7th Grade Math Unit 3 Module

Unit 3: “Rational Reasoning”

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Unit Overview

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OVERVIEW:



In this unit students will explore positive and negative numbers, extending many previously learned concepts to include the use of all

 rational numbers. Algorithms for computing with rational numbers should be investigated, discovered and formalized by students using models, diagrams, manipulatives, and patterns.



Students will:

 • investigate the use of positive and negative numbers in real contexts; • plot positive and negative numbers in the coordinate plane;

 • order positive and negative rational numbers and plot them on a number line;

• use absolute value to explore the relationship between a number and its additive inverse;

 • develop algorithms for computing with positive and negative rational numbers; • simplify and evaluate algebraic expressions involving positive and negative rational numbers;

 • extend properties of real numbers to include all rational numbers; • solve one- and two-step linear equations in one variable; and,

 • solve problems by defining a variable, writing and solving an equation, and interpreting the solution of the equation in the context of the original problem.



To assure that this unit is taught with the appropriate emphasis, depth and rigor, it is important that the task listed under “Evidence of



Learning” be reviewed early in the planning process. A variety of resources should be utilized to supplement, but not completely replace, the textbook.

Textbooks not only provide much needed content information, but excellent learning activities as well. The tasks in these units illustrate the type of learning activities that should be utilized from a variety of sources.

Reference:

Georgia Performance Standards Framework for Mathematics Grade 7.

(2006, September 20, 2006). Retrieved January 15, 2008 from http://public.doe.k12.ga.us/DMGetDocument.aspx/7th%20Grade%20Unit

%203%20Rational%20Reasoning%202-19-

07.pdf?p=6CC6799F8C1371F6F46BEA13164EAA7A1945391FE9C84C82B48CA

7E45D5A791C&Type=D

Number of Weeks to Teach Lesson

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5 weeks according to Georgia Performance Standards Curriculum Map,

Atlanta Public Schools Teaching Plans and Georgia Performance

Standards Framework for Mathematics

GPS Standards to be covered

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

KEY 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 a 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.



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.



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.

Reference:





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Notes

The standards found in both the APS Teaching Plans and the GPS

Framework are the same. The Georgia State Framework also includes related standards which I have excluded for simplicity sake. Please decide whether or not the related standards should be included.

In the Atlanta Public Schools Teaching Plans, the GPS standards change slightly from week to week.

Essential Questions

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ESSENTIAL QUESTIONS:



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?

Reference:





7E45D5A791C&Type=D

Notes

*The Atlanta Public Schools Teaching Plans is delineated by week, and so there are essential questions for each week. In contrast, the GPS

Framework for Mathematics is delineated by unit.

*Essential Questions change slightly week to week.

Enduring Understandings

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ENDURING UNDERSTANDINGS:



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.

Reference:


(2006, September 20, 2006). Retrieved January 15, 2008 from http://public.doe.k12.ga.us/DMGetDocument.aspx/7th%20Grade

%20Unit%203%20Rational%20Reasoning%202-19-

07.pdf?p=6CC6799F8C1371F6F46BEA13164EAA7A1945391FE9C

84C82B48CA7E45D5A791C&Type=D

Assessment

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GPS Framework, Grade 7, Unit 2, Patterns and Relationships,

Culminating Task “A Poster,”

Georgia Online Assessment System (GOAS)

Student Tests available for this unit:

Math, Grade 7, A Test 63



Reference:

Georgia Online Assessment System developed by the Georgia State

Department of Education. (2007). Retrieved from https://www.georgiaoas.org/servlet/a2





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Literacy GPS Connection

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APS Teaching Plans Unit 2 Literacy GPS Connection

ELA7R2 The student understands and acquires new vocabulary and uses it correctly in reading and writing.

ELA7RC3 The student acquires new vocabulary in each content area and uses it correctly.

ELA7RC4 The student establishes a context for information acquired by reading across subject areas.

Note

The GPS Framework for Mathematics – Grade 7 does not include Literacy

GPS connections.

