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Fifth-Grade English Language Arts

The Georgia Department of Juvenile Justice

7th Grade Mathematics

Units of Instruction Resource Manual

Table of Contents

7th Grade Mathematics

Acknowledgments

Superintendents Letter

Mission and Vision Statements

Chapter 1: Introduction

Chapter 2: Teachers Guide

Chapter 3: Instructional Rotation

Chapter 4: Georgia Performance Standards

Chapter 5: Curriculum Map

Chapter 6: Essential Questions and Enduring Understandings

Chapter 7: Units of Instruction

Chapter6: Units of Instruction

Unit 1: Algebraic Reasoning

Task 1

Task 2

Task 3

Task 4

Task 5Focus CAPs

Unit 2: Integers & Rational Numbers, Applying Rational Numbers,

Patterns & Functions

Task 1

Task 2

Task 3

Task 4

Task 5

Task 6

Task 7

Task 8

Task 9 Focus CAPs

Unit 3: Proportional Relationships

Task 1

Task 2

Task 3

Task 4

Task 5

Task 6

Task 7

Task 8

Task 9

Task 10

Task 11 Focus CAPs

Unit 4: Percents

Task 1

Task 2

Task 3

Task 4 Focus CAPs

Unit 5: Collecting Displaying and Analyzing Data

Task 1

Task 2

Task 3

Task 4

Task 5 Focus CAPs

Unit 6: Geometric Figures

Task 1

Task 2

Task 3

Task 4

Task 5

Task 6

Task 7 Focus CAPs

Unit 7: Measurements Two Dimensional Figures

Task 1

Task 2

Task 3

Task 4

Task 5

Task 6 Focus CAPs

Unit 8: Three Dimensional Figures, Probability &

Multi-Step Equations & Inequalities

Task 1- Culminating Task

Task 2 Focus CAPs

Chapter 8: Task websites

Acknowledgements

The Georgia Department of Juvenile Justice Department of Education would like to thank the many educators who have helped to create this 7th Grade Math Units of Instruction Resource Manual. The educators have been particularly helpful in sharing their ideas and resources to ensure the completion and usefulness of this manual.

Students served by the DJJ require a special effort if they are to become contributing and participating members of their communities. Federal and state laws, regulations, and rules will mean nothing in the absence of professional commitment and dedication by every staff member.

The Georgia Department of Juvenile Justice is very proud of its school system. The school system is Georgias 181st and is accredited by the Southern Association of Colleges and Schools (SACS). The DJJ School System has been called exemplary by the US Department of Justice. This didnt just happen by chance; rather it was the hard work of many teachers, clerks, instructors and administrators that earned DJJ these accolades and accreditations. The DJJ education programs operate well because of the dedicated staff. These dedicated professionals are the heart of our system.

These Content Area Units of Instruction were designed to serve as a much needed tool for delivering meaningful whole group instruction. In addition, this resource will serve as a supplement to the skills and knowledge provided by the Georgia Department of Juvenile Justice Curriculum Activity Packets (CAPs).

I would like to thank all the DJJ Teaching Staff, the Content Area Leadership Teams, Kimberly Harrison, DJJ Special Education/Curriculum Consultant and Martha Patton, Curriculum Director for initiating this project and seeing it through. Thank you all for your hard work and dedication to the youth we serve.

Sincerely yours,

James Jack Catrett, Ed.D.

Associate Superintendent

Mission

The mission of Department of Juvenile Justice Math Consortium (DJJMC) is to build a multiparty effort statewide to achieve continuous, systemic and sustainable improvements in the education system serving the Math students of the Department of Juvenile Justice (DJJ).

Vision

To achieve the mission of the DJJMC, members work collaboratively in examining the Georgia Performance Standards. These guidelines speak specifically to teachers being able to: deliver meaning content pertaining to the Characteristics of Math and its content standards across the Math Units of Instruction Resource Manual. The DJJMC will master and develop whole-group unit lessons built around Curriculum Activity Packets (CAPs), critique student work, and work as a team to solve the common challenges of teaching within DJJ. Additionally, the DJJMC jointly analyzes student test data in order to: develop strategies to eradicate common academic deficits among students, align curriculum, and create a coherent learning pathway across grade levels. The DJJMC also reviews research articles, attends workshops or courses, and invites consultants to assist in the acquisition of necessary knowledge and skills. Finally, DJJMC members observe one another in the classroom through focus walks.

Introduction

The 7th Grade Math Units of Instruction Resource Manual is a tool that has been created to serve as a much needed tool for delivering meaningful whole group instruction. This manual is a supplement to the skills and knowledge provided by the Georgia Department of Juvenile Justice Curriculum Activity Packets (CAPs). It is imperative that our students learn to reason mathematically, to evaluate mathematical arguments both formally and informally, to use the language of mathematics to communicate ideas and information precisely, and to work in cooperative learning groups. Best practices in education indicate that teachers should first model new skills for students. Next, teachers should provide opportunities for guided practice. Only then should teachers expect students to successfully complete an activity independently. The 7th Grade Math Units of Instruction meets that challenge.

The Georgia Department of Juvenile Justice

Office of Education

Direct Instruction Lesson Plan

Teacher:

Subject:______________________________

Date:_____________to__________________

Period

1st

2nd

3rd

4th

5th

6th

Students will engage in:

Independent activities pairing

Cooperative learning hands-on

Peer tutoring Visuals

technology integration Simulations

a project centers

lecture Other

Essential Question(s):

Standards:

CAPs Covered:

Grade Level:____ Unit:______

RTI Tier for data collection: 2 or 3

Tier 2 Students:

Tier 3 Students:

Time

Procedures Followed:

Material/Text

_______

Minutes

Review of Previously Learned Material/Lesson Connections:

Recommended Time: 2 Minutes

_______

Minutes

Display the Georgia Performance Standard(s) (project on

blackboard via units of instruction located at

http://thevillage411.weebly.com/units-of-instruction3.html, or print on blackboard) Read the Georgia Performance Standard(s) aloud and explain it to your students. You can rephrase the Georgia Performance Standard to make sure your students understand it.

Display the Essential Question(s) (project on blackboard via units of instruction, or print on blackboard). Read the Essential Question (s) aloud and explain it to your students. You can rephrase the Essential Question (s) to make sure your students understand it.


_______

Minutes

Introduce task by stating the purpose of todays lesson.


_______

Minutes

Engage students in conversation by asking open ended questions related to the essential question(s).


_______

Minutes

Begin whole group instruction with corrective feedback:


_______

Minutes

Lesson Review/Reteach:


_______

Minutes

Independent Work CAPs:


Teacher Reflections:

The Instructional Rotation Matrix has been designed to assist language arts teachers in providing a balanced approach to utilizing the Math Units of Instruction across all grade levels on a rotating schedule.

Monday

Tuesday

Wednesday

Thursday

6th Grade Content

Middle School

9th Grade Content

High School

7th Grade Content

Middle School

10th Grade Content

High School

8th Grade Content

Middle School

11th Grade Content

High School

6th Grade Content

Middle School

12th Grade Content

High School

7th Grade Content

Middle School

9th Grade Content

High School

8th Grade Content

Middle School

10th Grade Content

High School

6th Grade Content

Middle School

11th Grade Content

High School

7th Grade Content

Middle School

12th Grade Content

High School

Georgia Performance Standards

M7A1 Students will represent and evaluate quantities using algebraic expressions.

a. Translate verbal phrases to algebraic expressions.

b. Simplify and evaluate algebraic expressions, using commutative, associative, and distributive properties as appropriate.

c. Add and subtract linear expressions.

M7A2 Students will understand and apply linear equations in one variable.

a. Given a problem, define a variable, write an equation, solve the equation, and interpret the solution.

b. Use the addition and multiplication properties of equality to solve one- and two-step linear equations.

