

33 Counting Coins Count various groups of coins. Practice spending money by deciding if there is enough money to but certain items. 300Q


NLVM Activity Essay: Problem Solving

34 Extra Information Identify unneeded information in word problems. 610Q Essay: Problem Solving

35 Spending Money Students see objects with price tags and groups of coins. Students decide if they have enough money to purchase the items. 350Q Study: Drag and Drop

Activity

36 Is There Enough? Learn how to spend money by counting coins to see if there is enough money to buy some things. 70Q Essay: Estimating

37 Estimating and Measuring Length

Measure the length of objects using a ruler. Estimate length using nonstandard measuring tools. 360Q



15

38 Types of Measurements

Learn about different types of measurement and measurement tools. Practice estimating and comparing the length and weight of different objects. Learn about quantitative change.

360Q Study: Drag and Drop

Activity Essay: Listing

39 Estimating and

Measuring Temperature

Estimate and measure temperature. Recognize different types of thermometers. 360Q


Essay: Estimation

40 Properties of Operations

Identify the Commutative and Identity properties of addition and subtraction. Review clue words for addition and subtraction word problems.

EM Essay: Writing Equations

41 Sums and Differences

Write addition and subtraction sentences from groups of objects. Observe how the number sentences you write create a fact family. EM


42 Telling Time Learn how to tell time to the hour and half hour on different types of clocks. 90Q


Essay: Listing

43 Telling Time Review Review telling time by the hour and half hour. 90Q

44 Combinations Make combinations of objects, numbers, and letters. 50Q Essay: Logic

45 Calendars Study the parts of a calendar and learn how to read it. EM Essay: Written Response

46 Shapes Recognize and name two- and three-dimensional shapes. Identify the shapes of different objects. 160Q

Study: Drag and Drop Activities (2)

Essay: Listing



16

47 Symmetry Identify objects that have two parts that match. Predict and investigate what happens when you take apart and put together different shapes.

EM

48 Fractions Understand the concept of dividing objects into equal parts called fractions. Recognize the fractions 1/2, 1/3, and 1/4. 100Q


EB Learning Material

NLVM Activity

49 Symbols Recognize symbols and solve code problems using numbers and letters. EM Essay: Logic


17

Mathematics II Grade Level 2

Mathematics II strengthens mathematical skills in the following areas: numbers and counting, odds and evens, money and money strategy, graphing, addition and subtraction, using a calculator, measurement, telling time, solving story problems, fractions, and estimating. It also introduces students to measuring perimeter, congruent and symmetrical objects, probability, problem-solving strategies, logic, ordered pairs, multiplication, and division. The lessons also review reading time on digital or analog clocks. The Mathematics II course includes other topics such as:

• advanced graphing

• ordinal numbers

• calculator usage

• measuring weight

• measuring temperature

• introduction to multiplication

• introduction to division

• coin values

A+nyWhere Learning System Mathematics II Scope & Sequence Grade Level 2


18


Learn about the many different ways in which numbers are used. Identify place value in the tens place and ones place. 30Q



2 Place Value Identify place value to the hundreds place using base ten blocks and write whole numbers in expanded and standard form. 30Q


Essay: Counting

3 Numbers and Counting

Use number lines to skip count by twos, fives, and tens. Identify even and odd numbers. 110Q



4 Odds and Evens Identify even and odd numbers. Complete even and odd number patterns. 110Q


Essay: Counting

5 Money Value Learn the value of pennies, nickels, dimes, quarters, and half dollars. Use different combinations of coins to make equal values. 70Q


NLVM Activity Essay: Problem Solving

6 Counting Money Learn how to make change by counting up. Compare values of items to determine which costs more and which costs less. 70Q Study: NLVM Activity


7 Introduction to Word Problems Learn the 4 important steps to use when solving a word problem. 250Q



19

8 Number Sense Put numbers in correct counting order. Determine which number comes before, between, or after certain numbers. Use the symbols >, <, and = to compare numbers.

50Q



Essay: Counting

9 Money Strategy Count money to determine if there is enough to purchase certain items. Decide which items to buy given a certain amount to spend. 70Q Study: NLVM Activity

Essay: Written Essay

10 Beginning Word Problems

Read word problems carefully to find needed information. Practice solving word problems. 50Q

11 Ordinal Numbers Learn about the ordinal numbers first through tenth. Use ordinal numbers to identify the position or order of objects. EM Study: EB Learning

Material

12 Beginning Graphing Answer questions about graphs, complete graphs with information, and use objects and graphs to represent addition and subtraction. 70Q Essay: Writing Equations

13 Advanced Graphing Define and give examples of bar graphs, tables, lists, and pictures. Answer questions, solve problems, and understand symbolic notions.

EM Study: NLVM Activity Essay: Listing

14 Addition and Subtraction Sentences

Learn the parts of addition and subtraction sentences. Write addition and subtraction number sentences. 300Q



15 Adding and

Subtracting With Two Digits

Add and subtract numbers with two digits. Learn how to line up place values when writing problems and find out how to solve addition problems with regrouping.

90Q Study: EB Learning

Material Essay: Writing Equations



20

16 Fact Families Learn how to identify fact families. Discover how to use related addition and subtraction facts to solve problems that have missing terms.

EM Study: EB Learning

Material Essay: Writing Equations

17 Intermediate Word Problems Determine the questions in word problems, and solve the problems. 50Q

18 Addition and Subtraction Clues

Look for clue words in math problems and choose the correct operation needed to solve problems. EM Study: EB Learning

Material

19 Using a Calculator Use a calculator to skip count, build numbers, and solve problems.

20 Adding Three Numbers Solve vertical and horizontal addition problems using three addends. EM



21 Extra Information Identify unneeded information in word problems. Solve word problems that contain extra information. 610Q

22 Choosing Operations

Learn how to solve word problems by first reading the information and then determining which operation to use. EM

23 Introduction to Measurement

Learn about types of measurement, tools used for measurement, and units of measurement. Identify quantitative change over time. 360Q



24 Measuring Length

Estimate and measure length using standard and nonstandard measuring instruments. Measure objects in centimeters and inches and choose appropriate units of measurement for different objects. Interpret and apply direction and distance in navigating space.

240Q Essay: Listing



21

25 Measuring Weight and Temperature

Learn about the measurements of weight and temperature. Compare the weight of different objects and identify the units of measure used for weight. Learn how to measure temperature by reading a thermometer.

360Q Study: Drag and Drop Activities (2)

26 Measuring Capacity Estimate and measure capacity. Compare the capacity of different containers and discover how the size of the objects in a container can affect the container’s capacity.

480Q Study: Drag and Drop Activity

27 Estimating Learn how to estimate sums and differences and then check the answers for reasonableness. 420Q

28 Telling Time Learn how to tell time on digital and analog clocks. Find out how to add and subtract time. 210Q Study: NLVM Activities (2)


29 Expressing and Estimating Time

Understand the terms half past, a quarter to, noon, and midnight. Learn how the write time in words. Estimate the length of time needed for certain activities.

10Q Study: NLVM Activity

30 Time Sequence Review the order of the days of the week and the months of the year. Learn how to use timelines and calendars to obtain information.

10Q Essay: Problem Solving

31 Shapes and Perimeter

Identify, describe, and compare two- and three-dimensional figures. Learn how to measure the perimeter of a 2-dimensional shape. 400Q


Essay: Listing

32 Congruent and Symmetrical

Objects

Understand the difference between congruent and similar figures. Identify symmetrical figures and lines of symmetry. EM Study: NLVM Activities (2)



22

33 Beginning Probability

Find out what the term probability means. Learn how to gather data and write probability statements. 70Q Study: NLVM Activity

Essay: Graphing

34 Using Probability to Predict

Use graphs to gather data and solve problems. Write probability statements and predict outcomes. 70Q

35 Missing Patterns Learn how to look for patterns in addition and subtraction problems and find missing numbers to complete patterns. 40Q Study: EB Learning

Material

36 Patterns and Ordered Pairs

Extend patterns and find what comes next. Learn how to locate a particular point using ordered pairs. 480Q Study: NLVM Activity

Essay: Logic

37 Sorting Sort and group objects according to similarities and differences. EM Study: NLVM Activity Essay: Logic

38 Strategy Learn how to use different strategies to solve the same problem. 70Q

39 Logic Use logical reasoning to solve problems. Essay: Written Essay

40 Identifying Fractions Learn what a fraction is. Identify fractions of various sizes. 190Q



41 Writing Fractions Identify and write fractional parts of a whole. 190Q Study: Drag and Drop

Activity NLVM Activity

42 Introduction to

Multiplication and Division

Use multiplication and division to share and group equally. 30Q Study: EB Learning Material


23

Mathematics III Grade Level 3

Mathematics III covers the following topics: addition and subtraction with regrouping; counting bills and coins; using a number line; using mental math; measuring length with standard and nonstandard measurements; using bar graphs; using a calculator; finding mean, median, mode, and range; estimating and measuring capacity, time, and weight; reading temperatures in Celsius and Fahrenheit; multiplying three numbers; measuring area; dividing by tens and hundreds; adding and subtracting fractions; solving problems using pictographs; decimals; probability; plane figures; ordered pairs; identifying faces, edges, and corners; and using logical reasoning. The Mathematics III course includes other topics such as:

• nonstandard measurement

• patterns and calculations

• conclusions and predictions

• multi-step word problems

• lines, rays, and segments

A+nyWhere Learning System Mathematics III Scope & Sequence Grade Level 3


24

1 Addition and

Subtraction with Regrouping

Learn about the concept of regrouping. Find out when regrouping must occur and how to add and subtract with regrouping. 250Q


Identify place value as well as odd and even numbers. Skip count by 2, 3, 4, 5, and 10. Use ordinal numbers to show order. 360Q Study: EB Learning Material

NLVM Activities (2)

3 Counting Money Identify and count money. Learn how to round money to the nearest dollar, ten dollars, and one hundred dollars. 600Q Study: NLVM Activity

4 Adding and Subtracting Money Learn how to add and subtract money amounts. 600Q


NLVM Activity

5 Ordering Numbers Discover how place value is used when comparing, ordering, and rounding numbers. Use place value when writing Roman numerals. 410Q


EB Learning Material Essay: Writing Equations

6 Fact Families Identify fact families and determine missing numbers. 250Q Study: EB Learning Material

7 Using Mental Math Use mental math to add and subtract whole numbers and to regroup. 250Q Study: EB Learning Material

8 Choosing the Operation

Learn how to determine if you should use the operation of addition or subtraction in a word problem. 300Q

9 Adding Numbers Horizontally

Discover how to solve horizontal addition problems by grouping numbers with parentheses. 250Q Essay: Problem Solving

10 Extra Information Identify unneeded information in word problems. 610Q Essay: Logic



25

11 Standard and Nonstandard

Measurements

Measure length using nonstandard and standard measurements. Estimate length and apply appropriate units of length to measurement.

240Q Study: EB Learning Material Essay: Logic

12 Using Graphs and Charts

Gather information and apply the information to bar graphs. Solve problems using bar graphs, compare ways to organize and display data.

480Q Study: NLVM Activity Essay: Graphing

13 Addition With Regrouping

Review how to regroup numbers when adding. Practice adding 2 and 3 digit numbers with regrouping. 250Q Study: NLVM Activity


14 Understanding Word Problems

Learn the five steps needed to solve problems which involve mathematical operations. 480Q

15 Perimeter Measure the perimeter of given shapes and estimate the perimeter of irregular shapes. Learn how perimeter is affected when shapes change.

240Q Study: NLVM Activity Essay: Logic

16 Subtraction With Regrouping

Review how to regroup numbers when subtracting. Practice subtracting two and three digit numbers with regrouping. 250Q Study: NLVM Activity


17 Using a Calculator Use calculators to repeat and extend patterns. Follow multiple steps of instructions on the calculator. Essay: Problem Solving

18 Solving Word Problems Review the five steps to solving word problems. 250Q

19 Deciding When to Regroup

Solve subtraction word problems that require regrouping. Find out how to regroup twice in the same problem and solve problems with zeros.

250Q Study: NLVM Activities (2)



26

20 Mean, Median, Mode, and Range

Discover how to find the mean, median, mode, and range in a group of numbers. 850Q Essay: Problem Solving

21 Capacity, Time, and Weight

Understand how to estimate and measure capacity, time, and weight. 450Q


NLVM Activity

22 Measuring Temperature

Learn about the Fahrenheit and Celsius temperature scales. Estimate temperatures. 450Q

23 Finding Needed Facts Identify missing information in word problems. 250Q Essay: Problem Solving

24 Introduction to Multiplication

Convert addition sentences to multiplication sentences, and learn how to create models of multiplication. Understand the commutative property of multiplication.

280Q Study: EB Learning Material NLVM Activities (2)

25 Multiplying by 2, 3, 4, and 5

Use skip counting and number lines to multiply by 2, 3, 4, and 5. Study multiplication tables for 2, 3, 4, and 5. 180Q Study: NLVM Activity

26 Multiplying by 6, 7, 8, 9, 10, and 100

Identify patterns in multiplication tables. Study multiplication tables for 6, 7, 8, and 9. Learn how to multiply by tens and hundreds. 330Q Study: EB Learning Material

27 Multiplying Three Numbers

Find out how multiplication is like addition. Discover the associative and distributive properties of multiplication. Learn how to multiply three factors, and practice solving problems that have missing factors.

220Q

28 Area Learn the concept of area and discover how to calculate the area of rectangles and squares. Estimate the area of irregular shapes and find out how area is affected when a shape changes.




27

29 Introduction to Division

Learn about the concept of division. Find out how to divide numbers and how to write division sentences. 350Q Study: EB Learning Material

NLVM Activity

30 Division: The Opposite of

Multiplication

Discover how multiplication and division are related. Learn how to divide by tens and hundreds. 350Q Essay: Writing Equations

31 Dividing by 2, 3, 4, and 5

Divide using 2, 3, 4, and 5. Check division by multiplying, and identify fact families. 350Q

32 Multi-step Word Problems

Learn clue words for solving words. Practice solving word problems that have multiple steps. 390Q

33 Dividing by 6, 7, 8, and 9

Divide using 6, 7, 8, and 9. Make comparisons and identify patterns. Complete division problems with remainders. 350Q Study: EB Learning Material

NLVM Activity

34 Long Division Learn how to divide three and four digit numbers by a one digit number. 450Q

35 Conclusions and Predictions

Learn about taking surveys, comparing sets of data, drawing conclusions, and making predictions. Find out how to display the information obtained from a survey in a pictograph.

390Q

36 Expressing Numbers

Determine whether addition, subtraction, and multiplication problems will result in even or odd numbers. Compare numbers in expanded and standard forms.

250Q Study: NLVM Activity Essay: Writing Equations

37 Data Collection Learn how to create and gather information from charts, maps, and graphs. Determine how data collection affects problem solving. 480Q



28

38 Introduction to Fractions

Learn how to divide shapes into equal parts to form fractions. Identify various fractions made from whole objects. 190Q



39 Parts of a Set Learn how to write different types of fractions made by dividing whole sets into separate equal parts. Identify the numerator and denominator in fractions.

190Q Study: NLVM Activities (2)

40 Equivalent Fractions Find out how to compare fractions by creating common denominators. Recognize equivalent fractions. 630Q Study: EB Learning Material

41 Adding and Subtracting Fractions

Add and subtract fractions with like denominators. 670Q Study: EB Learning Material

42 Mixed Numbers Learn how to identify, create, and compare mixed numbers. 610Q Study: EB Learning Material

43 Decimals Learn how to write decimals and how to relate fractions to decimals. 710Q Study: EB Learning Material

44 Adding and Subtracting Decimals

Discover the skills necessary to add and subtract decimals. 580Q Study: EB Learning Material Study: NLVM Activity

45 Probability Learn how to gather data and write probability statements. Make predictions based on probability. 440Q Essay: Logic

46 Lines, Rays, and Segments

Identify lines, rays, segments, and angles, and learn how they relate to each other. 680Q Study: Drag and Drop

Activity



29

47 Plane Figures Identify different types of plane figures. Learn about congruent shapes and lines of symmetry. Make interesting designs using shapes.


Activity NLVM Activity

48 Solids Recognize three-dimensional shapes. Identify faces, edges, and corners on solid figures. Learn how to find volume. 530Q Study: NLVM Activity

49 Ordered Pairs Learn how to use ordered pairs to find a location on a map. 370Q Study: NLVM Activity

50 Logical Reasoning Use logic to solve problems and check for reasonability of answers.


30

Mathematics IV Grade Level 4

Mathematics IV contains lessons covering these concepts: the five-step process for problem solving; grouping addends; addition and subtraction; odd and even numbers; multiplication and division problems using money; using a calendar; temperature; writing decimals to the tenths and hundredths positions; line segments and angles; comparing maps and grids; comparing graph types; and formulating information into a story problem. The Mathematics IV course includes other topics such as:

• using logic to solve problems

• identifying missing information

• using pictographs

• mental math computations

A+nyWhere Learning System Mathematics IV Scope & Sequence Grade Level 4


31

1 Story Problems 1 The five-step process for problem solving.


Essay: Writing Word Problems

2 Number Sense 1 Order of addends in relation to the sum. Adding zero. Students rewrite addition facts in inverse order. Grouping addends. 280Q Study: EB Learning

Material

3 Number Sense 2 Rounding numbers and money amounts to nearest 10, 100, and 1000. 410Q Study: EB Learning

Material

4 Addition & Subtraction 1

Given an addition fact, students write a related subtraction fact. Given a subtraction fact, students write a related addition fact. 300Q



5 Addition & Subtraction 2

Addition and subtraction fact families. Students complete fact families using missing elements. Students read story problems and choose the correct operation to solve the problem.

300Q Essay: EB Learning

Material Written Response

6 Patterns Odd and even numbers. Students recognize the relationships between numbers to determine patterns. 550Q Study: EB Learning

Material

7 Ordinal Numbers Cardinal and ordinal numbers. 90Q Study: EB Learning Material

8 Money 1

Review counting bills and coins. Addition and subtraction problems using money. Students make change for dollar amounts up to $20 and coin change. Students are given a dollar amount to spend and choose which items they could purchase.





32

9 Money 2 Multiplication and division problems using money. Locating the dollar sign and decimal points when multiplying and dividing money. 700Q Study: EB Learning

Material

10 Measurement 1 Estimating, determining, and measuring time to the nearest minute. Finding elapsed time. Using the appropriate time units to measure time. Writing time using AM or PM. Adding and subtracting time.

450Q Study: EB Learning Material

11 Measurement 2 Using a calendar. Looking for patterns on a calendar. Learning the number of days in each month. EM Study: EB Learning

Material

12 Measurement 3 Estimating and measuring capacity, mass, temperature, and distance. Using customary units of measurement. Standard and Metric measurements.


Material Essay: Listing

13 Multiplication 1 Multiplication by two and three digit numbers. Multiplication problems with missing factors. 530Q

14 Multiplication 2 Commutative, associative, zero, one, and distributive multiplication properties. Mental math techniques. Estimating. 560Q Essay: Written Response

15 Multiplication 3 Estimation and mental math. Decimal multiplication. Zero in decimals. 700Q

16 Division 1 Dividing whole numbers by one and two digit numbers. Estimating quotients. 690Q

17 Division 2 Dividing decimals. Estimation and mental math. 830Q

18 Fractions 1 Review of fractions. Reading, writing, and renaming mixed numbers. Comparing and ordering fractions and mixed numbers. 630Q

Study: EB Learning Material NLVM Activities (2)



33

19 Fractions 2 Addition and subtraction of fractions and mixed numbers. Equivalent fractions. Students read story problems, choose strategies, and solve problems.


Material NLVM Activities (2)

20 Decimals 1 Students read and write decimals to tenths and hundredths positions. Relating decimals and fractions. Relating decimals and money. Writing mixed numbers as decimals.



21 Decimals 2 Adding and subtracting decimals of the same place value. The use of zero in decimals. Writing decimals as fractions. 470Q


22 Geometry 1 Identifying faces, edges, and corners of solid and plane figures. Comparing solid figures. Comparing plane figures. Differences in solid and plane figures.

680Q


Essay: Listing

23 Geometry 2 Line segments and angles. Identifying right angles. Using greater than and less than with right angles. Definition and examples of intersecting lines, parallel lines, and perpendicular lines.


Material Essay: Written Response

24 Geometry 3 Definition and examples of congruence and symmetry. Open and closed figures. Identifying parts of angles. Identifying and continuing patterns using geometrical shapes.


25 Geometry 4 Measuring perimeter and area of polygons. Finding the volume of solid figures. Difference in area and volume. 480Q Study: EB Learning

Material

26 Graphs 1 Students locate and name ordered pairs on a coordinate grid. Comparing maps and grids. Students locate points by writing ordered pairs.





34

27 Graphs 2 Students gather data from visual aid and complete bar graph. Students make predictions from bar graph information. Students complete line graph.


Material Essay: Graphing

28 Graphs 3

Using tables and pictographs. Gathering information with a pictograph. Using pictograph information to create line and bar graphs. Comparing graph types. Students decide what type graph to use.



29 Probability

Definition and examples of probability. Identifying possible outcomes. The probability equation. Students predict if outcome of given situation is probable, certain, or impossible. Using graphs to chart probability and predict outcomes.



30 Using Mental Math Examples of different methods of computation using mental math. 530Q

31 Choosing the Operation Students decide which operation to use and solve problems. 450Q Essay: Problem Solving

32 Extra Information Students read problems, identify unneeded information, and solve problems. 610Q Essay: Problem Solving

33 Story Problems 2 Students are given information to formulate into problems. Students check for reasonableness of answers. Study: EB Learning

Material

34 Finding Needed Facts Students read information and identify missing information. 610Q

35 Story Problems 3 Students read information, form a plan, choose the correct operation, and solve problems having more than two steps.





35

36 Logical Reasoning Students use logic to solve problems. Students check for reasonability of answers.




36

Mathematics V Grade Level 5

Mathematics V covers these concepts: exponents; standard, expanded, and word forms of numbers; writing decimals; adding and subtracting decimals; the properties of addition; the five-step thinking plan; multiplying two- and three-digit numbers; surveys; uses of line and circle graphs; Venn diagrams; least common multiples; units of length; elapsed time; lines and angles; circles; perimeter; circumference; pyramids; and probability. The Mathematics V course includes other topics such as:

• organizing data

• rounding whole numbers

• polygons

• vertices, edges, and prisms

A+nyWhere Learning System Mathematics V Scope & Sequence Grade Level 5


37

1 Whole Numbers 1 Place value through millions. Exponents. Standard, expanded, and word forms of numbers. 600Q



2 Whole Numbers 2 Comparing and ordering whole numbers. Rounding through millions. 600Q Study: EB Learning Material

3 Decimals 1 Place value through thousandths. Zero in decimals. Writing decimals. 620Q


Essay: Listing

4 Decimals 2 Comparing and ordering decimals. Patterns in decimals. Rounding decimals. 600Q


Essay: Written Response

5 Properties of Addition Commutative, associative, and zero properties of addition. 280Q Essay: Problem Solving

6 Addition & Subtraction 1 Inverse operations. Estimating sums. 420Q



7 Addition & Subtraction 2 Addition and subtraction across zeros. Regrouping with zeros. 250Q


8 Addition & Subtraction 3 Addition and subtraction of decimals. Clustering. 580Q Study: NLVM Activity



38

9 Problem Solving 1 The five step thinking plan. Using inverse operations. Study: EB Learning


10 Organizing Data 1 Using tables to organize, analyze, and solve problems. 390Q Study: EB Learning Material

11 Problem Solving 2 Multi-step problems. Using the five step thinking plan. Study: EB Learning


12 Multiplication 1 Commutative, associative, zero, one, and distributive multiplication properties. Mental math techniques. Estimating. 560Q



13 Multiplication 2 Multiplication by two and three digit numbers. Multiplication problems with missing factors. 330Q


14 Multiplication 3 Estimation. Mental math. Decimal multiplication. Zero in decimals. 700Q Study: EB Learning Material

15 Division 1 Dividing whole numbers by one and two digit numbers. Estimating quotients. 690Q


16 Division 2 Short division. Dividing larger numbers. Using zeros in division. Identifying patterns. Checking division. 690Q Study: EB Learning

Material

17 Division 3 Dividing decimals. Estimation and mental math. 700Q Study: EB Learning Material



39

18 Organizing Data 2 Surveys. Sample groups. Range, mode, mean, median. 850Q Study: EB Learning

Material Essay: Problem Solving

19 Bar Graphs & Pictographs

Parts of a bar graph. Selecting a scale. Displaying data on bar graph. Displaying data on pictograph. Reading data. Analyzing data. Making inferences.


20 Line Graphs Uses of line graphs. Parts of line graphs. Displaying data. 470Q Study: EB Learning


21 Circle Graphs Circle graphs. Venn diagrams. Analyzing data. 600Q Study: EB Learning


22 Ordered Pairs Ordered pairs. Graphing on a grid. 370Q Study: EB Learning


23 Fractions 1 Fraction parts. Equivalent fractions. Prime numbers. Composite numbers. Greatest common factors. Simplest form. Least common multiples.



