tutor math handbook - paadultedresources.org · tabe (9–10) scale score for total math: 313 and...

;,;:., Pennsylvania

a ~ Adult Education ~ Resources

TLC Tutor Handbook for Reading and Writing Instruction

Tutors of Literacy in the Commonwealth

September 2016

1

Tutor Handbook Math Instruction


~ Pennsylvania ,&~ Adu lt Education ~ Resources

This manual created by Tutors of Literacy in the Commonwealth, with special thanks and

acknowledgement to Dr. Sheila Sherow, under PDE Grant # 064-15-0000. The development

of this resource was supported in part by the U.S. Department of Education. However, it

does not necessarily reflect the position or policy of the U.S. Department of Education or

the Pennsylvania Department of Education and no official endorsement by these agencies

should be inferred.

Copyright 2016 Commonwealth of Pennsylvania


925 West College Ave

State College, PA 16801

Division of Adult Education

Pennsylvania Department of Education, Bureau of Postsecondary and Adult Education

333 Market Street, 12th Fl., Harrisburg, PA 17126‐0333

Phone: 717‐787‐5532

TLC Tutor Handbook for Reading and Writing Instruction 2 Tutors of Literacy in the Commonwealth

September 2016

~ Pennsylvania a ~ Adult Education ~ Resources

Table of Contents

The chapter and section titles below are interactive. Clicking on a title will take you directly

to that section in this document.

Page

How to Use This Resource 4

Introduction to This Resource 5

Chapter One: Overview of Math Instruction 7

Chapter Two: Beginning Level Math 8

Place Value 8

Addition and Subtraction 10

Mental Math 12

Algebraic Thinking 13

Measurement 14

Multiplication and Division 15

Fractions 17

Geometry 19

Data and Statistics 21

Whole Number Problem Solving 23

Chapter Three: Intermediate Level Math 24

Place Value 24

Number System 28

Measurement 31

Multiplication and Division 32

Fractions, Decimals, and Percents 34

Geometry 36

Coordinate Plane 39

Ratio, Rate, and Proportion 41

Statistics and Probability 43

Beginning Algebra 47

Appendix: College and Career Readiness Standards for Mathematics 51


September 2016

~ Pennsylvania ,&~ Adult Education ~ Resources

How to Use This Resource

The following chapters containing information about mathematics align with the College and Career

Readiness Standards for Adult Education (CCR). The CCR Standards reflect what experts believe is most

important for college and career readiness, and are organized by the National Reporting System for

Adult Education (NRS) educational functioning levels for adults. As such, these content standards

can help tutors understand where to focus instruction. The standards are included in the Appendix.

Content standards are not intended to dictate how to teach a particular subject. Rather, they define

what students should understand and be able to do.

This handbook offers teaching strategies, ideas, and assessments that tutors can use to inform and

guide instruction, guided practice exercises to enhance instruction, and suggested resources.

Sources

Pimentel, S. (2013). College and Career Readiness Standards for Adult Education. U.S. Department of Education, Office of

Vocational and Adult Education.

National Reporting System for Adult Education: http://www.nrsweb.org


September 2016

http://www.nrsweb.org/

~ Pennsylvania a ~ Adu lt Education ~ Resources

Introduction to This Resource

This Tutor Handbook for Math Instruction is organized by beginning and intermediate adult education

levels and aligns with the College and Career Readiness Standards for Adult Education (CCR Standards).

Secondary education levels of math instruction are not included in this handbook.

Chapter One is a general overview of how to teach math effectively. Chapters Two (Beginning Level)

and Three (Intermediate Level) focus on major math concepts related to the CCR Standards and

include tutoring tips, ideas, and strategies, as well as examples of visuals that can be easily created

to help students to understand and remember key concepts, symbols, and terminology.

This handbook is not intended to be a math curriculum or a complete list of math skills, nor does it

present the appropriate or recommended sequence of instruction. Rather, it lists the math content

and skills students should understand and be able to do at beginning and intermediate educational

functioning levels, as determined by the math experts who wrote the CCR Standards.

Note: Basic information about teaching adult learners is included in the Tutor Handbook for Reading

and Writing Instruction, but is not repeated in this resource.

Adult Education Levels The following levels have been defined by the National Reporting System for Adult Education (U.S.

Department of Education, 2013).

Beginning Level

Beginning Adult Basic Education Literacy (CCR Standards Level A)

Students at this level typically have little or no recognition of numbers or simple counting skills, or

may have only minimal skills, such as the ability to add or subtract single-digit numbers.

TABE (9–10) scale score for Total Math: 313 and below (grade level 0–1.9).

CASAS scale score of 200 and below.

Beginning Basic Education (CCR Standards Level B)

Students at this level typically can add and subtract three-digit numbers; can perform multiplication

through 12; can identify simple fractions; and can perform other simple arithmetic operations.

TABE (9–10) scale score for Total Math: 314–441 (grade level 2–3.9).

CASAS scale score of 201-210.

5

http://lincs.ed.gov/publications/pdf/CCRStandardsAdultEd.pdf

Introduction to This Resource

Intermediate Level

Low Intermediate Basic Education (CCR Standards Level C)

Students at this level typically can perform all four basic math operations using whole numbers up

to three digits, and can identify and use all basic mathematical symbols.



High Intermediate Basic Education (CCR Standards Level D)

Students at this level typically can perform all four basic math operations with whole numbers and

fractions; can determine correct math operations for solving narrative math problems; can convert

fractions to decimals and decimals to fractions; and can perform basic operations on fractions.



Advanced Level (not included in this handbook)

Low Adult Secondary and High Adult Secondary Education (CCR Standards Level E)

Low Adult Secondary Education: Students at this level typically can perform all basic math functions

with whole numbers, decimals, and fractions; can interpret and solve simple algebraic equations,

tables, and graphs; can develop tables and graphs; and can use math in business transactions.

TABE (9–10): scale score for Total Math: 566–594 (grade level 9–10.9)

CASAS scale score of 236-245

High Adult Secondary Education: Students at this level typically can make mathematical estimates of

time and space; can apply principles of geometry to measure angles, lines, and surfaces; and can

apply trigonometric functions.

TABE (9–10): scale score for Total Math: 595 and above (grade level 11–12).

CASAS scale score of 246 and above.

Sources:

U.S. Department of Education. (2013). “Measures and Methods for the National Reporting System for Adult Education.” Division of Adult Education and Literacy Office of Vocational and Adult Education U.S. Department of Education Contract No.

ED-VAE-10-O-0107.

Pimentel, S. (2013). “College and Career Readiness Standards for Adult Education.” U.S. Department of Education, Office of

Vocational and Adult Education.

TLC Tutor Handbook for Math Instruction


September 2016

6


Chapter One: Overview of

Math Instruction

Think about “How will students best learn this?” instead of thinking “How should I teach this math concept?”

Don’t rely on teaching the way you were taught.

Evaluate and address students’ attitudes and beliefs regarding both learning math and using

math.

Ask students why they want to learn math so you can find a relevant and meaningful context

for instruction.

Assess students’ understanding frequently so you can correct misunderstandings before

they create confusion.