M7A3 Students will understand relationships between two variables.

a. Plot points on a coordinate plane.

b. Represent, describe, and analyze relations from tables, graphs, and formulas.

c. Describe how change in one variable affects the other variable.

d. Describe patterns in the graphs of proportional relationships, both direct (y = kx) and inverse (y = k/x).

M7D1 Students will pose questions, collect data, represent and analyze the data, and interpret results.

a. Formulate questions and collect data from a census of at least 30 objects and from samples of varying sizes.

b. Construct frequency distributions.

c. Analyze data using measures of central tendency (mean, median, and mode), including recognition of outliers.

d. Analyze data with respect to measures of variation (range, quartiles, interquartile range).

e. Compare measures of central tendency and variation from samples to those from a census. Observe that sample statistics are more likely to approximate the population parameters as sample size increases.

f. Analyze data using appropriate graphs, including pictographs, histograms, bar graphs, line graphs, circle graphs, and line plots introduced earlier, and using box-and-whisker plots and scatter plots.

g. Analyze and draw conclusions about data, including a description of the relationship between two variables.

M7G1 Students will construct plane figures that meet given conditions.

a. Perform basic constructions using both compass and straight edge, and appropriate technology. Constructions should include copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

b. Recognize that many constructions are based on the creation of congruent triangles.

M7G2 Students will demonstrate understanding of transformations.

a. Demonstrate understanding of translations, dilations, rotations, reflections, and relate symmetry to appropriate transformations.

b. Given a figure in the coordinate plane, determine the coordinates resulting from a translation, dilation, rotation, or reflection.

M7G3 Students will use the properties of similarity and apply these concepts to geometric figures.

a. Understand the meaning of similarity, visually compare geometric figures for similarity, and describe similarities by listing corresponding parts.

b. Understand the relationships among scale factors, length ratios, and area ratios between similar figures. Use scale factors, length ratios, and area ratios to determine side lengths and areas of similar geometric figures.

c. Understand congruence of geometric figures as a special case of similarity: The figures have the same size and shape.

M7G4 Students will further develop their understanding of three-dimensional figures.

a. Describe three-dimensional figures formed by translations and rotations of plane figures through space.

b. Sketch, model, and describe cross-sections of cones, cylinders, pyramids, and prisms.

M7N1 Students will understand the meaning of positive and negative rational numbers and use them in computation.

a. Find the absolute value of a number and understand it as the distance from zero on a number line.

b. Compare and order rational numbers, including repeating decimals.

c. Add, subtract, multiply, and divide positive and negative rational numbers.

d. Solve problems using rational numbers.

M7P1 Students will solve problems (using appropriate technology).

a. Build new mathematical knowledge through problem solving.

b. Solve problems that arise in mathematics and in other contexts.

c. Apply and adapt a variety of appropriate strategies to solve problems.

d. Monitor and reflect on the process of mathematical problem solving.

M7P2 Students will reason and evaluate mathematical arguments.

a. Recognize reasoning and proof as fundamental aspects of mathematics.

b. Make and investigate mathematical conjectures.

c. Develop and evaluate mathematical arguments and proofs.

d. Select and use various types of reasoning and methods of proof.

M7P3 Students will communicate mathematically.

a. Organize and consolidate their mathematical thinking through communication.

b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.

c. Analyze and evaluate the mathematical thinking and strategies of others.

d. Use the language of mathematics to express mathematical ideas precisely.

M7P4 Students will make connections among mathematical ideas and to other disciplines.

a. Recognize and use connections among mathematical ideas.

b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.

c. Recognize and apply mathematics in contexts outside of mathematics.

M7P5 Students will represent mathematics in multiple ways.

a. Create and use representations to organize, record, and communicate mathematical ideas.

b. Select, apply, and translate among mathematical representations to solve problems.

c. Use representations to model and interpret physical, social, and mathematical phenomena.

M7RC1 Students will enhance reading in all curriculum areas by:

a. Reading in All Curriculum Areas

Read a minimum of 25 grade-level appropriate books per year from a variety of subject disciplines and participate in discussions related to curricular learning in all areas.

Read both informational and fictional texts in a variety of genres and modes of discourse.

Read technical texts related to various subject areas.

b. Discussing books

Discuss messages and themes from books in all subject areas.

Respond to a variety of texts in multiple modes of discourse.

Relate messages and themes from one subject area to messages and themes in another area.

Evaluate the merit of texts in every subject discipline.

Examine authors purpose in writing.

Recognize the features of disciplinary texts.

c. Building vocabulary knowledge

Demonstrate an understanding of contextual vocabulary in various subjects.

Use content vocabulary in writing and speaking.

Explore understanding of new words found in subject area texts.

d. Establishing context

Explore life experiences related to subject area content.

Discuss in both writing and speaking how certain words are subject area related.

DJJ 7th Grade Mathematics

Georgia Performance Standards: Curriculum Map

1st Semester

2nd Semester

Algebraic Reasoning

Integers & Rational Numbers

Applying Rational Numbers


Proportional Relationships

Percents

Collecting, Displaying & Analyzing Data

Geometric Figures

Measurements Two Dimensional Figures

Three Dimensional Figures, Probability

& Multi-Step Equations& Inequalities

Chapter

1

CAPs

1-7

Chapter

2

CAPs

8-14

Chapter

5

CAPs

26- 31

Chapter

6

CAPs

32-36

Chapter

7

CAPs 37-42

Chapter

8

CAPs 43-49

Chapter

9

CAPs

50-54

Chapter

10

CAPs

55-58

3

15-21

11

59-63

4

22-25

12

64-68

GPS:

M7P2.c,d

M7P5 a,b,c

M7P3 a,c,d

M7P1a,b,c

M7P4 a,b,c

M7A1a,b,c

M7A2a,b

GPS:

M7N1.a,b,c,d

M7P5.a,b,c

M7P3 a,b,c,d

M7P4a,b,c

M7P2c,d

M7A1 a,b

M7P1 a,b,c,d

M7A2a,b

M7A3a,b,c

GPS:

M7P1a,b,c,d

M7P4a,b, c

M7P5a,b,c,

M7P3a,c

M7A3a,b

M7N1b,d

M7P2.c

M7A2a,b

M7G3a,b

GPS:

M7P4a,b,c

M7P5a,b,c

M7P1b,c,d

M7P3a,b,c

M7A2a,b

M7A1.b

GPS:

M7D1a,b,c,f,g

M7P4c

M7P5a,b

M7P1a,b

M7P3a,c

M7A2a,b

M7A3a,b c

M7P2.c

GPS:

M7P1a,b,c,d

M7P3a,b,c,d

M7P5a,b

M7P4a,b,c

M7P2a,b,c,d

M7A1a

M7A2a,b

M7G1

M7D1.f

M7G3.c

M7G2a,b

GPS:

M7P1a,b,c,d

M7P4a,b,c

M7P2a,b,c,d

M7A3a

M7P5a,b

M7A1a,b

M7A2a,b

M7P3a,c

M7A3.b

GPS:

M7P3a,b,c,d

M7P5a,b,c

M7P1a,b,c,d

M7P4 c

M7P2c,d

M7A1a,b

M7G3a,b

M7N1.d

M7A2a,b

M7D1.g

Focus CAPs:

1&7

Focus CAPs:

Chapter 2

8&14

Chapter 3

15 & 21

Chapter 4

22 & 25

Focus CAPs:

26 & 31

Focus CAPs:

32 & 36

Focus CAPs:

37 & 42

Focus CAPs:

43 & 49

Focus CAPs:

50 & 54

Focus CAPs:

55 & 58

Enduring Understandings & Essential Questions

Algebraic Reasoning

Enduring Understandings:

In mathematics, letters are used to represent quantities that vary.

Collecting and examining data can sometimes help one discover patterns in the way in which two quantities vary.