24 Fractions 2 Comparing fractions. Reducing fractions. Improper fractions. Mixed numbers. 710Q


25 Fractions 3 Estimation. Adding and subtracting like fractions. Adding and subtracting unlike fractions. Adding and subtracting mixed numbers. 790Q




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26 Fractions 4 Multiplying fractions and mixed numbers. 920Q Study: EB Learning Material

27 Metric Measurement Units of length. Units of capacity. Units of mass. Changing metric units. 820Q


Essay: Listing

28 Customary Measurement Units of length. Units of capacity. Units of weight. 820Q Study: EB Learning

Material

29 Time Measurement Time. Elapsed time. Schedules. 820Q Study: EB Learning Material

30 Lines & Angles Point, line, line segment, ray, angle, plane. Intersecting lines. Parallel lines. Perpendicular lines. Acute, obtuse, and right angles. Angle measurement.



31 Polygons Plane figures. Triangles. Classifying by sides. Classifying by angles. Quadrilaterals. Classifying quadrilaterals. Symmetry. Congruent figures. Similar figures.


32 Circles Circles. Circle parts. Circumference, diameter, radius, chord. Compass. Relationships of parts. Pi. 810Q


33 Space Figures Faces, vertices, edges. Prisms. Pyramids. Cones. Cylinders. Spheres. 530Q Study: EB Learning


34 Geometric Measurement

Perimeter. Circumference. Area of rectangles, triangles, parallelograms, circles. Area of irregular shapes. Volume of space figures.





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35 Ratios Meaning of ratio. Equivalent ratios. Use of ratio in scale drawings. 590Q Study: EB Learning Material

36 Percent Ratio. Percent of decimals. Finding percent of a numbers. 860Q



37 Probability Definition and examples of probability. Prediction of outcomes. Fractions and probability. 440Q



42

Mathematics VI Grade Level 6

Mathematics VI strengthens mathematical knowledge and ability in the following areas: rounding numbers, estimation, place value; properties of numbers, multiplying decimals, dividing by one- or two-digit numbers, prime numbers; equivalent fractions, tallies, identifying variables, solving equations; length, capacity and weight units; temperature; lines and rays; parts of a circle; perimeter; positive and negative integers; and ordered pairs. The Mathematics VI course includes other topics such as:

• geometric measurements

• using scales

• coordinate planes

• formulas and pi

A+nyWhere Learning System Mathematics VI Scope & Sequence Grade Level 6


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1 Number Values Place value through billions. Decimal place value through thousandths. Word form. Standard form. Expanded form. 600Q Study: EB Learning

Material

2 Number Sense 1 Rounding numbers. Number Ranges. Estimation. Study: EB Learning Material

3 Number Operations Order of operations. Exponents and powers of numbers. 770Q Study: EB Learning Material

4 Number Sense 2 Square numbers and square roots. 850Q Study: EB Learning Material

5 Problem Solving 1 Five step thinking plan. Strategies solving problems.



6 Problem Solving 2 Review of 5 step plan. Selecting relevant information. Working with misleading or unnecessary data. Finding strategy in patterns. Choosing strategy to solve problems.

Study: EB Learning


7 Multiplication 1 Estimating products. Commutative, zero, associative, and one properties. 700Q Study: EB Learning

Material

8 Multiplication 2 Multiplying decimals. Zeros in multiplication. 700Q Study: EB Learning Material

9 Division Estimating quotients. Multiplying and dividing by powers of ten. Dividing by one-digit numbers. Dividing by two-digit numbers. Dividing decimals.




44

10 Number Sense 3 Factors. Divisibility. Prime numbers. Composite numbers. Factor trees. Prime factorization. Greatest common factors. Multiples. Least common multiples.



11 Fractions 1 Equivalent fractions. Comparing fractions. Simplest form. Relating fractions to decimals. 780Q Study: EB Learning

Material

12 Fractions 2 Estimating sums and differences. Adding and subtracting like fractions. Adding and subtracting unlike fractions. Adding and subtracting mixed numbers.



13 Fractions 3 Estimating products. Multiplying fractions and mixed numbers. Simplifying. Dividing fractions and mixed numbers. 920Q



14 Organizing Data Collecting data. Tallies, frequency tables. Mean, median, mode, and range. 850Q


15 Graphing Data Bar graphs. Pictographs, histograms, circle graphs, and stem and leaf plots. Analyzing graphed information. 850Q


16 Variables & Equations

Identifying variables. Numerical expressions. Algebraic expressions. Addition and subtraction equations. Multiplication and division equations.


17 Equations & Inequalities

Rational numbers in equations. Solving equations. Solving Inequalities. 800Q





45

18 Metric Measures Metric place value. Comparing metric units. Length units. Capacity units. Mass units. 820Q Study: EB Learning

Material

19 Customary Measures Length units. Capacity units, weight units. 820Q Study: EB Learning

Material

20 Other Measures Time and temperature. 820Q


Essay: Logic

21 Plane Geometry Points, lines, rays. Angles. Parallel, intersecting, perpendicular, and skew lines. Bisecting lines. 1020Q


22 Polygons Triangle classifications. Quadrilateral classifications. Other polygons. Congruence. Similar figures. Symmetry. Motion geometry. 1070Q


23 Circles Parts of a circle. Pi. Formulas. 1040Q



24 Space Figures Prisms. Pyramids. Faces, edges, and vertices. Bases. 530Q Study: EB Learning Material

25 Geometric Measurements Perimeter. Circumference. Area. Surface Area. Volume. 1040Q





46

26 Ratio & Proportion Ratio. Proportion. Solving proportions. Rates. Using scales. 830Q Study: EB Learning Material

27 Percent Percents and decimals. Percents and fractions. Estimation. Finding percents of whole numbers. 870Q


28 Probability Outcomes. Predicting outcomes. Identifying independent events. 440Q Study: EB Learning


29 Integers Positive integers. Negative integers. Zero. Comparing integers. Adding and subtracting integers. 800Q


30 Coordinate Graphing Ordered pairs. Coordinate planes. x-axis. y-axis. 850Q



47

Mathematics VII Grade Level 7

Mathematics VII covers these concepts: place value; commutative, associative, zero, one, and distributive properties; inverse operations; factors; number theory; mixed numbers; ratios; percent concepts; markups; commissions; steps to solving equations; measurement of length; mass/weight; metric units; points; angles; calculating perimeter; area; volume; using a number line; and graphing ordered pairs on a coordinate axis. The Mathematics VII course includes other topics such as:

• inverse operations

• steps to solving equations

• statistics and data collection

• adding and subtracting integers

A+nyWhere Learning System Mathematics VII Scope & Sequence Grade Level 7


48

1 Decimal Number Concepts

Place value, exponents, powers of ten, expanded notation, scientific notation. 910Q Study: EB Learning

Material

2 Number Operations Commutative, associative, zero, one, and distributive properties. Inverse operations, order of operations. 770Q Study: EB Learning

Material

3 Decimal Number Operations Expanding skills in addition, subtraction, multiplication, and division. 700Q


4 Problem Solving 1 Solving practical problems that deal with decimal numbers. Study: EB Learning Material

5 Number Theory Inverse operations. Prime and composite numbers. Divisibility rules, factors, greatest common factor. Multiples, least common multiple. Square numbers, square roots.



6 Problem Solving 2 Practical application of number theory in problem solving. Study: EB Learning Material

7 Fraction Concepts Simplest form, equivalent fractions. Comparing and ordering fractions. Improper fractions. Mixed numbers, relating fractions to decimals.



8 Fraction Operations Expanding skills in addition, subtraction, multiplication, and division with fractions and mixed numbers. 870Q


9 Problem Solving 3 Practical situations dealing with fractions in problem solving. Study: EB Learning Material



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10 Ratio & Proportion Ratios, equivalent ratios, cross products, rates, proportions, and solving proportions. 830Q



11 Problem Solving 4 Problem solving with ratios, rates, proportions, scale drawings. 830Q Study: EB Learning Material

12 Probability Properties of probability. Making predictions. Outcomes and permutations. 750Q Study: EB Learning

Material

13 Percent Concepts Percent, decimals, fractions, finding percents of a whole. Finding parts of a whole, finding the whole. 870Q



14 Problem Solving 5 Problem solving with percents. Percent of increase or decrease. Discounts, markups, commissions, interest, and sales tax. 900Q Study: EB Learning

Material

15 Algebra Concepts Numerical expressions, variables, equations, algebraic expressions, inequalities, inverse operations. 800Q



16 Expressions & Equations

Replacing variables to evaluate expressions. Steps to solving equations. 840Q





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17 Problem Solving 6 Writing equations to solve word problems. Solving multi-step equations. 800Q



18 Measurement Measurement of length, capacity, mass/weight, and time dealing with both metric and customary units. 820Q Study: EB Learning

Material

19 Problem Solving 7 Problem solving applications of measurement. 820Q Study: EB Learning Material

20 Geometric Concepts Definitions and examples of points, lines, planes, and angles. Bisecting lines and angles. Relationships of lines. Relationships of angles.



21 Plane Figures Polygons: triangles, quadrilaterals, pentagon, octagon, decagon; diagonal; circles: parts of a circle, calculate circumference and area 1040Q


22 Motion Geometry Similar and congruent figures. Rotation, translation, and reflection. 680Q


Essay: Listing

23 Space Figures Review and expansion of ideas about prisms, pyramids, cones, and cylinders. 1160Q


24 Geometric Measurement Length, area, volume, surface area of geometric figures. 1040Q





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25 Statistics Data collection. Mean, median, mode, and range. 850Q Study: EB Learning


26 Graphs and Plots Bar, circle, and line graphs; stem and leaf plots; box and whisker plots; frequency plots 960Q


Essay: Graphing

27 Integers Negative and positive numbers, comparing and ordering integers. Using a number line. Absolute value. 990Q Study: EB Learning

Material

28 Adding &

Subtracting Integers

Finding sums and differences with integers; additional properties of integers 800Q



29 Multiplying & Dividing Integers

Finding products and quotients with integers; using multiplication properties with integers 850Q


30 Coordinate Graphing Graphing ordered pairs on a coordinate axis. 850Q



52

Math Foundations IA Grade Levels 9-12 This course is designed to build foundational math skills for students—helping students progress at their optimum pace through interactive instruction and assessment spanning math skills typically found in third through fifth grade. Carefully paced, guided instruction is accompanied by interactive practice. When used in combination with Math Foundations II (addressing skills typically found in grade 6 to 8), the courses effectively remediate computational skills and conceptual understanding needed to undertake high school-level math courses with confidence.

Math Foundations IA is presented as a full semester course.

• All forty-six lessons contain a study guide, practice test, and mastery test. A math unit review assignment is presented at the end of each unit. Answer keys are available to assist teachers in grading the unit assignments.

• This course is split into five units; Understanding

Numbers, Adding and Subtracting, Measurement and Multiplication, Division and Data, and Fractions and Geometry.

• All lessons contain introduction and reinforcement of

mathematical skills and concepts.

A+nyWhere Learning Systems Math Foundations IA Teacher’s Guide Grade Levels 9–12

53

• The content in this course is based on Common Core Standards.

• Students learn about basic odd and even numbers, including solving by grouping, regrouping, word problems, identifying unnecessary information, skip counting, and mental math.

• Students review addition and subtraction, graphs and charts, and understanding and solving word problems.

• Students learn about the properties of division, multi-step word problems, conclusions and predictions, and more.

• Students learn about fractions, comparing like fractions, simplifying fractions, and finding common denominators. They learn about comparing and ordering mixed numbers and converting mixed numbers.

Course Name Number of

Lessons


Grade Levels

Unit Assignments

Math Foundations IA 46 1 9–12 5

A+nyWhere Learning Systems Math Foundations IA Scope & Sequence Grade Levels 9–12

Lesson Name Lesson Content Activities

54

Unit 1: Understanding Numbers

1.01 Addition and

Subtraction With Regrouping

Learn about the concept of regrouping. Find out when regrouping must occur and how to add and subtract with regrouping.

1.02 Understanding Numbers

Identify place value as well as odd and even numbers. Skip count by 2, 3, 4, 5, and 10. Use ordinal numbers to show order.

1.03 Ordering Numbers

Discover how place value is used when comparing, ordering, and rounding numbers. Use place value when writing Roman numerals.

1.04 Fact Families Identify fact families and determine missing numbers.

1.05 Using Mental Math Use mental math to add and subtract whole numbers and to regroup.

1.06 Choosing the Operation

Learn how to determine when to use the operation of addition or subtraction in a word problem.

1.07 Adding Numbers Horizontally

Discover how to solve horizontal addition problems by grouping numbers with parentheses.

1.08 Extra Information Identify unneeded information in word problems.

1.09 Unit 1 Assignment Math review of Unit 1. Study: Math Assignment



55

Unit 2: Adding and Subtracting

2.01 Standard and Nonstandard

Measurements

Measure length using nonstandard and standard measurements. Estimate length and apply appropriate units of length to measurement.

2.02 Using Graphs and Charts

Gather and apply information to bar graphs. Solve problems using bar graphs. Compare ways to organize and display data.

2.03 Addition with Regrouping

Review how to regroup numbers when adding. Practice adding 2- and 3-digit numbers with regrouping.

2.04 Understanding Word Problems

Learn the five steps needed to solve problems that involve mathematical operations.

2.05 Perimeter Measure the perimeter of given shapes and estimate the perimeter of irregular shapes. Learn how perimeter is affected when shapes change.

2.06 Subtraction With Regrouping

Review how to regroup numbers when subtracting. Practice subtracting 2- and 3-digit numbers with regrouping.

2.07 Solving Word Problems Review the five steps to solving word problems.

2.08 Deciding When to Regroup

Solve subtraction word problems that require regrouping. Find out how to regroup twice in the same problem and solve problems with zeros.




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Unit 3: Measurement and Multiplication

3.01 Mean, Median, Mode, and Range

Discover how to find the mean, median, mode, and range in a group of numbers.

3.02 Capacity, Time, and Weight Understand how to estimate and measure capacity, time, and weight.

3.03 Finding Needed Facts Identify missing information in word problems.

3.04 Introduction to Multiplication

Convert addition sentences to multiplication sentences. Learn how to create models of multiplication. Understand the commutative property of multiplication.

3.05 Multiplying by 2, 3, 4, and 5

Use skip counting and number lines to multiply by 2, 3, 4, and 5. Study multiplication tables for 2, 3, 4, and 5.

3.06 Multiplying by 6, 7, 8, 9, 10, and

100

Identify patterns in multiplication tables. Study multiplication tables for 6, 7, 8, and 9. Learn how to multiply by tens and hundreds.

3.07 Multiplying Three Numbers

Find out how multiplication is like addition. Discover the associative and distributive properties of multiplication. Learn how to multiply three factors. Practice solving problems that have missing factors.

3.08 Area Learn the concept of area and discover how to calculate the area of rectangles and squares. Estimate the area of irregular shapes and find out how area is affected when a shape changes.




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Unit 4: Division and Data

4.01 Introduction to Division

Learn about the concept of division. Find out how to divide numbers and how to write division sentences.

4.02 Division: The Opposite of

Multiplication

Discover how multiplication and division are related. Learn how to divide by tens and hundreds.

4.03 Dividing by 2, 3, 4, and 5

Divide using 2, 3, 4, and 5. Check division by multiplying and identify fact families.

4.04 Multi-step Word Problems

Learn clue words for solving problems. Practice solving word problems that have multiple steps.

4.05 Dividing by 6, 7, 8, and 9

Divide using 6, 7, 8, and 9. Make comparisons and identify patterns. Complete division problems with remainders.

4.06 Long Division Learn how to divide 3- and 4-digit numbers by a 1-digit number.

4.07 Conclusions and Predictions

Learn about taking surveys, comparing sets of data, drawing conclusions, and making predictions. Find out how to display the information obtained from a survey in a pictograph.

4.08 Expressing Numbers

Determine whether addition, subtraction, and multiplication problems will result in even or odd numbers. Compare numbers in expanded and standard forms.

4.09 Data Collection Learn how to create and gather information from charts, maps, and graphs. Determine how data collection affects problem solving.



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Unit 5: Fractions and Geometry

5.01 Introduction to Fractions

Learn how to divide shapes into equal parts to form fractions. Identify various fractions made from whole objects.

5.02 Parts of a Set Learn how to write different types of fractions made by dividing whole sets into separate equal parts. Identify the numerator and denominator in fractions.

5.03 Equivalent Fractions

Find out how to compare fractions by creating common denominators. Recognize equivalent fractions.

5.04 Adding and Subtracting Fractions

Add and subtract fractions with like denominators.

5.05 Mixed Numbers Learn how to identify, create, and compare mixed numbers.

5.06 Decimals Learn how to write decimals and how to relate fractions to decimals.

5.07 Adding and Subtracting Decimals

Discover the skills necessary to add and subtract decimals.

5.08 Probability Learn how to gather data and write probability statements. Make predictions based on probability.

5.09 Lines, Rays, and Segments

Identify lines, rays, segments, and angles. Learn how they relate to each other.



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5.10 Plane Figures Identify different types of plane figures. Learn about congruent shapes and lines of symmetry. Make interesting designs using shapes.

5.11 Solids Recognize three-dimensional shapes. Identify faces, edges, and corners on solid figures. Learn how to find volume.

5.12 Ordered Pairs Learn how to use ordered pairs to find a location on a map.

5.13 Logical Reasoning Use logic to solve problems and check for reasonability of answers.



60

Math Foundations IB Grade Levels 9-12 This course is the second semester of Math Foundations I. It is designed to build foundational math skills for students—helping students progress at their optimum pace through interactive instruction and assessment spanning math skills typically found in third through fifth grade. Carefully paced, guided instruction is accompanied by interactive practice. When used in combination with Math Foundations II (addressing skills typically found in grade 6 to 8), the courses effectively remediate computational skills and conceptual understanding needed to undertake high school-level math courses with confidence.

Math Foundations IB is presented as a full semester course.

• All sixty-eight lessons contain a study guide, a practice test, and a mastery test. A math unit review assignment is presented at the end of each unit. Answer keys are available to assist teachers in grading the unit assignments.

• This course is divided into five units: Addition,

Subtraction, and Measurement; Fractions and Geometry; Multiplication, Division and Decimals; Operations; and Data, Graphs, Fractions and Geometry.

• All lessons contain introduction and reinforcement of

mathematical skills and concepts.

A+nyWhere Learning Systems Math Foundations IB Teacher’s Guide Grade Levels 9-12

61


• Students learn about cardinal and ordinal numbers, estimating and measuring capacity, mass, temperature, time, and

days.

• Students learn about decimals, estimating, and logic in solving, commutative and associative properties, as well as place values and exponents

• Students learn about organizing data and displaying data with various methods, comparing, multiplying and estimating fractions, as well as points, segments, rays, geometric vocabulary, and more.

• Students practice word problems with fractions and decimals—adding and subtracting like and unlike fractions, mixed

numbers, and decimals, as well as segments, symmetry, ordered pairs, and more.


Lessons


Grade Levels

Unit Assignments

Math Foundations IB 68 1 9–12 5

A+nyWhere Learning Systems Foundations Math IB Scope & Sequence Grade Levels 9-12

Lesson Title Lesson Content Activities

62

Unit 1: Addition, Subtraction, and Measurement

1.01 Story problems: A 5-step process The five-step process for problem solving.

1.02 Number Sense:

Grouping Addends

Order of addends in relation to the sum. Adding zero. Students rewrite addition facts in inverse order. Grouping addends.

1.03 Number Sense:

Rounding Numbers

Rounding numbers and money amounts to the nearest 10, 100, and 1000.

1.04 Addition and Subtraction: Related Facts

Given an addition fact, students write a related subtraction fact. Given a subtraction fact, students write a related addition fact.

1.05 Addition and

Subtraction: Fact Families

Addition and subtraction in fact families. Students complete fact families using missing elements. Students read story problems and choose the correct operation to solve the problem.

1.06 Patterns Odd and even numbers. Students recognize the relationships between numbers to determine patterns.

1.07 Ordinal Numbers Cardinal and ordinal numbers.

1.08 Measurement: Time

Estimating, determining, and measuring time to the nearest minute. Finding elapsed time. Using the appropriate time units to measure time. Writing time using AM or PM. Adding and subtracting time.



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1.09 Measurement: Calendars

Using a calendar. Looking for patterns on a calendar. Learning the number of days in each month.

1.10 Measurement:

Units of Measurement

Estimating and measuring capacity, mass, temperature, and distance. Using customary units of measurement. Standard and Metric measurements.


Unit 2: Fractions and Geometry

2.01 Fractions: The Basics

Review of fractions. Reading, writing, and renaming mixed numbers. Comparing and ordering fractions and mixed numbers.

2.02 Fractions: Adding and Subtracting

Addition and subtraction of fractions and mixed numbers. Equivalent fractions. Students read story problems, choose strategies, and solve problems.

2.03 Decimals to the Hundredths

Students read and write decimals to tenths and hundredths positions. Relating decimals and fractions. Relating decimals and money. Writing mixed numbers as decimals.

2.04 Decimals:

Comparing and Ordering

Adding and subtracting decimals of the same place value. The use of zero in decimals. Writing decimals as fractions.

2.05 Geometry Identifying faces, edges, and corners of solid and plane figures. Comparing solid figures. Differences in solid and plane figures.



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2.06 Geometry: Angles

Line segments and angles. Identifying right angles. Using greater than and less than with right angles. Definition and examples of intersecting lines, parallel lines, and perpendicular lines.

2.07 Geometry:

Congruence and Symmetry

Definition and examples of congruence and symmetry. Open and closed figures. Identifying parts of angles. Identifying and continuing patterns using geometrical shapes.

2.08 Geometry:

Perimeter and Area

Measuring perimeter and area of polygons. Finding the volume of solid figures. Difference in area and volume.

2.09 Graphs: Ordered Pairs

Students locate and name ordered pairs on a coordinate grid. Comparing maps and grids. Students locate points by writing ordered pairs.

2.10 Graphs: Bar and Line Graphs

Students gather data from visual aid and complete bar graph. Students make predictions from bar graph information. Students complete line graph.

2.11 Graphs: Tables and Pictographs

Using tables and pictographs. Gathering information with a pictograph. Using pictograph information to create line and bar graphs. Comparing graph types. Students decide what type of graph to use.

2.12 Probability

Definition and examples of probability. Identifying possible outcomes. The probability equation. Students predict if outcome of given situation is probable, certain, or impossible. Using graphs to chart probability and predict outcomes.

2.13 Using Mental Math Examples of different methods of computation using mental math.

2.14 Unit 2 Assignment Math review of Unit 2 Study: Math Assignment



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Unit 3: Multiplication, Division, and Decimals

3.01 Choosing the Operation Students decide which operation to use and solve problems.

3.02 Extra Information Students read problems, identify unneeded information, and solve problems.

3.03 Story Problems:

Forming the Problem

Students are given information to formulate into problem. Students check for reasonableness of answers.

3.04 Finding Needed Facts Students read information and identify missing information.

3.05 Story Problems: From Problem to

Solution

Students read information, form a plan, choose the correct operation, and solve problems having more than two steps.

3.06 Logical Reasoning Students use logic to solve problems. Students check for reasonability of answers.

3.07 Multiplying Two and Three Digits

Multiplication by two and three digit numbers. Multiplication problems with missing factors.

3.08 Properties of Multiplication

Commutative, associative, zero, one, and distributive multiplication properties. Mental math techniques. Estimating.

3.09 Multiplication: Estimating Estimation and mental math. Decimal multiplication. Zero in decimals.



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3.10 Dividing Whole Numbers Dividing whole numbers by one and two digit numbers. Estimating quotients.