Be sure you know the math concepts you’re going to teach. If necessary, review and refresh

your memory on each math concept. This includes using the correct math vocabulary.

Anticipate what students may have difficulty understanding and be prepared to explain

concepts in different ways.

Be prepared to do more than tell students about a new math concept. Be ready to explain

why math concepts are important and should be learned.

Be excited about math and motivate students to feel the same excitement.

Use visuals and real-life examples as much as possible.

Make connections with things students already know and understand.

Use manipulatives and authentic materials.

Engage students in hands-on activities.

Encourage algebraic thinking.

Encourage the development and practice of estimation skills.

Emphasize the use of mental math.

Be as concrete as possible when explaining concepts.

Talk about when to use a strategy, not just how to use it.

Help students to recognize and use patterns.

Teach math as problem solving.

Encourage use of multiple solution strategies.

Situate problem-solving tasks within familiar, meaningful, and realistic contexts in order to

facilitate the transfer of learning.

Develop students’ skills in interpreting numerical or graphical information appearing within

documents and texts.

Develop students’ calculator skills and promote familiarity with computer technology.

Provide plenty of time for practice and reinforce concepts in different contexts.

TLC Tutor Handbook for Math Instruction 7 Tutors of Literacy in the Commonwealth

September 2016

~ Pennsylvania ,&~ Adult Education ~ Resources

Chapter Two: Beginning Level

Math (Levels A and B)

Place Value These levels align with the CCR Standards.

Level A

Students need to be able to:

Read and write numbers.

Count by ones, twos, and tens to 100.

Count to determine the number of objects.

Understand that the two digits of a two-digit number represent amounts of tens and ones.

Compare two-digit numbers based on ones and tens using the following signs: >, =, and <

Understand that:

o The number 10 can be thought of as a bundle of 10 ones — called a ten.

o The numbers from 11 to 19 are composed of a ten and 1, 2, 3, 4, 5, 6, 7, 8, or 9 ones.

o The numbers 10, 20, 30, 40, 50, 60, 70, 80, and 90 refer to 1, 2, 3, 4, 5, 6, 7, 8, or 9 tens (and

0 ones).

Level B


Count, read, and write numbers within 1,000.

Count by fives, tens, and hundreds to 1,000.

Understand that the three digits of a three-digit number represent amounts of hundreds,

tens, and ones. For example, 706 equals 7 hundreds, 0 tens, and 6 ones.

Understand that 100 can be thought of as a bundle of 10 tens — called a hundred.

o The numbers 100, 200, 300, 400, 500, 600, 700, 800, and 900 refer to 1, 2, 3, 4, 5, 6, 7, 8,

or 9 hundreds (and 0 tens and 0 ones).

Compare two three-digit numbers using the signs: >, =, and <.

Round whole numbers to the nearest 10 or 100.


September 2016

• •

Chapter Two: Beginning Level Math (Levels A and B)

Tutoring Strategies

Create informal visuals or help students to create graphics to help them to understand and

remember key concepts and symbols.

Symbols that Compare

> greater than

< less than

= equal to

≠ not equal to

≥ greater than or equal to

≤ less than or equal to

Talk about numbers in terms of their order, position, and amount.

Help students develop a sense of ten.

Place counters in a 10-frame (like the one below) to illustrate numbers to help students to

see the relationship between 10 and a number less than and equal to 10.



September 2016

9


Addition and Subtraction

Level A


Add and subtract within 1,000.

Add a two-digit number and a one-digit number.

Add a two-digit number and a multiple of 10.

Subtract multiples of 10 from multiples of 10 (10-90).

Relate counting to addition and subtraction.

Understand the relationship between addition and subtraction.

Understand that, when adding two-digit numbers, you add tens and tens, and ones and

ones.

Level B


Add and subtract within 100 to solve one- and two-step problems.

Know from memory all sums of two one-digit numbers.

Add up to four two-digit numbers.

Add and subtract three-digit numbers.

o Understand that in adding or subtracting three-digit numbers, you add or subtract

hundreds and hundreds, tens and tens, ones and ones.

o Understand that it is sometimes necessary to compose or decompose tens or hundreds.

Work with addition and subtraction equations, using the equal sign.

Determine the unknown whole number in an addition or subtraction equation:

6 + 6 = ?, 8 + ? = 11, 5 = ? - 3.

Tutoring Strategies

Introduce meaning of term inverse.

Use understanding of place value.

Use concrete models or drawings.

Present addition as putting together or adding to.

Present subtraction as taking apart or taking from.

Use strategies:

o Using the relationship between addition and subtraction.

o Making tens by memorizing the number combinations that make 10, such as

3 + 7 and 6 + 4.

o Decomposing and leading to 10.

o Creating equivalent, but easier, unknown sums.



September 2016

10

�

�


Tutoring Strategies (continued)

Properties of Addition

Commutative property: When two numbers are added, the sum is the same regardless of

the order of the addends. For example: 4 + 2 = 2 + 4.

Associative property: When three or more numbers are added, the sum is the same

regardless of the grouping of the addends. For example: (2 + 3) + 4 = 2 + (3 + 4).

Additive identity property: The sum of any number and zero is the original number. For

example: 5 + 0 = 5.

Distributive property: The sum of two numbers times a third number is equal to the sum of

each addend times the third number. For example: 4 x (6 + 3) = 4 x 6 + 4 x 3. (Explain when

teaching multiplication.)

Composing a number

349 is composed of 3 hundreds, 4 tens, and 9 ones.

300 + 40 + 9 349

Decomposing a number (breaking a number apart)

349 300 + 40 + 9



September 2016

11


Mental Math

Level A


Use mental math to find 10 more or 10 less than a given two-digit number.

Level B


Mentally add or subtract 10 or 100 to or from a given number (100-900).

Add and subtract within 20 using mental strategies.

Assess the reasonableness of their answer using mental math, estimation, and rounding.

Tutoring Strategy

Explain that mental math is doing math in your head, and that research has found that 75

percent of all calculations done by adults are done mentally.

Teach the following mental math strategies:

o Counting on to the biggest number by counting up the second number.

o Making tens by memorizing the number combinations that make 10, such as

3 + 7 and 6 + 4.

o Adding 9 by adding 10 and counting down one.

o Rearranging numbers and operations.

o Memorizing double numbers such as 7 + 7 and 5 + 5.

o Balancing numbers before adding them such as 70 + 55 instead of 68 + 57.



September 2016

12


Algebraic Thinking

Level A

Relate counting to addition and subtraction.

Solve word problems that call for addition of three whole numbers whose sum is less than

or equal to 20, using objects, symbols, drawings, or equations with a symbol for the

unknown number.

Apply properties of operations as strategies to add and subtract (commutative and

associative properties).

Understand subtraction as an unknown addend problem; subtract 10 – 8 by finding the

number that makes 10 when added to 8.

Understand the meaning of the equal sign.

Determine if equations involving addition and subtraction are true or false.

Determine an unknown whole number in an addition or subtraction equation relating three

whole numbers.

Level B

Use addition and subtraction within 100 to solve one- and two-step word problems involving

situations of adding to, taking from, putting together, taking apart, and comparing with

unknowns in all positions.

