Lesson 10

Equations and Relationships

These materials, when encountered before Algebra 1, Unit 2, Lesson 10 support success in that lesson.

10.1: Which One Doesn't Belong: Slopes and Intercepts (5 minutes)

Warm-up

This warm-up prompts students to carefully analyze and compare features of the graphs of linear equations. In making comparisons, students have a reason to use language precisely (MP6). The activity also enables the teacher to hear the terminologies students know and how they talk about characteristics of a linear equation, slope, and \(y\)-intercept.

Launch

Arrange students in groups of 2–4. Display the representations for all to see. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, ask each student to share their reasoning why a particular item does not belong, and together find at least one reason each item doesn't belong.

Student Facing

Coordinate plane, x, negative 12 to 12 by 3, y, negative 12 to 12 by 3. Line drawn through 0 comma 3 and 1 point 5 comma 10 point 5.

A.

Coordinate plane, x, negative 12 to 12 by 3, y, negative 12 to 12 by 3. Horizontal line drawn at y = 3.

B.

C. \(y=\text -2.5x - 7.5\)

Coordinate plane, x, negative 16 to 16 by 4, y, negative 16 to 16 by 4. LIne drawn through 0 comma 8 and 16 comma 0.

D.

 

Student Response

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Activity Synthesis

Ask each group to share one reason why a particular item does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which one does not belong, attend to students’ explanations and ensure the reasons given are correct.

During the discussion, ask students to explain the meaning of any terminology they use, such as rate of change, slope, linear equation, and \(y\)-intercept. Also, press students on unsubstantiated claims.

10.2: What’s the Same? What’s Different? (15 minutes)

Activity

In this activity, students practice recognizing key features of graphs of linear relationships and generating equivalent equations. Allow students to use graphing technology to check their thinking. The work in this activity will be useful in the associated Algebra 1 lesson when students use situations to build connections with two-variable linear equations and graphs.

Student Facing

Here are the graphs of four linear equations.

Graph A

Coordinate plane, x, negative 10 to 8 by 2, y, negative 10 to 8 by 2. Line through negative 2 0 and 0 comma 6.

Graph B

Coordinate plane, x, negative 10 to 8 by 2, y, negative 10 to 8 by 2. Line through negative 1 comma negative 4 and 1 comma 2.

Graph C

Coordinate plane, x, negative 10 to 8 by 2, y, negative 10 to 8 by 2. Line through negative 2 comma negative 2 and 2 comma 0.

Graph D

Coordinate plane, x, negative 10 to 8 by 2, y, negative 10 to 8 by 2. Line through negative 4 comma negative 1 and 0 comma 1.
  1. Which graphs have a slope of 3?
  2. Which graphs have a slope of \(\frac12\)?
  3. Which graphs have a \(y\)-intercept of -1?
  4. Which graphs have an \(x\)-intercept of -2?
  5. Graph A represents the equation \(2y - 6x = 12\). Which other equations could graph A represent?
    1. \(y - 3x = 6\)
    2. \(y = 3x + 6\)
    3. \(y = -3x + 6\)
    4. \(2y = -6x + 12\)
    5. \(4y - 12x = 12\)
    6. \(4y - 12x = 24\)
  6. Write three equations that graph B could represent.

Student Response

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Activity Synthesis

Discuss the connection between linear equations and key features of their graphs. Allow students to share their strategies for matching graphs with the appropriate key features. Here are sample questions to promote class discussion:

  • "How can you use the structure of a linear equation to determine the slope and \(y\)-intercept of the graph that matches it?" (Considering slope-intercept form of a linear equation, I know that the slope of a graph is represented by the rate of change in the equation. The rate of change is the coefficient of the input variable, and the \(y\)-intercept is represented by the constant term in the linear equation.)
  • “Another graph represents the equation \(y = 2x - 3\). What strategies could you use to find additional equations that would be represented by the same graph?” (I can re-write the original equation in terms of \(x\), or I can multiply each term in the equation by the same number and use the products to create an equivalent equation.)
  • "Among the equivalent equations for graph A, which is easiest to recognize the slope?” (B is the equation in which the slope is easiest to recognize. The equation in A is written in terms of \(x \), which makes it a little more difficult to easily recognize the slope, and one could think the slope is -3. The equation in F is written as a multiple of the original equation, so the slope is not written as a unit rate and one could think the slope is -12.)

10.3: Situations and Graphs (20 minutes)

Activity

The purpose of this activity is for students to practice using the key features of a graph of a linear equation to interpret key characteristics of the situation that it represents. Previously, students have made connections between linear equations, graphs, tables, and situations. The work in this activity prepares students to consider how parts of two-variable linear equations relate to features of the graphs of those equations in an associated Algebra 1 lesson. 

Launch

Allow students to work individually. 

Student Facing

For each situation, find the slope and intercepts of the graph. Then, describe the meaning of the slope and intercepts. Determine if the values you come up with are reasonable answers for the situation.

  1. The printing company keeps an inventory of the number of cases of paper it has in stock.
    Coordinate plane, horizontal, number of orders, 0 to 10 by 2, vertical, cases of paper stock, 0 to 70 by 10. Line drawn from 0 comma 60 to 10 comma 20.
  2. The market value of a house is determined by the size of the house.

    Coordinate plane, horizontal, area in square feet, 0 to 300 by 50, vertical, market value in thousands of dollars, 0 to 70 by 5. Line drawn from 0 comma 10 through 250 comma 62 point 5.
  3. Tyler teaches paint classes in which the amount of money he makes depends on the number of participants he has.

    Coordinate plane, horizontal, number of participants, 0 to 7 by 1, vertical, income in dollars, 0 to 300 by 20. Line drawn from 0 comma 50 through 2 comma 120.
  4. Mai tracks the amount of money in her no-interest savings account.

    Coordinate plane, horizontal, weeks since Mai opened the account, 0 to 3 by 1, vertical, value of Mai’s account, 0 to 1,600 by 200. Line through 1 comma 600 and 2 point 5 1,300.
  5. Priya earns coins for each new level she reaches on her game. 

    Coordinate plane, horizontal, level, 0 to 4 by 1, vertical, amount of coins, 0 to 9,600 by 800. Line drawn through 0 comma 400 and 1 comma 2,400.

Student Response

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Activity Synthesis

Discuss how students make connections between graphs of linear equations and situations.