3.11 Dividing and Estimating Dividing decimals. Estimation and mental math.

3.12 Whole Numbers: Place Value

Place value through millions; exponents; standard, expanded, and word forms of numbers

3.13 Whole Numbers:

Compare and Order

Comparing and ordering whole numbers; rounding through millions

3.14 Decimals to the Thousandths Place value through thousandths; zero in decimals; writing decimals

3.15 Decimals:

Compare and Order

Comparing and ordering decimals; patterns in decimals; rounding decimals


Unit 4: Operations

4.01 Properties of Addition Commutative, associative, and zero properties of addition



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4.02

Addition and Subtraction:

Inverse operations

Inverse operations; estimating sums

4.03 Addition and Subtraction:

Zeros Addition and subtraction across zeros; regrouping with zeros

4.04 Addition and

Subtraction with Decimals

Addition and subtraction of decimals; clustering

4.05 Problem Solving: The 5-Step Plan The five step thinking plan; using inverse operations

4.06 Organizing Data: Using Tables Using tables to organize, analyze, and solve problems.

4.07 Problem Solving:

Multistep Problems

Multi-step problems; using the five step thinking plan

4.08 Properties of Multiplication

Commutative, associative, zero, one, and distributive multiplication properties; mental math techniques; estimating

4.09 Multiplication

with 2 and 3 digit numbers

Multiplication by two and three digit numbers; multiplication problems with missing factors

4.10 Multiplication: Estimation Estimation; mental math; decimal multiplication; zero in decimals



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4.11 Dividing Whole Numbers Dividing whole numbers by one and two digit numbers; estimating quotients

4.12 Dividing Larger Numbers

Dividing larger numbers; short division; using zeros in division; identifying patterns; checking division

4.13 Dividing Decimals Dividing decimals; estimation and mental math


Unit 5: Data, Graphs, Fractions and Geometry

5.01 Organizing Data:

Mean, Mode, Median, Range

Sample groups; range; mode; mean; median

5.02 Bar Graphs and Pictographs

Parts of a bar graph; selecting a scale; displaying data on a bar graph; displaying data on a pictograph; reading data; analyzing data; making inferences

5.03 Line Graphs Uses of line graphs; parts of line graphs; displaying data

5.04 Circle Graphs Circle graphs; Venn diagrams; analyzing data

5.05 Ordered Pairs Ordered pairs; graphing on a grid

5.06 Fractions Fraction parts; equivalent fractions; prime numbers; composite numbers; greatest common factor; simplest form; least common multiple; like fractions



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5.07 Comparing and

Reducing Fractions

Comparing fractions; reducing fractions; improper fractions; mixed numbers

5.08 Adding and Subtracting Fractions

Estimation; adding and subtracting like fractions; adding and subtracting unlike fractions; adding and subtracting mixed numbers

5.09 Multiplying Fractions Multiplying fractions and mixed numbers

5.10 Lines and Angles Point, line, line segment, ray, angle, plane; intersecting lines; parallel lines; perpendicular lines; acute, obtuse, and right angles; angle measurement

5.11 Polygons Plane figures; triangles; classifying by sides; classifying by angles; quadrilaterals; classifying quadrilaterals; symmetry; congruent figures; similar figures

5.12 Circles Circles; circumference; diameter; radius; chord; compass; relationships of parts; pi

5.13 Space Figures Faces, vertices, edges; prisms; pyramids; cones; cylinders; spheres

5.14 Geometric Measurement

Perimeter; circumference; area of rectangles, triangles, parallelograms, circles; area of irregular shapes; volume of space figures

5.15 Ratios Meaning of ratio. Equivalent ratios. Use of ratio in scale drawings.

5.16 Percent Ratio; changing percent to decimals and fractions; finding the percent of a number



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5.17 Probability Definition and examples of probability; prediction of outcomes; fractions and probability



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Math Foundations IIA Grade Levels 9-12 This full semester course guides students through foundational math skills typically found in sixth to eighth grade instruction. It is appropriate for use as remediation at the high school level, as a bridge to high school, or as middle school curriculum. This course builds computational skills and conceptual understanding needed to undertake high school-level math courses with confidence. Carefully paced, guided instruction is accompanied by interactive practice.

Math Foundations IIA is presented as a full semester course.

• All fifty-five lessons contain a study guide, a practice test, and a mastery test. A math unit review assignment is presented at the end of each unit. Answer keys are available to assist teachers in grading the unit assignments.

• This course is split into six units; Numbers and Operations;

Data, Measurement, and Geometry; Ration, Proportion, and Decimals; Numbers and Properties; Concepts in Algebra; and Geometry and Statistics.

• All lessons contain introduction and reinforcement of mathematical skills and concepts.

A+nyWhere Learning Systems Math Foundations IIA Teacher’s Guide Grade Levels 9–12

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• Students learn about rounding numbers, order of operations, square numbers and square roots, five-step thinking plan

to solving word problems, multiplication properties, division, factoring, comparing fractions, addition/subtraction of fractions, and multiplication/division of fractions.

• Students learn about types of graphs, solving equations, solving inequalities, metric measures, customary measures, time, temperature, planes, polygons, circles, space figures, and calculating perimeter, circumference, area, surface area, and volume.

• Students learn how to solve proportion problems, find percents of whole numbers, predict outcomes, add and subtract integers, graph ordered pairs, and express numbers in expanded and scientific notation.

• Students review commutative, associative, zero, one, and distributive properties as well as learn about solving problems

with decimals, writing fractions in simplest form, and performing operations on fractions and mixed numbers.

• Students learn about evaluating expressions, solving multistep equations and solving problems involving measurement.

• Students learn basic geometric concepts, as well as how to identify plane figures, calculate circumference and area of a circle, determine similarity or congruency, measure length, area, volume, and surface area of geometric figures, find mean, median, mode, and range in a set of data, and organize data on various graphs and plots.

A+nyWhere Learning Systems Math Foundations IIA Teacher’s Guide Grade Levels 9–12

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Lessons


Grade Levels

Unit Assignments

Math Foundations IIA 55 1 9–12 6

A+nyWhere Learning Systems Foundations Math IIA Scope & Sequence Grade Levels 9-12


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Unit 1: Numbers and Operations

1.01

Number Sense: Rounding,

Estimating, and Range

Rounding numbers; number ranges; estimation

1.02 Number Operations Order of operations; exponents and powers of numbers

1.03 Number Sense:

Squares and Square Root

Square numbers and square roots

1.04 Problem Solving: The 5-Step Plan Five-step thinking plan; strategies solving problems

1.05 Problem Solving: Application

Review of 5-step plan; selecting relevant information; working with misleading or unnecessary data; finding strategy in patterns; choosing strategy to solve problems

1.06 Multiplication: Properties Estimating products; commutative, zero, associative, and one properties

1.07 Multiplication: Decimals Multiplying decimals; zeros in multiplication

1.08 Division Estimating quotients; multiplying and dividing by powers of ten; dividing by one-digit numbers; dividing by two-digit numbers; dividing decimals

1.09 Number Sense: Factors

Factors; divisibility; prime numbers; composite numbers; factor trees; prime factorization; greatest common factors; multiples; least common multiples



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1.10 Fractions: Equivalent Fractions

Equivalent fractions; comparing fractions; simplest form; relating fractions to decimals

1.11 Fractions:

Estimating with Fractions

Estimating sums and differences; adding and subtracting like fractions; adding and subtracting unlike fractions; adding and subtracting mixed numbers

1.12 Fractions:

Multiplying & Dividing Fractions

Estimating products; multiplying fractions and mixed numbers; simplifying; dividing fractions and mixed numbers


Unit 2: Data, Measurement, and Geometry

2.01 Organizing Data Collecting data; tallies; frequency tables; mean, median, mode, and range

2.02 Graphing Data Bar graphs; pictographs, line graphs, circle graphs, and stem and leaf plots; analyzing graphed information

2.03 Metric Measures Metric place value; comparing metric units; length units; capacity units; mass units

2.04 Customary Measures Length units; capacity units; weight units

2.05 Other Measures Time and temperature



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2.06 Plane Geometry Points, lines, rays; angles; parallel, intersecting, perpendicular, and skew lines; bisecting lines

2.07 Polygons Triangle classifications; quadrilateral classifications; other polygons; congruence; similar figures; symmetry; motion geometry

2.08 Space Figures Prisms; pyramids; faces, edges, vertices, and bases

2.09 Geometric Measurements Perimeter; circumference; area; surface area; volume

2.10 Circles Parts of a circle; pi; formulas


Unit 3: Ratio, Proportion, & Decimals

3.01 Ratio and Proportion Ratio; proportion; solving proportions; rates; using scales

3.02 Percent Percents and decimals; percents and fractions; estimation; finding percents of whole numbers

3.03 Probability: Predicting Outcomes

Outcomes; predicting outcomes; identifying independent events

3.04 Integers Positive integers; negative integers; zero; comparing integers; adding and subtracting integers



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3.05 Coordinate Graphing Ordered pairs; coordinate planes; x-axis; y-axis

3.06 Number Values Place value through billions; decimal place value through thousandths; word form; standard form; expanded form; comparing and ordering numbers

3.07 Decimal Number Concepts Place value, exponents, powers of ten, expanded notation, scientific notation


Unit 4: Numbers and Properties

4.01 Number Operations

Commutative, associative, zero, one, and distributive properties; inverse operations; order of operations

4.02 Decimal Number Operations

Expanding skills in addition, subtraction, multiplication, and division using decimals

4.03 Problem Solving: Decimals Solving practical problems that deal with decimal numbers

4.04 Number Theory Prime and composite numbers; prime factorization; divisibility rules; common factors; greatest common factor; common multiples; least common multiple; square numbers; square roots

4.05 Problem Solving: Number Theory Practical application of number theory in problem solving



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4.06 Fraction Concepts Simplest form; equivalent fractions; like and unlike fractions, comparing fractions; improper fractions; mixed numbers; relating fractions to decimals; reciprocals

4.07 Fraction Operations

Expanding skills in addition, subtraction, multiplication, and division with fractions and mixed numbers

4.08 Problem Solving: Fractions Practical situations dealing with fractions in problem solving

4.09 Ratio and Proportion

Ratios; equivalent ratios; cross products; rates; proportions; solving proportions


Ratio, Rates, Proportions

Problem solving with ratios, rates, proportions, and scale drawings

4.11 Probability: Properties Properties of probability; making predictions; outcomes and permutations

4.12 Percent Concepts Percent; decimals; fractions; finding percents of a whole; finding parts of a whole; finding the whole

4.13 Problem Solving: Percents

Problem solving with percents; percent of increase or decrease; discounts, markups, commissions, interest, and sales tax




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Unit 5: Concepts in Algebra

5.01 Algebra Concepts Numerical expressions, variables, equations, algebraic expressions, inequalities, inverse operations

5.02 Variables and Equations

Identifying variables; numerical expressions; algebraic expressions; addition and subtraction equations; multiplication and division equations

5.03 Equations and Inequalities Rational numbers in equations; solving equations; solving Inequalities

5.04 Expressions and Equations Replacing variables to evaluate expressions; steps to solving equations

5.05 Problem Solving: Equations Writing equations to solve word problems; solving multi-step equations

5.06 Problem Solving: Measurement Problem solving applications of measurement


Unit 6: Geometry and Statistics

6.01 Geometric Concepts

Definitions and examples of points, lines, planes, and angles; bisecting lines and planes; relationships of lines; relationships of angles

6.02 Plane Figures Polygons: triangles, quadrilaterals, pentagon, octagon, decagon; diagonal; circles: parts of a circle, calculate circumference and area



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6.03 Motion Geometry Similar and congruent figures; rotation; translation; reflection

6.04 Space Figures: A Review Review and expansion of ideas about prisms, pyramids, cones, and cylinders

6.05 Geometric

Measurement: A Review

Length, area, volume, surface area of geometric figures

6.06 Statistics Data collection; mean, median, mode, and range

6.07 Graphs and Plots Bar, circle, and line graphs; stem and leaf plots; box and whisker plots; frequency plots



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Math Foundations IIB Grade Levels 9-12 This full semester course is the second semester of Math Foundations II. It is designed to guide students through foundational math skills typically found in sixth to eighth grade instruction. It is appropriate for use as remediation at the high school level, as a bridge to high school, or as middle school curriculum. Carefully paced, guided instruction is accompanied by interactive practice. When used in combination with Math Foundations I (addressing skills typically found in grade 3 to 5), the course effectively remediates the computational skills and conceptual understanding needed to undertake high school-level math courses with confidence.

Math Foundations IIB is presented as a full semester course.

• All fifty-two lessons contain a study guide, a practice test, and a mastery test. A math unit review assignment is presented at the end of each unit. Answer keys are available to assist teachers in grading the unit assignments.

• This course is split into seven units; Operations and Real

Numbers; Ratio, Proportion, and Probability; Concepts in Algebra; Geometric Concepts; Expressions and Equations; Elements of Algebra; and Geometry.

• All lessons contain introduction and reinforcement of mathematical skills and concepts.

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• The content in this course is based on Common Core Standards. • Students learn about absolute value and how to add, subtract, multiply, and divide integers. • Students will understand numbers, ways of representing numbers, relationships among numbers, and number systems.

Students review simplifying expression and solving multistep equations and inequalities. They also review plane and space figures and learn formulas for the area of plane figures and volume of space figures.

• Students use number lines to identify, locate, and compare integers, as well as determine absolute value. • Students learn rules for solving equations by adding/subtracting/multiplying/dividing integers with the same or different

sign. Students review the definitions of point, line, ray, line segment, and angle. Students find area of polygons and surface area of pyramids, prisms, and cones.


Lessons


Grade Levels

Unit Assignments

Math Foundations IIB 52 1 9–12 7

A+nyWhere Learning Systems Foundations Math IIB Scope & Sequence Grade Levels 9-12


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Unit 1: Operations and Real Numbers

1.01 Integers Negative and positive numbers; comparing and ordering integers; using a number line; absolute value

1.02 Adding and Subtracting

Integers Finding sums and differences with integers; additional properties of integers

1.03 Multiplying & Dividing Integers

Finding products and quotients with integers; using multiplication properties with integers

1.04 Coordinate Graphing Graphing ordered pairs on a coordinate axis

1.05 Number Concepts Review Place value; rounding; estimation; compatible numbers

1.06 Properties Commutative property of addition; associative property of addition; commutative property of multiplication; associative property of multiplication; identity properties; distributive property

1.07 Problem Solving: The 5-Step Plan Five-step thinking plan; problem solving strategies

1.08 Basic Operations

with Rational Numbers

Positive and negative rational numbers; equivalent fractions; cross products; absolute value; order of operations; fractions to decimals; comparing and ordering.

1.09 Basic Operations with Fractions Addition; subtraction; multiplication; division



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Unit 2: Ratio, Proportion, and Probability

2.01 Ratio and Proportion Ratio; rates; equivalent ratios; proportion


Ratio and Proportion

Practical problems dealing with ratio, proportion, unit rates, rates, and scale

2.03 Percent Concepts Percent to decimals; percent to fractions; finding percents

2.04 Percent: Practical uses Percent of increase or decrease; sales tax and discounts

2.05 Probability

Probability; finding samples; randomly occurring events; compound events; independent and dependent events; making predictions; capture and recapture method; tree diagrams; counting principle factorials; Venn diagrams


Unit 3: Concepts in Algebra

3.01

Algebra: Expressions,

Equations, Order of Operations

Numerical expressions; variables; variable expressions; order of operations; equations; solutions



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3.02 Algebra: Simplifying

Simplifying numerical and variable expressions; coefficients; constants; evaluating algebraic expressions

3.03 Algebra:

Equalities and Inequalities

Properties of equality; inverse operations; inequalities; solving inequalities

3.04 Algebra: Multi-Step Equalities and Inequalities

Guides for practice in solving multi-step equations and inequalities

3.05 Algebra:

Translating Data to Equations

Translating data into equations and inequalities; using equations and inequalities to solve practical problems


Unit 4: Geometric Concepts

4.01

Geometric Concepts: Points,

Lines, Rays, Angles

Point, line, ray; line relationships; angles; angle relationships; angle measurement and classifications; bisecting line segments and angles

4.02 Plane Figures: Polygons and

Circles Regular polygons; triangles; quadrilaterals; congruent figures; circles



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4.03

Space Figures: Polyhedrons, Cones, and

Spheres

Polyhedrons; prisms; pyramids; cylinders; cones; spheres

4.04 Geometric Measurement

Perimeter and circumference; formulas for area of plane figures; surface area of polyhedrons; volume of space figures

4.05 Square Roots and Triangles

Square roots; principal square roots; perfect squares; irrational numbers and real numbers; Pythagorean Theorem; similar triangles; special right triangles; trigonometry

4.06 Coordinate Graphing

Coordinate planes; ordered pairs; solutions; graphing equations; linear equations; slope; slope formula

4.07 Coordinate Graphing to

Solve Problems

Using coordinate graphing to solve problems; solving and graphing equations and inequalities; quadrants


Unit 5: Expressions and Equations

5.01 Number Notation Writing numbers in standard, written, expanded, and factor form; scientific notation

5.02 Expressions and Equations

Writing numerical expressions; introduction to variables; writing algebraic expressions; evaluating algebraic expressions



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5.03 Properties Review Review of commutative, associative, distributive, zero, and identity properties and examples of steps in solving problems using each one; multiplicative property of zero

5.04 Simplifying Expressions

Terms; operational symbols; numerical coefficients; identifying like terms and constants; using various properties to simplify expressions

5.05 Solving Equations using Properties

Properties of equality; using inverse operations; using equivalent equations and inverse operations to solve problems

5.06

Solving Equations with

Multiplication and Division

Inverse operations of multiplication and division; the multiplication and division properties of equality


Unit 6: Elements of Algebra

6.01 Integers Integers and their opposites; using number lines to identify, locate, and compare integers; absolute value of integers; using integers in real life situations

6.02

Equations with Integers by Adding and Subtracting

Rules for solving equations by adding and subtracting integers with the same sign or different signs; examples of strategies to solve equations



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6.03

Equations with Integers by

Multiplying and Dividing

Rules for solving equations by multiplying and dividing integers with the same or different signs; examples of strategies to solve equations

6.04 Inequalities Definition and examples of inequalities; signs that indicate inequality

6.05

Solving Inequalities with

Inverse Operations

Solving inequalities by reversing signs

6.06

Solving Inequalities with Multiplication and

Division

Other strategies for solving inequalities; checking for reasonability

6.07 Factors ad Exponents

Identifying laws of exponents; writing numbers in exponential form; writing exponents and variables; rules for determining exponents when adding, subtracting, multiplying, and dividing

6.08 Rational Numbers Identifying parts of fractions; relating fractions to rational numbers; locating rational numbers on the number line; comparing rational numbers; writing rational numbers in decimal form; algebraic fractions

6.09

Equations-Inequalities by

Adding and Subtracting

Solving equations and inequalities with addition and subtraction; two-step equations; solving equations with variables on both sides



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6.10

Equations-Inequalities by Multiplying and

Dividing

Solving equations and inequalities with multiplication and division; two-step equations; solving equations with variables on both sides with combined operations

6.11 Graphing on the Coordinate Plane

Graphing on the coordinate plane; graphing linear equations; slope and intercept; graphing linear equalities

6.12 Ratio and Proportion

Definition and examples of ratios; writing ratios in lowest terms; rates; unit rates; equivalent ratios; cross products; definition of proportions; solving proportions

6.13 Percent Decimals and percents; figuring percents; place value in percent problems; relating percent to wholes

6.14 Problem Solving with Percent

Examples of solving percent problems in everyday life; discounts; commission; checking answers for reasonableness; numbers larger than one hundred percent

6.15 Probability:

Permutations and Combinations

The counting principle; permutations; combinations; independent events; dependent events


Unit 7: Geometry

7.01 Algebra with Geometry

Definitions of point, line, ray, line segment, angles, and angle measures; parallel and perpendicular lines



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7.02 Polygons and Circles

Triangles, quadrilaterals, and other polygons; congruent figures; similar triangles; circles

7.03 Area and Volume Finding area of polygons; finding volume and surface area of pyramids, prisms, and cones

7.04 Special Triangles Square roots; Pythagorean theorem; tangents; sine and cosine ratios

7.05 Statistics and Graphing

Frequency tables; range, mean, mode, and median; bar, line, and circle graphs; scatter plots; stem and leaf plots



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Mathematics for Grade Levels 8-12 A+LS Mathematics are comprehensive, completely integrated courses for grades 8–12. The Mathematics courses develop strategies to solve real-world problems and are designed to move the student beyond the level of basic knowledge into critical thinking and learning activities.

• Mathematics is a collection of one- and two- semester long

courses.

• All lessons in these titles contain a study guide, a practice test, and mastery test.

• Some courses are enriched by National Library of Virtual Manipulatives (NLVM) applets or Encyclopædia Britannica Online School Edition (EB) workspaces that contain learning materials. Learning materials may contain articles, games, images, maps, and/or videos.


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The National Library of Virtual Manipulatives is a library of uniquely interactive, web-based virtual manipulatives or concept tutorials for mathematics instruction supported by the National Science

Foundation. Each applet provides interactive math lessons and activities.



93

The Encyclopædia Britannica Online School Edition has teacher resources and student learning materials. The materials include a wide range of interactive lessons, research projects, animations, and worksheets that support many A+LS lessons.




94

Course Name

Number of Lessons


Grade Levels

Pre-Algebra IA 43 1 8

Pre-Algebra IB 37 1 8

Algebra IA 63 1 9–10

Algebra IB 54 1 9–10

Algebra I: A Functional Approach Part 1 37 1 9–10

Algebra I: A Functional Approach Part 2 37 1 9–10

Algebra II: Part 1 44 1 10–12

Algebra II: Part 2 33 1 10–12

Geometry IA 45 1 9–10

Geometry IB 46 1 9–10

Trigonometry 29 1 11–12

Calculus I 30 1 12

Calculus II 26 1 12


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Pre-Algebra IA Grade Level 8

Pre-Algebra IA introduces students to the following concepts and functions:

• number notation

• decimals • operational symbols • inverse operations of multiplication and division • rules for solving equations by adding and subtracting integers • exponents • fractions • graphing on the coordinate plane • algebraic expressions • statistics • scatter plots • definitions of basic geometric terms • polygons and circles

A+nyWhere Learning System Pre-Algebra IA Teacher’s Guide Grade Level 8

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A+LS™ Pre-Algebra IA is a full semester course designed to help students bridge the gap from elementary studies into the more advanced and rigorous studies of high school mathematics courses. The Pre-Algebra IA content is designed to exceed the standards of Common Core Math, Grade 8, which many schools today refer to as the Pre-Algebra course. Students are provided foundational coursework in an easy-to-use interactive style, which encourages them to work problems thoroughly and check their work before moving on to the next task.

Pre-Algebra IA is presented as a one-semester core course.

• All forty-four lessons contain a study guide, a practice test, and a

mastery test.

• Each lesson contains interactive learning objects. Students click through each step of a problem in order to advance to the next step, providing students the opportunity to read and comprehend at the pace best suited for their individual needs.

• A glossary of terms and a calculator are provided in each study guide by clicking the “Tools” menu at the upper right. The built-in glossary helps students by reinforcing the key terms that are used across the title. They can also find key terms from previous lessons when they need them most.

• A virtual coach, Greek mathematician and philosopher Pythagoras, appears throughout the course to provide students with additional information about key concepts taught in the lesson. Students simply point at the image of Pythagoras in the lower left corner to read the information.


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• Lessons are designed in units. Each unit groups related material as a foundation for the upcoming unit. Students should

master a complete unit prior to moving to the next.

• Interactive sample problems are provided in each lesson. This allows students to immediately check their understanding as they move throughout the lesson. If they get the problems wrong, they can easily navigate to the previous page in order to review the module again.

• Pre-Algebra IA has an overall objective of laying the foundation for upper-level high school mathematics courses. Course materials are presented in a foundational method, and lessons become more complex as the student progresses.

• By the end of the course, students should have a solid working knowledge of the following key concepts: expressions, graphs, integers, exponents, rational numbers, square roots, equations, inequalities, statistics, and geometric shapes.


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Key Terms Key terms, as listed in the following table, indicate new concepts that are introduced in each lesson. These concepts are listed as key terms in the course’s glossary. When a lesson has no key terms listed, it typically means that the student will be working with concepts already introduced.


Lessons


Grade Level

Built-in Calculator

Glossary of Key Terms

Pre-Algebra IA 44 1 8 Yes Yes

A+nyWhere Learning System Pre-Algebra IA Scope & Sequence Grade Level 8

Lesson Name Lesson Content

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Unit 1: Expressions and Graphs

01-01 Variables and Expressions

• Identify the parts of an expression, including terms, coefficients, constants, and variables. • Evaluate algebraic expressions for given values of the variables. Key Terms: expression, numerical expression, algebraic expression, variable, term, coefficient, constant, substitution

01-02 Writing Algebraic

Expressions

• Translate words to numbers and numbers to words. • Write an equation from a word problem. • Solve a word problem.

01-03 Combining Like Terms

• Identify like terms in an expression. • Simplify expressions by combining like terms. • Simplify expressions using the distributive property. Key Terms: like terms, constant

01-04 Graphing on a

Coordinate Plane

• Graph a point given an ordered pair. • Construct a line on a coordinate plane given a table. • Solve an equation with two variables by using a table and coordinate plane. Key Terms: coordinate, plane, axes, x-axis, y-axis, ordered pair

01-05 Interpreting

Graphs and Tables • Interpret information from a table. • Interpret information from a graph.



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Unit 2: Integers and Exponents

02-01 Integers

• Define and recognize integers. • Use a number line to compare two integers. • Find the absolute value of an integer. Key Terms: integer, positive integer, negative integer, opposites, absolute value, zero pair

02-02 Adding Integers • Perform addition with positive and negative integers. Key Term: zero pair

02-03 Subtracting

Integers • Perform subtraction with positive and negative integers.

02-04 Multiplying and

Dividing Integers • Multiply positive and negative integers. • Divide positive and negative integers.

02-05 Exponents

• Evaluate expressions with exponents. • Find the prime factorization of a number. • Write a factorization with exponents. Key Terms: exponent, base, PEMDAS, prime number, factor tree

02-06 Properties of Exponents

• State the laws of exponents. • Simplify expressions with exponents. • Multiply and divide terms with exponents.