Understand how to use a formula.

Tutoring Strategies

Use drawings.

Help students to create equations with a symbol for the unknown number to represent the

problem.

Help students to identify and explain arithmetic patterns.

Use the following prompts as ways to encourage algebraic thinking:

o Tell me what you were thinking.

o Can you solve this in a different way?

o How do you know this is true?

o Does this always work?

Teach number sentences to help students to articulate mathematical generalizations.

65 + 0 = 65 When you add 0 to a number, you get the number you started with.

43 - 43 = 0 When you subtract a number from itself, you get 0.



September 2016

13


Measurement

Level A


Solve problems involving dollar bills, quarters, dimes, nickels, and cents.

Measure lengths in whole numbers.

Tell and write time in hours and half-hours.

Level B


Measure and estimate lengths using inches, feet, centimeters, and meters.

Relate addition and subtraction to length.

Tell and write time to the nearest minute and measure time intervals in nearest five minutes,

using AM and PM.

Tutoring Strategies

Help students to solve word problems involving addition and subtraction of time intervals in

minutes by representing the problem on a number line diagram.

Teach measurement in three phases:

o Phase 1: Identify the measurement attribute, discuss its concept, and compare it to other

measurement attributes.

o Phase 2: Teach how to measure it using formal units and by estimating.

o Phase 3: Teach how to calculate by using formulas and converting from one unit to

another.



September 2016

14


Multiplication and Division

Level B


Know from memory all products of two one-digit numbers.

Multiply one-digit whole numbers by multiples of 10.

Multiply and divide within 100 to solve word problems.

Understand the relationship between multiplication and division.

Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the

number that makes 32 when multiplied by 8.

Determine the unknown whole number in a multiplication or division equation relating three

whole numbers. For example, determine the unknown number that makes the equation

true, such as 8 x ? = 48, 5 = ? ÷ 3, and 6 x 6 = ?

Tutoring Strategies

Make sure students understand the concept of multiplication and division and how they

relate to each other.

When introducing products of whole numbers, explain 5 x 7 as the total number of objects in

5 groups of 7 objects each. For example, describe a context in which a total number of

objects can be expressed as 5 x 7.

When introducing whole-number quotients of whole numbers, explain 56 ÷ 8 as the number

of objects in each share when 56 objects are partitioned equally into 8 shares, or as a

number of shares when 56 objects are partitioned into equal shares of 8 objects each. For

example, describe a context in which a number of shares or a number of groups can be

expressed as 56 ÷ 8.

Help students to learn the multiplication tables.

Use fact triangles, like the one below, to help students see the relationship between

multiplication and division.

56

x ÷ 7 8

Vocabulary

Product

Quotient



September 2016

15



Teach number sentences to help students to articulate mathematical generalizations.

27 x 0 = 0 When you multiply a number times 0, you get 0.

55 x 64 = 64 x 55 When multiplying two numbers, you can change the order of the numbers.

Properties of Multiplication

Commutative property of multiplication: If 6 x 4 = 24 is known, then 4 x 6 = 24 is also known.

Associative property of multiplication: 3 x 5 x 2 can be found by 3 x 5 = 15 and then 15 x 2 =

30, or by 5 x 2 = 10 and then 3 x 10 = 30.

Distributive property of multiplication: Knowing that 8 x 5 = 40 and 8 x 2 = 16, you can find 8

x 7 as 8 x (5 + 2) = (8 x 5) + (8 x 2) = 40 + 16 = 56. Another example: 8 x 13 = (8 x 10) + (8 x 3).

Identity property of multiplication: 1 x 5 = 5 and 0 x anything = 0.



September 2016

16

I I I I •.::11=r1IJ1 I ••c�1 1 I I I I I I I I I I I I I

- 11 - 1111 11


Fractions

Level B


Understand a fraction as a part of a whole.

Understand two fractions as equivalent (equal) if they are the same size or the same point

on a number line.

Recognize and generate simple equivalent fractions (1/2 = 2/4, 4/6 = 2/3).

Compare two fractions with same numerator or same denominator.

Tutoring Strategies

Explain a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b

equal parts.

Explain a fraction a/b as the quantity formed by a parts of 1/b.

Explain a fraction as a number on the number line:

o Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as

the whole and partitioning it into b equal parts.

o Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0.

Create and use simple visuals.

(Source: https://sites.google.com/a/ccpsd.k12.va.us/mrs-anderson-s-project-site/home)



September 2016

17

https://sites.google.com/a/ccpsd.k12.va.us/mrs-anderson-s-project-site/home



Have students divide a rectangle into parts to represent a fraction. For example, have them

divide a rectangle into eight parts and shade two of the parts. Then talk about what is the

same and different about 2/8 and 1/4, and 3/4 and 6/8.

Use real-life examples.

Vocabulary

Denominator

Numerator

Reciprocal



September 2016

18


Geometry

Level A


Recognize, analyze, and compare two-dimensional shapes (rectangles, squares, trapezoids,

triangles, half-circles, quarter-circles).

Recognize, analyze, and compare three-dimensional shapes (cubes, right rectangular prisms,

right circular cones, right circular cylinders).

Compose two- and three-dimensional shapes.

Level B


Recognize and draw shapes having specified attributes, such as angles and equal faces.

Understand that shapes in different categories (rhombuses, rectangles, etc.) may share

attributes (e.g., having four sides), and that the shared attributes can define a larger

category, such as quadrilaterals.

Recognize rhombuses, rectangles, and squares as examples of quadrilaterals.

Measure and estimate area.

Measure and estimate perimeter.

Measure and estimate volume.

o Solve problems involving the volume of a cube, cylinder, rectangular solid, cone,

pyramid, or right rectangular prism.

o Apply formulas V = l x w x h and V = b x h for rectangular prisms to find the volume.

Solve problems involving the area and circumference of a circle.

Tutoring Strategies

Work on terminology, but use informal language to describe similarities, differences, parts,

and other attributes of shapes.

Teach perimeter before area.

Give students 2”, 4”, and 6” pieces of straws and pipe cleaners to explore the perimeters of

polygons.

Use cubes (such as Legos) to help students to understand square units.

Show students how to measure volume by counting unit cubes using cm, cubic inch, and

cubic foot.

Use drawings to represent the problem.



September 2016

19



Area and Perimeter

On an index card, write/draw definitions of area and perimeter like the ones below for students to

study. You can include perimeter formulas for other shapes as well.

Area = w x h

w = width (5 inches)

h = height (3 inches)

Area = 5 inches x 3 inches = 15 inches

Perimeter is the distance around a two-dimensional shape.

Perimeter = 2 x (w + h) for a rectangle.

(Source: http://pefourth.weebly.com/math/category/all)

(Source: Pierce, Rod. (29 Jul 2016). Math is fun. Retrieved 2016 from http://www.mathsisfun.com)

Vocabulary

Area

Attributes

Circumference

Perimeter

Plane figures

Polygons

Prisms

Quadrilaterals

Rhombuses

Three-dimensional shapes

Trapezoids

Two-dimensional shapes

Volume



September 2016

20

http://pefourth.weebly.com/math/category/all

http://www.mathsisfun.com/


Data and Statistics

Level A


Organize objects, and represent and interpret data with up to three categories.