Changes in varying quantities are often related by patterns which, once discovered, can be used to predict outcomes and solve problems.

Written descriptions, tables, graphs and equations are useful in representing and investigating relationships between varying quantities.

Different representations (written descriptions, tables, graphs and equations) of the relationships between varying quantities may have different strengths and weaknesses.

The properties of real numbers are true for algebraic as well as numeric expressions.

Algebraic expressions may be used to represent and generalize real situations.

Inverse operations are helpful in understanding and solving equations.

Essential Questions:

What does the data tell me?

How does a change in one variable affect the other variable in a given situation?

Which tells me more about the relationship I am investigating, a table, a graph or a formula?

What strategies can I use to help me understand and represent real situations using algebraic expressions and equations?

What properties and conventions do I need to understand in order to simplify and evaluate algebraic expressions?

How is an equation like a balance?

How can the idea of balance help me solve an equation?

Integers & Rational Numbers, Applying Rational Numbers,


Enduring Understanding:

Negative numbers are used to represent quantities that are less than zero such as temperatures, scores in games or sports, and loss of income in business.

Absolute value is useful in ordering and graphing positive and negative numbers.

Computation with positive and negative numbers is often necessary to determine relationships between quantities.

Models, diagrams, manipulatives and patterns are useful in developing and remembering algorithms for computing with positive and negative numbers.

Properties of real numbers hold for all rational numbers.

Positive and negative numbers are often used to solve problems in everyday life.


When are negative numbers used and why are they important?

Why is it useful for me to know the absolute value of a number?

What strategies are most useful in helping me develop algorithms for adding, subtracting, multiplying, and dividing positive and negative numbers?

What properties and conventions do I need to understand in order to simplify and evaluate algebraic expressions?

Why is graphing on a coordinate plane helpful?

Proportional Relationships


A dilation is a transformation that changes the size of a figure, but not the shape.

The notation used to describe a dilation includes a scale factor and a center of dilation. A dilation of scale factor k with the center of dilation at the origin may be described by the notation (kx, ky).

Two similar figures are related by a scale factor, which is the ratio of the lengths of the corresponding sides.

The sides and perimeters of similar figures are related by a scale factor and the areas are related by the square of the scale factor.

Scale factors, length ratios, and area ratios may be used to determine missing side lengths and areas in similar geometric figures.

Three-dimensional objects can be created from two-dimensional plane figures through transformations such as translations and rotations.

Cross-sections of three-dimensional objects can be formed in a variety of ways, depending on the angle of the cut with the base of the object.

All three-dimensional objects can be built by stacking congruent or similar plane figures.

Parallel cross sections of the same solid may be similar or congruent plane figures.


What is dilation and how does this transformation affect a figure in the coordinate plane?

How can I tell if two figures are similar?

In what ways can I represent the relationships that exist between similar figures using the scale factors, length ratios, and

area ratios?

What strategies can I use to determine missing side lengths and areas of similar figures?

Under what conditions are similar figures congruent?

What plane figures can I make by slicing a cube by planes? What about when I use cones, prisms, cylinders, and

pyramids instead of cubes?

How can I be sure I have found all possible cross-sections of a solid?

When rotating or translating plane figures through space, what solids can I form?

If I stack congruent or similar plane figures, what kinds of solids are formed?

Percents


Percent means per hundred. The symbol % also means percent or per hundred.

Percent is a business term going back to the Roman times which is still used to measure interest rates, commission rates,

tips, mark-ups, discounts, and tax rates.

Percent is another way of writing a fraction with a denominator of 100.

A percentage represents a part of a whole. When all fractional parts are included the result is equal to the whole and to one.

Decimals, fractions, and percents are three equivalent ways of saying the same amount:

0.30 is the same as saying thirty hundredths

30/100 is also the same as saying thirty hundredths

30% so when you say thirty percent, it means thirty per hundred

When money is borrowed, you must pay to use it because someone else is losing an opportunity to use it while you have it. What you pay to use the money is called interest. The rate of interest is a percent. The money you borrow is called the principal. The formula used for interest is

Interest = Principal x Rate of Interest x Time (I = P x R x T)

Models, diagrams, manipulatives and patterns are useful in understanding, developing and remembering percentages.

Essential Question:

What are the advantages of using percents as a standardized means of making comparisons?

Mental math lends itself to "solutions at the ready." What benefit might this provide for one when dealing with

percents in a store, restaurant, bank, or school?

How does understanding percents make one a "smart consumer"?

Collecting Displaying and Analyzing Data


Data can be represented graphically in a variety of ways. The type of graph is selected to best represent a particular data set.

Measures of center (mean, median, mode) and measures of variation (range, quartiles, interquartile range) can be used to analyze data.

Larger samples are more likely to be representative of a population.

Conclusions can be drawn about data sets based on graphs, measures of center, and measures of variation.

We can use graphs to investigate the relationship between data sets.


What is meant by the center of a data set, how is it found and how is it useful when analyzing data?

How can I describe variation within a data set?

In what ways are sample statistics related to the corresponding population parameters?

How do I choose and create appropriate graphs to represent data?

What conclusions can be drawn from data?

Geometric Figures


Coordinate geometry can be a useful tool for understanding geometric shapes and transformations.

Reflections, translations and rotations are actions that produce congruent geometric objects.

Many geometric constructions are based upon congruent triangles.


How can the coordinate plane help me understand properties of reflections, translations and rotations?

What is the relationship between reflections, translations and rotations?

Why is the definition of a circle a foundation for geometric constructions?

In what ways can I use congruent triangles to justify many geometric constructions?

Measurements Two Dimensional Figures


Double number lines, models and manipulatives are helpful in recognizing and describing proportional relationships.

The equation y = kx describes a proportional relationship in which y varies directly as x.

The equation y = k/x describes a proportional relationship in which y varies inversely as x.

Proportional relationships can be represented using words, rules, tables and graphs.

Many problems encountered in everyday life can be solved using proportions.


How can I tell the difference between an inverse proportion and a direct proportion?

In an inverse proportion, how do quantities vary in relation to each other?

How can I decide if data varies directly or inversely?

In what real world situations can I find direct and inverse variation?

How can I determine the constant of proportionality in a proportional relationship by looking at a table, graph, or equation?

Three Dimensional Figures, Probability & Multi-Step Equations & Inequalities


Proficiency in computing with positive and negative rational numbers is vital to success in real life.

Algebraic expressions may be used to represent relationships between variables when solving problems.

Proportional relationships can be represented using words, rules, tables and graphs.

Inverse operations and the basic properties of our number system are essential to understanding and solving linear equations.

Constructions and transformations can be used to verify congruence.

Essential Question:

How do I add, subtract, multiply and divide using positive and negative rational numbers?

In what ways are natural numbers, whole numbers, integers, and rational numbers alike?

In what ways are they different?

How can I use congruent triangles to justify geometric constructions?

How could using the coordinate plane help me understand properties of reflections, translations and rotations?

How can I be sure I have found all possible cross-sections of a solid?

What information do I need to know in order to be assured that two figures are similar and/or congruent?

How does an inverse proportion differ from a direct proportion?

What role does solving equations play in solving real world problems?

How can I formulate questions, gather data, display and decipher the results?

Unit 1: Algebraic Reasoning

Georgia Performance Standards:






















b. Simplify and evaluate algebraic expressions, using commutative, associative, and distributive properties as appropriate.

c. Add and subtract linear expressions.




Teachers Place:

Prior to beginning the performance activity, the teacher should implement the following steps using teaching techniques you have found to be effective for your students.

1. Explain the activity (activity requirements)

2. Display the Georgia Performance Standard(s) (project on blackboard via units of instruction located at http://thevillage411.weebly.com/units-of-instruction3.html .

3. Read the Georgia Performance Standard(s) aloud and explain it to your students. You can rephrase the Georgia Performance Standard to make sure your students understand it.