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02-07 Scientific Notation • Convert from scientific notation to standard form. • Convert from standard form to scientific notation.

02-08 Operations with

Scientific Notation

• Add, subtract, multiply and divide numbers written in scientific notation. • Add, subtract, multiply and divide when numbers in standard form and numbers in scientific notation

are present in a problem

Unit 3: Rational Numbers and Square Roots

03-01 Rational Numbers

• Write fractions in simplest form. • Convert fractions to decimals. • Graph fractions on a number line. • Compare two fractions or decimals. Key Terms: rational number, simplest form, mixed number, improper fraction

03-02 Adding and Subtracting

Rational Numbers

• Add and subtract rational numbers in decimal form. • Add and subtract rational numbers in fraction form with like denominators.

03-03 Multiplying and

Dividing Rational Numbers

• Multiply and divide rational numbers in decimal form. • Multiply and divide rational numbers in fraction form. Key Terms: reciprocal, multiplicative inverse

03-04

Adding and Subtracting with

Unlike Denominators

• Add and subtract rational numbers in fraction form with unlike denominators.



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03-05 Squares, Cubes,

and Roots

• Find squares of numbers. • Identify perfect squares. • Find the square roots of numbers. • Find the cubes of numbers • Identify perfect cubes. • Find the cube roots of numbers.

Key Terms: exponent, base, square, perfect square, radical symbol, square root, principal square root, cube, perfect cubes, cube root

03-06 The Real Numbers

• Describe and give examples of irrational numbers. • Define sets of numbers, including natural numbers, whole numbers, integers, rational numbers,

irrational numbers, and real numbers, and classify numbers by one or more number sets. Key Terms: set, natural numbers, whole numbers, integers, rational number, terminating decimal, repeating decimal, irrational number, real numbers

Unit 4: Solving Equations and Inequalities

04-01 Solving Equations

by Adding and Subtracting

• Identify the variable in an equation. • Solve a simple equation using addition. • Solve a simple equation using subtraction. Key Terms: variable, equation, inverse operation

04-02 Solving Equations by Multiplying or

Dividing

• Solve a simple equation using multiplication. • Solve a simple equation using division.



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04-03 Solving Two-Step

Equations • Solve two-step equations involving addition, subtraction, multiplication, and division.

04-04 Solving Multi-Step

Equations

• Use the properties of equality to solve multi-step equations that require the use of the distributive property. • Use the properties of equality to solve multi-step equations that require combining like terms. • Use the properties of equality to solve multi-step equations with fractional coefficients. Key Terms: like terms, constant terms

04-05 Solving Equations with Variables on

Both Sides

• Use the properties of equality to solve two-step equations with variables on both sides of the equation. • Use the properties of equality to solve multi-step equations with variables on both sides of the equation.

04-06 Solving for a

Variable

• Use properties of equality to rearrange formulas to solve for a quantity of interest. • Solve an equation in two or more variables for a specified variable. Key Term: formula

04-07 Systems of Equations

• Determine whether an ordered pair is a solution of a system of equations. • Solve a system of linear equations. Key Terms: system of equations, solution of a system of equations

04-08 Inequalities

• Identify inequalities. • Sketch inequalities on a number line. • Describe the solutions of an inequality. Key Terms: inequality, number line



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04-09

Solving Inequalities with

Addition and Subtraction

• Solve a simple inequality with addition. • Solve a simple inequality with subtraction. • Graph the solution to a simple inequality. Key Terms: inequality, solution set

04-10

Solving Inequalities with Multiplication and

Division

• Solve an inequality with multiplication. • Solve an inequality with division.

04-11 Solving Two-Step

and Multi-Step Inequalities

• Evaluate a multi-step inequality. • Evaluate a multi-step inequality involving negative numbers. • Graph the solution for a multi-step inequality on a number line.

Unit 5: Statistics

05-01 Samples and

Surveys

• Classify data and sampling methods. • Identify possible sources of bias in sampling methods. • Choose which sampling method is most appropriate for a given situation. Key Terms: population, sample, survey, simple random sampling, systematic sampling, stratified random sampling, convenience sampling, self-selected sampling, bias

05-02 Organizing Data

• Organize data in tables. • Organize data in stem-and-leaf plots. Key Terms: data, frequency table, stem-and-leaf plot, back-to-back stem-and-leaf plot



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05-03 Measures of

Central Tendency

• Calculate the mean, median, and mode of a set of data. • Determine which measure of central tendency is most appropriate for a given data set. Key Terms: average, normal, median, measure of central tendency, mean, median, mode, outlier

05-04 Variability

• Calculate the range of a data set. • Find the five-number summary of a data set. • Create a box plot of a data set. Key Terms: measure of variation, range, quartiles, first quartile, second quartile, third quartile, five-number summary, box plot

05-05 Displaying Data

• Display data in bar graphs, pictographs, and histograms. • Display data in line graphs. • Recognize misleading graphs. Key Terms: bar graph, pictograph, histogram, line graph

05-06 Scatter Plots

• Create a scatter plot. • Describe a correlation for paired data sets. • Use a scatter plot to make a prediction. Key Terms: scatter plot, correlation, positive correlation, negative correlation, no correlation



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Unit 6: Geometric Shapes and Measurements

06-01 Points, Lines, Planes, and

Angles

• Identify basic geometric figures. • Name and classify angles. • Determine angle relationships. • Use the properties of geometric figures to solve problems. Key Terms: point, line, plane, line segment, ray, angle, vertex, acute, right angle, obtuse, straight angle, congruent, complementary angles, supplementary angles, vertical angles, parallel lines, intersecting lines, perpendicular lines

06-02 Parallel and

Perpendicular Lines

• Identify congruent and supplementary angles formed by parallel lines intersected by a transversal. • Find measures of angles formed by parallel lines intersected by a transversal, including cases in which the transversal is perpendicular to the parallel lines. Key Terms: transversal, alternate interior angles, alternate exterior angles, corresponding angles, same-side interior angles, congruent alternate interior angles, congruent alternate exterior angles, congruent corresponding angles, supplementary same-side interior angles

06-03 Triangles

• Classify triangles by their side lengths and angle measures. • Find unknown measures of angles in triangles. Key Terms: triangle, isosceles triangle, equilateral triangle, scalene triangle, right triangle, acute triangle, obtuse triangle, triangle sum theorem



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06-04 Polygons and

Circles

• Identify a polygon by its properties. • Find unknown measures of angles in polygons. • Identify special types of quadrilaterals. • Identify the special segments contained in a circle. Key Terms: polygon, regular polygon, parallelogram, rhombus, square, trapezoid, quadrilaterals, pentagons, octagons, circle, center, radius, chord, diameter

06-05 Coordinate Geometry

• Graph and classify polygons in the coordinate plane. • Find the coordinates of vertices of polygons in the coordinate plane.

06-06 Congruence

• Determine whether two polygons are congruent. • Use properties of congruent polygons to find unknown measures. Key Terms: correspondence, congruent figures, congruent angles, congruent segments, congruent polygons

06-07 Transformations

• Identify translations, rotations, and reflections. • Transform plane figures by using translations, rotations, and reflections. Key Terms: transformation, preimage, image, translation, reflection, rotation

06-08 Symmetry and Tessellations

• Identify line symmetry in figures. • Identify rotational symmetry in figures. • Use symmetry to complete figures. • Use patterns to create tessellations. Key Terms: line symmetry, rotational symmetry, tessellation


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Pre-Algebra IB Grade Level 8

Pre-Algebra IB introduces students to the following concepts and functions:

• area and perimeter of geometric shapes

• the Pythagorean theorem • ratios, rates, and proportions • scale drawings and scale models • percents • theoretical and experimental probability • odds • linear equations • graphing inequalities • arithmetic and geometric sequences • exponential and quadratic functions • inverse variation • adding, subtracting, and multiplying polynomials

A+nyWhere Learning System Pre-Algebra IB Teacher’s Guide Grade Level 8

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A+LS Pre-Algebra IB is a full semester course designed to follow the Pre-Algebra IA course. Students will progress through lessons that lay the foundation for algebraic and geometric concepts. The Pre-Algebra IB content is designed to exceed the standards of Common Core Math, Grade 8, which many schools today refer to as the Pre-Algebra course. Students are provided foundational coursework in an easy-to-use interactive style, which encourages them to work problems thoroughly and check their work.

Pre-Algebra IB is presented as a one-semester core course.

• All thirty-seven lessons contain a study guide, a practice test, and a

mastery test.

• Each lesson contains interactive learning objects. Students click through each step of a problem in order to advance to the next step, providing students the opportunity to read and comprehend at the pace best suited for their individual needs.

• A glossary of terms and a calculator are provided in each study guide and can be accessed by clicking the “Tools” menu at the upper right. The built-in glossary helps students by reinforcing the key terms that are used across the title. They can also find key terms from previous lessons when they need them most.

• A virtual coach, Greek mathematician and philosopher Pythagoras, appears throughout the course to provide students with additional information about key concepts taught in the lesson. Students simply point at the image of Pythagoras in the lower left corner to read the information.


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• Lessons are designed in units. Each unit groups related material as a foundation for the upcoming unit. Students should

master a complete unit prior to moving to the next.

• Interactive sample problems are provided in each lesson. This allows students to immediately check their understanding as they move throughout the lesson. If they get the problems wrong, they can easily navigate to the previous page in order to review the module again.

• Pre-Algebra IB has an overall objective of familiarizing students with the foundational concepts that will take them into advanced math studies of the high school level.

• By the end of the course, students should have a solid working knowledge of the following key concepts: area and perimeter, proportions and ratios, percents, probability and statistics, linear equations, sequences, functions, and polynomials.


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Unit 7: Area and Perimeter of Geometric Shapes

7-1 Perimeter and Circumference

• Calculate the perimeter of a polygon. • Calculate the circumference of a circle. • Calculate the diameter and radius of a circle when given its circumference. Key Terms: perimeter, circumference, pi

7-2 Area of Polygons

and Circles

• Calculate the area of various polygons. • Calculate the area of circles. Key Terms: area, base, height

7-3 The Pythagorean

Theorem

• Use the Pythagorean theorem to find the unknown side length of a right triangle. • Use the Pythagorean theorem to find the distance between two points on the coordinate plane. Key Terms: legs, hypotenuse, Pythagorean theorem

7-4 Drawing Three-

Dimensional Figures

• Identify parts of three-dimensional figures. • Draw three-dimensional figures using perspective and isometric grids. Key Terms: solids, polyhedron, face, edge, vertex, prism, pyramid, cylinder, cone, sphere, perspective drawing, horizon line, vanishing point, one-point perspective, two-point perspective, isometric drawing

7-5 Surface Area and Volume of Prisms

and Cylinders

• Calculate surface area of prisms and cylinders. • Calculate the volume of prisms and cylinders. Key Terms: solid, surface area, cylinder



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7-6

Surface Area and Volume of Pyramids and Cones

• Calculate surface area of pyramids and cones. • Calculate the volume of pyramids and cones. Key Terms: solid, surface area, regular pyramid, vertex, height, slant height, square pyramid, regular triangular pyramid, cone, lateral surface, right cone, height, slant height

7-7 Spheres

• Find the surface area of a sphere. • Find the volume of a sphere. Key Terms: sphere, center, radius of a sphere, diameter of a sphere, hemisphere, great circle

Unit 8: Proportions and Ratios

8-1 Ratios, Rates,

and Proportions

• Write ratios in simplest form. • Compare rates by calculating unit rates. • Use conversion factors to convert units of measurement. • Use cross products to determine whether two ratios form a proportion. Key Terms: ratio, rate, unit rate, conversion factor, proportion, cross products

8-2 Solving

Proportions

• Use cross products to solve proportions. • Write proportions to model real-world situations. Key Terms: ratio, proportion



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8-3 Dilations

• Identify dilations of plane figures. • Identify dilations in the coordinate plane. • Dilate figures in the coordinate plane. Key Terms: dilation, center of dilation, scale factor, enlargement, reduction

8-4 Similar Figures

• Determine whether figures are similar. • Find unknown dimensions in similar figures. Key Term: similar

8-5 Scale Drawings

and Scale Models

• Find the scale factor in a scale drawing or scale model. • Use the scale factor in a scale drawing or scale model to find unknown dimensions. • Compare lengths, areas, and volumes in scale drawings and scale models. Key Terms: scale drawing, scale model, scale

Unit 9: Percents

9-1 Percents

• Represent a percent as a decimal and as a fraction. • Use a proportion to solve problems involving percents. Key Term: percent

9-2 Problem Solving with Percents

• Use various methods to calculate the percent of a number. • Solve percent problems in varying contexts. Key Term: percent proportion



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9-3 Percent Increase

and Decrease

• Find percent increase. • Find percent decrease. • Solve problems involving percent increase and decrease. Key Terms: percent change, percent increase, percent decrease, percent discount, percent markup

9-4 Applications of Percents

• Solve real-world percent problems, including commissions, interest, taxes, and tips. • Estimate percents in real-world situations. Key Terms: commission, commission rate, interest, principal, simple interest

Unit 10: Probability and Statistics

10-1 Theoretical Probability

• Determine the theoretical probability of an event. • Find the probability of mutually exclusive events. • Find the probability of independent and dependent events. Key Terms: experiment, outcome, sample space, event, probability, theoretical probability, mutually exclusive events, independent events, dependent events

10-2 Experimental Probability

• Find the experimental probability of an event. • Use experimental probability to estimate the likelihood of an event. • Use experimental probability to make predictions. Key Terms: experimental probability, trials



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10-3 Counting Sample

Spaces

• Calculate permutations and combinations. • Use permutations and combinations to find probabilities. Key Terms: fundamental counting principal, combination, permutation, factorial

10-4 Odds

• Determine the odds in favor of an event and the odds against an event. • Estimate odds from an experiment. • Convert between probability and odds. Key Terms: odds in favor, odds against

Unit 11: Linear Equations

11-1 Linear Equations

and Slope

• Determine whether equations are linear. • Find the slope of a line. Key Term: linear equation

11-2 Slope-Intercept

Form

• Determine the y-intercept and slope of a line from its equation in slope-intercept form. • Graph a line from an equation in slope-intercept form. • Determine the slope-intercept form of the equation of a line from a graph. Key Terms: slope-intercept form, y-intercept



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11-3 Point-Slope Form

• Determine the coordinates of a point on a line and the slope of the line from its equation in point-slope form. • Graph a line from an equation given in point-slope form. • Determine the point-slope form of the equation of a line from a graph. Key Term: point slope form

11-4 Direct Variation • Given data that varies directly, find the constant of variation. • Set up and solve direct variation problems using slope as the constant of variation. • Represent direct variation relationships as linear functions in graph, table, and equation form.

11-5 Graphing

Inequalities in Two Variables

• Solve and graph linear inequalities in two variables. • Write a linear inequality in two variables from a given graph. Key Term: inequality

11-6 Lines of Best Fit

• Use a scatter plot to show the relationship between two quantities. • Determine whether the correlation between two quantities is positive or negative or whether there is no correlation. • Interpret scatter plots and lines of best fit and use them to make predictions. Key Terms: scatter plot, correlation, positive correlation, negative correlation, no correlation, line of fit, trend line, line of best fit



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Unit 12: Sequences and Functions

12-1 Arithmetic Sequences

• Determine whether a sequence is arithmetic. • Identify and use a common difference in an arithmetic sequence. • Find a given term of an arithmetic sequence. Key Terms: sequence, term, arithmetic sequence, common difference

12-2 Geometric Sequences

• Determine whether a sequence is geometric. • Identify and use a common ratio in a geometric sequence. • Find a given term of a geometric sequence. Key Terms: geometric sequence, common ratio

12-3 Other Sequences

• Use second differences to find terms in sequences. • Write a rule for a given sequence or find a term given the rule for a sequence. • Find a given term of the Fibonacci sequence. Key Terms: first differences, second differences, Fibonacci sequence, Fibonacci numbers

12-4 Functions

• Determine whether a relationship between two quantities is a function. • Represent a function in multiple ways. • Evaluate a function. Key Terms: function, domain, range, discrete graph, continuous graph, function notation, function rule, evaluate



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12-5 Linear Functions

• Identify linear functions from a table, rule, or graph. • Graph linear functions and interpret their graphs. Key Term: linear function

12-6 Exponential Functions

• Determine whether a rule or table represents an exponential function. • Evaluate exponential functions for given values of the variable. Key Terms: exponential function, base

12-7 Quadratic Functions

• Determine whether a rule represents a quadratic function. • Evaluate quadratic functions for given values of the variable. • Graph quadratic functions and identify characteristics of their graphs. Key Terms: quadratic function, quadratic parent function, parabola, axis of symmetry

12-8 Inverse Variation

• Find the constant of variation given data that varies inversely. • Set up and solve inverse variation problems. • Represent inverse variation relationships as functions in graph, table, and equation form. Key Terms: inverse variation, constant of variation, direct variation



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Unit 13: Polynomials

13-1 Polynomials

• Find the degree of a polynomial. • Classify polynomials by number of terms. • Evaluate polynomials for given values of the variables. Key Terms: monomial, constant, degree of a monomial, coefficient, polynomial, term, binomial, trinomial, degree of a polynomial, FOIL

13-2 Adding and Subtracting Polynomials

• Simplify polynomials. • Add two or more polynomials. • Subtract polynomials.

13-3 Multiplying Polynomials

• Multiply two monomials. • Multiply a polynomial by a monomial. • Multiply two binomials. Key Term: FOIL


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Algebra IA Grade Levels 9-10 A+LS Algebra IA introduces students to the following mathematical concepts:

• operations with integers, rational numbers, and real numbers

• properties of numbers

• solving equations and inequalities

• development of concepts related to functions

• linear equations and their graphs

• linear functions and systems of linear equations

A+nyWhere Learning System Algebra IA Teacher’s Guide Grade Levels 9–10

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A+LS Algebra IA is a full-semester course designed to teach students the first level of algebra studies in lesson units. Course content is designed to meet and exceed Common Core Standards for Mathematics: High School Algebra. Algebra IA uses interactivity and real-world applications to engage students and to transition from a basic level to a deeper understanding of mathematical concepts, such as analyzing and explaining the process of solving equations and inequalities, developing function concepts, and graphing linear equations.

Algebra IA is presented as a one-semester core course.

• All sixty-three lessons contain a study guide, a practice test,

and a mastery test.

• Each lesson contains interactive learning objects. These learning objects allow students to progress through a lesson at the pace best suited for their individual needs.

• A glossary of terms and a calculator are provided in each study guide and can be easily accessed by clicking the “Tools” menu at the upper right corner of the screen.

• A virtual coach, French mathematician René Descartes, appears throughout the course to provide students with additional information about key concepts taught in the lesson.


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• The content of the Algebra IA course is based upon the Common Core State Standards for Mathematics.

• The course is designed to provide comprehensive, core instruction in a traditional instructional style with an emphasis on

procedural fluency, conceptual understanding, and strategic competence.

• The Algebra IA course contains six units of study. Each unit is grouped by topic to provide seamless instruction from one lesson to the next. Topics covered include the following:

o Unit 1 reviews operations with integers, rational numbers, and real numbers as well as properties of these

numbers.

o Units 2 and 3 focus on analyzing and explaining the process of solving equations and inequalities.

o Unit 4 develops concepts related to functions.

o Units 5 and 6 cover linear equations and their graphs, including linear functions and systems of linear equations.

• Through completion of the Algebra IA course, students are provided with the foundations of algebraic concepts and properties that are necessary for advancement in upper-level mathematics in the course of their studies throughout high school and beyond.


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Lessons


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Unit 1: Real Numbers

1.1.1 Integers

• Define integers. • Classify integers as positive or negative • Identify the opposite of an integer. • Order and compare integers. Key Terms: positive integer, negative integer

1.1.2 Adding Integers

• Add integers. Key Term: absolute value

1.1.3 Subtracting Integers • Subtract integers.

1.1.4 Multiplying

and Dividing Integers

• Multiply two or more integers. • Divide two integers.

1.2.1 Rational and

Irrational Numbers

• Write rational numbers as decimals or fractions in simplest form. • Compare and order rational and irrational numbers. • Classify real numbers as rational or irrational. Key Terms: rational, simplified form, improper fraction, terminating decimal, repeating decimal, irrational number, real number

1.2.2 Fractions, Decimals,

and Percents

• Convert rational numbers among fraction, decimal, and percent forms. • Use rational numbers in fraction, decimal, and percent forms to solve problems.



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1.2.3 Real Numbers

• Define sets of numbers, including natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers, and classify numbers by one or more number sets. • Use set notation to describe the union or intersection of two sets of numbers. • Compare and contrast sets of numbers. Key Terms: set, element, set notation, subset, union, intersection, empty set

1.2.4 The Number Line

• Use a number line to plot real numbers. • Identify the coordinates of points on a number line. Key Terms: coordinate, pi

1.3.1 Exponents

• Define and use exponents and the parts of an exponential expression. • Define and use negative exponents and zero exponents. • Define and use properties of exponents. Key Terms: expanded form, exponential form

1.3.2 Scientific Notation

• Convert numbers between scientific and standard notation. • Add and subtract numbers written in scientific notation. • Multiply and divide numbers written in scientific notation. Key Term: scientific notation

1.3.3 Roots

• Use square and cube roots, and use fractional exponents to represent roots. • Use the properties of exponents and the fractional form to write properties of roots. • Simplify numerical expressions involving square and cube roots. Key Terms: principal square root, radical symbol, perfect square, perfect cube, rationalizing the denominator, conjugate



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1.3.4 Order of Operations

• Use the order of operations to simplify numerical expressions containing grouping symbols. • Use the order of operations to simplify numerical expressions containing exponents. • Use the order of operations to simplify numerical expressions containing the four basic operations: addition, subtraction, multiplication, and division. Key Terms: order of operations, grouping symbol

1.4.1 Number Properties

• Define closure of real numbers and other number sets within the real numbers and determine whether a set is closed under a given operation. • Define and use the additive and multiplicative identities for real numbers. • Define and use the additive and multiplicative inverses for real numbers. Key Terms: closure, closed, counterexample, additive inverse, additive identity, multiplicative inverse, multiplicative identity

1.4.2 Commutative Property

• Identify the commutative property of addition and use it to justify steps in simplifying an expression. • Identify the commutative property of multiplication and use it to justify steps in simplifying an expression. Key Terms: commutative property of addition, commutative property of multiplication

1.4.3 Associative Property

• Identify the associative property of addition and use it to justify steps in simplifying an expression. • Identify the associative property of multiplication and use it to justify steps in simplifying an expression. Key Terms: associative property of addition, associative property of multiplication

1.4.4 Distributive Property

• Identify the distributive property and use it to justify steps in simplifying an expression. Key Terms: order of operations, distributive property



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Unit 2: Algebraic Expressions and Equations

2.1.1 Algebraic Expressions

• Identify the parts of an algebraic expression, including variable, constant, coefficient, factor, and term. • Combine like terms to simplify algebraic expressions. • Use the distributive property to simplify algebraic expressions. • Use substitution and the order of operations to evaluate algebraic expressions. Key Terms: expression, variable, numerical expression, algebraic expression, term, constant, coefficient, like terms

2.1.2 Equations

• Classify equations as true, false, or open sentences. • Use substitution to check whether a given value of the variable is a solution of an equation. Key Terms: open sentence, equation, open equation, solution, identity

2.1.3 Verbal Statements

• Identify key verbal phrases that can be used to translate words to algebraic expressions and equations. • Convert verbal phrases and statements into algebraic expressions and equations. • Describe patterns and real-world situations using expressions and equations.

2.2.1 Solving One-

Step Equations

• Use the addition and subtraction properties of equality to solve one-step equations. • Use the multiplication and division properties of equality to solve one-step equations. Key Terms: subtraction property of equality, addition property of equality, inverse operation, one-step, multiplication property of equality, division property of equality

2.2.2 Solving Two-

Step Equations

• Use the addition, subtraction, multiplication, and division properties to solve two-step equations. • Explain the order of the steps used to solve two-step equations. Key Term: two-step equation



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2.2.3 Solving Multi-

Step Equations

• Use the addition, subtraction, multiplication, and division properties to solve multi-step equations that require the use of the distributive property. • Use the properties of equality to solve multi-step equations that require combining like terms. • Use the properties of equality to solve multi-step equations with decimal or fractional coefficients.

2.2.4

Solving Equations

with Variables on Both Sides

• Use the properties of equality to solve two-step equations with variables on both sides of the equation. • Use the properties of equality to solve multi-step equations with variables on both sides of the equation.