Ask and answer questions regarding total number of data points in categories.

Level B


Draw a picture graph and a bar graph to represent a data set with up to four categories.

Draw a scaled picture graph and a scaled bar graph to represent a data set with several

categories.

Solve simple put-together, take-apart, and compare problems using information from a bar

graph.

Solve one-step and two-step how many more and how many less problems using information

from scaled bar graphs.

Read data from a line plot.

Tutoring Strategies

Help students to interpret graphs in newspapers and magazines.

Show students how to construct a scaled graph.

o Identify variables.

o Determine variable range.

o Number and label each axis.

o Determine data points and plot on graph.

Help students to predict outcomes of events, based on data.

Introduce the concept of probability by discussing whether a real-world event is likely or

unlikely.

Vocabulary

Bar graph

Data

Picture graph

Line graph

Line plot

Pie chart/circle graph



September 2016

21

Graph Tide SubTille

300

250

200 " " i2

i2 ... '; 150

., 'i(

·;. < < N

� � 100

50

0 Radial Angle

Ill X Axis Title

� Line 1 � Line 2 � Line 3 � Line 4


Tutoring Strategies (continued) Help students make their own graphs.

(Sources: http://www.bbc.co.uk/bitesize/standard/maths_i/relationships/data_graphs/revision/1/;

http://www.omnis.net/support/newsletter/mar_06.html)


September 2016


Whole Number Problem Solving

Level B


Solve two-step word problems using the four operations (addition, subtraction,

multiplication, and division).

Represent problems as equations with a letter standing for the unknown quantity.

Apply appropriate strategies for solving whole number word problems.

o Read a problem several times.

o Personalize the problem.

o Draw a picture or diagram to help solve the problem.

o Eliminate extraneous information.

o Simplify the problem with easier numbers.

o Determine the number of steps and operations needed to solve the problem (students

will often stop after the first step leading them to choose the wrong answer).

o Solve the problem and check the answer.

Assess the reasonableness of their answer using mental math, estimation, and/or rounding.



September 2016

23

~ Pennsylvania & ~ Adult Education ~ Resources

Chapter Three: Intermediate

Level Math (Levels C and D)

Place Value

These levels align with CCR Standards.

Levels C and D


Recognize that in a multi-digit number, a digit in one place represents ten times as much as

it represents in the place to its right and 1/10 of what it represents in the place to its left.

Round multi-digit numbers to any place.

Compare two multi-digit numbers, based on the meanings of the digits in each place, using

>, =, and < symbols.

Understand decimal place value.

Understand that exponents show how many times a number is multiplied times itself.

Understand powers of 10, and use whole number exponents to denote powers of 10.

o Understand patterns in the number of zeros of the product when multiplying a number

by powers of 10.

o Understand patterns in the placement of the decimal point when a decimal is multiplied

or divided by a power of 10.

o Use numbers expressed in the form of a single digit times an integer power of 10 to

estimate very large or very small quantities, and to express how many times as much

one is than the other. For example, estimate the population of the United States as 3 x

108 and the population of the world as 7 x 109, and determine that the world population

is more than 20 times larger.

Understand that scientific notation uses exponents and powers of 10 to write very large and

very small numbers.

o Perform operations with numbers expressed in scientific notation, including problems

where both decimal and scientific notation are used.

o Use scientific notation and choose units of appropriate size for measurements of very

large or very small quantities; for example, use millimeters per year for seafloor

spreading.

Understand radicals.


September 2016

Coefficient \ / ponent

8.39 X 105

~ base

Convert to Scienti fic Notation 3~

9 units to the LEFT

LEFT -+ posit ive exponent

3.25 X 109

0~

7 units to the RIGHT

RIGHT -+ negative exponent

4 X 10-7

Chapter Three: Intermediate Level Math (Levels C and D)

Tutoring Strategies

Construct a decimal place value chart.

(Source: http://www.coolmath.com/prealgebra/02-decimals/01-decimals-place-value-0)

Help students to create their own visuals to help them to understand and remember

important concepts and terminology.

(Source: https://www.tes.com/lessons/EreV9eU6MeqMzw/scientific-notation-and-significant-figures)

(Source: http://msroymaths7.weebly.com/powers-scientific-notation--square-roots.html)

Vocabulary

Exponent

Integer

Powers of 10

Scientific notation



September 2016

25



Decimal Place Value Card Game

The goal of this game is to form the highest decimal number using playing cards. You can also play

this game at a lower level by replacing the place value columns with hundred thousands, ten

thousands, thousands, hundreds, tens, and ones.

Number of players: 2-4

Materials needed:

Score sheet - one per player

Deck of regular playing cards (remove all jacks, queens, kings, and jokers)

Pencil for each player

Score Card: Circle any rounds in which you earned one point.

Round H T O Tths Hths Thths

1

2

3

4

5

6

7

8

9

10

How to play the game:

1. Shuffle the deck of cards and place it face down in the center of the table.

2. Each player takes a turn taking one card from the top of the pile. Players place the card face-up

directly in front of them so that all players can see each card that has been drawn.

3. After drawing a card from the pile, the player looks at the number on the card and decides

which place value to assign that number. The player can only write that number under one place

value column (hundreds, tens, ones, tenths, hundredths, or thousandths). Players should not

show their opponents where they wrote the number on their score sheet. Once each player has

written down the number in a place value column, it cannot be changed at any point during the

game.

4. Players repeat steps two and three until all six place value columns have been filled in (which

means that six cards have been drawn by each player).

5. Players show their final number to the other players to determine who has written the highest

number.



September 2016

26



6. The player with the highest number must read his/her decimal aloud correctly (i.e. one hundred

and seven thousandths, not one hundred point zero zero seven) in order to score the points.

(The tutor can read it correctly or not.)

If the player with the highest number reads his/her decimal aloud correctly, he/she gets one

point for the round. The other players do not get any points.

If the player with the highest number does not read his/her decimal aloud correctly, he/she

does not get any points. Every other player gets one point for the round.

7. Place all of the used cards back into the pile of cards and shuffle the entire pile again. Begin the

next round by repeating the same steps. Play ends after ten rounds have been played.

(Source: http://games4gains.com/blogs/teaching-ideas/41379652-decimal-place-value-with-playing-cards)



September 2016

27

http://games4gains.com/blogs/teaching-ideas/41379652-decimal-place-value-with-playing-cards

iples of 3 3 6 9 , 12 15 18 -

Q 1 ,2 3 4 5 6 7 8 9 10 11 12 13 14 15 11$ 17 18 1!l 20

5,- - 10 - 15 - � 20 Multiples o·f 5


Number System

Level C


Recognize that a whole number is a multiple of each of its each of its factors.

Find all factor pairs for a whole number within the range of 1-100.

Find the greatest common factor of two whole numbers less than or equal to 100.

Find the least common multiple of two whole numbers less than or equal to 12.

Determine if a whole number within the range of 1-100 is a multiple of a given one-digit

number.

Determine if a given whole number within the range of 1-100 is prime or composite.