4. Display the Essential Question(s) (project on blackboard via units of instruction, or print on blackboard)

5. Read the Essential Question (s) aloud and explain it to your students. You can rephrase the Essential Question (s) to make sure your students understand it.

6. Engage students in conversation by asking open ended questions related to the Essential Question (s) display answers on the blackboard.

7. Discuss answers with the students using the following questioning techniques as applicable:

Questioning Techniques:

Memory Questions

Signal words: who, what, when, where?

Cognitive operations: naming, defining, identifying, designating

Convergent Thinking Questions


Cognitive operations: explaining, stating relationships, comparing and

contrasting

Divergent Thinking Questions

Signal words: imagine, suppose, predict, if/then

Cognitive operations: predicting, hypothesizing, inferring, reconstructing

Evaluative Thinking Questions

Signal words: defend, judge, justify (what do you think)?

Cognitive operations: valuing, judging, defending, justifying

8. Guide students into the activity utilizing the web-based activities listed under the resource section below the task number.

9. Complete the activity with the students (some tasks may require students to work independently, peer to peer, learning circles [2-3 students] or as a whole group [the entire class]. Therefore the teacher may serve as activity leader and or facilitator. When an activity calls for students to work in learning circles you should assign roles to students individually i.e. recorder, discussion leader or presenter)

10. At the end of the *whole group learning session, students will transition into independent CAP assignments.

*The phrase, whole group learning session is utilized rather than, the end of the activity because all of the activities may not be completed in one day.

Task: 1

Resources:

http://nces.ed.gov/nceskids/createagraph/

http://www.mathleague.com/help/data/data.htm#linegraphs

Graph paper

Colored pencils

Activity

Have you ever tried to walk on the beach when the sand was so hot it burned your feet? The following table contains data collected by heating dry sand with a heat lamp simulating what happens when the sun begins to heat the sand on a beach.

Time (seconds)

Temperature of Dry Sand (C)

0

23

30

23.9

60

24

90

24.9

120

26.4

150

27.2

180

27.6

210

29

240

29.8

270

30.9

300

31.9

330

32.8

1. What variable did you put on the x-axis? Explain your reasoning.

2. Would it make sense to connect the points on this graph? Why or why not?

3. Discuss how the temperature of the sand changed over time.

The table below contains data collected by heating wet sand with the same heat lamp over the same period of time.

Time (seconds)

Temperature of Wet Sand (C)

0

23

30

23.3

60

23.4

90

23.9

120

24.0

150

24.6

180

24.7

210

25

240

25.2

270

25.3

300

25.7

330

26

4. Graph this data on the same coordinate plane you used for the dry sand.

5. Compare the graphs. Discuss how they are alike and how they are different. What does this tell you?

6. Can you give a scientific explanation for the differences in the heating of wet and dry sand?

7. What does this data suggest about taking a walk on the beach?

Discussion, Suggestions, Possible Solutions

1.

Solution

Time: Time is the independent variable in this case. Temperature depends on time.

Make sure that students have labeled their axes and used appropriate scales.

2.

Solution

Yes. It would make sense to connect the points. Time passes between points and temperature continues to increase between points. If we connect the points, it is easier to approximate what the temperature might be between the 30 second intervals. It may also be easier to see the trends in the graph. If we connect the points with a line segment, we are assuming a constant rate of change in the temperature over the time interval. Students may or may not be ready for this idea but it is worth discussing.

3.

Solution

The temperature of the sand increased over time. The total amount of increase was 32.8-23 = 9.8 degrees centigrade. The relationship between time and temperature in this case is fairly linear. This is not the place to be formal about linear functions but it is good for students to notice that the graph has a somewhat linear shape.

Questions to access knowledge:

Explain the difference in temperature between 60 and 90 seconds.

Explain the difference in temperature between 120 and 180 seconds.

How long did it take for the temperature to increase one degree? Explain how this is illustrated in the table and on the graph.

Possible Graph

5.

Solution

Both graphs start at the point (0, 23) which means that the temperatures of the wet and dry sand were the same when the heating began. Both graphs are increasing which means that in both cases the temperatures increase over time. The graph for the dry sand lies above the graph for the wet sand. At each plotted point after time is 0, the temperature of the dry sand is greater than the temperature of the wet sand. This means that the dry sand is heating faster than the wet sand. Students may say that the graph for the dry sand is steeper than the graph for the wet sand. This is not the place for a formal discussion of slope but it can be mentioned.

6.

Possible solution

Evaporation occurs when water changes from a liquid to a gas. This requires an input of energy, usually heat energy. When evaporation occurs, the matter from which the heat energy is derived is cooled. Seventh grade students have not yet studied physical science but they may be able to relate this idea to the life science idea of sweating to cool the body or of a dog panting. Dogs pant to evaporate water from their tongues thereby lowering their body temperature.

7.

Possible Solution

This suggests that sand closer to the ocean still wet, or more recently wet, from the tide coming in and out, will be cooler than sand further from the shore than has not been wet by the tide. So, if you dont want your feet burned, walk by the water!

Questions to access knowledge:

Explain the difference in temperature between 60 and 90 seconds for both dry sand and wet sand.

Compare the difference in temperature between 60 and 90 seconds for both dry sand and wet sand. How is this difference illustrated in the table and on the graph?

Explain the difference in temperature between 120 and 180 seconds for both dry sand and wet sand. How is this difference illustrated in the table and on the graph?

Compare the difference in temperature between 120 and 180 seconds for both dry sand and wet sand.

How long did it take for the temperature to increase one degree? Explain how this is illustrated in the table and on the graph.

((Time, Temp)(120, 24)) ((Time, Temp)(120, 26.4))

Comment

Students may benefit from drawing vertical and horizontal lines to connect data points to the corresponding x-value and y-value on a graph. Students may also benefit from labeling the data points as a means of interpreting the data. Students should select, apply, and translate among mathematical representations. In this case, students should make a connection between the data in the graph and the data in the table.

Time (seconds)

Temperature of Dry Sand (C)

Temperature of Wet Sand (C)

0

23

23

30

23.9

23.3

60

24

23.4

90

24.9

23.9

120

26.4

24.0

150

27.2

24.6

180

27.6

24.7

210

29

25

240

29.8

25.2

270

30.9

25.3

300

31.9

25.7

330

32.8

26

Task: 2

Resources:

http://www.mathgoodies.com/lessons/vol1/perimeter.html

http://www.helpingwithmath.com/printables/worksheets/geo0701perimeter01.htm

http://www.mathleague.com/help/geometry/area.htm

http://www.onlinemathlearning.com/composite-figures-rectangles-2.html

Activity

3a + 4

4 6

2a

A corner has been removed from this rectangle.

Find an expression for the perimeter of the rectangle. Write your expression as simply as possible. What is the perimeter of the rectangle if a = inch? Show your calculations step-by-step.

Find an expression for the area of the rectangle. Write your expression as simply as possible. What is the area of the rectangle if a = 1.8 feet? Show your calculations step-by-step.


1.

Solution

The perimeter of the rectangle is found by adding all of the sides. The sum of the sides is 4 + 3a + 4 + 6 + 2a + 2 + (3a+ 4 -2a). In simplest form this expression becomes:

3a +2a + a + 4 + 4 + 6 + 2 + 4 subtraction and commutative property of addition

(3a +2a + a )+( 4 + 4 + 6 + 2 + 4) associative property of addition

6a + 20

If a = , then 6() + 20 = 24.5 inches.

2.

Solution

There are two ways to find this area. One is to treat the shape as the composition of two rectangles. The area of the smaller rectangle is 2 x 2a or 4a. The area of the larger rectangle is 4(3a + 4). Thus the area is:

4a + 4(3a + 4)

4a + 12a + 16distributive property

16a + 16

If a = 1.8 feet, then the area is 16(1.8) + 16 = 28.8 + 16 = 44.8 square feet.