2.3.1

Solving Absolute

Value Equations

• Represent absolute value algebraically and geometrically and use these representations to simplify expressions containing absolute value symbols. • Solve one-step absolute value equations and check the validity of possible solutions. • Identify extraneous solutions. • Solve multi-step absolute value equations, including equations that contain extraneous solutions. Key Terms: absolute value, absolute value equation, extraneous solution

2.3.2 Literal Equations

• Use properties of equality to rearrange formulas to solve for a quantity of interest. • Solve an equation in two or more variables for a specified variable. Key Terms: literal equation, formula

2.3.3 Ratios, Rates,

and Unit Conversions

• Write and use ratios and rates, including unit rates. • Use ratios equivalent to 1 to convert units. Key Terms: ratio, equivalent ratio, rate, unit rate, unit analysis, dimensional analysis, conversion factor

2.3.4 Solving Proportions

• Write and solve proportions. • Use proportions to solve real-world problems, such as those involving percents and similar figures. Key Terms: proportion, cross products, cross products property, percent



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Unit 3: Inequalities

3.1.1 Inequalities and Their Graphs

• Represent the solutions of an inequality as a graph on a number line. • Write an inequality from a graph on a number line. • Use substitution to check whether a given value of the variable is a solution of an inequality. Key Term: inequality

3.1.2 Verbal

Statements of Inequalities

• Identify inequality symbols that represent key verbal phrases. • Convert verbal statements to algebraic inequalities. • Describe patterns and real-world situations using inequalities.

3.2.1

Solving Inequalities by Adding or Subtracting

• Use the addition and subtraction properties of inequality to solve one-step inequalities and graph the solution sets. Key Terms: addition property of inequality, subtraction property of inequality

3.2.2

Solving Inequalities

by Multiplying or Dividing

• Use the multiplication and division properties to solve one-step inequalities and graph the solution sets. • Explain why the inequality symbol is reversed when multiplying or dividing both sides by a negative number.

3.2.3

Solving Two-Step and Multi-Step Inequalities

• Use the addition, subtraction, multiplication, and division properties of inequality to solve two-step and multi-step inequalities and graph the solution sets. • Explain the order of the steps used to isolate the variable when solving two-step and multi-step inequalities.



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3.2.4

Solving Inequalities

with Variables on Both Sides

• Use the properties of inequality to solve two-step inequalities with variables on both sides and graph the solution sets. • Use the properties of inequality to solve multi-step inequalities with variables on both sides and graph the solution sets. Key Terms: contradiction, identity

3.3.1 Solving

Compound Inequalities

• Use the properties of inequality to solve compound inequalities containing AND and graph the solution set. • Use the properties of inequality to solve compound inequalities containing OR and graph the solution set. Key Term: compound inequality

3.3.2

Solving Absolute

Value Inequalities

• Solve one-step absolute value inequalities and check the validity of possible solutions. • Solve multi-step absolute value inequalities. Key Terms: absolute value equation, absolute value inequality

Unit 4: Functions and Graphs

4.1.1 The

Coordinate Plane

• Identify the parts of the coordinate plane, including the origin, axes, and quadrants. • Plot points on a coordinate plane and identify the coordinates of points in the plane. • Determine the location of a point in the coordinate plane based on the signs of the coordinates. Key Terms: coordinate, plane, axes, coordinate plane, origin, ordered pair, x-coordinate, y-coordinate, quadrant

4.1.2 Using Graphs

to Show Relationships

• Match a situation to a graph. • Analyze a graph that matches a given situation. • Draw a graph that represents a given situation. Key Terms: continuous graph, discrete graph



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4.1.3 Relations and Functions

• Describe relations and functions and use them to represent relationships between quantities. • Represent a relation using multiple representations, including a written explanation, a mapping diagram, a table, a graph, or an equation. • Determine whether a relation in the form of a table, equation, graph, or written expression is a function. Key Terms: relation, discrete, continuous, function

4.2.1 Function Notation and Rules

• Write a function rule from a table of data. • Write a function rule from a graph. • Identify independent and dependent variables. Key Terms: function notation, function rule, independent variable, dependent variable

4.2.2 Domain and Range

• Determine the domain and range of a function expressed as a table. • Determine the domain and range of a function expressed as a graph. • Determine the domain and range of a function expressed as an equation. Key Terms: domain, range

4.2.3 Evaluating Functions

• Determine the output value of a function expressed as a table for a given input value in the domain. • Determine the output value of a function expressed as a graph for a given input value in the domain. • Determine the output value of a function expressed as an equation for a given input value in the domain.

4.2.4 Graph of a Function

• Graph a function given in table form by plotting points. • Graph a function given in equation form by generating a table of values and plotting points. Key Terms: linear function, constant function, absolute value function, quadratic function, cubic function, exponential function



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4.2.5 Analyzing Function Graphs

• Analyze the graph of a function. • Use the graph of a function to make predictions. Key Terms: x-intercept, y-intercept, minimum, maximum, domain, range, end behavior, symmetric

4.3.1 Function Operations

• Create new functions by adding, subtracting, multiplying, or dividing existing functions expressed as equations. • Create new functions by adding, subtracting, multiplying, or dividing existing functions expressed as tables. • Create new functions by adding, subtracting, multiplying, or dividing existing functions expressed as graphs. Key Terms: revenue, cost, profit

4.3.2 Inverse of a Function

• Given a function in table form, write the inverse and determine whether it is a function. • Given the graph of a function, graph the inverse and determine whether it is a function. • Given the rule for a function, find the rule for the inverse and determine whether it is a function. Key Term: inverse of a function

4.3.3 Sequences

• Write a sequence as a function whose domain is the set of natural numbers and describe the range of this function. • Use an explicit rule to find a specified term in a sequence. • Use a recursive rule and a given term to find the next term in a sequence. Key Terms: sequence, term, explicit rule, Fibonacci sequence

Unit 5: Linear Functions and Equations

5.1.1 Slope

• Define slope as a constant rate of change and recognize whether a slope is positive, negative, zero, or undefined. • Find the slope of a line using a graph or table. • Use the slope formula to calculate the slope of a line, given two points on the line. Key Terms: slope, rise, run, rate of change



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5.1.2 Direct Variation

• Given data that varies directly, find the constant of variation. • Set up and solve direct variation problems using slope as a constant of variation. • Represent direct variation relationships as linear functions in graph, table, and equation form. Key Terms: direct variation, constant of variation

5.1.3 Applying Rates of Change

• Interpret the slope as a rate of change in real-world situations. • Find the rate of change in a real-world situation and use it to make predictions. Key Terms: speed, rate of change, rise, run, slope, rate

5.2.1 Linear Functions

• Identify slope-intercept form, point-slope form, and standard form of a linear function. • Determine whether a function is linear based on a graph, equation, or table of values. • Identify the characteristics of the graph of a linear function given in equation, graph, or table form. Key Terms: linear function, x-intercept, y-intercept, rise, run, slope, slope-intercept form, point-slope form, standard form

5.2.2 Converting Between Forms

• Convert between slope-intercept and point-slope form of a line. • Convert between slope-intercept and standard form of a line. • Convert between point-slope and standard form of a line. Key Terms: slope-intercept form, point-slope form, standard form

5.2.3

Writing the Equation of

a Line

• Write the equation of a line in slope-intercept form, point-slope form, or standard form by using given information. • Write the equation of a line given the graph of the line. Key Term: Parameter



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5.2.4 Graphing

Linear Functions

• Graph linear functions.

5.2.5 Vertical and Horizontal

Lines

• Determine the slope of a vertical or horizontal line. • Write the equation of a vertical or horizontal line. Key Term: constant function

5.3.1 Parallel and

Perpendicular Lines

• Determine whether two lines are parallel, perpendicular, or neither. • Write the equation of a line that is parallel to another line and passes through a given point. • Write the equation of a line that is perpendicular to another line and passes through a given point. Key Terms: parallel, perpendicular, opposite reciprocals

5.3.2 Graphing

Linear Inequalities

• Solve and graph linear inequalities in two variables. • Write a linear inequality in two variables from a given graph. Key Term: inequality

5.3.3 Line of Best Fit

• Use a scatter plot to show the relationship between two quantities. • Determine whether the correlation between two quantities is positive or negative or whether there is no correlation. • Determine the equation of the line of best fit based on tables of data or scatter plots that relate to real-world situations. • Interpret scatter plots and lines of best fit and use them to make predictions. Key Terms: scatter plot, correlation, positive correlation, negative correlation, no correlation, line of fit, trend line, line of best fit, median-fit line



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5.3.4 Arithmetic Sequences

• Use a recursive rule and a given term to find the next term in an arithmetic sequence and relate the parts of the rule to the slope and a point on the line. • Use an explicit rule to find a specified term in an arithmetic sequence and relate the parts of the rule to the slope and y-intercept of the line. • Write an arithmetic sequence as a linear function whose domain is the set of natural numbers and describe the range of this function. Key Terms: sequence, arithmetic sequence, common difference, recursive rule, explicit rule

Unit 6: Systems of Equations and Inequalities

6.1.1

Solving Systems of

Linear Equations by

Graphing

• Determine whether an ordered pair is a solution of a system of equations. • Use a graph to determine the number of solutions of a system of linear equations. • Use a graph to estimate the solution of a system of linear equations and verify by using substitution. Key Terms: system of equations, system of linear equations, consistent and independent, consistent and dependent, coincide, inconsistent

6.1.2

Solving Systems of

Linear Equations by Substitution

• Solve consistent, independent systems of linear equations by substitution. • Use substitution to show that systems of linear equations are inconsistent or dependent and interpret the meaning of each case. Key Terms: consistent and independent, consistent and dependent, inconsistent, substitution

6.1.3

Solving Systems of

Linear Equations by Elimination

• Solve consistent, independent systems of linear equations by elimination. • Use elimination to show that systems of linear equations are inconsistent or dependent and interpret the meaning of each case. • Prove that replacing an equation in a system of linear equations with a linear combination of the two equations produces a system with the same solution set. Key Terms: consistent and independent, consistent and dependent, inconsistent, elimination



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6.2.1

Solving Systems of

Linear Inequalities

• Graph systems of linear inequalities. • Use the graph of a system of linear inequalities to interpret the solution. Key Terms: system of linear inequalities, solution region

6.2.2 Linear Programming

• Graph the feasible region of a linear programming problem. • Use the graph of the feasible region to solve linear programming problems. Key Terms: linear programming, constraint, feasible region, objective function


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Algebra IB Grade Levels 9–10

Algebra IB introduces students to the following concepts and functions:

• exponential expressions and equations

• graphing exponential functions

• geometric sequences

• adding, subtracting, and multiplying polynomials

• factoring polynomials

• quadratic functions, equations, and inequalities

• square-root functions and graphs

• radical expressions and equations

• the Pythagorean theorem

• inverse variation

• rational expressions and equations

• measures of central tendency

• matrices

• graphing data

• theoretical and experimental probability

• counting methods

A+nyWhere Learning System Algebra IB Teacher’s Guide Grade Levels 9–10

139

A+LS Algebra IB is a full-semester course designed to teach students the second level of Algebra studies in lesson units. Course content is designed to meet and exceed Common Core Standards for Mathematics: High School Algebra. The course uses interactivity and real-world applications to engage students and to transition from a basic level to a deeper understanding of mathematical concepts, such as transforming graphs of exponential functions, factoring polynomials, transforming and graphing quadratic and square-root functions, and collecting and analyzing data sets to make predictions.

Algebra IB is presented as a one-semester core course.

• All fifty-four lessons contain a study guide, a practice test, and

a mastery test.

• Each lesson contains interactive learning objects. These learning objects allow students to progress through a lesson at the pace best suited for their individual needs.

• A glossary of terms and a graphing calculator are provided in each study guide and can be easily accessed by clicking the “Tools” menu at the upper right corner of the screen.

• A virtual coach, French mathematician René Descartes, appears throughout the course to provide students with additional information about key concepts taught in the lesson.


140

• The content of the Algebra IB course is based upon the Common Core State Standards for Mathematics.

• The course is designed to provide comprehensive, core instruction in a traditional instructional style with an emphasis on

procedural fluency, conceptual understanding, and strategic competence.

• The Algebra IB course contains six units of study. Each unit is grouped by topic to provide seamless instruction from one lesson to the next. Topics covered include the following:

o Unit 7 introduces the properties of exponents, exponential functions, and exponential equations, including exponential growth and decay.

o Unit 8 focuses on polynomials: classifying, adding, subtracting, multiplying, dividing, and factoring polynomials.

o Unit 9 covers quadratic equations: identifying, graphing, and solving quadratic equations using several methods.

o Unit 10 concentrates on radical functions and equations: square-root function graphs, simplifying and adding radical expressions, the Pythagorean theorem, and the distance formula.

o Unit 11 focuses on rational functions and equations, finding an inverse of a function, graphing, adding, and multiplying rational expressions, and solving rational equations.

o Unit 12 introduces basic probability and statistics topics, such as measures of central tendency, graphs and charts that represent data, and theoretical and experimental probability.

• Through completion of the Algebra IB course, students are provided with the foundations of algebraic concepts and properties that are necessary for advancement in upper-level mathematics in the course of their studies throughout high school and beyond.


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Lessons


Grade Levels

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Algebra IB 54 1 9–10 Yes Yes

A+nyWhere Learning System Algebra IB Scope & Sequence Grade Levels 9–10

Lesson Title Lesson Content

142

Unit 7: Exponential Functions and Equations

7.1.1 Exponential Expressions

• Use properties of exponents to simplify algebraic expressions. • Multiply and divide algebraic expressions that contain exponents. Key Terms: product of powers property, quotient of powers property, power of a power property, power of a product property, power of a quotient property, exponential expression

7.1.2 Solving

Exponential Equations

• Solve exponential equations with the bases equal on both sides. • Solve exponential equations by transforming one or both sides so that the bases are equal. Key Terms: exponential equation, compound interest, half-life

7.2.1 Exponential Functions

• Determine whether a rule or table represents an exponential function. • Evaluate exponential functions for given values of the variable. Key Terms: exponential function, base, undefined, indeterminate

7.2.2 Graphing

Exponential Functions

• Identify the characteristics of the graph of an exponential function. • Graph exponential functions. • Write a rule for an exponential function, given a graph. Key Term: asymptote

7.2.3 Transforming Exponential Graphs

• Transform the graph of an exponential function. • Write the rule for the transformed graph of an exponential function. Key Terms: parent function, transformation, translation, reflection, vertical stretch, vertical compression



143

7.2.4 Geometric Sequences

• Write a geometric sequence as an exponential function whose domain is the set of natural numbers and describe the range of this function.

• Use an explicit rule to find a specified term in a geometric sequence and relate the parts of the rule to the function.

• Use a recursive rule and a given term to find the next term in a geometric sequence and relate the parts of the rule to the function.

Key Terms: sequence, geometric sequence, common ratio, recursive rule, explicit rule, power of a quotient property, exponential function, base

7.3.1 Exponential Growth and Decay

• Determine the rate of exponential growth or decay. • Solve real-world problems involving exponential growth and decay. Key Terms: exponential growth, exponential growth function, exponential decay, exponential decay function, compound interest, half-life

7.3.2 Comparing Linear and Exponential

Functions

• Compare linear and exponential functions. • Determine whether a linear or exponential model is a better fit for a given real-world situation and use

the models to make predictions.

Unit 8: Polynomials and Factoring

8.1.1 Polynomials

• Classify polynomials by degree and number of terms. • Write polynomials in standard form. • Evaluate polynomials for given values of the variables. Key Terms: monomial, coefficient, degree of the monomial, polynomial, term, binomial, trinomial, degree of a polynomial, standard form, leading coefficient, constant polynomial, linear polynomial, quadratic polynomial, cubic polynomial, quartic polynomial, quintic polynomial



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8.1.2 Adding and Subtracting Polynomials

• Add two or more polynomials. • Subtract two or more polynomials. Key Term: like terms

8.1.3 Multiplying Polynomials

• Use the distributive property to multiply a polynomial by a monomial. • Use the distributive property to multiply two or more polynomials. • Use the FOIL method to multiply two binomials. Key Term: FOIL

8.2.1

Factoring Polynomials Using

the Greatest Common Factor

• Find the greatest common factor of two or more monomials. • Find the greatest common factor of the terms of a polynomial. • Factor a polynomial using the greatest common factor. Key Terms: factoring, completely factor, greatest common factor, factor tree, prime polynomial

8.2.2 Factoring x2 + bx + c

• Factor expressions of the form x2 + bx + c where c > 0. • Factor expressions of the form x2 + bx + c where c < 0. Key Term: quadratic trinomial

8.2.3 Factoring ax2 + bx + c

• Factor expressions of the form ax2 + bx + c where a ≠ 1 and c > 0. • Factor expressions of the form ax2 + bx + c where a ≠ 1 and c < 0.

8.3.1 Factoring Special Products

• Factor perfect square trinomials. • Factor the difference of squares. • Factor the sum or difference of cubes. Key Terms: perfect square trinomial, difference of squares, sum of cubes, difference of cubes



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8.3.2 Factoring by Grouping

• Factor polynomials with four terms by grouping. • Factor trinomials by grouping. Key Term: factoring by grouping

8.3.3 Choosing a Factoring Method

• Determine which methods are most appropriate for factoring a given polynomial. • Factor polynomials using a combination of factoring methods.

Unit 9: Quadratic Functions and Equations

9.1.1 Quadratic Functions

• Determine whether a rule or table represents a quadratic function. • Evaluate quadratic functions for given values of the variable. Key Terms: monomial, polynomial, degree of a polynomial, quadratic function, standard form of a quadratic function, parent quadratic function, first differences, second differences

9.1.2 Graphing Quadratic Functions

• Identify characteristics of the graph of a quadratic function. • Graph quadratic functions. • Write a rule for a quadratic function, given a graph. Key Terms: parent quadratic function, parabola, axis of symmetry, vertex, end behavior, standard form

9.1.3 Transforming Quadratic Graphs

• Transform the graph of a quadratic function. • Write the rule for a transformed graph of a quadratic function.

9.1.4 Maximum and Minimum

• Locate the vertex of a quadratic function and use it to find the maximum or minimum of the function. • Use the maximum or minimum of a quadratic function to solve real-world problems. Key Terms: standard form, transformation form



146

9.2.1 Solving Quadratic

Equations by Graphing

• Solve quadratic equations by graphing. Key Terms: quadratic equation, related function, solutions of an equation, roots of an equation, zeros of a function

9.2.2 Solving Quadratic

Equations by Factoring

• Use the zero product property to solve quadratic equations by factoring. Key Term: zero product property

9.2.3 Completing the Square

• Solve quadratic equations using square roots. • Complete the square for an algebraic expression to create a perfect square trinomial. • Solve quadratic equations by completing the square.

9.2.4 Quadratic Formula

• Use the discriminant to determine the number of real solutions of a quadratic equation. • Solve quadratic equations by using the quadratic formula. Key Term: discriminant

9.3.1 Solving Quadratic Inequalities

• Solve quadratic inequalities by graphing. • Solve quadratic inequalities by factoring. • Solve quadratic inequalities by completing the square or using the quadratic formula.

9.3.2 Solving Non-Linear Systems

• Solve non-linear systems by graphing. • Solve non-linear systems algebraically. Key Term: non-linear system of equations



147

9.3.3 Comparing Linear, Exponential, and Quadratic Models

• Compare linear, exponential, and quadratic functions. • Determine whether a linear, exponential, or quadratic model is a better fit for a given real-world

situation and use the models to make predictions. Key Terms: linear function, exponential function

Unit 10: Radical Functions and Equations

10.1.1 Square-Root Functions

• Determine whether a rule represents a square-root function. • Evaluate square-root functions for given values of the variable. • Determine whether a table represents a square-root function.

Key Terms: square-root function, parent square-root function

10.1.2 Graphing Square-Root Functions

• Identify characteristics of the graph of a square-root function. • Graph square-root functions. Key Term: parent square-root function

10.1.3 Transforming Square-Root

Graphs

• Transform the graph of a square-root function. • Write the rule for a transformation of the graph of a square-root function.

10.2.1 Simplifying Radical Expressions

• Simplify algebraic expressions that contain radicals. • Simplify algebraic expressions by rationalizing the denominator. • Identify excluded values of the variables for rational expressions.

Key Terms: radical expression, perfect square, perfect cube, simplest radical form, rationalizing the denominator



148

10.2.2 Operations with

Radical Expressions

• Add and subtract radical expressions. • Multiply radical expressions. • Divide radical expressions. Key Terms: like radicals, product property of square roots, quotient property of square roots

10.2.3 Solving Radical Equations

• Solve radical equations and check the validity of possible solutions. • Identify extraneous solutions of radical equations. • Estimate solutions of radical equations from graphs. Key Terms: radical equation, equivalent equations, extraneous solution

10.2.4 Rational Exponents

• Convert expressions between radical and rational exponent forms. • Simplify expressions containing rational exponents.

Key Terms: rational number, rational exponent

10.3.1 Pythagorean Theorem

• Use the Pythagorean theorem to find side lengths of right triangles. • Apply the Pythagorean theorem and its converse to solve real-world problems. Key Terms: right triangle, legs, hypotenuse, Pythagorean theorem, converse of Pythagorean theorem

10.3.2 Distance and Midpoint Formulas

• Use the distance formula to find the distance between two points in the coordinate plane. • Use the midpoint formula to find the midpoint of the segment connecting two points in the coordinate

plane.



149

Unit 11: Rational Functions and Equations

11.1.1 Inverse Variation

• Find the constant of variation given data that varies inversely. • Set up and solve inverse variation problems. • Represent inverse variation relationships as rational functions in graph, table, and equation forms. Key Terms: inverse variation, constant of variation, direct variation, rational function

11.1.2 Rational Functions

• Determine whether a rule or table represents a rational function. • Evaluate rational functions for given values of the variable. Key Terms: rational function, excluded values

11.1.3 Graphing Rational Functions

• Identify the characteristics of the graph of a rational function.

• Graph and describe the parent rational function, , and transform its graph.

• Graph rational functions. • Write a rule for a rational function, given a graph. Key Terms: rational function, discontinuous function, asymptote

11.2.1 Simplifying

Rational Expressions

• Simplify rational expressions. • Identify excluded values of the variable for rational expressions. Key Terms: rational expression, excluded values, simplified rational expression, opposite binomials

11.2.2 Multiplying and

Dividing Rational Expressions

• Multiply rational expressions. • Divide rational expressions.

( ) 1f xx

=



150

11.2.3

Adding and Subtracting

Rational Expressions

• Add and subtract rational expressions with like denominators. • Add and subtract rational expressions with unlike denominators.

11.2.4 Rational Equations

• Solve rational equations and check the validity of possible solutions. • Identify extraneous solutions of rational equations. • Estimate solutions of rational equations from a graph. Key Term: rational equation

Unit 12: Probability and Statistics

12.1.1 Samples and Surveys

• Classify data and sampling methods. • Identify possible sources of bias in sampling methods. • Choose which sampling method is most appropriate for a given situation. Key Terms: population, census, survey, sample, simple random sampling, systematic sampling, stratified random sampling, cluster sampling, convenience sampling, self-selected sampling, bias

12.1.2 Measures of Central Tendency

• Calculate the mean, median, and mode of a set of data. • Determine which measure of central tendency is most appropriate for a given data set. Key Terms: data, numerical data, categorical data, parameter, statistic, measure of central tendency, mean, median, mode, outlier, symmetric distribution, distribution, skewed distribution, bimodal distribution

12.1.3 Measures of Variation and

Boxplots

• Calculate the range of a data set. • Calculate the interquartile range of a data set. • Use the five-number summary of a data set to make a box plot. Key Terms: measure of variation, range, interquartile range, quartile, first quartile, third quartile, five-number summary, boxplot, whiskers



151

12.2.1 Matrices

• Use matrices to organize data. • Perform operations with matrices. • Use operations with matrices to solve real-world problems involving data. Key Terms: matrix, dimension of a matrix, element, entry, scalar, scalar multiplication, scalar product

12.2.2 Graphing Data

• Organize and display data in stem-and-leaf plots. • Organize and display data in bar graphs. • Organize and display data in line graphs. • Choose an appropriate type of graph to display data. Key Terms: stem-and-leaf plot, stem, leaf, bar graph, pictograph, line graph

12.2.3 Frequency and Histograms

• Choose appropriate intervals and create a frequency table for a set of data. • Use a frequency table to make a histogram. • Summarize data in two-way frequency tables and determine relative frequencies. Key Terms: frequency, frequency table, histogram, symmetric distribution, skewed distribution, bimodal distribution, two-way frequency table, joint frequencies, marginal frequencies, relative frequency, conditional relative frequency

12.3.1 Theoretical Probability

• Determine the theoretical probability of an event. • Use theoretical probability to make predictions. • Find the expected value of a random variable. • Convert between probability and odds. Key Terms: experiment, trial, outcome, sample space, event, probability, impossible event, certain event, theoretical probability, equally likely events, fair, probability distribution, random variable, expected value, odds in favor, odds against

12.3.2 Experimental Probability

• Use experimental probability to estimate the likelihood of an event. • Use experimental probability to make predictions. Key Terms: experiment, trial, outcome, sample space, event, probability, experimental probability



152

12.3.3 Counting Methods

• Use the multiplication principle to determine the number of outcomes in a sample space. • Use formulas for permutations and combinations to determine the number of outcomes in a sample

space. • Use the number of outcomes in a sample space to determine the probability of an event. Key Terms: tree diagram, multiplication principle, factorial, permutation, combination

12.3.4 Probability of Compound Events

• Find probabilities of mutually exclusive and overlapping events. • Find probabilities of independent events. • Find conditional probabilities of dependent events. Key Terms: compound event, mutually exclusive events, overlapping events, independent events, dependent events


Algebra I: A Function Approach Part 1 Grade Levels 9–10

A+LS Algebra I: A Function Approach Part 1 introduces students to:

• independent and dependent variables • verbal and algebraic models to write algebraic

expressions • patterns as algebraic expressions • the relationship between variables in a situation • domain and range for an application problem • positive and negative correlations in a real-world context • graphs to represent a described situation • functions from tables, graphs, descriptions, and

mappings • conversion between fractions, decimals, and percents • direct variation relationship between diameter and

circumference • proportional reasoning to create tables, graphs, and

equations to make predictions from data • interpretation of constant of variation in real-world

context • proportions as linear functions • slope as a rate of change from a graphical and physical

point of view • lines with no slope, zero slope, positive slope, and

negative slope • graphing an equation from slope-intercept form • drawing graphs of lines in y = mx + b form • writing the equation of a line

A+nyWhere Learning System Algebra I: A Function Approach Part 1 Teacher’s Guide Grade Levels 9–10

154

Algebra I: A Function Approach Part 1 is designed to provide students with varied approaches to solving real-world application problems. This course focuses on identifying functional relationships including determining dependence, identifying and analyzing rate of change, making predictions from data, and using data to generalize and develop equations to predict trends. The primary focus is on developing linear functions and solving linear equations, linear inequalities, and linear systems. Developing quadratic functions and solving quadratic equations are covered to a lesser extent and exponential functions are introduced.