Vocabulary

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GPS Framework for Mathematics – Grade 7

Distributive property: The sum of two addends multiplied by a number will be the sum of the product of each addend and the

 number. Example: a(b + c) = ab + ac Integers: The set of whole numbers and their opposites {... -3, -2, -1, 0, 1, 2, 3, ...}



Natural numbers: The set of numbers {1, 2, 3, 4, ...}. Natural numbers can also be called counting numbers. Negative Numbers: The set of numbers less than zero.



Opposite Numbers: Two different numbers that have the same absolute value. Example: 4 and –4 are opposite numbers because both have an absolute value of 4.



Positive Numbers: The set of numbers greater than zero. Rational Numbers:

The set of numbers that can be written in the form a/b where a and b are integers and b ≠ 0.



Sign: a symbol that indicates whether a number is positive or negative. Example: in –4, the

(–) sign shows this number is read “negative four”.



Whole numbers: The set of all natural numbers and the number zero.



You may visit www.intermath-uga.gatech.edu and click on dictionary to see definitions and specific examples of terms and symbols

 used in this seventh grade GPS unit.

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

APS Teaching Plans Unit 2 Week 1 Vocabulary

Discrete data: a variable with a limited number of values (e.g.), gender (male/female), college class

(freshman/sophomore/junior/senior)

Continuous data: data that can take on any value (subject to the resolution of the measuring process).



Independent Variable: a variable that is manipulated, measured, or selected by the researcher as an antecedent condition to an observed behavior. In a hypothesized cause-and-effect relationship, the independent variable is the cause and the dependent variable is the outcome or effect.

Dependent Variable: a variable that is not under the experimenter's control -- the data. It is the variable that is observed and measured in response to the independent variable.

Narrative: (a message that tells the particulars of an act or occurrence or course of events; presented in writing or drama or cinema or as a radio or television program) "his narrative was interesting"; "Disney's stories entertain adults as well as children"




Equation: A mathematical sentence that contains an equals sign.

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



Proportion: An equation that states two ratios are equal.


Ratio: A comparison of two quantities that have the same unit of measure.





Ratio: A comparison of two quantities that have the same unit of measure.


References

Organization of Terms produced by Bob Hall of the University of Texas.

(2007). Retrieved September 25, 2007 from http://bobhall.tamu.edu/FiniteMath/Module8/Introduction.html

The Glossary of Global Productivity Solutions. (2006). Retrieved

September 25, 2007 from http://www.gpsqtc.com/library/abc.shtml

The Interactive Mathematics Dictionary produced by University of Georgia

Mathematics Education Program. (2007). Retrieved September 25, 2007 from www.intermath-uga.gatech.edu

Miller, G. (2006). Wordnet. Retrieved September 25, 2007 from http://wordnet.princeton.edu/

Notes

 APS Teaching Plans include vocabulary but not definitions.

 GPS Framework for Mathematics use definitions from www.intermathuga.gatech.edu

and so I have defined remaining terms from this site as well.

 The four vocabulary words in red did not appear on the www.intermath-uga.gatech.edu

or the Holt Mathematics course

Multilingual Glossary. I used the Bob Hall of the University of Texas statistics site, the global site for the purple and Princeton for narrative.

Problem Solving

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APS Teaching Plan Unit 2 Week 1 Problem Solving

Solve non-routine problems involving the Find a Pattern strategy from

Holt Mathematics Course 2, Problem Solving Handbook

Solve multi-step routine problems


Solve non-routine problems involving the Make a Table strategy from Holt

Mathematics Course 2, Problem Solving Handbook



Solve non-routine problems involving the Solve a Simpler Problem strategy from Holt Mathematics Course 2, Problem Solving Handbook,



Solve routine and non-routine problems using the Guess and Check strategy

Notes:

GPS Framework for Mathematics – Grade 7 does not include a Problem

Solving section.

Focus Lessons

Unit 2 Week 1

Unit 2 Week 2

Unit 2 Week 3

Unit 2 Week 4

Note:

See Focus Lessons on APS Teaching Plans

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Technology Connections

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UNITEDSTREAMING VIDEOS

GPS Addressed this unit: M7A1a-c, M7A2a-b, M7A3a-c, M7D1

GPS Addressed in the video: M7A1.b

Video Title: Discovering Math: Algebra Solving Linear Equations

Introduction

Reference:

Discovering Math: Algebra (Grades 6-8). Discovery Channel School (2006).