Task: 3

Resources:

http://www.algebra-class.com/algebra-word-problems.html (see example 3)

http://www.mathplayground.com/wpdatabase/MDLevel2_6.htm

Square tiles or cubes

Activity

Jamal wants to buy a Sony PlayStation 3 with accessories. The entire package costs $234.10. Jamal has already saved $39.

Every Saturday night Jamals Aunt Eunice comes over for dinner. Aunt Eunice has no children and is always interested in what Jamal is doing. He told her about the PlayStation and she made him a deal. Since she believes that saving money for the things that you want is a virtue, she will match every dollar Jamal saves 3 to 1 beginning at that moment. She will not match the $39 he has already saved. When Jamal has saved all of the money he needs, Aunt Eunice will take him, after Saturday night dinner, to buy the PlayStation.

Jamal figures that he can save $4 per week. The Saturday that Aunt Eunice came for dinner was April 1st. On what day can Jamal pick up his play station? How much more do you need? If you save {1,2,3}, how much would she give you? At what point did he figure out it was April Fools Day?

Show how you figured it out. You may use models, pictures, tables, etc. but you must also write and solve an equation, labeling your variables. Give a written explanation of your work.


Comment

For every dollar Jamal saves, Aunt Eunice will give him $3. Students might want to draw a model or use manipulatives to represent this situation.

For example:

Jamal

Aunt Eunice

Total

1

$4

2

$4 + $4 = $8

$4(2) = $8

3

$4 + $4 + $4 = $12

$4(3) = $12

4

$4 + $4 + $4 + $4 = $16

$4(4) = $16

x

$4(x) = 4x

This means that for every dollar Jamal saves, he actually saves four dollars. Students should be able to recognize that if x represents Jamals money, 3x will represent the amount of money that Aunt Eunice contributes.

Jamals money plus Aunt Eunices money = x + 3x = 4x

Jamal has already saved $39. So, the $39 he has already saved plus the money he gets by saving and matching (4x) needs to equal $234.10.

Possible solution

Jamals money plus Aunt Eunices money plus $39 must add up to $234.10.

Our equation is:

Since x represents the amount of money that Jamal must save, Jamal must save $48.78. If he saves $4 per week, as he thinks he can, it will take him or just a little more than 12 weeks to save the money. Decimals here should lead to some good discussion. What do we do about 12.195 weeks? Since we know that Jamal must have all of his money and he will buy the PlayStation on a Saturday night, he and Aunt Eunice can buy the PlayStation on a Saturday night 13 weeks from April 1st. This means he will buy the PlayStation on July 1st.

Task: 4

Resources:

http://www.mytestbook.com/worksheet.aspx?test_id=1364&grade=7&subject=Math&topics=Algebraic Reasoning

http://www.onlinemathlearning.com/algebraic-expression-word-problem.html

Large Styrofoam cups with lips approximately of an inch

Graph paper

Activity

You will be given a stack of identical paper cups similar to those shown in the picture below. The paper cups shown here are identical. By making appropriate measurements, you are to represent the relationship between the number of cups in a stack and the height of the stack using a table, a coordinate graph, a formula and a written description. In the case of each representation, discuss the advantage of that representation over the other three (e.g. What does the table tell you that the graph does not?)

Height of stack

Lip of cup

Base of cup

Use your representations to answer each of the following questions.

What is the height of 60 cups? Show how you know.

How many cups will fit on a shelf that is 18 inches tall? Show how you know.


Comments

Teachers should get real cups for students to measure. The answers in this problem will depend on the size of the paper cups. Large styrofoam cups with lips approximately of an inch work well.

For purposes of discussion, we will use cups for which the base is approximately 7 inches and the lip is of an inch.

Students should measure and then begin working on their representations. Students may choose to measure the cups in centimeters.

Number of Cups

Height of Lips (in)

Height of Base (in)

Height of Stack (in)

1

.75

7

7.75

2

1.5

7

8.5

3

2.25

7

9.25

4

3

7

10

5

3.75

7

10.75

6

4.5

7

11.5

7

5.25

7

12.25

The table needs enough values to show a pattern. Students may or may not include 0 cups but should understand that if the number of cups is 0, the height of the stack is 0. Some students may include a variable representation for the height of the stack given n cups.

Advantages of the table over other representations include that the table shows exactly how much the height of the stack increases with each successive cup.

A coordinate graph of the relationship should have the number of cups on the x-axis and the height of the stack on the y-axis since height depends on the number of cups. Make sure that students do not connect the points on the line since having a piece of a cup makes no sense in this case. Students should recognize that all the points lie on the same line. The graph shows that the height of the stack increases by a constant amount with each additional cup.

((cups, height)(4, 10))

The formula for this relationship is h = 7 + .75n where h represents the height of the stack and n represents the number of cups in the stack. One of the advantages of the formula over other representations is that it allows you to find the height of the stack for any number of cups or the number of cups given any height of the stack.

One verbal description of this relationship is as follows:

The base of the bottom cup is 7 inches high. Each lip is .75 or of an inch. Only one base is part of the height of the stack because all other cups are stacked inside of it. The lip of each cup contributes of an inch to the height, including the lip of the first cup. Therefore, the total height of the stack is the measure of the base of the bottom cup (7 inches) plus .75 inch (or ) times the number of cups in the stack. One advantage of a verbal explanation is that it helps clarify other representations.

Number of Cups

Height of Stack (in)

1

(0.75 in)7.75

2

(0.75 in)8.5

3

(0.75 in)9.25

4

10

5

10.75

6

11.5

7

12.25

The formula for this relationship is h = 7 + .75n where h represents the height of the stack and n represents the number of cups in the stack.

The height of 60 cups is inches.

The number of cups that can fit on a shelf that is 18 inches high is found by substituting 18 in our equation for h.

Since there is no such thing as or .67 parts of a cup in this situation, 14 cups can be stacked on a shelf that is 18 inches high. Discuss the remainder in the context of the problem. You may want to ask students why you would not round up in this situation.

Course Title:

7th Grade

GaDJJ

State Code:

27.0220000

CAP:

1

Georgia Performance Standard(s):

M7P2.c Develop and evaluate mathematical arguments and proofs.

M7P2.d Select and use various types of reasoning and methods of proof.

M7P5.a Create and use representations to organize, record, and communicate mathematical ideas.

M7P5.b Select, apply, and translate among mathematical representations to solve problems.

M7P3.a Organize and consolidate their mathematical thinking through communication.

M7P3.c Analyze and evaluate the mathematical thinking and strategies of others.

M7P1.b Solve problems that arise in mathematics and in other contexts.

M7P4.c Recognize and apply mathematics in contexts outside of mathematics.

Objective(s):

The student identifies and extends patterns. Students will represent numbers by using exponents.

Instructional Resources:

Holt Mathematics Course 2 Pgs. 6-13

Chapter 1 Resource Book (CRB)

One-Stop Planner

Activities:

Complete Are You Ready? In textbook pg. 3

Read textbook pgs. 6-9.

Complete Think and Discuss, pg.7 in textbook.

Complete Practice and Problem Solving, Problems 1-4, 8-11, and 30-41 on pgs. 8-9 in textbook.

Complete Practice A 1-1 CRB, pg. 3.

Complete Reading Strategies 1-1 CRB, pg. 9.

Read textbook pgs. 10-13.

Complete Think and Discuss, pg. 11 in textbook.

Complete Practice and Problem Solving, Problems 1-5. 11-20, 36-43, 53, and 60-69 on pgs. 12-1 3 in textbook.

Complete Practice A 1-2 CRB, pg. 11.

Complete Reading Strategies 1-2 CRB, pg. 17.

Evaluation:

Complete Power Presentations Lesson Quiz 1-1 and 1-2.