• Algebra I: A Function Approach Part 1 is presented as a

semester-long high school course.

• All lessons contain a study guide, a practice test, and a mastery test.

• Many lessons include graphing calculator activities

designed to expand the student's understanding of functional relationships and data representation. A virtual, fully functional graphing calculator designed specifically for this course is included. A graphing calculator tutorial is also included.

• Enriched interactive features include animations, off-line investigations, drag and drop, layered sequence activities, and other processes that allow the student to respond to the material in a variety of ways. Each feature is designed to fully engage the student with the course content and provide the student varied methods to master the skills presented.


155

• Study guides contain practice problems with immediate feedback and additional explanation of algebraic applications, allowing students to self-correct as they progress through the course.

• This course is enriched by National Library of Virtual Manipulatives (NLVM) applets.

• The content in this course addresses many objectives of the National Council of Teachers of Mathematics Principles and

Standards for School Mathematics.

• The Algebra I: A Function Approach Part 1 course is designed to help the student understand that functions represent a relationship of dependence, which can be represented in a variety of ways including tabular, graphical, and symbolic. The student will integrate the mechanics of algebra with real-life applications and gain an appreciation of the many ways algebra is used in everyday situations.

• The student will gain an understanding of how algebra can be used to express generalizations, learn to recognize and use the power of symbols to represent situations, and understand different algebraic methods used to solve problems.

• The student will learn to identify and solve proportional or non-proportional linear relationships in application situations. They will obtain an understanding for the meaning of the slope, intercepts, and zeros of the graphs of linear functions. They will learn to interpret and describe the effects of parameter changes for linear functions in real world and mathematical situations.

• The student will formulate systems of linear equations and inequalities from problem situations, use a variety of methods to solve the system, and analyze the system's solution in terms of the situation.

• The student will understand that graphs of quadratic functions are affected by the parameters of the function and interpret and describe the effects of changes to these parameters. They will learn different methods to solve quadratic equations and determine which method is appropriate for a given situation.

• The student will understand there are situations modeled by functions that are neither linear nor quadratic and model those situations.


156





157

The lessons in Algebra I: A Function Approach Part 1 are divided into three units of study. These units are designed to guide the student through mastery of algebra concepts. Each lesson contains a primary skill and utilizes multiple skills from the group skill set.

Units Lessons

Unit 1: Foundations for Functions Lessons: 1–15

Unit 2: Introduction to Linear Functions Lessons: 16–22

Unit 3: Linear Functions Lessons: 23–37

The following lessons contain Reflect and Apply worksheets:

• Independent and Dependent Variables • Domain and Range • Interpreting Graphs • Slope and Rate of Change • The Slope-Intercept Equation • Linear Functions and Parameter Changes.

These worksheets provide additional problems that can be completed independently or presented for whole-class instruction. A Solutions Manual for instructor use is provided as a separate document in .PDF form. This manual contains the questions and answers for all Reflect and Apply worksheet problems.


158

A+ VLabs Some lessons contain links to the A+ VLabsTM. These virtual laboratories are interactive experiments in which students observe, record, and enter data using methods similar to live laboratories. Please note that the A+ VLabs module is purchased separately, as a supplement to Algebra I, A Functional Approach, Parts 1 and 2.

The icon for A+ VLabs, , is found on the pages of a lesson specific to the concept being taught.

A+nyWhere Learning System Algebra I: A Function Approach Part 1 Scope & Sequence Grade Levels 9–10


159

Unit 1: Foundations for Functions

1 Mathematical Models 1 Identify independent and dependent variables; organize and represent independent and dependent values in a graph or tabular model

1190Q

Study: Graphing Calculator, Drag and Drop

Activities (2), Layered Sequence

2 Mathematical Models 2 Convert between verbal and algebraic models to write algebraic expressions; use the model to make predictions; find specific function values


Activity, Graphing Calculator

3 Identifying Patterns 1 Express a pattern as an algebraic expression, use the algebraic expression to make predictions about the pattern 800Q Study: Fill in Charts (2)

4 Identifying Patterns 2 Find a rate of change; find an algebraic rule for a pattern, determine the constant value in a pattern, use the rule to make predictions about the pattern

930Q


Activity, Fill in Charts (2), Layered Sequence

A+ VLabs: Functions and Patterns

5 Independent and Dependent Variables

Determine the relationship between variables in a situation; determine independent and dependent quantities on a graph, table, situation, or equation

1190Q Study: Fill in Chart, Reflect

and Apply Worksheet

6 Domain and Range

Identify a reasonable domain and range for an application problem; determine domain and range from graph or table; express domain and range in set-builder notation; identify discrete and continuous graphs

1190Q

Study: Drag and Drop Activity, Layered Sequence, Reflect and Apply Worksheet



160

7 Graphing a Story Identify independent/dependent quantities for a situation; determine reasonable domain and range for a situation; develop a graph for a functional relationship

780Q

Study: Drag and Drop Activity, Reflect and Apply Worksheet

A+ VLabs: Matching Graphs to Data

8 Domain and Range from Equations

Calculate a range for a given domain; calculate a domain for a given range 1190Q

9 Correlations 1 Recognize positive correlations in a real-world context; develop a scatter plot; determine domain and range for a scatter plot

1190Q Study: Graphing Calculator

Drag and Drop Activity, Fill in Chart

10 Correlations 2 Recognize negative correlations in a real-world context; recognize negative correlations from a graph; develop a scatter plot


11 Correlations 3 Recognize no correlation, positive correlation, and negative correlation in varied real-world contexts 800Q

Study: Graphing Calculator Drag and Drop Activities (2)

12 Graphs of Real-World Situations

Create graphs to represent a described situation; read a graph representing a relationship between two variables, e.g., height vs. time

780Q

13 Interpreting Graphs

Match or create real-world descriptions of graphs; determine reasonable domain and range for a situation; distinguish between continuous and discrete graphs; identify graphs with a constant rate of change

950Q

Study: Layered Sequences (3)

A+ VLabs: Matching Graphs to Data

14 Functions 1 Develop the definition of function; identify a relation; identify a mapping 1190Q Study: Animation (2), Drag

and Drop Activity



161

15 Functions 2

Identify functions from tables, graphs, descriptions, and mappings; vertical line test 1190Q

Study: Animation (3), Layered Sequence

A+ VLabs: Vertical Line

Unit 2: Introduction to Linear Functions

16 Proportions as Functions 1

Convert between fractions, decimals, and percents; identify ways to express a ratio; collect and organize data to use in proportions; use data to determine functional relationships; solve problems using percents and proportions

860Q Study: Fill in Charts (5)


Function concepts in proportions; relate direct variation to constant of variation; solve problems involving proportional change

890Q Study: Drag and Drop Activity, Fill in Chart


Use simple proportions to solve one-step equations; determine constant of variation; convert proportions to direct variation functions

890Q

Study: Graphing Calculator (6)

Drag and Drop Activity

19 Investigations with the Constant of Variation

Learn direct variation relationship between diameter and circumference; introduce slope as a constant of variation; determine the constant of variation for linear data; use proportional change to solve problems

1140Q

Study: Graphing Calculator (11) Fill in Charts (4), Off-Line Investigation

A+ VLabs: Time and Distance



162

20 Predictions from Data Use proportional reasoning to create tables, graphs, and equations to make predictions from data; create a scatter plot; identify a line of best fit

800Q

Study: Graphing Calculator (10) Fill in Charts (3), Off-Line Investigation

A+ VLabs: Line of Best Fit

21 Direct Variation Interpret constant of variation in real-world context; identify direct variation relationship from a table or graph; interpret meaning of axes intercepts in real-world contexts


(6) Fill in Chart

22 Summary of Direct Variation Functions

Summary of proportions as linear functions; represent data in graphs, tables, and scatter plots; calculate the constant of variation; generate a direct variation equation to solve a problem or make a prediction about data


(10) Fill in Charts (4)

Unit 3: Linear Functions

23 Introduction to Rate of Change

Introduce slope as a rate of change from a graphical and physical point of view; introduce formal definition of slope; determine slope of a line from graphs and tables

1140Q

Study: Animation, Fill in Charts (3), Layered Sequences (4), NLVM Activities (2), Plot the Line

24 Slope and Rate of Change

Slope in additional real-world situations; slope as a rate of change between two quantities; interpret how changes in the rate affect changes in a graph

1190Q


Activity, Fill in Charts (2), Layered Sequences (2), Plot the Point, Reflect and Apply Worksheet



163

25 Rate of Change Given 2 Points Slope formula 1140Q



26 Rate of Change and Slope Summary

Slope in additional real-world situations; slope as rise/run; find slope from graph of a line; relate slope formula to graph 1140Q

Study: Drag and Drop Activity, Layered Sequence

A+ VLabs: Rise and Run

27 Graphing with Slope 1 Identify lines with no slope, zero slope, positive slope, and negative slope; use slope formula to calculate no slope, zero slope, positive and negative slope

1140Q

28 Graphing with Slope 2 Slope as a parameter change of the equation y = mx; identify how the value of the slope affects the steepness of a line 1150Q

Study: Graphing Calculator (6) Drag and Drop Activities (3), Layered Sequences (2)

29 Constant of Variation Real-world situations for y = mx; identify how changes in rate of change affect the possible outcome for a situation 1140Q

Study: Graphing Calculator (4) Fill in Charts (3)



164

30 The Slope-Intercept Equation Introduce meaning of y-intercept in real-world contexts 1140Q

Study: Graphing Calculator (4) Plot the Line, Reflect and Apply Worksheet

A+ VLabs: Slope-Intercept

31 Linear Functions and Parameter Changes

Slope and y-intercept as parameter changes; graph an equation from slope-intercept form; evaluate a function using function notation

1190Q

Study: Graphing Calculator (2) Animations (4), Fill in Charts (2), Layered Sequences (3), Plot the Line, Reflect and Apply Worksheet

32 Applications of y = mx + b

Advanced real-world contexts for linear functions; identify slope and y-intercept to develop a linear function for the real-world context; identify the domain and range for the real-world context

1190Q

Study: Animation, Graphing Calculator, Drag and Drop

Activities (2) A+ VLabs: Rate of

Change

33 Parameter Changes in y = mx + b

Identify numerical and graphical parameter changes for slope and y-intercept; write an equation of a line in slope-intercept form from a graph

1050Q

Study: Graphing Calculator (6) Animation, Drag and Drop Activities (2), Layered Sequences (3)



165

34 Linear Equations from Two Points

Write an equation of a line in slope-intercept form when given two points on the line, a graph of the line, or data in an application problem

1110Q


A+ VLabs: Linear Equations from Two Points

35 The Point-Slope Equation

Write the point-slope equation for a line given the slope and point on a line; use the equation to answer questions about the function

1130Q


Drag and Drop Activity

36 Graphs of y = mx + b Draw graphs of lines in y = mx + b form; name slope-intercept equation from a graph; change an equation from point-slope to slope-intercept form

1140Q Study: Layered Sequences

(3), NLVM Activity, Plot the Line

37 Summary of Linear Equations Review of concepts in lessons 29–36 1140Q

Study: Fill in Charts (3), Graphing Calculator

(3) Layered Sequence


166

Algebra I: A Function Approach Part 2 Grade Levels 9–10

A+LS Algebra I: A Function Approach Part 2 introduces students to:

• the slope and y-intercept for linear functions in the context of real-life applications

• solving compound inequalities and graphing compound inequalities

• algebraically determining the number of solutions to a system

• determining the solution region for an inequality in standard form

• identifying the parameters of a quadratic function • using the power rule of exponents to simplify

exponential expression • identifying the graph and the equation of an inverse

variation function • writing linear equations using two points. • determining which linear equation form is most

appropriate for a given situation • writing and graphing compound inequalities • using matrices to represent real-world data • factoring polynomials by the greatest common factor

and difference of squares • solving equations using the quadratic formula • simplifying expressions using negative exponents • identifying exponential growth and decay • analyzing data involving inverse variation


167

Algebra I: A Function Approach Part 2 is a continuation of Algebra I: A Function Approach Part 1. Part 2 provides students with more approaches to the real-world application of algebra. The continued focus of this course is on functional relationships and the various uses of a rate of change. This course moves on to writing and solving equations, linear models in two variables, linear inequalities, and systems of equations and inequalities. Polynomials, their applications, and the factoring of polynomials are examined. Quadratics, their roots, factors, zeros, and solutions are introduced, followed by the quadratic formula, laws of exponents, exponential functions, and functions of inverse variation.

• Algebra I: A Function Approach Part 2 is presented as a

semester-long high school course.

• All lessons contain a study guide and practice and mastery test.

• Many lessons include graphing calculator activities designed

to expand students' understanding of functional relationships and data representation. A virtual, fully functional graphing calculator designed specifically for this course is included. A graphing calculator tutorial is also included.

• Enriched interactive features include Reflect and Apply worksheets, drag and drop activities, layered sequence activities, and other processes that allow the student to respond to the material in a variety of ways. Each feature is designed to fully engage the student with the course content and provide varied methods to help the student master the skills presented.


168

• Study guides contain practice problems with immediate feedback and additional explanation of algebraic applications,

allowing the students to self-correct their learning as the course progresses.

• This course is enriched by National Library of Virtual Manipulatives (NLVM) applets.

• The content in this course addresses many objectives of the National Council of Teachers of Mathematics Principles and

Standards for School Mathematics.

• The Algebra I: A Function Approach Part 2 course is designed to help students understand that functions represent a relationship of dependence that can be represented in a variety of ways, including tabular, graphical, and symbolic. Students will integrate the mechanics of algebra with real-life applications and gain an appreciation of the many ways algebra is used in everyday situations. Students will write and solve equations for real-world situations.

• Students will learn to analyze the slope and y-intercept for linear functions in the context of real-life applications and to translate a real-life application problem into an equation.

• Students will develop a point-slope linear equation given a rate and a data point and identify which linear equation form is most appropriate for a given situation. They will solve and graph inequalities and compound inequalities.

• Students will develop a system of equations, generate a table of values for a linear system, and solve a system of equations by graphing. They will solve a system of equations by graphing from slope-intercept form and from standard form. Students will understand the process used when solving a system of equations by matrices.

• Students will identify the graph and parameters of a quadratic function. They will understand monomials and

polynomials and how to arrange polynomials in ascending and descending order. Students will find the greatest common factor for a set of terms, factor a trinomial into two binomial factors, and solve a quadratic equation by using the quadratic formula.


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• Students will use the product rule and the power rule of exponents to simplify exponential expressions. They will simplify expressions using the quotient rule for exponents and those with negative exponents. Students will identify an exponential growth or decay function and the graph of an inverse variation function. They will analyze data and situations involving inverse variations.





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The lessons in Algebra I: A Function Approach Part 2 are divided into four units of study. These units are designed to guide the student through mastery of algebra concepts. Each lesson contains a primary skill and utilizes multiple skills from the group skill set.

Algebra I: A Function Approach Part 2 Units Lessons Unit 1: Linear Functions Lessons: 1–9 Unit 2: Linear Systems Lessons: 10–23 Unit 3: Quadratic Functions Lessons: 24–33 Unit 4: Introduction to Other Functions Lessons: 34–37

The following lessons contain Reflect and Apply worksheets: Linear Inequalities Applications 1, Applications of Linear Systems 2, Solving Systems Using Matrices, Solving Inequalities by Graphing, and Solving Systems of Inequalities and Applications. These worksheets provide additional problems that can be completed independently or presented for whole-class instruction. A Solutions Manual for instructor use is provided as a separate document in .PDF form. This manual contains the questions and answers for all Reflect and Apply worksheet problems.


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A+ VLabs Some lessons contain links to the A+ VLabs. These virtual laboratories are interactive experiments in which students observe, record, and enter data using methods similar to live laboratories. Please note that the A+ VLabs module is purchased separately, as a supplement to Algebra I, A Functional Approach, Parts 1 and 2.

The icon for A+ VLabs, , is found on the pages of a lesson specific to the concept being taught.



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Unit 1: Linear Functions

1 Linear Models: Function Notation

Write a linear equation model given the slope and the y-intercept and use the model to solve problems 1190Q Study: Layered Sequence

2 Linear Models Write a linear equation model given the slope and the y-intercept and use the model to solve problems; express linear functions as equations

1140Q Study: Layered Sequence

3 Writing Equations Write linear equations to solve word problems; solve multi-step equations 950Q Study: Layered Sequence

4 Real-World Models Using 2 Points

Write a slope-intercept equation for real-world models using two data points; use a slope-intercept equation to answer questions about real-world models

1140Q

Study: Layered Sequences (3),

A+ VLabs: Slope-Intercept, Linear Equations from Two Points

5 Real-World Models Using Point-Slope

Generate a point-slope equation for real-world models using the slope and point; use the equation to answer questions about the model


6 Choosing a Linear Model

Determine an appropriate linear form (slope-intercept, point-slope, or standard form) to model a situation 1130Q

Study: Drag and Drop Activity, Layered Sequences (4)

7 Solving Two-Step Equations

Write and solve 2-step equations for real-world situations; solve equations with variables on both sides of an equation 950Q

Study: Graphing Calculator Drag and Drop

Activity, NLVM Activity



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8 Linear Inequalities Applications 1

Use linear inequalities to represent situations and solve problems; use compound inequalities to represent situations 840Q Study: Reflect and Apply

Worksheet

9 Linear Inequalities Applications 2

Use linear inequalities to represent situations and solve problems, graph linear inequalities; write and graph compound inequalities


Unit 2: Linear Systems

10 Introduction to Linear Systems

Write a system of linear equations for an applied situation; understand the meaning of the solution; interpret the system's data represented in a graph or table

900Q

Study: Graphing Calculator (3),

Layered Sequences (4)

11 Solving Linear Systems Using a Table

Write and solve a system using a table; identify a solution to a linear system from a table; generate a table of values for a linear system

930Q

12 Solving Systems by Graphing 1

Solve systems by graphing; determine the number of possible solutions for a system of equations 900Q

Study: Graphing Calculator (7), Layered Sequences (3)

13 Solving Systems by Graphing 2

Solve systems by graphing; convert the equations in the system to y = or slope-intercept form 1000Q Study: Graphing Calculator

(6)

14 Writing Systems of Equations

Write a system to solve a problem; assign variables for a system of equations, write a system of equations 810Q Study: Drag and Drop

Activity, Fill-in Chart

15 Applications of Linear Systems 1

Write a system to solve a problem; translate a problem with two unknowns into a system of equations 810Q Study: Graphing Calculator

(2)



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16 Applications of Linear Systems 2

Write a system to solve a problem; translate an application problem with two unknowns into a system of equations 810Q

Study: Layered Sequence, Reflect and Apply Worksheet

17 Review: Solving Systems by Graphing

Solve systems by graphing using slope-intercept form and x- and y-intercepts for standard form 1110Q

Study: Graphing Calculator (10), Layered Sequences (5), Plot the Lines (3)

A+ VLabs: Rate of Change

18 Solving Systems by Substitution

Write a system and solve by substitution; algebraically determine the number of solutions to a system 990Q


19 Solving Systems by Elimination

Write a system and solve by elimination using addition and multiplication; use the elimination method to solve a system of equations

990Q Study: Layered Sequences (4), Fill-in Chart

20 Solving Systems Using Matrices

Write a system in matrix notation; identify the matrix product that will produce the solution to a system if solved by matrices 1100Q

Study: Drag and Drop Activity, Layered Sequence, Reflect and Apply Worksheet

21 Exploring Systems of Linear Inequalities

Write a system of linear inequalities and explore reasonable solutions; determine if an ordered pair is a solution of linear inequalities


22 Solving Inequalities by Graphing

Graph a linear inequality on a coordinate plane; determine the solution region for an inequality in slope-intercept and standard form

1050Q

Study: Plot the Lines (3), Layered Sequences (2), Reflect and Apply Worksheet



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23 Solving Systems of

Inequalities and Applications

Write and solve linear systems of inequalities; write and solve a system of inequalities for an application problem 1150Q

Study: Plot the Line, Reflect and Apply Worksheet

Unit 3: Quadratic Functions

24 Exploring Quadratic Functions

Identify a parabola; analyze graphs of quadratic functions in real-life applications; identify the graph of a quadratic function 1090Q

Study: Graphing Calculator (2), Layered Sequence

25 Graphs of Quadratic Functions

Examine the effects of parameter changes for a, b, and c on the graph of a quadratic function 1220Q

Study: Graphing Calculator (8), Drag and Drop Activities (3), Layered Sequences (3)


26 Adding Polynomials Define monomial and polynomial; add and subtract polynomials; write polynomials in descending order 1050Q


27 Naming Polynomials Identify degree of a monomial and polynomial; arrange polynomials in descending or ascending order by specific variable

820Q

Study: Drag and Drop Activities (5), Layered Sequences (2)

28 Polynomials in Perimeter and Area

Multiply polynomials; find perimeter and area of plane figures where length of a side is represented as a polynomial expression, Box and FOIL method

1050Q

Study: Graphing Calculator (3), Drag and Drop Activity, Fill-in Charts (2), Layered Sequences (6), NLVM Activity



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29 Multiplying Polynomials Multiply polynomials using the distributive property in a linear and box format 1050Q

Study: Drag and Drop Activities (2), Layered Sequences (5), NLVM Activity

30 Factoring Polynomials: GCF and Difference of

Squares

Factor quadratic polynomials by the greatest common factor and by difference of squares 1140Q

Study: Drag and Drop Activity, Fill-in Chart, Layered Sequences (2)

31 Factoring Trinomials Factor quadratic trinomials; factor quadratic trinomials to solve area problems 1130Q

Study: Fill-in Charts (4), Layered Sequences (8)

32 Roots, Factors,

Zeros, and Solutions of Quadratics

Define and make connections between roots, factors, zeros, and solutions of quadratics; solve quadratic equations by factoring and graphing

1200Q

Study: Graphing Calculator (13),

Drag and Drop Activities (2),

Layered Sequences (5)

33 The Quadratic Formula

Find solutions of a quadratic equation by the quadratic formula; identify, write and solve quadratic equations 1200Q

Study: Graphing Calculator Drag and Drop

Activity, Layered Sequence



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Unit 4: Introduction to Other Functions

34 Laws of Exponents 1 Use laws of exponents to find the product of monomial expressions; simplify monomial expressions raised to a power 1000Q

Study: Check for Understanding, Fill-in Chart

35 Laws of Exponents 2 Use laws of exponents to simplify and combine monomial expressions; learn quotient rule, definition of negative exponent and definition of zero as an exponent


36 Introduction to Exponential Functions

Introduce exponential functions in practical applications; define, write, and recognize functions 1050Q

Study: Graphing Calculator (9), Drag and Drop Activity, Layered Sequences (3)

37 Inverse Variation Functions

Introduce inverse variations functions in practical applications; identify equations for inverse variation; identify graph of inverse variation functions

1040 Study: Graphing Calculator (2)


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Algebra II: Part 1 and 2 Grade Levels 10–12

A+LS Algebra II: Part 1 and 2 introduces students to the following concepts and functions and more: Algebra II: Part 1

• rational numbers • rules for combining and multiplying real numbers • order of operations • connecting words and numbers through expressions • grids • quadrants • defining linear equations • three-variable equations • matrix multiplication • polynomial types

Algebra II: Part 2

• review of square roots • radicals • solving and factoring • identifying and evaluating the discriminant of a

quadratic equation • graphing parabola • circle parts and formulas • hyperbola • graphing quadratic relations and inequalities

I A+nyWhere Learning System Algebra II: Part 1 Scope & Sequence Grade Levels 10–12


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1 Algebra II: Part 1

Rules of Algebra

Review of the real number system including rational numbers, integers, whole numbers, counting numbers, and irrational numbers; rules for combining and multiplying real numbers and order of operations.