Tutoring Strategies

Multiples and Least Common Multiple

On an index card, write/draw definition of multiples and least common multiples like the one below

for students to study.

Multiples are the result of multiplying a number by another number.

Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36. . . .

Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40. . . . .

The least common multiple is the smallest number that is a multiple of two or more numbers.

The least common multiple of 3 and 5 is 15.

(Source: Pierce, Rod. (29 Jul 2016). Math is fun. Retrieved 2016 from http://www.mathsisfun.com)

Prime Numbers, Composite Numbers, and Factors

On an index card, write/draw definitions of prime numbers, composite numbers, and factors like the

ones below for students to study.

A factor is a number that divides exactly into a larger number, such as the numbers 1, 2, 3, 4,

and 6 for the number 12.

A prime number has only two factors: 1 and itself, such as the number 7.

A composite number has more than two factors, such as the number 12 (1, 2, 3, 4, 6, and 12)



September 2016

28



Level D


Find and position integers and other rational numbers on a horizontal or vertical number

line diagram.

Understand ordering and absolute value of rational numbers.

o Write, interpret, and explain statements of order for rational numbers in real-world

contexts.

o Understand the absolute value of a rational number as its distance from 0 on the

number line.

Understand that 6 is 6 away from zero, and that −6 is also 6 away from zero, so the

absolute value of 6 is 6 and the absolute value of −6 is also 6.

o Interpret absolute value as magnitude for a positive or negative quantity in a real-world

situation.

o Distinguish comparisons of absolute value from statements about order.

Know that there are numbers that are not rational, and approximate them by rational

numbers.

o Use rational approximations of irrational numbers to compare the size of irrational

numbers, locate them approximately on a number line diagram, and estimate the value

of expressions.

Understand that positive and negative numbers are used together to describe quantities

having opposite directions or values, such as credits/debits.

o Recognize opposite signs as indicating locations on opposite sides of 0 on the number

line.

o Solve multi-step problems posed with positive and negative rational numbers in any

form (whole numbers, fractions, and decimals).

o Understand signs of numbers in ordered pairs as indicating locations in quadrants of the

coordinate plane; recognize that when two ordered pairs differ only by signs, the

locations of the points are related by reflections across one or both axes.



September 2016

29

A.b~,tc AJ,sef4,~ "°"wt -. \) .....

I ( •I) ( ., , . . .

-'I - 3 · Z · I I ,-I z 3 1


Tutoring Strategies

Ordering and absolute value.

o Show students how a rational number can be represented as a point on a number line

and the number line can be used as a tool to order rational numbers.

o Explain that absolute value can be described in more than way, depending upon the

real-world context. It can be distance, or it can be size (magnitude).

Use models, diagrams, manipulatives, number lines, and patterns in developing and

remembering algorithms for computing with positive and negative numbers.

o Explain that computation with positive and negative numbers is often necessary to

determine relationships between quantities, and that positive and negative numbers are

often used to solve problems in everyday life.

o Explain that a positive quantity and negative quantity of the same absolute value add to

make 0.

o Describe situations in which opposite quantities combine to make 0. For example, if a

check is written for the same amount as a deposit, made to the same checking account,

the result is a zero increase or decrease in the account balance.

Explain when irrational numbers can be used in solving problems.

Create and use visuals.

(Source: http://tjhomeschooling.blogspot.com/2015/08/absolute-value-graphic-organizer.html)

Questions to discuss with students:

How do you use positive and negative numbers to describe quantities having opposite

values?

What is absolute value?

Is it always true that multiplying a negative factor by a positive factor always produces a

negative product?

How are rational and irrational numbers related?



September 2016

30

http://tjhomeschooling.blogspot.com/2015/08/absolute-value-graphic-organizer.html

-

-


Measurement

Levels C and D


Understand the concept of distance and how to measure it.

Understand how to measure liquid volume.

Understand the difference between mass and weight, and how they are measured.

Solve problems involving distances, intervals of time, liquid volumes, masses of objects, and

money, including problems involving simple fractions or decimals, and problems that

require expressing measurements given in a larger unit in terms of a smaller unit.

Convert like measurement units within a given measurement system.

Tutoring Strategies

Represent measurement quantities using diagrams such as number line diagrams that

feature a measurement scale.


Measurement customary units Example

Convert between inches, feet and yards 147 in = _____ ft _____ in

Convert between miles, feet and yards 8800 yd = ____ mi _____ yd

Convert between ounces, pounds and tons 172 oz = _____ lb _____ oz

Convert between cups, pints, quarts and

gallons 6 C = _____ qt ______ C

Convert between ounces, cups, quarts and

gallons 230 oz = ____ gal ____ qt ____ oz

Measurement metric units Example

Convert between mm, cm, m and km 4896 cm = ____ m ____ cm

Convert between mm, cm, m and km - using

decimals 0.4 cm = ______ mm

Convert between ml & l and g & kg 44305 g = ____ kg ____ g

Convert between ml & l and g & kg - using

decimals 6.095 L = ______ ml

(Source: http://metricssixth.wikispaces.com)



September 2016

31

http://www.k5learning.com/free-math-worksheets/sixth-grade-6/measurement/convert-units-length

http://www.k5learning.com/free-math-worksheets/sixth-grade-6/measurement/conversion-miles-feet-yards

http://www.k5learning.com/free-math-worksheets/sixth-grade-6/measurement/convert-units-mass

http://www.k5learning.com/free-math-worksheets/sixth-grade-6/measurement/convert-units-volume

http://www.k5learning.com/free-math-worksheets/sixth-grade-6/measurement/convert-units-volume

http://www.k5learning.com/free-math-worksheets/sixth-grade-6/measurement/conversion-ounces-cups-quarts-gallons

http://www.k5learning.com/free-math-worksheets/sixth-grade-6/measurement/conversion-ounces-cups-quarts-gallons

http://www.k5learning.com/free-math-worksheets/sixth-grade-6/measurement/conversion-metric-length

http://www.k5learning.com/free-math-worksheets/sixth-grade-6/measurement/conversion-metric-length-decimals

http://www.k5learning.com/free-math-worksheets/sixth-grade-6/measurement/conversion-metric-length-decimals

http://www.k5learning.com/free-math-worksheets/sixth-grade-6/measurement/conversion-metric-volume-mass

http://www.k5learning.com/free-math-worksheets/sixth-grade-6/measurement/conversion-metric-volume-mass-decimals

http://www.k5learning.com/free-math-worksheets/sixth-grade-6/measurement/conversion-metric-volume-mass-decimals

http://metricssixth.wikispaces.com/


Multiplication and Division

Levels C and D


Apply properties of operations as strategies to multiply and divide rational numbers.

Multiply two two-digit numbers.

Multiply up to a four-digit whole number by one-digit number.

Find whole number quotients of whole numbers with up to four-digit dividends and two-digit

divisors.

Understand that integers can be divided, provided that the divisor is not zero, and every

quotient of integers (with non-zero divisor) is a rational number. If p and q are integers,

then –(p/q) = (–p)/q = p/(–q).

Convert a rational number to a decimal using long division.

Tutoring Strategies

Explain products of rational numbers by describing real-world contexts.

