Source: Wiggins, G., & McTighe, J. Understanding by Design . Merrill Prentice Hall: 1998.
For further information about Backward Design refer to http://www.ubdexchange.org/
Title: Problem Solving Using the Pythagorean Theorem Subject/Course: 7
Topic: Linear Equations Grade(s): 7 th Designer(s): Dottie Mitcham th grade GPS
Established Goals:
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.
Understandings:
Students will understand…
 relationships can be linear.
 characteristics of linear equations such slope, independent and dependent variable
Essential Questions:

Is the relationship linear?

How do you identify linear relationships?

How do you represent linear relationships in tables, graphs, and equations?

How do you interpret linear relationships in tables, graphs, and equations?
Students will know…


the connection between linear equations and patterns
How to make decisions about linear relationships using information given in tables and graphs to solve problems
 Recognize problem situations in which two or more variables have a linear
Students will be able to…
 Construct tables, graphs, and equations
 to express linear relationships.
Translate information about linear relations given in a table, a graph, or an equation to one of the other forms
Performance Tasks:
Theater Seating
The Mathematics Theater has twenty-five seats in the first row, twenty-seven seats in the second row, twenty-nine seats in the third row, and so on. How many seats are in the theater if there are fifteen rows in all?
Other Evidence:

Teacher observation


Quizzes
Journal entries

Performance task rubric
Learning Activities: from intermath
Arranging Toothpicks
Toothpicks are used to build a rectangular grid that is 20 toothpicks long and 10 toothpicks wide. The grid is filled with squares that have 1 toothpick on each side. What is the total number of toothpicks used?
If a represents the number of toothpicks in the length of a grid and b represents the number of toothpicks in the width of a grid (again, the grid is filled with squares that have 1 toothpick on each side), write an expression representing the total number of toothpicks in any

rectangular grid of this sort.
Completing the Sequence the Sequence
Find the next three numbers in each of these sequences:
-4 -1 2 5 8 11 14 17 20
-1 0 3 8 15 24 35 48 63
-128 -54 -16 -2 0 2 16 54 128
Number Machine
I have a "number machine" that always affects in the same way whatever number I put in it.
For example, when I put in 1, the machine gives me 6; when I put in 3, it gives me10; when
I put in 6, it gives me 16; and when I put in 9, it gives me 22. What will the machine give me if I put in 100?
OVERVIEW: In this unit we will investigate relationships between variables. We will look at relationships in forms of tables, graphs, and equations. Students will determine which relationships are linear. Students will learn characteristics of linear equations such slope, independent and dependent variable
ENDURING UNDERSTANDINGS: To understand the connection between linear equations and the patterns in table and graphs of those equations. To solve problems and make decisions about linear relationships using information given in tables and graphs.
Translate information about linear relations given in a table, a graph, or an equation to one of the other forms. Construct tables, graphs, and equations to express linear relationships. Recognize problem situations in which two or more variables have a linear

relationship to each other.
ESSENTIAL QUESTIONS:
Is the relationship linear?
How do you identify the independent variable?
How do you identify the dependent variable?
What are the variables in the problem?
How do you identify linear relationships?
Do the variables in this problem have a linear relationship to each other?
How do you represent linear relationships in tables, graphs, and equations?
How do you interpret linear relationships in tables, graphs, and equations?
What patterns in the problem suggest that it is linear?
How do you write linear equations in slope-intercept form?
How do you interpret slope as the ration of vertical change to horizontal change?
How do changes in one variable affect changes in a related variable?
How are these changes captured in a table, graph, or equation?
How do you identify the x- and y-intercept from a graph or equation?
How can tables, graphs, and equations of linear relationships be used to answer questions?
STANDARDS ADDRESSED IN THIS UNIT
Mathematical standards are interwoven and should be addressed throughout the year in as many different units and activities as possible in order to emphasize the natural connections that exist among mathematical topics.
KEY STANDARDS:
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.
RELATED STANDARDS:
M7N1. Students will understand the meaning of positive and negative rational numbers and use them in computation. 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.
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.
CONCEPTS/SKILLS TO MAINTAIN:
It is expected that students will have prior knowledge/experience related to the concepts and skills identified below. It may be necessary to pre-assess in order to determine if time needs to be spent on conceptual activities that help students develop a deeper understanding of these ideas.
*graphing data in the coordinate plane
*using symbols to represent relationships between variables
*expressing relationships between variables in words, symbols, graphs, and tables
*understanding the meaning of parallel and intersecting lines
*computing and interpreting ratios
*finding rates of change in relationships between variables
*understanding positive and negative numbers
*graphing relationships between variables
SELECTED TERMS AND SYMBOLS:
Linear equation
Slope-intercept form
Tables
Graphs
Charts