Retrieve January 15, 2008, from unitedstreaming: http://streaming.discoveryeducation.com/

PODCASTS

Data in Action Podcast, http://georgiastandards.org/math_vc.aspx

Related Materials:

Data in Action PowerPoint Presentation

Centimeter Graph Paper PDF

Reference:

Data in Action. Georgia Department of Education. (2007). Retrieved

September 25, 2007, from http://georgiastandards.org/math_vc.aspx

PROMETHEAN FLIPCHARTS

Algebra

In this lesson students will review basic algebraic problem types (ex: values of variables, inequalities) used on standardized tests. http://www.prometheanplanet.com/us/server/show/ConResource.7490

Reference:

Fussillo, S. (2006). The Coordinate Plane. Promethean Planet. Retrieved on September 26, 2007 from http://www.prometheanplanet.com/us/server.php?show=conResource.82

23

Strohl, J. (2006). The Coordinate Plane. Promethean Planet. Retrieved on

September 26, 2007 from http://www.prometheanplanet.com/us/server/show/ConResource.7490

Wagoner, K. (2006). The Coordinate Plane. Promethean Planet. Retrieved on September 26, 2007 from http://www.prometheanplanet.com/us/server/show/ConResource.163

WEBQUESTS

Brown, L. Solving Linear Systems of Equations. InstantProjects.Org

Educational Technology Portal and Services. Retrieved on September 26,

2007 from http://instantprojects.org/webquest/webquest.php?AuthorID=5087

Clark, C. (2003). The Tortise and the Hare: A Webquest developed for

Algebra I or Algebra II on Systems of Equations. Retrieved on September

26, 2007 from http://edtech.tph.wku.edu/~ppetty/cclark.htm

Giguere, S. (1999). Graphing and Applying Linear Equations. Retrieved on

September 26, 2007 from http://www.gmpdc.org/webquests/sgiguere/linearequations/#Introduction

Womick, A. A Functional Housing Market: Linear Equations. Retrieved on

September 26, 2007 from http://www.spa3.k12.sc.us/house.html

INTERACTIVE LEARNING ACTIVITIES

National Council of Teachers of Mathematics. (2000-2007). Line of Best Fit.

Retrieved on September 26, 2007 from http://illuminations.nctm.org/ActivityDetail.aspx?ID=146

National Council of Teachers of Mathematics. (2000-2007). Pan Balance -

Expressions. Retrieved on September 26, 2007 from http://illuminations.nctm.org/ActivityDetail.aspx?ID=10

National Council of Teachers of Mathematics. (2000-2007). Pan Balance –

Shapes (Fixed Values). Retrieved on September 26, 2007 from http://illuminations.nctm.org/ActivityDetail.aspx?ID=131 eMath Textbook at www.go.hrw.com

 Homework Help: Course 2 Chapter 4 Patterns and Functions

 http://go.hrw.com/hrw.nd/gohrw_rls1/pKeywordResults?keywor

d=ms7+hwhelp4

 Multilingual Glossary

 http://my.hrw.com/math06_07/nsmedia/tools/glossary/msm/gl ossary.html

 Chapter Projects: Project Recording Sheet Patterns and Functions

 http://go.hrw.com/resources/go_mt/hm2/cp/7sech04.pdf

 Lab Resources: Chapter 4 Lab Resources


d=ms7+lab4

 Parent Resources


d=mr7+parent

 Career Resources


d=MS7+Career

 Web Extra!: Course 2 Chapter 4


d=ms7+webextra4

 Game Time


d=MS7+Games

 No game for chapter 4

Reference:

Holt, Rinehart and Winston. (2007). Holt Mathematics: Course 2.

Retrieved on September 26, 2007 from http://www.go.hrw.com/gopages/ma/msm2_07.html

NETS for STUDENT STANDARDS

Leave as is

Sample Tasks

Sample Task 1: Connect the Dots

Sample Task 2: Helicopters and Submarines

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Culminating Tasks

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GPS Framework, Grade 7, Unit 2, Patterns and Relationships,

“Culminating Task: Designing a Poster,”

Evidence of Learning

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GPS Framework for Mathematics – Grade 7 EVIDENCE OF LEARNING:

By the conclusion of this unit, students should be able to demonstrate

the following competencies:



Distinguish between natural numbers, whole numbers, integers and other rational numbers. • Find the absolute value of a number.