Explain quotients of rational numbers by describing real-world contexts.

Help students to illustrate and explain their calculations.

Divisibility Tricks

If a number is divisible by 2, it is even.

If a number is not divisible by 2, the number is odd.

A number is divisible by 3 if the sum of the digits is divisible by 3. For example, 243 is

divisible by 3 because 2 + 4 + 3 = 9.

A number is divisible by 9 if the sum of the digits is divisible by 9. For example, 891 is

divisible by 9 because 8 + 9 + 1 = 18.

A number is divisible by 4 if the last two digits are divisible by 4. For example, 132 is divisible

by 4 because 32 is divisible by 4.



September 2016

32



Multiplication Tricks

To multiply by Trick

2 Add the number to itself (example 2×9 = 9+9).

The last digit goes 5, 0, 5, 0, ...

5 Is always half of 10× (Example: 5x6 = half of 10x6 = half of 60 = 30).

Is half the number times 10 (Example: 5x6 = 10x3 = 30).

6 When you multiply 6 by an even number, they both end in the same digit.

Example: 6×2=12, 6×4=24, 6×6=36, etc.

9

The last digit goes 9, 8, 7, 6, ...

Example: To multiply 9 by 8, hold your 8th finger down, and count "7" and "2",

the answer is 72.

Is 10× the number minus the number. Example: 9×6 = 10×6−6 = 60−6 = 54

When you add the answer's digits together, you get 9. Example: 9×5=45 and

4+5=9. (But not with 9×11=99).

10 Put a zero after it.

Up to 9x11: Just repeat the digit (Example: 4x11 = 44).

11

For 10x11 to 18x11: write the sum of the digits between the digits. Example:

15x11 = 1(1+5)5 = 165.

Note: This works for any two-digit number, but when the sum of the digits is

more than 9, you need to carry the one. Example: 75x11 = 7(7+5)5 = 7(12)5 =

825.

12 Is 10× plus 2×



September 2016

33

http://www.mathsisfun.com/numbers/addition-column.html


Fractions, Decimals, and Percents

Level C


Draw, read, and write proper and improper fractions and mixed numbers.

o Find fractions and mixed numbers on a number line.

o Solve word problems by adding and subtracting fractions with like denominators.

o Add and subtract fractions with unlike denominators, including mixed numbers.

o Add and subtract mixed numbers with like denominators.

o Decompose a fraction into a sum of fractions with the same denominator.

o Compare two fractions with different numerators and different denominators.

o Solve word problems by multiplying a fraction by a whole number or a fraction.

o Divide fractions by whole numbers and whole numbers by fractions.

o Divide fractions by fractions.

Understand that a decimal shows a fraction of a number using the place value system.

o Order decimals.

o Understand, read, and write decimals to thousandths.

o Compare two decimals to hundredths.

o Compare decimals based on the meanings of the digits in each place, using >, =, and <

symbols.

o Add, subtract, multiply, and divide decimals to hundredths.

o Round decimals to any place.

o Solve word problems involving decimals.

Understand that percent means per 100.

o Understand that a percent can be expressed as a fraction or decimal.

Change decimals to fractions and percents.

Interpret multiplication as scaling (resizing), by:

o Comparing the size of a product to the size of one factor on the basis of the size of the

other factor, without performing the indicated multiplication.

o Explaining why multiplying a given number by a fraction greater than 1 results in a

product greater than the given number (recognizing multiplication by whole numbers

greater than 1 as a familiar case).

o Explaining why multiplying a given number by a fraction less than 1 results in a product

smaller than the given number; and relating the principle of fraction equivalence a/b = (n

x a)/(n x b) to the effect of multiplying a/b by 1.



September 2016

34


Tutoring Strategies

Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b) by using visual fraction

models, with attention to how the number and size of the parts differ even though the two

fractions themselves are the same size. Use this principle to recognize and generate

equivalent fractions.

Explain a fraction as division of the numerator by the denominator (a/b = a ÷ b).

Explain addition and subtraction of fractions as joining and separating parts referring to the

same whole.

Relate decimals to money.

Use percents to calculate taxes, commissions, and money.

Show students how to calculate percent of increase or decrease.

Vocabulary

Decimal

Percent (Percents are also addressed in Ratio, Rate, and Proportion.)

Improper fraction

Mixed number

Proper fraction

Scaling



September 2016

35


Geometry

Level C


Draw points, lines, line segments, rays, angles, and perpendicular, and parallel lines.

Recognize acute, obtuse, straight, and right angles as geometric shapes that are formed

wherever two rays share a common endpoint.

o Understand concepts of angle measurement; an angle is measured with reference to a

circle with its center at the common endpoint of the rays, by considering the fraction of

the circular arc between the points where the two rays intersect the circle.

o Measure angles in whole-number degrees using a protractor.

o Sketch angles of specified measure.

o Solve addition and subtraction problems to find unknown angles on a diagram in real-

world and mathematical problems.

Solve problems involving area, volume, and surface area of two- and three-dimensional

objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.

Level D


Draw, construct, and describe geometric figures and describe the relationship between

them.

Use facts about supplementary, complementary, vertical, and adjacent angles to write and

solve equations about unknown angles.

Understand that a two-dimensional figure is congruent to another if the second can be

obtained from the first by a sequence of rotations, reflections, and translations.

Understand that a two-dimensional figure is similar to another if the second can be obtained

from the first by a sequence of rotations, reflections, translations, and dilations.

Solve problems involving scale drawings of geometric figures.

Understand the Pythagorean Theorem: That when a triangle has a right angle (90°) and

squares are made on each of the three sides, then the biggest square has the exact same

area as the other two squares put together; understand that It is can be written in one short

equation: a2 + b2 = c2.

o Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in

real-world and mathematical problems in two and three dimensions.

o Apply the Pythagorean Theorem to find the distance between two points in a coordinate

system.



September 2016

36

a

�

�


Tutoring Strategies Explain that attributes belonging to a category of two-dimensional figures also belong to all

subcategories of that category. For example, all rectangles have four right angles and

squares are rectangles, so all squares have four right angles.

Illustrate a sequence of transformations that lead to congruent or similar figures.

Apply the area and perimeter formulas for rectangles in real-world and mathematical

problems. For example, help students to find the width of a rectangular room given the area

of the flooring and the length, by viewing the area formula as a multiplication equation with

an unknown factor.

Relate volume to the operations of multiplication and addition and solve real-world and

mathematical problems involving volume.


Pythagorean Theorem

a2 + b2 = c2

(Source: Pierce, Rod. (29 Jul 2016. Math is fun. Retrieved 2016 from https://www.mathsisfun.com/pythagoras.html)

Congruent and Similar Figures

When we The shapes are

Only rotate (turn), reflection (flip),

or translate (slide)

Congruent

(Figures are the same shape and size.)

Also need to resize Similar

(Figures are the same shape, but not the same size.)


How do you find the surface area and volume of a three-dimensional figure?

What is the total number of degrees in supplementary and complementary angles?

What is the relationship between vertical and adjacent angles?