Dependent variable
Independent variable
Slope
Scale for graph
Coordinate pair
X- and y-intercept
Vertical change
Horizontal change
Patterns
Parallel lines
Intersecting lines
EVIDENCE OF LEARNING:
By the conclusion of this unit, students should be able to demonstrate the following competencies:

The connection between linear equations and the patterns in tables and graphs of those equations.

To solve problem and make decisions using linear relationships expressed in tables and graphs.

Translate information about linear relations given in a table, a graph, or an equation to one of the other forms.

Construct tables, graphs, and equations to express linear relationships.

Recognize problem situations in which two or more variables have a linear relationship to each other.
STRATEGIES FOR TEACHING AND LEARNING:

Students should be actively engaged by developing their own understanding.

Mathematics should be represented in as many ways as possible by using graphs, tables, pictures, symbols and words.

Appropriate manipulatives and technology should be used to enhance student learning.

Students should be given opportunities to revise their work based on teacher feedback, peer feedback, and metacognition which includes self-assessment and reflection.
The collection of the following tasks represents the level of depth, rigor and complexity expected of all seventh grade students to demonstrate evidence of learning.
Wasting Water
Letter Perfect
Walking for Charity
Looking Back and Looking Ahead – Culminating Tasks


In this experiment, you will simulate a leaking faucet and collect data about the volume of water lost at 5-second intervals.
You will then use the patterns in your results to predict how much water is wasted when a faucet leaks for one month.
Equipment: a paper cup, water, a sharp object (such as a paper clip or a small nail), a clear measuring container, and a watch or clock with a second hand
Directions: You will need to figure out how to divide the work among the members of your group.
1.
Make a table with columns for recording time and amount of water lost. Fill in the time column with values from 0 seconds to 60 seconds in 5-second intervals (that is 5, 10, 15, and so on).
2.
Use the sharp object to punch a small hole in the bottom of the cup. Cover the hole with your finger.
3.
Fill the paper cup with water.
4.
Hold the paper cup over the measuring container.
5.
When you are ready to begin timing, uncover the hole so that the water drips into the measuring container.
6.
In a table, record the amount of water in the measuring container at 5-second intervals, up to a total of 60 seconds.
Investigation:
A.
Make a coordinate graph of the data you collected.
B.
What variables did you investigate in this experiment? Describe the relationship between the variables.
C.
If a faucet dripped at the same rate as your cup does, how much water would be wasted in 2 minutes? In 2.5 minutes? In 3 minutes and 15 seconds? Explain how you made your predictions. Did you use the table, the graph, or some other method? What clues in the data helped you?
Standards Addressed in this Task
M7N1. Students will understand the meaning of positive and negative rational numbers and use them in computation.
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.
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.

Ms. Chang’s class decides to participate in a walkathon to raise money for a local hospital. Each participant in the walkathon must find sponsors to pledge a certain amount of money for each mile the participant walks.
Ma. Change says that some sponsors might ask the students to suggest a pledge amount. The class wants to agree on how much they will ask for. Leanne says that $1 per mile would be appropriate. Gilberto says that $2 per mile would be better because it would bring in more money. Alana points out that if they ask for too much money, not as many people will want to be sponsors. She suggests that they ask each sponsor for a $5 donation plus $.50 per mile.
A.
1. Make a table showing the amount of money a sponsor would owe under each pledge plan if a student walked distances between 0 and 10 miles.
2.
Graph the three pledge plans on the same coordinate axes. Use a different color for each plan.
3.
For each pledge plan, write an equation that can be used to calculate the amount of money a sponsor owes, given the total distance the student walks.
B.
What effect does increasing the amount pledged per mile have on the table? On the graph? On the equation?
C.
If a student walks 8 miles in the walkathon, how much would a sponsor owe under each pledge plan? Explain how you got your answer.
D.
For a sponsor to owe a student $10, how many miles would the student have to walk under each pledge plan?
Explain how you got your answer.
E.
Alana suggested that each sponsor make a $5 donation and then pledge $.50 per mile. How is this fixed $5 donation represented in the table? In the graph? In the equation?
Individual Problem
1.
a. On the graph of a pledge plan, the point (2,6) means that a student who walks 2 miles earns $6 from each sponsor.
On which of the graphs is the point (2,6)? b.
Find a point on each graph, and describe what the coordinates of the point mean in the context of the walkathon.
2.
a. Write an equation for a pledge plan whose graph is steeper line than any of the lines you graphed in the problem.
Check your equation by graphing it on the coordinate axes with the other three lines. b. Write an equation for a pledge plan whose graph is less steep than any of the lines you graphed in the problem.