Modifications:

Performance Tasks: IDEA works CD

Course Title:

7th Mathematics

GaDJJ:

State Code:

27.0220000

CAP:

7

Georgia Performance Standard(s):

M7P2.d Select and use various types of reasoning and methods of proof.

M7P3.a Organize and consolidate their mathematical thinking through communication.

M7P3.c Analyze and evaluate the mathematical thinking and strategies of others.

M7P2.c Develop and evaluate mathematical arguments and proofs.

M7P4.c Recognize and apply mathematics in contexts outside of mathematics.

M7P1.b Solve problems that arise in mathematics and in other contexts.

M7P1.c Apply and adapt a variety of appropriate strategies to solve problems.

M7P3.d Use the language of mathematics to express mathematical ideas precisely.

M7P5.b Select, apply, and translate among mathematical representations to solve problems.

Objective(s):

The student organizes and reviews key concepts and skills presented in Chapter One.

The student assesses mastery of concepts and skills in Chapter One.

Instructional Resources:

Holt Mathematics Course Two Textbook

Chapter 1 Resource Book (CRB)

One Stop Planner

Activities:

Complete Study Guide: Review 1-51 on pgs. 64-66 in textbook.

Have teacher check your work on review. If score is at least 80%, then go on to Chapter Test. If score is less than 80%, then teacher will give Reteach worksheets in CRB to cover concepts not understood.

Complete Chapter Test 1-45 on pg. 67 in textbook.

Have teacher check your work on test. If score is at least 80%, then go on to next CAP. If score is less than 80%, then teacher will give worksheets in CRB covering concepts still not understood.

Evaluation:

Complete Chapter Test in textbook on pg. 67 with 80% accuracy.

Modifications:

Performance Tasks: IDEA Works

Unit 2: Integers & Rational Numbers, Applying Rational Numbers,

Georgia Performance Standards:

M7N1 Students will understand the meaning of positive and negative rational numbers and use them in computation.

a. Find the absolute value of a number and understand it as the distance from zero on a number line.

b. Compare and order rational numbers, including repeating decimals.

c. Add, subtract, multiply, and divide positive and negative rational numbers.

d. Solve problems using rational numbers.







b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.












b. Simplify and evaluate algebraic expressions, using commutative, associative, and

distributive properties as appropriate.





d. Monitor and reflect on the process of mathematical problem solving.




M7A3 Students will understand relationships between two variables.

a. Plot points on a coordinate plane.

b. Represent, describe, and analyze relations from tables, graphs, and formulas.

c. Describe how change in one variable affects the other variable.

Selected Terms and Symbols:

Variable: A symbol (often a letter) that represents a number.

Proportion: An equation that states two ratios are equal.

Ratio: A comparison of two quantities that share a fixed, multiplicative relationship.

Rational Number: A number that can be written as a/b where a and b are integers, but b is not equal to 0.

Equation: A mathematical sentence that contains an equal sign.

Algebraic Expression: A mathematical phrase involving at least one variable and sometimes numbers and operation symbols.

Teachers Place:

Prior to beginning the performance activity, the teacher should implement the following steps using teaching techniques you have found to be effective for your students.

1. Explain the activity (activity requirements)

2. Display the Georgia Performance Standard(s) (project on blackboard via units of instruction located at http://thevillage411.weebly.com/units-of-instruction3.html .

3. Read the Georgia Performance Standard(s) aloud and explain it to your students. You can rephrase the Georgia Performance Standard to make sure your students understand it.

4. Display the Essential Question(s) (project on blackboard via units of instruction, or print on blackboard)

5. Read the Essential Question (s) aloud and explain it to your students. You can rephrase the Essential Question (s) to make sure your students understand it.

6. Engage students in conversation by asking open ended questions related to the Essential Question (s) display answers on the blackboard.

7. Discuss answers with the students using the following questioning techniques as applicable:

Questioning Techniques:

Memory Questions


Cognitive operations: naming, defining, identifying, designating

Convergent Thinking Questions


Cognitive operations: explaining, stating relationships, comparing and

contrasting

Divergent Thinking Questions

Signal words: imagine, suppose, predict, if/then

Cognitive operations: predicting, hypothesizing, inferring, reconstructing

Evaluative Thinking Questions

Signal words: defend, judge, justify (what do you think)?

Cognitive operations: valuing, judging, defending, justifying

8. Guide students into the activity utilizing the web-based activities listed under the resource section below the task number.

9. Complete the activity with the students (some tasks may require students to work independently, peer to peer, learning circles [2-3 students] or as a whole group [the entire class]. Therefore the teacher may serve as activity leader and or facilitator. When an activity calls for students to work in learning circles you should assign roles to students individually i.e. recorder, discussion leader or presenter)

10. At the end of the *whole group learning session, students will transition into independent CAP assignments.

*The phrase, whole group learning session is utilized rather than, the end of the activity because all of the activities may not be completed in one day.

(-10051015-5) Task: 1

Resources: http://www.purplemath.com/modules/absolute.htm

http://www.mathscore.com/math/practice/Absolute%20Value%202/

Activity

1. Which temperature is colder, -10 (ten below zero) or 0 (zero)?

Plot both numbers on the number line.


2. Which temperature is colder, -5 (five below zero) or 0 (zero)?


3. Which temperature is warmer, -10 (ten below zero) or -5 (five below zero)?

4. Which temperature is warmer, -10 (ten below zero) or 15 (fifteen degrees)?


5. What do you notice about negative numbers?


1.

-10 degrees is colder than 0 degrees

2.

-5 degrees is colder than 0 degrees.

3

-5 degrees is warmer than -10 degrees

4.

15 degrees is warmer than -10 degrees

5.

Students should recognize all numbers below zero are located to the left of zero and all numbers above zero are located to the right of zero on the number line. Students should understand that when reading the number line from left to right, the numbers are ordered least to greatest; therefore, negative numbers are less than positive numbers and any positive number is greater than any negative number. Students should also understand the number to the left of any number is smaller in value. Consequently, the number to the right of any number is greater in value.

Part II

(Georgia AvenuePeach RoadPulaski StreetBroad StreetPiedmont Avenue)

1. If the park is located at zero on the number line, plot the location of the house and school if they are located one unit from the park. What do you notice about the placement of your plots on the number line?

2. Plot the location of the house and school if they are two units from the park. What do you notice about the placement of your plots on the number line?

3. Plot the location of the house and school if they are five units from the park. What do you notice about the placement of your plots on the number line?

4. Plot the location of the house and school if they are nine units from the park. What do you notice about the placement of your plots on the number line?

The distance between a number and zero on the number line is called absolute value. The symbol for absolute value is shown in this equation |8| = 8 and |-8| = 8.

5. Explain |4|.

6. Explain |-7|.

7. Explain |8|.

8. Explain |-21|.

9. Explain |d|.


Part II.

1.

Both plots are one unit from zero. The distance between 0 and 1 is 1. The distance between -1 and 0 is also 1. There is one plot on positive one and one plot on negative one.

2.

Both plots are two units from zero. The distance between 0 and 2 is 2. The distance between -2 and 0 is also 2. There is one plot on positive two and one plot on negative two.

3.

Both plots are five units from zero. The distance between 0 and 5 is 5. The distance between -5 and 0 is also 5. There is one plot on positive five and one plot on negative five.

4.

Both plots are nine units from zero. The distance between 0 and 9 is 9. The distance between -9 and 0 is also 9. There is one plot on positive nine and one plot on negative nine.

5.

The absolute value of 4 is 4 because the distance between 0 and 4 is 4.

6.

The absolute value of -7 is 7 because the distance between 0 and -7 is 7.

7.

The absolute value of 8 is 8 because the distance between 0 and 8 is 8.

8.

The absolute value of -21 is 21 because the distance between 0 and -21 is 21.

9.

The absolute value of d is d because the distance between 0 and d is d, where d is any value.