Vocabulary

Angles (acute, obtuse, straight, right, adjacent, complementary, supplementary, and vertical)

Protractor

Rotation, reflection, translation, dilation

Scale factor

Surface area

Transversal

Vertex



September 2016

37

https://www.mathsisfun.com/pythagoras.html



Scale Drawings

Explain that a scale drawing shows a real object with accurate sizes that have been either

reduced or enlarged by a certain amount (called the scale).

Explain that the scale is shown as the ratio of the length in the drawing to the length of the

real thing. For example: 1:10, which means in the drawing anything with the size of 1 would

have a size of 10 in the real world.



September 2016

38


Coordinate Plane

Levels C and D


Understand that a coordinate plane is a visual representation of points (x-value and y-value)

that are known as ordered pairs or coordinates.

Understand that axes define a coordinate system with the intersection of the lines (the

origin) arranged to coincide with zero on each line.

Understand that slope is a number that refers to the steepness of a line.

o Determine the slope between two points.

Graph points on a coordinate plane.

Determine the intercepts of a graphed line.

Graph a two-variable equation.

Determine the equation of a line when given the graph of a line.

Draw polygons in a coordinate plane, given the coordinates for the vertices.

Solve real-world and mathematical problems by graphing points in all four quadrants of a

coordinate plane.

Tutoring Strategies

Have students practice:

o Plotting points on a coordinate plane.

o Reading coordinates of a point on a coordinate plane.

o Finding the distance between points on a coordinate plane by counting.

o Moving a point and determining its location on a coordinate plane.

o Determining the shortest distance between two points on a coordinate plane.

o Drawing a shape on a coordinate plane.

Use a visual to identify parts of a coordinate plane.

(Source: https://en.wikibooks.org/wiki/Geometry_for_Elementary_School/Rectangular_coordinate_system)



September 2016

39

https://en.wikibooks.org/wiki/Geometry_for_Elementary_School/Rectangular_coordinate_system




What is the meaning of the slope and intercept of a line?

How can similar triangles be used to show that the slope is the same, given two distinct sets

of points on a graph?

Is the slope between any two points on the same line the same?

How can an equation be created with given information from a graph?

Vocabulary

Coordinate plane

Intercept/point of interception

Origin

Point

Quadrant

Slope

Vertical

x axis, y axis

x coordinate, y coordinate

x intercept, y intercept



September 2016

40


Ratio, Rate, and Proportion Level C


Understand that a ratio compares two numbers.

o Describe the ratio relationship between two quantities.

Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0.

Use rate language in the context of a ratio relationship.

Understand that when two ratios are written as equal, the equation is a proportion.

Level D


Recognize, analyze, and represent proportional relationships between quantities.

Use proportional relationships to solve multistep ratio and percent problems, such as simple

interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase

and decrease, and percent error.

Represent proportional relationships by equations. For example, if total cost (t) is

proportional to the number (n) of items purchased at a constant price (p), the relationship

between the total cost and the number of items can be expressed as t = pn.

Use ratio reasoning to solve problems.

Compare ratios.

Use ratio reasoning to convert measurement units.

Solve unit rate problems.

Make tables of equivalent ratios relating quantities with whole-number measurements; find

missing value from table; plot values on coordinate plane.

Tutoring Strategies

Explain that the relationship between two quantities can often be expressed as ratios and

can be explained using ratio language.

Explain that proportional relationships express how quantities change in relation to each

other.

Show students how multiplication and division can be used to solve ratio and rate problems.

Apply ratios, rates, and percents to real-life situations.


What is a ratio and how does it describe a relationship between two quantities?

What is a unit rate and how do you use it in the context of a ratio relationship?

How would you use ratio and rate reasoning in real-world situations?

How can proportions be used to solve problems?

When is a relationship proportional?

How can proportions increase our understanding of the real world?



September 2016

41

: 1



Vocabulary

Per

Percent

Proportional relationship

Rate

Ratio

Unit

Unit rate

Ratio

On an index card, write/draw a definition of ratio like the one below for students to study. Include

examples that are meaningful to students.

A ratio states how two values compare.

Ratios can be expressed as:

3 : 1 or a : b

3 to 1 or a to b

As a fraction: 3/1 or a/b

Source: Pierce, Rod. (29 Jul 2016). Math is fun. Retrieved 2016 from http://www.mathsisfun.com.



September 2016

42



Statistics and Probability

Level C


Recognize a statistical question that anticipates variability in the data.

Represent data with plots on the real number line (dot plots, histograms, and box plots).

Use statistics appropriate to the shape of the data distribution to compare center (median,

mean, mode, and range) of two or more different data sets.

Recognize that a measure of center for a numerical data set summarizes all of its values with

a single number, while a measure of variation describes how its values vary with a single

number.

Level D


Summarize and describe distributions.

Contrast measures of center and measures of variability for numerical data.

Use measures of center and measures of variability for numerical data from random

samples to draw informal comparative inferences about two populations.

Construct and interpret a two-way table summarizing data on two categorical variables

collected from the same subjects.

o Use relative frequencies calculated for rows or columns to describe possible association

between the two variables. For example, collect data from friends on whether or not

they like to cook and whether they participate actively in a sport. Is there evidence that

those who like to cook also tend to play sports?

Understand that the probability of a chance event is a number between 0 and 1 that

expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood.

A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event

that is neither unlikely nor likely, and a probability near 1 indicates a likely event.

Approximate the probability of a chance event by collecting data on the chance process that

produces it and observing its long-run relative frequency, and predict the approximate

relative frequency given the probability. For example, when rolling a number cube 600 times,

predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.

Develop a probability model and use it to find probabilities of events; compare probabilities

from a model to observed frequencies; if the agreement is not good, explain possible

sources of the discrepancy.

Understand that, just as with simple events, the probability of a compound event is the

fraction of outcomes in the sample space for which the compound event occurs.

Represent sample spaces for compound events using methods such as organized lists,

tables, and tree diagrams.



September 2016

43

0 0 0 0 0 0 0 000000 0 (X) 0

0 5 10 2S 30 35

I I I I 0

130 140 150 160 170 H i ht


Define and differentiate between univariate data and bivariate data.

Construct and interpret scatter plots for bivariate measurement data to investigate patterns

of association between two quantities.

o Describe patterns such as clustering, outliers, positive or negative association, linear

association, and nonlinear association.

Tutoring Strategies Use a four-step investigative process for statistical reasoning:

o Formulate questions that can be answered with data.

o Design and use a plan to collect relevant data.

o Analyze the data with appropriate methods.

o Interpret results and draw valid conclusions from the data that relate to the questions

posed.

Use visuals.



September 2016

44

USA l'i$togrem ;

�50

�00 ---- -350 ...._ ~

- --300 -- -

Cl ~

j: ...._

--lo ... ~-150 -

100 -so

~ 0 20 40 60 80 100

Age



(Source: http://math.serpmedia.org/poster_problems/dot-plots,-histograms,-and-box-plots.html)

Probability

On an index card, write/draw a definition of probability like the one below for students to study.

Include examples that are meaningful to students.

Probability does not tell us exactly what will happen, but rather how likely an event is to happen.

Probability = the number of ways it can happen divided by the total number of outcomes.