Check your equation by graphing it on the coordinate axes with the other line.
Standards Addressed in this Task
M7N1. Students will understand the meaning of positive and negative rational numbers and use them in computation.
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.
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.

Paul invented three tricks for a block of four numbers. Find which of Paul’s tricks are true for every possible two-by-two block of numbers on the calendar.

Which tricks always worked? Which did not? How did you find out?

For any tricks that did not always work, how can you revise them so that they do always work?

Trick One: The sums of opposite pairs of numbers will be equal. For example: 2 + 10 = 3 + 9.
Trick Two: If you add all four numbers, the sum will always be evenly divisible by 8. For example: 2 + 3+ 9 + 10 = 24;
24/8 = 3.
Trick Three: If you multiply opposite pairs of numbers, the two answers will always differ by 7. For example: 2 X 10 = 20;
3 X 9 = 27; 27 – 20 = 7
Invent Your Own Tricks
Make up at least two tricks for diagonals of three numbers – using a calendar format.

Will your tricks be true for every diagonal of three numbers? How do you know?

Which of our tricks will still be true if the diagonal goes in the opposite direction like the numbers in the red outline?
Make up at least two more tricks using your own shapes. Your shapes should be different from the ones used so far.

Are your tricks always true, no matter where on the calendar you put your shape?

How do you know your tricks will always be true?
Write About Finding Patterns

How did you check whether your tricks will work everywhere on the calendar?

What suggestions would you give to help another student find patterns?
Standards Addressed in this Task
M7N1. Students will understand the meaning of positive and negative rational numbers and use them in computation.
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.
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.
This culminating task represents the level of depth, rigor and complexity expected of all 7 th
grade students to demonstrate evidence of learning.

1.
Suppose that a theater charges a school group $4.50 per student to show a special film. Suppose that the theater’s operating expenses include $130 for the staff and a film rental fee of $1.25 per students. a.
What equation relates the number of students x to the theater’s income
I ?
b.
What equation relates the theater’s operating expenses
E to x ?
c.
Copy and complete the table below.
d.
On the same set of axes, graph the theater’s income and operating expenses for any number of students from 0 to 100.
e.
Describe the patterns by which income and operating increase as the number of students increases.
f.
Write and solve an equation whose solution will answer the question “How many students need to attend the movie so that the theater’s income will equal its operating expenses?
2.
At another theater, the income and expenses combine to give the equations y = 3 x – 115 relating operating profit y to the number of students in a group x . a.
What do the numbers 3 and -115 tell about i.
The relationship between the number of students in a group and the theater’s profit? ii.
The pattern of entries that would appear in a table of sample (students, profit) pairs? iii.
A graph of the relation between the number of students and the profit? b.
Use the equation to find the number of students necessary for the theatre to i.
Break even (make 0 profit). ii.
Make a profit of $100. c.
Write and solve an equation that will find the number of students for which the theaters in Problem 1 and Problem 2 will make the same profit. Then find the amount of that profit.
Explaining Your Reasoning
3.
Consider the variables and relationships. a.
What are the variables? b.
Which pairs of variables are related to each other?

c.
In each pair of related variables, how does changes in the value of one variable cause change in the value of the other?
4.
Which relationships are linear and which are not? What pattern in the tables, graphs, and symbolic equations support your conclusions?
5.
For those relationships that are linear, what do the slopes and intercepts of the graphs indicate about the relationships involved?
6.
How do the slopes and intercepts relate to data patterns in the various tables of values?
7.
Consider the strategies for solving linear equations such as those in Problem 1, part (f), and Problem 2, part
(c). a.
How can the equations be solved using tables of values? b.
How can you solve those equations by using graphs? c.
How can you solve the equations by reasoning about the equations alone?
8.
Suppose you were asked to write a report describing the relationships among number of students, theater income, and operating costs. What value might be gained by including the table? Including the graph?
Including the equation? What are the limitations of each type of display?