2 Real Number Properties

Review of properties of real numbers; associative property of multiplication and division, distributive property, substitution property; terms associated with real number properties and operations, and review of inequalities.


3 Algebraic Expressions

Connecting words and numbers through expressions, students practice writing and simplifying expressions.

4 Algebraic Equations Difference between expressions and equations, symbols used in writing equations, identifying unknowns.

5 Solving Equations Rules for solving equations, combining like terms, step-by-step examples of simplifying and solving equations.


6 Problem Solving 1

Developing equations to solve for unknowns, developing a plan to solve problems, and working related problems that develop from one original problem and checking answers for reasonability.


7 Rewriting Formulas Solving for variables with more than one unknown, converting Celsius to Fahrenheit and vice versa, isolating variables, multiplying by reciprocals. Study: NLVM Activity

8 Solving & Graphing Definition and examples of ordered pairs, x and y axes, and the coordinate plane, students write equations from information on grids, positive and negative slope.

9 Properties of Inequality

Rules and properties of inequalities, review of divisibility and multiplication properties.



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10 Inequalities Relating inequalities to variables, intersection and union, examples of solving and graphing inequalities.

11 Absolute Value Equations

Review of absolute values, determining absolute values as related to equations, comparing absolute values as solutions to equations, checking answers for reasonableness.

12 Absolute Value Inequality

Examples of positive and negative numbers in inequalities, inequalities having no solution.

13 Problem Solving 2

Converting words in problems into symbols, converting answers to similar terms, various problem solving examples and strategies.


14 Relations & Functions

Review of coordinate plane, quadrants, identifying origin, abscissa, ordinate, domain, range, and function, representing relations on graphs.



15 Graph Linear Functions

Defining linear equations, rise, run, slope, writing linear equations in standard form, graphs as linear functions, constant functions, x and y intercepts.


16 Slope of a Line Identification of positive, negative, zero, and undefined slopes, rise, run, relating slope to graphs.


17 Graph Linear Inequalities

Half planes and boundaries, writing equations and graphing in slope-intercept form, double-checking linear equality graphs.

18 Parallel & Perpendicular

Defining and graphing parallel and perpendicular lines on the coordinate plane, solving for parallel lines from points and slope, negative reciprocals as slopes. Study: NLVM Activities (2)



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19 Identify Linear Equations

Difference of slope-intercept form and standard form for linear equations, determining when to use point-slope, slope-intercept, x-intercept or y-intercept to graph linear equations, review of relations and functions,


20 Problem Solving 3

Identifying relationships between variables, checking answers for reasonableness, using equations to solve problems, using charts or other visual tools as aids in solving problems.

21 Direct Variation Definition of direct variations and examples of graphs of direct variations, proportionality constants, means as a product of extremes, using proportions to solve problems.

Study: NLVM Activity Essay: Written Response

22 Graphing Equation Systems

Characteristics of intersecting, coinciding, and parallel planes and systems of equations for each, comparing equations that have same slope, different slope, and different intercepts,

Study: NLVM Activities (2) Essay: Written Response

23 Graphing Systems Solving equations by graphing intersecting, coinciding, and parallel lines in planes, equations with infinite solutions, equations that have no solution. Study: NLVM Activities (3)

24 Addition & Substitution

Solving linear systems by addition and substitution, comparing solutions to problems worked using both methods, practicing using linear equations to solve everyday problems, hints for evaluating problems to find the best way to solve.


Essay: Logic

25 Solving Inequalities Illustrating inequalities with graphs and using them to find solutions, the effect of absolute value on graphs, adding and subtracting numbers inside and outside absolute value symbols.



26 Linear Programming Identifying variables, various constraints, and feasible regions in graphs, determining maximum and minimum values within feasible regions, the importance of linear programming as it relates to various careers,




182

27 Three-Variable Equations

Using matrices, Cramer’s rule, and/or addition to solve equations with three variables, graphing ordered triples, three-dimensional thinking in solving problems.



28 Data in Matrices Identifying and labeling data in matrices, performing operations using matrices, dimensions of matrices.



29 Matrix Multiplication Checking the dimensions of matrices before multiplication, products of matrices, step-by-step examples of multiplying matrices.



30 Size & Reflections Changes in size or magnitude and scale factor, examples using matrices in everyday life situations, coordinates of reflected images, graphing reflections. Study: NLVM Activities (2)

31 Transformation Definition of transformation, formula, point, and matrix transformations, commutative, associative, and identity properties with matrix multiplication, closed sets.


32 Rotation Definition and examples of rotation, relating rotation to angles, negative and positive magnitude, algebraic formulas for rotation, finding the images of rotations.


33 Matrix Addition

Discussion of rules of matrix addition and subtraction of elements, addition properties in matrices, adding three matrices, multiplying elements in matrices, subtracting matrices, using matrices to solve problems in everyday life.



34 Exponents How to utilize exponents as a shortcut method when multiplying variables and simplifying fractions.





183

35 Polynomial Types Definition and examples of monomials, binomials and polynomials, examples of like and unlike terms, determining the degree of polynomials.



36 Polynomial Operations

Graphing and factoring quadratic trinomials, linear terms, ascending and decreasing order of polynomials. Essay: Written Response

37 Factoring Quadratics

Graphing and factoring quadratic trinomials, linear terms, ascending and decreasing order of polynomials. Essay: Written Response

38 Polynomial Equations

Solving problems using polynomials equations. 5-step approach to solving problems. Formulas and computations for solving problems.



39 Negative Exponents Review of exponents and their uses, zero as an exponent, negative exponents, simplifying problem using positive and negative exponents.



40 Scientific Notation Definition and examples of scientific notation, using negative and positive exponents. Converting expressions from decimal form to scientific notation, significant digits.


41 Rational Operations 1

Common denominators, finding higher variables, step by step factoring and solving, adding subtracting, rationals by simplifying.



42 Rational Operations 2

Products of rational expressions, factoring numerators, and denominators of polynomials solving problems using rational expressions to solve practical problems.





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43 Simplifying Rationals

Formula for quotient of 2 polynomials, factoring polynomials review of ACF, quadratic trinomials, perfect squares and difference of squares.



44 Complex Rationals Definition and examples of complex rationals using shortcuts to simplify and solve complex rationals.





185

1 Algebra II: Part 2 Roots and Radicals

Review of square roots and perfect squares and radicals; solving problems containing radicals; inverse of squaring numbers, irrational numbers, principal square roots

Study: EB Learning Material Essay: Draw and Label

2 Real Number Properties 1

Multiplication and division of radicals, simplifying radical expressions, radical exponents, irrational numbers, product property of rationals


3 Real Number Properties 2

Addition and subtraction of radicals, like radicals and like terms, using the distributive property to solve problems

4 Rational Exponents Using addition, subtraction, multiplication, and division and combinations of operations to solve problems with rational exponents Essay: Written Response

5 Equations Identify radicals and solving equations with radicals Study: EB Learning Material

6 Imaginary Numbers Identification and problem solving using imaginary numbers Study: EB Learning Material

7 Complex Numbers 1 Solving addition and subtraction problems of complex pure and imaginary numbers


8 Complex Numbers 2 Multiplication and division of complex numbers, using the commutative property to solve problems, the FOIL method of factoring and solving


9 Quadratic Equations 1

Solving quadratic equations by completing the square, solving and factoring, completing the square to solve equations



10 Quadratic Equations 2

Using the quadratic formula to solve problems, checking for reasonableness of all solutions


Essay: Logic



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11 The Discriminant Identifying and evaluating the discriminant of a quadratic equation; using the discriminant to determine the number of solutions to an equation


12 Roots Equations involving the sum and products of roots and their connection to the coordinate plane



13 Quadratic Equations 3 Rewriting equations in quadratic form to solve Study: EB Learning

Material

14 Problem Solving Solving problems using quadratic equations Study: EB Learning Material

15 Quadratic Relations

Identifying and illustrating distance and midpoint, solving problems with number lines, absolute value, the Pythagorean Theorem



16 Parabolas Characteristics and definition of parabola



17 Graphing Parabola Plotting parabola on the coordinate plane



18 Circles Circle characteristics; solving problems involving identification of circle parts and formulas





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19 Ellipses Characteristics of ellipses; plotting ellipses on the coordinate plane, identification and illustration of fixed points


20 Hyperbola Characteristics of hyperbola, visual illustrations of hyperbola, intersection of planes and cones, identifying the difference between ellipses and hyperbola


21 Graphing Relations Identifying relations; identifying functions; graphing quadratic relations and inequalities


22 Graphing Inequalities

Intersections of graphs of quadratic relations, graphing conic inequalities and intersections Essay: Problem Solving

23 Variations Inverse and joint variations of linear functions; combined variation

24 Exponential Functions

Different strategies for simplifying and solving equations and expressions with rational positive and negative exponents


25 Inverse Functions Ordered pairs, coordinates, the domain, identification and illustrations of the inverse function

26 Logarithmic Functions

Identification and explanation of logarithmic functions, the exponential/logarithmic scale, definition and examples of logarithms


27 Exponential Equations

Definition and examples of exponential equations, solving problems using the graphing calculator, properties of logarithms, significant digits, compound interest problems

28 Arithmetic Sequence

Definition and examples of arithmetic sequences, difference of numbers, finite sequences of numbers




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29 Arithmetic Series Definition and examples of arithmetic series in real world situations, identification of sigma, solving problems using arithmetic series



30 Geometric Sequence

Definition and examples of geometric sequence, geometric progression, terms of geometric sequences


Essay: Logic

31 Geometric Series Definition and examples of geometric series, formulas for solving problems with geometric series



32 Infinite Geometric Series

Examples and definition of common ratios, formulas, convergent geometric series, solving problems with geometric series


33 Binomial Theorem Identification of patterns and integral powers, finite series, coefficients, variable powers, factorials, solving factorial problems Essay: Writing Equations


189

Geometry IA Grade Level 9–10

Geometry IA introduces students to the following concepts and functions and more:

• basic geometric concepts

• geometric reasoning including converse, inverse, and contrapositive of a conditional statement

• parallel and perpendicular lines and parallel planes

• congruent triangles

• triangle inequalities

• theorems and formulas

A+nyWhere Learning System Geometry IA Teacher’s Guide Grade Levels 9–10

190

A+LS Geometry IA is a full semester course designed to teach students the first level of Geometry studies with lessons that are based on the Common Core Standards for Mathematics. All lessons have been designed to help students understand key concepts by applying real-world knowledge. Topics covered in Geometry IA begin with the basic geometric concepts of points, lines, planes, segments, and angles, then progress into increasingly complex studies that include formulas, proofs, theorems, congruence theorems, ratios and proportions, and polygons.

Geometry IA is presented as a one-semester core course.

• All forty-five lessons contain a study guide, a practice test, and

a mastery test.

• Students are provided a number of practice questions within each lesson. These are designed to show the students correct and incorrect answers so that they may check their progress.

• An extensive toolkit is provided for the students in the upper right-hand corner of each page of the study guide, which includes a calculator, glossary, chart of symbols, and a complete list of formulas, theorems, and postulates.

• Many of the lessons contain a built-in protractor and a virtual

construction tool, which includes a pencil, straightedge, compass, and eraser.

• A virtual coach, Greek mathematician Euclid, appears throughout the lessons to provide additional information regarding

key concepts.


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• The content of the Geometry IA course is based upon the Common Core State Standards for Mathematics.

• Lessons are designed in a foundational structure. Each lesson provides material that the student will utilize in upcoming

lessons.

• Practice problems are demonstrated via interactive design concepts in each lesson. Students are then provided sample problems to work on their own. This allows students to immediately check their understanding as they move throughout the lesson. If they get the problems wrong, they can easily navigate to the previous page in order to review the lesson again. This type of interactivity also allows students to work at their own pace, moving forward when they feel they have mastered a new concept.

• Geometry IA is an in-depth look at the foundations of geometric concepts, postulates, formulas, and theorems that are

necessary to move on to the second semester of Geometry as well as other upper-level mathematics courses.


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Lessons


Grade Levels

Built-in Calculator

Crossword Puzzles


Geometry IA 45 1 9–10 Yes Yes Yes

A+nyWhere Learning System Geometry IA Scope & Sequence Grade Levels 9–10


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Unit 1: Geometry Basics

01 Points, Lines, and Planes

• Identify and name points, lines, and planes. • Define, identify, and name segments and rays. • Determine if points are collinear and coplanar. • Identify the intersection of lines and planes. • Understand and use the point, line, and plane postulates.

Key Terms: undefined, point, line, plane, space, line segment, endpoints, ray, collinear points, noncollinear, intersection, coplanar points, axiom, postulate

02 Segments

• Find the length of a segment whose endpoints are on a number line. • Identify congruent segments. • Use the Segment Addition Postulate. • Find the midpoint of a segment whose endpoints are on a number line. • Use the definition of a segment bisector to find unknown values. Key Terms: coordinate, congruency, between, midpoint, segment bisector

03 Distance and

Midpoint on the Coordinate Plane

• Identify and plot points on a coordinate plane. • Find the distance between two points on a coordinate plane. • Find the coordinates of the midpoint of a segment on a coordinate plane. Key Terms: coordinate plane, x-axis, y-axis, origin, quadrant, ordered pair, Distance Formula, radicand, Midpoint Formula



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04 Angles

• Name angles. • Use a protractor to measure angles. • Classify angles. • Use the Angle Addition Postulate to find missing angle measures. Key Terms: angle, vertex, side, interior, exterior, degrees, opposite rays, acute angle, right angle, obtuse angle, straight angle, congruent angle, angle bisector

Study: Protractor Tool

05 Basic Constructions

• Construct a segment congruent to another segment. • Construct a segment bisector. • Construct an angle congruent to another angle. • Construct an angle bisector. Key Terms: compass, straightedge

Study: Virtual Construction Tool

06 Angle Pairs

• Identify and find missing measures in linear pairs and vertical angles. • Solve equations containing expressions representing angle measures in

linear pairs and vertical angles. • Identify complementary and supplementary angles. • Solve equations containing expressions representing angle measures in

complementary and supplementary angles. • Solve problems involving complementary and supplementary angles. Key Terms: adjacent angles, linear pair, vertical angles, complementary angles, supplementary angles



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07 Formulas

• Find the perimeter and area of a rectangle, square, and triangle. • Find the circumference and area of a circle. • Determine unknown lengths in rectangles, squares, triangles, and circles

given the perimeter, area, or circumference. Key Terms: perimeter, formulas, radius, diameter, circumference, pi, irrational number, area, base, height

08 Problem Solving Strategies

• Use the Guess and Check strategy to solve geometry problems. • Use the Make a Table strategy to solve geometry problems. • Use the Solve a Simpler Problem strategy to solve geometry problems. • Use the Write an Equation strategy to solve geometry problems. Key Term: problem-solving strategy

Unit 2: Geometric Reasoning

09 Conditional Statements

• Write conditional statements in if-then form. • Identify the hypothesis and conclusion in a conditional statement. • Determine whether a conditional statement is true or false. • Provide counterexamples for false conditionals. • Identify and write the converse, inverse, and contrapositive of a

conditional statement. Key Terms: conditional statement, if-then form, hypothesis, conclusion, counterexample, related conditionals, converse, inverse, contrapositive, logically equivalent statements



196

10 Logic

• Write and interpret statements in symbolic notation. • Write truth tables for compound statements. • Draw and interpret Venn diagrams. • Draw and interpret Euler diagrams. Key Terms: statement, negation, compound statement, conjunction, disjunction, conditional statement, truth table

11 Inductive Reasoning

• Make a conjecture based on given information. • Explain why a conjecture is invalid. • Find a counterexample to show a conjecture is invalid. • Identify a numeric or geometric pattern. • Use inductive reasoning to predict the next item in a pattern. Key Terms: conjecture, inductive reasoning

12 Deductive Reasoning

• Use the Law of Detachment to form a conclusion. • Use the Law of Syllogism to form a conclusion. • Determine if a conclusion is valid. Key Term: deductive reasoning

13 Definitions and Biconditional Statements

• Define and write biconditional statements. • Write biconditional statements as two separate statements. • Determine whether biconditional statements are true. • Recognize poor definitions and explain why they are poor. • Write definitions as biconditional statements. Key Terms: biconditional statement, iff, definition

14 Algebraic Proof • Identify properties of equality. • Justify steps in an algebraic proof. • Prove algebraic statements true.



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15 Geometric Proof - Part 1

• Write and complete two-column geometric proofs. • Identify and use the Congruent Complements and Congruent

Supplements Theorems. Key Terms: theorem, two-column proof

Study: Protractor Tool

16 Geometric Proof - Part 2

• Write and complete geometric paragraph proofs. • Write and complete geometric flow proofs. • Identify and use properties of angle and segment congruence. Key Terms: paragraph proof, flow proof

Unit 3: Parallel and Perpendicular Lines

17 Lines and Transversals

• Identify parallel, perpendicular, and skew lines and parallel planes. • Understand the Parallel Postulate and its related theorems. • Classify angle pairs formed by lines intersected by a transversal. Key Terms: geometry, parallel lines, perpendicular lines, skew lines, parallel planes, perpendicular planes, parallel segments, perpendicular segments, transversal, interior, exterior, corresponding angles, alternate interior angles, alternate exterior angles, consecutive interior angles

Study: Crossword Puzzle (.PDF)

18 Angle Pair Theorems

• Determine measures of angles formed by parallel lines and a transversal. • Prove theorems related to measures of angles formed by parallel lines

and a transversal.

19 Converses of

Angle Pair Theorems

• Use measures of angles formed by parallel lines and a transversal to prove lines parallel.

• Prove the converses of theorems about parallel lines. Key Terms: plumb bob, auxiliary line



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20 Perpendicular Lines

• Define the perpendicular bisector of a segment. • Prove lines are perpendicular. • Use perpendicular lines to prove lines are parallel. Key Terms: perpendicular bisector, distance from a point to a line, corollary

21 More Geometric Constructions

• Construct the perpendicular bisector of a segment. • Construct a line perpendicular to a given line through a given point on

the line. • Construct a line perpendicular to a given line through a given point not

on the line. • Construct a line parallel to a given line through a given point. Key Terms: constructible number, perpendicular bisector, equidistant, rhombus method


22 Lines in the Coordinate Plane

• Classify slope as positive, negative, zero, or undefined. • Determine the slope of a line in the coordinate plane. • Write equations of lines in slope-intercept, point-slope, and standard

forms. • Graph lines in the coordinate plane. Key Terms: slope, positive slope, negative slope, zero slope, undefined, rate of change, y-intercept, slope-intercept form, point-slope form, standard form



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23

Parallel and Perpendicular Lines in the

Coordinate Plane

• Determine whether lines in the coordinate plane are parallel, perpendicular, or neither.

• Write the equation of a line parallel to another line through a given point. • Write the equation of a line perpendicular to another line through a given

point. Key Term: opposite reciprocals

24 Systems of Linear Equations

• Define systems of linear equations and their solutions. • Determine the possible number of solutions for a system of linear

equations. • Solve systems of linear equations by graphing. Key Terms: system of equations, system of linear equations, solution of a system of equations, consistent independent, inconsistent system, consistent dependent system, coincide


Unit 4: Congruent Triangles

25 Triangles

• Classify triangles by angle measures. • Classify triangles by side lengths. • Use slopes and the Distance Formula to classify triangles in the

coordinate plane. Key Terms: equilateral triangle, isosceles triangle, scalene triangle, acute triangle, right triangle, obtuse triangle, equiangular triangle

Study: Virtual Construction Tool, Crossword Puzzle (.PDF)



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26 Angles of Triangles

• Prove the sum of the angle measures in a triangle is 180 degrees. • Use the Triangle Sum Theorem to find angle measures in a triangle. • Prove the Exterior Angles Theorem. • Use the Exterior Angles Theorem to find angle measures. Key Terms: auxiliary line, logically equivalent, interior angles, exterior angle, remote interior angles, corollary, equiangular

27 Congruence

• Define congruent triangles. • Prove triangles congruent. • Identify corresponding parts of triangles. • Use the Third Angle Theorem to prove triangles congruent. Key Term: correspondence

28

Congruence Theorems and

Postulates: SSS and SAS

• Prove triangles congruent using the Side-Side-Side (SSS) Congruence Postulate.

• Prove triangles congruent using the Side-Angle-Side (SAS) Congruence Postulate.

• Construct congruent triangles using SSS and SAS. • Use corresponding parts of congruent triangles to prove properties of

figures. Key Terms: triangle rigidity, compass rose, included angle


29

Congruence Theorems and

Postulates: ASA and AAS

• Prove triangles congruent using the Angle-Side-Angle (ASA) Congruence Postulate.

• Prove triangles congruent using the Angle-Angle-Side (AAS) Congruence Theorem.

Key Terms: triangulation, included side




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30 Congruence

Theorems and Postulates: HL

• Identify the parts of a right triangle. • Prove triangles congruent using the Hypotenuse-Leg (HL) Theorem. Key Terms: ambiguous case, legs, hypotenuse

31 Isosceles and

Equilateral Triangles

• Identify parts of isosceles triangles. • Prove the Isosceles Triangle Theorem and its converse. • Prove theorems about isosceles and equilateral triangles. • Find angle measures and side lengths of isosceles and equilateral

triangles. Key Terms: legs, vertex angle, base angles, auxiliary line

32 Congruence Transformations

• Translate figures in the coordinate plane. • Rotate figures in the coordinate plane. • Reflect figures in the coordinate plane. • Verify that transformed figures in the coordinate plane are congruent. Key Terms: transformation, translation, rotation, reflection, preimage, image, translation vector, center of rotation, line of reflection

Study: Virtual Construction Tool, Crossword Puzzle (.PDF)

33 Segments and

Angles in Triangles

• Define concurrent, circumcenter, and incenter. • Understand and use the Perpendicular Bisector Theorem and its

converse. • Understand and use the Angle Bisector Theorem and its converse. • Use the Circumcenter Theorem and Incenter Theorem to find missing

measures and identify segments of equal measure. Key Terms: equidistant, concurrency, circumcenter, angle bisector, incenter



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34 Medians and Altitudes of Triangles

• Define median, altitude, centroid, and orthocenter. • Use the Centroid Theorem to find missing lengths. • Understand the Orthocenter Theorem and how to find an orthocenter. • Apply the Triangle Midsegment Theorem to find distance in real-life

situations. Key Terms: median of a triangle, centroid of a triangle, altitude of a triangle, midsegment triangle


Unit 5: Triangles

35 Constructing

Special Points in Triangles

• Construct the centroid, circumcenter, orthocenter, and incenter of a triangle.

• Construct the circumscribed and inscribed circle of a triangle. • Construct the Euler line of a triangle. Key Terms: circumcenter, incenter, centroid, orthocenter, concurrent, circumscribed circle, median, inscribed circle, foot, Euler’s line


36 Triangle

Inequalities in One Triangle

• Prove and use the Exterior Angle Inequality Theorem. Determine the order of angles in triangles when given side lengths.

• Determine the order of side lengths in triangles when given angle measures.

• Understand and use the Triangle Inequality Theorem.

37 Triangle

Inequalities in Two Triangles

• Understand and write an indirect proof. • Use the Hinge Theorem and its converse.



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Unit 6: Similarity

38 Ratio and Proportion

• Write and simplify ratios. • Solve proportions by using cross products. Key Terms: ratio, proportion, means, extremes

39 Similar Polygons

• Identify corresponding parts of similar polygons. • Identify similar polygons. • Apply properties of similar polygons to find unknown lengths. Key Terms: similar, similarity statement, corresponding sides, corresponding angles, proportional, similar polygons, similarity ratio

40 Proving Triangles Similar: AA, SSS,

and SAS

• Understand the Reflexive, Symmetric, and Transitive Properties of Similarity.

• Use the Angle-Angle (AA) Similarity Postulate to determine whether triangles are similar.

• Prove and use the Side-Side-Side (SSS) and Side-Angle-Side (SAS) Similarity Theorems to show that triangles are similar.

• Find missing measures in similar triangles.

41 Right Triangle Similarity

• Understand how the altitude of a right triangle divides the right triangle into similar triangles.

• Calculate and understand the geometric mean. • Understand and use the Geometric Mean Altitude Theorem. • Understand and use the Geometric Mean Leg Theorem. Key Terms: altitude of a triangle, geometric mean



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42 Segments in Similar Triangles

• Prove and use theorems about segments in similar triangles. • Find missing measures and distances using the corresponding segments

theorems. Key Terms: median of a triangle, altitude of a triangle, angle bisector

43 Proportionality Theorems

• Prove and use the Triangle Proportionality Theorem and its converse. • Calculate distances using proportionality theorems. • Prove and use the Triangle Angle Bisector Theorem to find missing

measures.