Task: 2

Resources:

http://www.purplemath.com/modules/absolute.htm

Activity

You are an engineer in charge of testing new equipment that can detect underwater submarines from the air.

Part 1: The first three hours

During this part of the test, you are in a helicopter 250 feet above the surface of the ocean. The helicopter moves horizontally to remain directly above a submarine. The submarine begins the test positioned at 275 feet below sea level.

After one hour, the submarine is 325.8 feet below sea level.

After two hours, the submarine dives another 23 feet.

After three hours, the submarine dives again, descending by an amount equal to the average of the first two dives.

Make a table/chart with five columns (Time, Position of Submarine, Position of Helicopter, Distance between Helicopter and Submarine, and a Mathematical Sentence showing how to determine this distance) and four rows (start, one hour, two hours, three hours).

Make a graphical display which shows the positions of the submarine and helicopter using the information in your table/graph.

Part 2: The next three hours

The equipment in the helicopter is able to detect the submarine within a total distance of 750 feet.

For each scenario, determine the maximum or minimum location for the other vehicle in order for the helicopter to detect the submarine; and write a mathematical sentence to show your thinking.

Determine the ordered pairs for these additional hours and include them on your graph.

At the end of the fourth hour, the helicopter remains at 250 feet.

At the end of the fifth hour, the submarine returns to the same depth that it was at the end of the third hour.

At the end of the sixth hour, the submarine descends to three times its second hour position.


1.

The Mathematical Sentence column shows two different approaches that students might use.

Should a student receive a negative answer in this column, they should recognize the distance between the vehicles as absolute value.

For students who still have difficulty with absolute value, http://www.purplemath.com/modules/absolute.htm has a variety of explanations.

Time

Position of

Submarine

Position of

Helicopter

Distance between

Submarine and

Helicopter

Mathematical

Sentence

Start

-275

250

525

-275 250 = -525

1 hour

-325.8

250

575.8

250 (-325.8) = 575.8

2 hours

-348.8

250

598.8

-348.8 250 = - 598.8

3 hours

-385.7

250

635.7

250 (-385.7) = 635.7

For the graph, the following information could be useful.

The helicopter would be plotted at +250 and the submarine at 275. Points above sea level should be noted as a positive number while positions below sea level would be negative numbers.

In line with the first hour, the submarine would be plotted at 325.8 and the helicopter would remain at 250.

The new position of the submarine is 348.8 feet. Helicopter remains at 250.

The position of the submarine will be 385.7 feet. Helicopter remains at 250.

Note: Students can simulate the submarine and helicopter before completing the table.

(635.7 ft) (525 ft) (750 ft)

2.

The mathematical sentence could be 750 250 = 500. If the helicopter is at 250 feet, the minimum level of the submarine I where the helicopter can detect the submarine is at

-500 feet. Therefore, the ordered pair for the fourth hour is (4, -500).

Comment

This would be a good place for teachers to point out that depth refers to distance rather than position. Students should recognize at the end of the fourth hour they need to determine the minimum position of the submarine.

At the end of the third hour, the submarine was at -385.7 feet. Therefore, a mathematical sentence could be 750 + (-385.7) = 364.3 feet.

Comment

Students should recognize at the end of the fifth hour they need to determine the maximum position of the helicopter.

This one is impossible because the submarine would be at a depth of 3(348.8) = 1046.4 feet which is out of the range of the equipment. A sample mathematical sentence is 750 3(348.8) = 750 1046.4 = -296.4 which shows that the helicopter would need to be under water.

Graphs

It is important for students to make connections between different representations including vertical and horizontal graphs.

(250 feet)

(750 feet)

(500 feet)

(750 feet) (250 feet) (500 feet)

Task: 3

Resources:

http://www.funbrain.com/linejump/index.html

http://www.helpingwithmath.com/by_subject/numbers/integers/int_comparing.htm

http://www.onlinemathlearning.com/integer-games.html

Activity

Graph these numbers on the number line and then answer questions 1 4.

1. How did you scale your number line? Explain why you chose this increment.

2. Which number has the larger absolute value 1.8 or ? How do you know?

3. Look at the fractions and mixed numbers in this list. Which of these numbers, when written as decimals, are repeating decimals? Which form terminating decimals? Can you tell, without dividing, which fractions will repeat and which will terminate? How do you know?

4. Compare your number line with a partner.

a. Did you both use the same increment? Is one choice better than the other? Why or why not?

b. Explain how you placed your numbers. Are your numbers in the same order? If not, decide who is correct and why.


From least to greatest:

1.

Students may scale their number lines using different increments. Tenths work well when plotting these numbers but scaling in tenths is not the only correct possibility.

2.

The best way to compare the two numbers is to write them both in the same form, is -1.75 in decimal form; therefore, 1.8 has the larger absolute value. Students should notice that 1.8 is farther away from 0 on the number line than is . They should also be able to compare the decimal forms 1.8 (1.80) and 1.75 realizing that 1.8 is larger

3.

The fractions and form repeating decimals.

The fractions and for terminating decimals.

Students need to see that if the denominator of a fraction, in lowest terms, is a factor of (or divides evenly into) 10, 100, 1000, etc., (powers of 10) then the fraction will terminate. If not, it will repeat. Numbers whose only prime factors are 2 and/or 5 will divide evenly into powers of 10.

4.

The fractions and form repeating decimals.

The fractions and for terminating decimals.

Students need to see that if the denominator of a fraction, in lowest terms, is a factor of (or divides evenly into) 10, 100, 1000, etc., (powers of 10) then the fraction will terminate. If not, it will repeat. Numbers whose only prime factors are 2 and/or 5 will divide evenly into powers of 10.

Task: 4

Resources:

www.education.ti.com

http://www.shodor.org/interactivate/activities/coords/index.html

http://www.funbrain.com/co/

Activity

Do you remember the activity books from elementary school? There were dots on a page numbered one to a larger number and you connected the dots to form a picture. Your task is to create a series of points on the coordinate plane that, when connected, will form a picture. You must then create a script that tells someone how to connect the dots. Your picture must have a minimum of eight points and there must be at least one point in each of the four quadrants.

The example below is a simple version of what you could create.

Script:

Start at (-1,1) and draw a line segment to (-4, 1).

Connect (-4, 1) to (-3,-1).

Connect this dot to (3, -1) and continue with a line segment to (4,1).

Connect this dot to (-1, 1).

Continuing from (-1, 1), connect to (-1, 6).

This will then be connected to (-4, 2).

Connect this to (0,2).

Finish the drawing by drawing a line segment from here to (-1,6).


This task helps students apply their knowledge of points on a coordinate plane. It also can be used as a partner task. Each student could create a script and then have the partner create the image from this script. Teachers may find it beneficial to have students draw their pictures first and then identify the points to use in the script.

For students who need extra practice, there are many commercially prepared activity books that have series of points listed for graphing that when completed, create a picture. Another activity would have teacher-created scripts for the vertices of basic geometric plane figures for students to plot and identify.

A common game used in classrooms is a version of Hangman. Twenty-six points are located on the coordinate plane and are labeled with the letters of the alphabet. Students then ask their partners to identify a word clue by its coordinates. Teachers can also make up practice problems by posing silly riddles with the letters to the answers spelled out in point coordinates.

This task can also be used in conjunction with most graphing calculators. Many calculators can be programmed to create the students pictures once the coordinate points have been listed into the program. Texas Instruments has published several activities using this skill. Going to the website www.education.ti.com may yield other activities suited to extension of this concept.

By going to the site http://www.shodor.org/interactivate/activities/coords/index.html, students can demonstrate their knowledge of coordinate graphing by navigating a robot through a minefield. Also http://www.funbrain.com/co/ allows students to identify points by their coordinates interactively. The site also has many other games that students can use to increase their skill level in identifying and placing points on the coordinate plane.