Compare two positive and/or negative rational numbers using <, >, or =.



Order a set of positive and/or negative rational numbers from least to greatest or greatest to least and plot the numbers on a number line.



Illustrate computation of positive and negative numbers using models, diagrams, manipulatives or patterns, including fact families.



Compute with positive and/or negative rational numbers.



Use the properties of real numbers (commutative, associative, distributive, inverse and identity) and the order of operations to simplify and evaluate simple numeric and algebraic expressions.



Solve problems involving applications of positive and/or negative rational numbers.



The following task represents the level of depth, rigor, and complexity expected of all 7th grade students. This task or a task of similar depth and rigor should be used to demonstrate evidence of learning.

Curriculum Map

Note:

Scan this in.

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Concept Map

6/20/2008 8:01:00 AM

 http://public.doe.k12.ga.us/DMGetDocument.aspx/G7-CM-

U2.pdf?p=6CC6799F8C1371F6D266E4B28571448133B78FD2FA1A841E3

2775F9A835269A0&Type=D

Parent Letter(s)

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Parent Letter from Math textbook at: http://georgiastandards.org/DMGetDocument.aspx/grade_7_unit_3_parent_letter.p

df?p=6CC6799F8C1371F65664838EA6545E94BD640990D0E54AD3D0F3C905

75E7A048&Type=D

Bloom’s Taxonomy

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“Patterns and Relationships” for Grade Seven should be at ???

Bloom's Taxonomy of Educational Objectives Cognitive Domain

1. Knowledge (Remembering previously learned material)

Mathematics: Using the Rectangle Activity Sheet, students will describe communicative rectangles with 90% accuracy.

2. Comprehension (Grasping the meaning of material)

Mathematics: Using the “Fibonacci in Nature Overhead”, students will

compare the Golden Ratio in nature, architecture, and art with 90% accuracy.

3. Application (Using information in concrete situations)

Mathematics: Using graphic calculators, students will illustrate the process of plotting points on a coordinate plane with 90% accuracy.

4. Analysis (Breaking down material into parts)

Mathematics: Using computer graphing software, students will examine the slopes, equations and x-intercept of linear functions data with 90% accuracy.

5. Synthesis (Putting parts together into a whole)

Mathematics: Using a rubric, students will construct a sample algebra examination with 90% accuracy.

6. Evaluation (Judging the value of a product for a given purpose, using definite criteria)

Mathematics: Using the culminating activity, students will debate Jamal’s mathematical reasoning for purchasing a PlayStation 2 with 90% accuracy.

Reference:

Bloom’s Taxonomy of Educational Objectives. Retrieved on September

26, 2007 from http://education.calumet.purdue.edu/vockell/edpsybook/Edpsy3/edpsy3

_bloom.htm

Note:

 Adapted from http://education.calumet.purdue.edu/vockell/edpsybook/Edpsy3/edp sy3_bloom.htm

and GPS standards.

7th Grade Math Unit 3 Module

Unit 3: “Rational Reasoning”

Unit Overview

Number of Weeks to Teach Lesson

GPS Standards to be covered

Essential Questions

Enduring Understandings

Assessment

Literacy GPS Connection

Vocabulary

Problem Solving

Focus Lessons

Technology Connections

Sample Tasks

Culminating Tasks

Evidence of Learning

Curriculum Map

Concept Map

Parent Letter(s)

Bloom’s Taxonomy

Related documents

Products

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7th Grade Math Unit 3 Module

Unit 3: “Rational Reasoning”

Unit Overview

Number of Weeks to Teach Lesson

GPS Standards to be covered

Essential Questions

Enduring Understandings

Assessment

Literacy GPS Connection

Vocabulary

Problem Solving

Focus Lessons

Technology Connections

Sample Tasks

Culminating Tasks

Evidence of Learning

Curriculum Map

Concept Map

Parent Letter(s)

Bloom’s Taxonomy

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