The sample space is all the possible outcomes. For example, if you choose a card from a deck of

cards, the sample space is the 52 possible cards.



September 2016

45

http://math.serpmedia.org/poster_problems/dot-plots,-histograms,-and-box-plots.html

coin 2nd coin 3rd coin


Tutoring Strategies (continued) Probability Tree Diagram

A probability tree diagram illustrates all the possible outcomes of an event and can help calculate

their probability. To find out the probability of a particular outcome, you need to look at all the

available paths (set of branches).

The sum of the probabilities for any set of branches is always 1.

To find a probability of an outcome, you multiply along the branches and add vertically.

For example, if you toss two coins, getting heads with the first coin will not affect the

probability of getting heads with the second. The following tree diagram illustrates a coin

being tossed three times–there are eight possible outcomes.

(Source: http://www.bbc.co.uk/schools/gcsebitesize/maths/statistics/probabilityhirev1.shtml)


How are lists, tables, tree diagrams, or simulation used to find the probability of an event?

How is probability used to predict frequency of an event?

Vocabulary

Bivariate data

Distribution

Mean, median, mode

Probability

Sample

Statistics

Univariate data

Variability



September 2016

46

http://www.bbc.co.uk/schools/gcsebitesize/maths/statistics/probabilityhirev1.shtml


Beginning Algebra

Level C


Perform arithmetic operations, including those involving whole-number exponents, in the

conventional order when there are no parentheses to specify a particular order (Order of

Operations).

Identify parts of an expression using mathematical terms.

Write, interpret, and evaluate numerical expressions involving whole-number exponents and

in which letters stand for numbers.

Identify when two expressions are equivalent.

Write expressions in equivalent forms to solve problems.

Understand that a variable can represent an unknown number.

Use variables to represent two quantities in a problem that change in relationship to one

another.

o Analyze the relationship between the dependent and independent variables using

graphs and tables and relate this to the equation.

Understand that an equation is a math statement that shows two expressions are equal and

can be solved by isolating the variable through inverse operations.

Understand that an inequality states that two expressions are unequal.

Understand solving an equation or inequality as a process of answering a question: Which

values from a specified set, if any, make the equation or inequality true?

o Use substitution to determine whether a given number in a specified set makes an

equation or inequality true.

Level D


Create and reason with equations and inequalities.

Understand that the square of a number is the result of multiplying the number times itself.

o Understand that finding the square root of a number requires finding a second number

that, when multiplied times itself, gives the first number.

Understand that the cube of a number is the result if multiplying that number by itself three

times.

o Understand that the cube root of a number is a number that, when cubed, gives the

original number.

Represent and solve equations and inequalities graphically.

Solve linear equations in one variable with one solution, infinitely many solutions, or no

solutions.



September 2016

47


Tutoring Strategies

Researchers recommend:

(Source: https://ies.ed.gov/ncee/wwc/Docs/practiceguide/wwc_algebra_040715.pdf)

Use solved problems that reflect the learning objectives to engage students in analyzing

algebraic reasoning and strategies.

o Select problems with varying levels of difficulty and arrange them from simplest to most

complex examples of the same concept.

o Use multiple examples of solved problems simultaneously to encourage students to

recognize patterns in the solution steps across problems.

o Ask students to describe the steps taken in the solved problem and to explain the

reasoning used. What were the steps? Why do they work in this order? Would they work

in a different order? Could the problem have been solved with fewer steps?

o Ask students specific questions about the solution strategy, and whether that strategy is

logical and mathematically correct. Will this strategy always work? Why or why not?

o Select some problems that illustrate common errors. Once students have an

understanding of correct strategies and problems, show students an incorrect solved

problem by itself or side-by-side with a correct version of the same problem. Have

students explain why identified errors led to an incorrect answer so they can better

understand the correct strategies. Clearly label correct and incorrect examples, so

students do not confuse correct and incorrect strategies.

Teach students that different algebraic representations can convey different information

about an algebra problem.

Teach students to notice the structure of equations, including quantities, variables,

operations, the presence of an equality or inequality, and relationships between quantities,

operations, and equalities or inequalities. Explain that complex structures are built out of

simple ones.

o Make a diagram of the expression or equation and discuss.

o Explain that they should ask themselves questions about the problem, such as: What am

I being asked to do in this problem? Is this problem structured similarly to another

problem I’ve seen before? How many variables are there? What am I trying to solve for?

What are the relationships between quantities in this expression or equation? How will

the placement of quantities and operations impact what I should do first?

Teach students to recognize and generate strategies for solving problems.

o Give students examples of problems that illustrate the use of multiple algebraic

strategies. Include strategies that students commonly use, as well as alternative

strategies that may be less obvious. Have students analyze the effectiveness and

efficiency of different strategies for solving a problem. What are the similarities? What

are the differences? What is the reasoning to use each strategy? Do different strategies

get the correct solution, and will they always work?



September 2016

48

https://ies.ed.gov/ncee/wwc/Docs/practiceguide/wwc_algebra_040715.pdf

= 1 since 12 = 1 ,----

\ 1 4 = 2 since 2 2= 4

,' 9 = 3 since 3 2 = 9 , 116 = 4 stnce 4 2 = 16

, 125 = 5 5ince 5 2 = 25 , 136 = 6 since 6 2 = 36 , 149 = 7 since 7 2 = 49 ,,64 = 8 since e,2 = 64 ,, 81 = 9 since 9 2 = 81

\.:'wo = 10 since 102 = 100

1 is the first cube number, because 1 x 1 x 1 = 1

8 is the second cube number, because 2 x 2 x 2 = 8

27 is the third cube number, because 3 x 3 x 3 = 27

64 is the fourth cube number, because 4 x 4 x 4 = 64


Tutoring Strategies (continued) Turn equations into word problems.

Use visuals to help students understand concepts and learn terminology.

(Source: http://msroymaths7.weebly.com/powers-scientific-notation--square-roots.html)


How can you apply the properties of operations to generate equivalent expressions?

Which values from a specified set, if any, make an equation or inequality true?

In what ways can you reason and solve one-variable equations and inequalities?

How do expressions and equations apply to real-life situations?

How might an inequality describe a real-life problem?

How can you show that inequalities can have infinitely many solutions?

In what ways can you show the relationship between dependent and independent variables?

How would you determine that a relationship is a function?



September 2016

49

http://msroymaths7.weebly.com/powers-scientific-notation--square-roots.html



Vocabulary

Coefficient

Equation

Equivalent

Expression

Function

Inequality

Input

Radical

Terms

Variables

Order of Operations (PEMDAS)

Parentheses first

Exponents (powers and square roots, etc.)

Multiplication and Division (left-to-right)

Addition and Subtraction (left-to-right)



September 2016

50


Appendix

The Appendix contains the College and Career Readiness Standards for Mathematics referenced in

this handbook and are excerpted from the following source:

Pimentel, S. (2013). College and Career Readiness Standards for Adult Education. U.S. Department of

Education, Office of Vocational and Adult Education.


September 2016

tutor math handbook - paadultedresources.org · tabe (9–10) scale score for total math: 313 and...

Documents