44 Proportional Relationships

• Use indirect measurement techniques to find inaccessible measurements. • Find lengths using scale drawings. • Use proportions to find perimeters and areas of similar polygons. Key Terms: indirect measurement, scale drawing

45 Similarity Transformations

• Draw a dilation of a figure centered at a point. • Find the center and scale factor of a dilation, given a figure and its

image. • Verify that the image of a dilated figure is similar to the preimage. Key Terms: transformation, dilation, center of dilation, enlargement, reduction, scale factor

Study: Geometric Construction (.PDF)


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Geometry IB Grade Levels 9–10

Geometry IB introduces students to the following concepts and functions:

• theorems, proofs, postulates, and corollaries

• sine and cosine • right triangles • area and perimeter • vectors • parallelograms, polygons, trapezoids, and kites • isometries • translations, rotations, and reflections • tessellations • geometric probability • surface area and volume • tangents, secants, and chords • arcs, angles, and segments

A+nyWhere Learning System Geometry IB Teacher’s Guide Grade Levels 9–10

206

A+LS Geometry IB is a full semester course designed as the follow-up course to A+LS Geometry IA, which is based on the Common Core Standards for Mathematics. All lessons have been designed to help students understand key concepts by applying real-world knowledge. Geometry IB continues to expand student knowledge of theorems, formulas, and proofs. The course then expands to include more complex topics such as sine and cosine, special geometric shapes, geometric measurements, and multi-dimensional figures and their translations.

A+LS Geometry IB is presented as a one-semester core course.

• All forty-six lessons contain a study guide, a practice test, and a

mastery test.

• Students are provided a number of practice questions within each lesson. These are designed to show the students correct and incorrect answers so that they may check their progress as they move along.

• An extensive toolkit is provided for the students in the upper right-hand corner of each page of the study guide, which includes a calculator, glossary, chart of symbols, and a complete list of formulas, theorems, and postulates.

• Many of the lessons contain a built-in protractor and a virtual

construction tool, which includes a pencil, straightedge, compass, and eraser.

• A virtual coach, Greek mathematician Euclid, appears throughout the lessons to provide additional information regarding key concepts.


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• The content of the Geometry IB course is based upon the Common Core State Standards for Mathematics.

• Lessons are designed in a foundational structure. Each lesson provides material that the student will utilize in upcoming

lessons.

• Practice problems are demonstrated via interactive design concepts in each lesson. Students are then provided sample problems to work on their own. This allows students to immediately check their understanding as they move throughout the lesson. If they get the problems wrong, they can easily navigate to the previous page in order to review the lesson again. This type of interactivity also allows students to work at their own pace, moving forward when they feel they have mastered a new concept.

• Geometry IB is designed to help students move forward into more complex geometric studies. A virtual construction tool

allows students to practice transformations of geometric shapes.

• The overall objective for students taking the Geometry IB course is to prepare them to move into more advanced mathematical studies, like Algebra II and Trigonometry.


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Lessons


Grade Levels

Built-in Calculator


Geometry IB 46 1 9–10 Yes Yes

A+nyWhere Learning System Geometry IB Scope & Sequence Grade Levels 9–10 Lesson Name Lesson Content Activities

209

Unit 7: Right Triangles and Geometry

46 The Pythagorean Theorem

• Use the Pythagorean Theorem to find the length of an unknown side in a right triangle.

• Use the Pythagorean Theorem to find the distance between two points on the coordinate plane.

Key Terms: legs, hypotenuse

47 The Converse of the Pythagorean Theorem

• Determine if a triangle is a right triangle. • Determine if a set of numbers forms a Pythagorean triple. • Create Pythagorean triples from a given Pythagorean triple. • Classify a triangle as acute, right, or obtuse given the side lengths of

the triangle. Key Term: Pythagorean triple

48 Special Right Triangles

• Find a missing side length of a 45°-45°-90° triangle. • Find a missing side length of a 30°-60°-90° triangle. • Use the properties of special right triangles to find perimeters and areas of figures. Key Terms: special right triangles, short leg of a triangle, long leg of a triangle

49 The Tangent Ratio

• Write tangent ratios as fractions and decimals. • Use tangent ratios to find missing side lengths in right triangles. Key Terms: trigonometry, trigonometric ratio, tangent ratio


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50 The Sine and Cosine Ratios

• Write sine ratios as fractions and decimals. • Write cosine ratios as fractions and decimals. • Use sine ratios to find missing side lengths in right triangles. • Use cosine ratios to find missing side lengths in right triangles. Key Terms: sine, cosine

51 Solving Right Triangles

• Use an inverse trigonometric function to find the measure of an angle. • Solve a right triangle. Key Term: inverse trigonometric function

52 Right Triangle Applications

• Identify angles of elevation and angles of depression. • Use right triangle trigonometry to solve real-world problems. Key Terms: angle of elevation, angle of depression, ground distance

53 Law of Sines and Law of Cosines

• Use the Law of Sines to find missing measures in triangles. • Identify when the Law of Sines can be used. • Use the Law of Cosines to find missing measures in triangles. • Identify when the Law of Cosines can be used. Key Terms: true bearing, conventional bearing

54 Vectors

• Write vectors in component form. • Find the magnitude and direction of vectors. • Identify equal and parallel vectors. • Add vectors geometrically and algebraically. Key Terms: vector, magnitude, component form, standard position, direction, velocity vector, parallel vectors, equal vectors, resultant vector


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Unit 8: Polygons

55 Exploring Polygons

• Identify and classify polygons. • Understand and use the Angle Sum Theorem and its corollaries. • Understand and use the Exterior Angle Sum Theorem. • Calculate angle measures and side lengths using polygon classifications. Key Terms: polygon, sides of a polygon, vertex, diagonal, consecutive, nonconsecutive, regular polygon, irregular polygon, convex, concave, interior angle, exterior angle

56 Parallelograms

• Define parallelogram. • Understand the properties of parallelograms. • Construct proofs to prove properties of parallelograms. • Construct proofs using properties of parallelograms. • Use properties of parallelograms to find measures. Key Terms: parallelogram, consecutive angles, diagonal

57 Conditions for Parallelograms • Prove a quadrilateral is a parallelogram.

58 Special Parallelograms

• Define rectangle, rhombus, and square. • Understand and prove properties of rectangles, rhombuses, and

squares and use these properties to find missing measures. Key Terms: rhombus, biconditional statement, rectangle, square, rhombus corollary, rectangle corollary, square corollary


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59 Trapezoids and Kites

• Define trapezoid and kite and identify their parts. • Calculate the length of the midsegment of a trapezoid. • Use properties of trapezoids and kites to find missing measures. Key Terms: trapezoid, base, leg, base angle, isosceles trapezoid, midsegment, kite

60 Coordinate Geometry

• Place figures in a coordinate plane and use variables or numbers to name the coordinates of the vertices.

• Complete a coordinate proof. • Classify a quadrilateral as a parallelogram, rhombus, rectangle, or

square, based on the coordinates of its vertices. • Use diagonals to show whether a parallelogram on a coordinate plane

is a rhombus, rectangle, or square.

Unit 9: Transformations

61 Isometries

• Understand the properties of an isometry. • Determine whether a transformation is an isometry. • Translate, rotate, and reflect figures in a coordinate plane. Key Terms: isometry, fixed point

62 Translations

• Translate figures in a plane. • Use matrices to translate figures in a coordinate plane. • Find the translation vector given the image and preimage. Key Terms: translation, translation vector, matrix, element, address



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63 Rotations

• Rotate figures in a plane. • Use matrices to rotate figures in a coordinate plane. • Determine the center and angle of a given rotation. Key Terms: rotation, center of rotation, angle of rotation, identity matrix


64 Reflections

• Reflect figures in a plane. • Use matrices to reflect figures in a coordinate plane. • Find the line of reflection given the preimage and image. Key Terms: reflection, line of reflection, orientation, inverse


65 Compositions of Transformations

• Find the image of a figure after two or more compositions. • Find the image of a figure after a glide reflection. • Determine the composition of transformations that will produce a

given image. Key Terms: glide reflection, composition of transformation, identity transformation, inverse


66 Dilations

• Dilate figures in a plane. • Use matrices to dilate figures in the coordinate plane. • Determine the scale factor and center of a dilation. Key Terms: dilation, center of dilation, scale factor, enlargement, reduction, similar, matrix multiplication, scalar multiplication, scalar


67 Symmetry

• Identify the type of symmetry of a figure. • Find the lines of symmetry of a figure with line symmetry. • Find the center and order of a figure with rotational symmetry. Key Terms: symmetry, line symmetry, reflection symmetry, line of symmetry, rotation symmetry, center of rotational symmetry, order of rotational symmetry, angle of rotational symmetry, point symmetry



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68 Tessellations

• Determine whether a shape tessellates. • Draw tessellations of a given shape. • Determine the symmetry of a given tessellation. Key Terms: tessellation, regular tessellation, semi-regular tessellation


Unit 10: Perimeter and Area

69 Perimeter and Area

of Triangles and Parallelograms

• Understand and use the Area Addition Postulate. • Use formulas to find perimeters and areas of parallelograms and

triangles. • Calculate perimeter and area of triangles and parallelograms using the

Pythagorean Theorem.

70 Area of Trapezoids, Rhombuses, and Kites

• Use formulas to find areas of trapezoids, rhombuses, and kites. • Understand and use the Area Congruence Postulate. Key Terms: area of a trapezoid, area of a kite, area of a rhombus

71 Area of Circles and Sectors

• Define circle, center of a circle, and sector. • Develop the formula for the area of a circle. • Calculate the area of a circle. • Calculate the area of a sector of a circle. Key Terms: circle, center of a circle, central angle of a circle, circumference, arc, sector of a circle, area of a circle, area of a sector


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72 Perimeter and Area of Regular Polygons

• Define center, apothem, radius, and central angle of a regular polygon. • Calculate the apothem, radius, and central angle of a regular polygon. • Calculate the areas and perimeters of regular polygons. Key Terms: center of a regular polygon, radius of a regular polygon, central angle of a regular polygon, apothem, area of a regular polygon

73 Composite Figures

• Find the area of a composite figure. • Find the perimeter of a composite figure. • Estimate the area and perimeter of an irregular figure. Key Terms: composite figure, irregular shape, lattice polygon, boundary point, interior point

74 Perimeter and Area in the Coordinate Plane

• Find the perimeter of a figure in the coordinate plane. • Find the area of a figure in the coordinate plane. • Estimate the area and perimeter of an irregular figure in the

coordinate plane. • Estimate the area under a curve. Key Term: lattice polygon

75 Changing Dimensions

• Determine the effect on the area of a figure when one or more dimensions is changed.

• Determine the effect on the perimeter of a figure when the dimensions are changed proportionally.


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76 Geometric Probability

• Use length models to determine probability. • Use angle measure models to determine probability. • Use area models to determine probability. Key Terms: experiment, sample space, outcome, probability, theoretical probability, probability model, probability distribution, geometric probability, event, complements, mutually exclusive, independent

Unit 11: Surface Area and Volume

77 Solids

• Identify the parts of a polyhedron. • Use Euler’s formula to find the number of faces, edges, and vertices of

a polyhedron. • Define Platonic solids. • Represent a solid in two dimensions by using a net. • Describe the cross section of a solid intersected by a plane. Key Terms: face, edge, vertex, polyhedron, convex, concave, regular polyhedron, Platonic solid, net, cross section

78 Two-Dimensional Representations of Solids

• Represent views of solids by using orthographic drawings. • Use isometric drawings to represent solids. • Draw solids in one-point, two-point, and three-point perspective. Key Terms: orthographic drawing, net, isometric grid, isometric drawing, perspective drawing, vanishing point, horizon, picture plane, one-point perspective, two-point perspective


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79 Surface Area of Prisms and Cylinders

• Identify parts of prisms and cylinders. • Use nets to represent prisms or cylinders. • Determine the lateral area and surface area of right prisms and right

cylinders. Key Terms: base, lateral face, lateral edge, right prism, oblique prism, cylinder, lateral surface, right cylinder, oblique cylinder, lateral area of a prism, surface area, lateral area of a cylinder, surface area of a cylinder

80 Surface Area of Pyramids and Cones

• Identify parts of pyramids and cones. • Use nets to represent pyramids or cones. • Determine the lateral area and surface area of right pyramids and

right cones. Key Terms: pyramid, base of a pyramid, lateral face, vertex, lateral edge, regular pyramid, right pyramid, oblique pyramid, height of a pyramid, slant height, cone, base of a cone, lateral surface, vertex, right cone, oblique cone, height of a cone, slant height of a cone, lateral area of a pyramid, sector, lateral area of a cone

81 Volume of Prisms and Cylinders

• Define the volume of a solid. • Determine the volume of a prism. • Determine the volume of a cylinder. Key Terms: unit cube, volume, Cavalieri’s principle

82 Volume of Pyramids and Cones

• Determine the volume of a pyramid. • Determine the volume of a cone. Key Terms: Euclid, frustum of a pyramid, frustum of a cone


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83 Surface Area and Volume of Spheres

• Identify the parts of a sphere. • Determine the volume of a sphere. • Determine the surface area of a sphere. Key Terms: sphere, center of a sphere, radius of a sphere, chord of a sphere, diameter of a sphere, hemisphere, great circle

84 Similar Solids

• Determine whether solids are similar. • Find the scale factor of two similar solids. • Compare the surface areas and volumes of similar solids. Key Terms: similar solids, scale factor, surface area to volume ratio

Unit 12: Circles

85 Tangents, Secants, and Chords

• Identify parts of circles. • Use theorems about tangents of circles to solve problems. • Identify and use congruent, tangent, and concentric circles. Key Terms: circle, radius, chord, diameter, secant, tangent, point of tangency, common tangent, common external tangent, common internal tangent, congruent circles, tangent circles, concentric circles

86 Measuring Arcs and Angles

• Identify minor arcs, major arcs, and semicircles. • Find measures of arcs. • Find measures of central angles. • Find arc lengths. Key Terms: central angle, arc, minor arc, semicircle, major arc, central angle, adjacent arc, congruent arc


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87 Arcs and Chords

• Understand theorems about chord and angle relationships in circles. • Use theorems about chord and angle relationships in circles to solve

problems. • Construct the center of a circle with a compass and straightedge

given a chord of the circle. Key Terms: arc of a chord, corresponding chord


88 Inscribed Angles and Polygons

• Identify inscribed angles and inscribed polygons. • Find measures of inscribed angles and intercepted arcs of inscribed angles. • Use theorems about inscribed angles and polygons to solve problems. Key Terms: inscribed angle, intercepted arc, inscribed polygon, circumscribed

89 Angle Measures

• Find missing measures when a tangent and secant intersect on the circle.

• Find missing measures when two secants intersect in the interior of the circle.

• Find missing measures when two tangents, two secants, or a secant and a tangent intersect in the exterior of the circle.

90 Segment Lengths

• Find a missing segment length formed by two intersecting chords. • Find a missing segment length formed by two secants with a common

exterior point. • Find a missing segment length formed by a secant and a tangent with

a common exterior point. Key Terms: secant segment, external secant segment, tangent segment


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91 Equations of Circles

• Find the radius and center of a circle given its equation. • Solve problems involving circles in the coordinate plane. • Write the equation of a circle in the coordinate plane. • Graph a circle in the coordinate plane given its equation. Key Term: standard equation of a circle


221

Trigonometry Grade Levels 11–12

A+LS Trigonometry introduces students to the following concepts, skills, and more:

• angles and angle terminology • reference angles • definition of sine, cosine, and tangent • definition and value of secant, cosecant, and

cotangent • calculating sides of right triangles • using trig to solve real world problems • the Law of Sines and Cosines • symmetry identities • verifying trigonometric identities • sum and difference for sine • cosine and tangent • using cofunction identities • graphing trig functions • principal values • arclength • area of circular sectors • simple harmonic motion • frequency

A+nyWhere Learning System Trigonometry Scope & Sequence Grade Levels 11–12


222

1 Angles Angles; angle terminology; radians; reference angles

2 Sine, Cosine, and Tangent

The unit circle; finding values from endpoint, definition of sine, cosine, and tangent


3 Values of Sin, Cos, and Tan

Values of sine, cosine, and tangent of various angles; using a calculator to find values



4 Reciprocal Functions Definition and value of secant, cosecant, and cotangent Study: EB Learning

Material

5 The Pythagorean Theorem The Pythagorean Theorem; calculating sides of right triangles


Essay: Problem Solving/ Written Response

6 Review 1

7 Inverse Functions Definition and value of inverse trig functions Study: EB Learning Material

8 Solving Right Triangles Using trig functions and the Pythagorean Theorem to solve right triangles Study: EB Learning

Material

9 Trigonometry Applications Using trig to solve real world problems Study: EB Learning

Material

10 Law of Sines 1 Definition of law of sines and applications Study: EB Learning Material



223

11 Law of Sines 2 Further application of the Law of Sines; determining the number of triangles and solving triangles



12 Law of Cosines Definition of law of cosines and solving triangles Study: EB Learning Material

13 Solving Triangles Using trig functions and the Laws of Sines and Cosines to solve triangles Study: EB Learning Material

14 Triangle Applications Solving triangles in word problems

15 Areas of Triangles Formulas for areas of triangles Essay: Written Response

16 Review 2

17 Trigonometric Identities

Definition of identity; reciprocal identities; quotient identities; Pythagorean identities; symmetry identities


18 Verifying Trig Identities Verifying trigonometric identities and manipulating identities for verification Study: EB Learning

Material

19 Sum Difference Identities Sum and difference for sine, cosine, and tangent; using cofunction identities

20 Graphing Trig Functions 1

Graphs of sine, cosine, tangent and reciprocal functions; finding values using graphs

21 Graphing Trig Functions 2

Analyzing amplitude, period, and phase shift; graphing functions and compound functions



224

22 Graphing Trig Functions 3 Graphs of inverse functions; finding values using graphs

23 Review 3

24 Double- and Half-Angles Double-angle and half-angle identities Study: EB Learning

Material

25 Solving Trig Equations

Solving equations involving trig functions; principal values; solving for principal and all values

26 Central Angle Application

Applications of central angles; arclength; linear and angular velocity; area of circular sectors

27 Simple Harmonic Motion

Writing equations for simple harmonic motion; using equations for information; frequency


28 Review 4

29 Comprehensive Exam Comprehensive exam of course content


225

Calculus I and II Grade Level 12

A+LS Calculus I and II introduces students to the following concepts and functions and more: Calculus I

• calculating x-values and corresponding values • continuous functions • asymptotes • graphing tangents • secants, and cosecants • sign graphs • antiderivatives with negative exponents

Calculus II

• notations of integrals • the fundamental theorem of calculus • indefinite integrals and antiderivatives • natural logarithms • arclength • surface area and work • hydrostatic force • inverse functions including natural exponent

functions • exponential growth and decay • inverse trigonometric functions

I A+nyWhere Learning System Calculus I Scope & Sequence Grade Level 12


226

1 Limits Calculating x-values and corresponding values; approaching function values, limits, and notation

Study: EB Learning Material Essay: Written Response

2 Continuous Functions

Definition of continuous function; continuous graphs of polynomial functions, sine, and cosine; evaluating the limits of continuous function


3 Discontinuous Functions 1

Examining various types of discontinuities: holes, asymptotes, jumps, and their graphs


4 Discontinuous Functions 2 Approaching negative and positive infinities Study: EB Learning Material


5 Discontinuous Functions 3 One-sided limits Study: EB Learning Material


6 Special Trig Functions

Trigonometric limits of sine and cosine functions; graphing tangents, cotangents, secants, and cosecants

7 Limits at Infinity Polynomials as they approach infinity, negative infinity, and infinity squared; definition of infinity squared; examples of how changing the argument of the function changes the limit


8 Limit Unit Review Review of limit lessons

9 Derivatives Derivatives and determining the slope of a tangent at a given point, using the derivative as a velocity, the derivative as a function; Leibniz notation


10 Derivative Shortcuts 1

Using the mathematical definition of derivatives; the Power Rule and coefficients of sums and differences Essay: Problem Solving

11 Derivative Shortcuts 2

Negative exponents; derivatives of sine and cosine; derivatives at specific points Study: EB Learning Material



227

12 Some Derivative Rules

Functions that are products; the Product Rule; rational functions and the Quotient Rule; the derivative as a reciprocal of sine Study: EB Learning Material

13 The Chain Rule Derivatives of composite functions; definition of the chain rule; extending the chain rule

14 Higher Derivatives Acceleration as a derivative of velocity; notation and use of higher derivatives

15 Implicit Differentiation Examples of finding the derivative implicitly without solving for y

16 Derivative Unit Review Review of derivatives

17 Maximum / Minimum Values 1

Determining maximum and minimum values of given functions on closed intervals


18 Maximum / Minimum Values 2

Using zero-slope to determine maximum and minimum values, critical points, and relative extrema

19 Maximum / Minimum Tests 1

The first derivative tests; increasing and decreasing slopes; finding relative extrema Study: EB Learning Material

20 Maximum / Minimum Tests 2 Second derivative tests; finding relative extrema

21 The Second Derivative

Concavity and inflection points of graphs; definition and determination of inflection points; sign graphs Essay: Written Response

22 Application Unit Review 1 Review of maximum and minimum values and tests



228

23 Applications of Extrema

Determining the need to find maximum and minimum values in real life situations Essay: Written Response

24 Related Rates 1 Problems with derivatives that are related; problems involving related rates and spheres

25 Related Rates 2 Using related rates to determine the volume of cones; using the Pythagorean relationship in related rate problems

26 Graphing Using Extremes 1 Understanding the nature of graphing; determining graphing data

27 Graphing Using Extremes 2 Asymptotes as related to graphs

28 Application Unit Review 2 Review of related rates and graphing

29 Antiderivatives Determining the original function from the derivative; definition of antiderivatives; proving antiderivatives; antiderivatives with negative exponents.


30 Comprehensive Exam Review of all material presented in Calculus I

A+nyWhere Learning System Calculus II Scope & Sequence Grade Level 12


229

1 Definite Integrals Definition of integral; discussion of finding integrals; notations for integrals; discussion of definite integrals


2 Fundamental Theorem

Implications of the fundamental theorem of calculus; evaluating definite integrals; addition property



3 Indefinite Integrals

The integral as a function; antiderivatives; integrating a constant multiple of a function Essay: Written Response

4 Integrals by Substitution

Inverse of the chain rule; determining u and du; integrals of the squares of sine and cosine; substitution for definite integrals/change limits

5 Natural Logarithms

Definition of natural logarithm as an integral; review of laws of logarithms; derivatives of natural logarithms; finding the integral of tangent using logarithms


6 Area Between Two Graphs

Discussion of implications of areas between graphs; points of intersection for region; comparing functions for subtraction direction Essay: Written Response

7 Integral Unit Review

Review of integrals, antiderivatives, chain rule, logarithms, laws of logarithms, and other material covered in previous lessons

8 Volumes 1 Volumes of rotation of F(x) about the x-axis; the disc method; the washer method (two functions)

9 Volumes 2 Volumes of rotation of F(x) about the y-axis; the shell method, the shell method with two functions Essay: Written Response

10 Arclength Definition and examples of arclength; finding the length of a curve Study: EB Learning Material



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11 Surface Area Finding the area of a rotational surface

12 Work Definition of work; finding work with a variable force; work to empty a tank Essay: Written Response

13 Application Unit Review Review of volumes, surface area, work, and hydrostatic force

14 Exponent Function The natural exponent function; inverse of the natural logarithm; laws of exponents; derivative of the natural exponent; integral of the natural exponent


15 Exponents and Logarithms

Exponential and logarithmic functions of other bases; rewriting exponentials with the natural exponential; derivative of general exponential functions; logarithms of different bases; derivative of general logarithms

16 Growth and Decay Exponential growth and decay; function of exponential growth and decay; half-lives and doubling times



17 Inverse Trig Functions

Arcsine and arccosine; arctangent and arccotangent; arcsecant and arccosecant; derivatives and integrals of all six functions

18 Inverse Functions Review Review of exponents, logarithms, and inverse trig functions

19 Integration by Parts Breaking up the function to be integrated; nth powers of sine and cosine Essay: Written Response

20 Trigonometric Integrals

Integrals involving trigonometric functions; products of sines and cosines; products of tangents and secants; changing to sines and cosines; using trig identities




231

21 Trig Substitutions Substitution trig functions in for x; forms containing square roots and x squared

22 Partial Fractions Preparing the fraction; division; factoring; breaking the fraction into its component parts

23 Approximations Using trapezoidal rule to approximate area; using Simpson’s rule to approximate area


24 Improper Integrals Unbounded integrands; unbounded intervals; convergent or divergent integrals Essay: Written Response

25 Techniques Unit Review Review of previous materials

26 Comprehensive Exam

a+ls mathematics curriculum planning manual · pre-algebra ia 43 1 8 pre-algebra ib 37 1 8 algebra...

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