Task: 5

Resources:

http://www.mathleague.com/help/integers/integers.htm

http://www.gradeamathhelp.com/math-properties.html

http://www.purplemath.com/modules/numbprop.htm

Activity

In the table below, you are given ten statements. You are to choose whether each statement is always true (A), sometimes true (S), or never true (N).

If the statement is always true, you should be able to give a rule or a property that justifies your claim.

If the statement is sometimes true, you should be able to give an example showing that it can be true and a counterexample showing that it can be false.

If the statement is never true, you should again be able to give a rule or a property that is contradicted by the statement.

1. 3a > 3

2. a + 2 = 2 + a

3. -2b < 0

4. a + -a = 0

5. a 6 > a

6. 5a + 5b = 5(a + b)

7. a - 2 = 2 - a

8. a + 10 > 10

9. If two different numbers have the same absolute value, their sum is zero.

10. If the sum of two numbers is negative, their product is negative.


1.

Sometimes true. If a is negative or a fractional value less than 1, 3 times a will be less than 3. Any example showing that the statement is sometimes true and a counterexample showing that the statement is sometimes false is a sufficient student response. However, you may want to discuss exactly when the statement is true and when it is false as a group.

2.

Always true. This is the commutative property for addition.

3.

Sometimes true. -2b < 0 if b is a positive number and -2b > 0 if b is a negative number. If b = 0, the statement is also false because -2 x 0 = 0 which is not less than zero. Some students may come up with this example. If not, it is worth asking the students to investigate what happens when b = 0.

4.

Always true. This is the property of additive inverses.

5.

Never true. If we take 6 away from a no matter what number a might be, the result will be smaller. If a is -2, -2 - 6 = -8. -8 is smaller than -2. If a is 0, 0 6 = -6. -6 is less than -2, etc.

6.

Always true. The distributive property.

7.

Sometimes true but only when a = 2. Then we would get 0 = 0. For all other values of a, this statement is false. It is important to bring out here that subtraction is not commutative.

8.

Sometimes true. If a is any positive number, a + 10 >10. If a is 0 or a negative number a + 10 < 10. Any example showing that the statement is sometimes true and a counterexample showing that the statement is sometimes false is a sufficient student response. However, you may want to discuss exactly when the statement is true and when it is false as a group.

9.

Always true. The key word here is different. If two different numbers have the same absolute value, the numbers will always be opposites and therefore by the property of additive inverses their sum will be 0.

10.

Sometimes true. If the numbers have opposite signs, the product will be negative. -6 + 3 = -3. -6 x 3 = -18. However, if both numbers are negative, the sum will be positive. If one of the numbers is 0, the sum may be negative but the product will be 0 which is neither positive nor negative. 0 + -5 = -5. 0 x -5 = 0.

Task: 6

Resources:

http://www.purplemath.com/modules/evaluate.htm

http://www.purplemath.com/modules/simparen.htm

http://www.onlinemathlearning.com/simplify-algebraic-expression.html

http://education.ti.com/educationportal/activityexchange/Activity.do?cid=US&aId=11614

Activity

In this task, your job is to create a CAP multiple choice test. You are given ten problems. For each problem, you are to write four possible answer choices. Your choices should include exactly one correct answer and at least one choice that contain common errors or misconceptions.

On a separate sheet of paper, for each problem, show how you got the correct answer; and explain the choices you made based on common errors or misconceptions.

So, here is the test!

Evaluating and Simplifying Algebraic Expressions

1. Which of the following is the value of 3a 2b when a = 2 and b = -6?

2. Which of the following is the value of 3a2 4 when a = -5?

3. Which of the following is the value of 2(4 b)2 b when b = 5?

4. Which of the following is the value of when ?

5. Which of the following is the value of (a + 7) 3b when a = -3 and b = 2?

6. Which of the following shows the expression -3m 2 + 5m 8 in simplest possible form with no parentheses and no like terms?

7. Which of the following shows the expression 4(3a 5) + 2(-3a + 7) in simplest possible form with no parentheses and no like terms?

8. Which of the following shows the expression in simplest possible form with no parentheses and no like terms?

9. Which of the following shows the expression in simplest possible form with no parentheses and no like terms?

10. Which of the following shows the expression 2ab + a ab - 7 in simplest possible form with no parentheses and no like terms?


1. The correct answer is 32 - 2(-6) = 6 + 12 = 18. Students may select any other choices they choose. One of the most common mistakes is to misunderstand that you are subtracting -12. Missing the sign, a student might get 6 12 or -6 rather than 18.

2. The correct answer is 3 (-5)2 -4 = 325 4 = 75 4 = 71.

One mistake student often make is to square the product of 3 and -5. This would give an answer of 221.

3. The correct answer is 2(4 5)2-5 = 2(-1)2 5 = 21 5 = 2 5 = -3.

4. The correct answer is (102/5 6)/4 = (20/5 6)/4 = (4 6)/4 = -1/2. A common mistake is to divide part of the numerator by the denominator and not the whole numerator. For example, if a student calculated and divided only the first term in the numerator (4) by 4, the result would be 6 = 1 6 = -5. Of course, there are also lots of mistakes that students might make in using the fractional value for a.

5. The correct answer is (-3 + 7) 32 = 4 32 = 4 6 = -2. One common mistake at the step 4 32 is to subtract 3 from 4 before multiplying 3 times 2. This would yield an answer of 2.

6. The correct answer is: -3m 2 + 5m 8 = 2m 10. The commutative and associative properties of addition are used to simplify this expression. The signs when collecting the constants in a problem of this nature are often missed.

7. The correct answer is 4(3a 5) + 2(-3a + 7) = 12a 20 + -6a + 14 = 6a 6. The distributive property and the commutative and associative properties of addition are used to simplify this expression.

8. The correct answer is . Common mistakes include using the distributive property incorrectly and multiplying fractions incorrectly.

9. The correct answer is . Mistakes with fractions are common.

10 The correct answer is 2ab + a ab 7 = ab + a 7. Common mistakes include combining the unlike term a with the terms 2ab and ab and not recognizing ab as having the same result as adding -1ab.

Task: 7

Resources:

http://www.purplemath.com/modules/numbprob.htm

http://www.onlinemathlearning.com/consecutive-integer-problems.html

http://www.purplemath.com/modules/fcnnot3.htm

Activity

Even and Odd Numbers

1. The algebraic expression 2n is often used to represent an even number. Why do you think this is true? Illustrate your explanation with pictures and/or models.

If 2n represents an even number, could n represent any number? Why or why not.

2. Write an algebraic expression that could represent an odd number. Explain your thinking and illustrate with a picture or a model.

3. What kind of number do you get when you add two even numbers? Justify your answer two different ways, by using models and by using algebraic expressions.

4. What kind of number do you get when you add two odd numbers? Justify your answer two different ways, by using models and by using algebraic expressions.

Consecutive Integers I

The lengths of the sides of the triangle below are consecutive integers.

1. What are consecutive integers? Give examples.

2. How can you represent the lengths of the sides of the triangle in terms of one variable? Explain your thinking.

Consecutive Integers II

Suppose three consecutive integers have a sum of -195.

1. If x denotes the middle integer, how can you represent the other two in terms of x?

2. Write an equation in x that can be used to find the integers.

3. Show that the sum of any three consecutive integers is always a multiple of three.

Task: 8

Resources:

http://www.algebra-class.com/algebra-readiness-test.html

http://www.mathta.com/

Activity

You are to make and present a poster showing what you have learned from your study of positive and negative rational numbers. Choose a theme for your poster. Be creative!

Choose four rational numbers. At least two of your numbers should be between 1 and 1, one of which should be written as a decimal and the other should be written as a fraction. Two of the numbers should be positive and two of the numbers should be negative. Make your poster using the rubric below:

Comparing

Use the >,

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