Illustrative Mathematics Grade 8, Unit 3, Lesson 7: Representations of Linear Relationships

Learning Targets:

  • I can use patterns to write a linear equation to represent a situation.
  • I can write an equation for the relationship between the total volume in a graduated cylinder and the number of objects added to the graduated cylinder.

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Illustrative Math
Grade 8

Lesson 7: Representations of Linear Relationships

Let’s write equations from real situations.

Illustrative Math Unit 8.3, Lesson 7 (printable worksheets)

Lesson 7 Summary

The following diagram shows how to use patterns to write a linear equation to represent a situation.
Represent Linear Relationships

Lesson 7.1 Estimation: Which Holds More?

Which glass will hold the most water? The least?

Lesson 7.2 Rising Water Levels

Move the green circle to set the starting level of water to a value you or your teacher choose.

Open Applet

  1. What is the volume, V, in the cylinder after you add: a. 3 objects? b. 7 objects? c. x objects? Explain your reasoning.
  2. If you wanted to make the water reach the highest mark on the cylinder, how many objects would you need?
  3. Plot and label points that show your measurements from the experiment.

Open Applet

  1. Plot and label a point that shows the depth of the water before you added any objects.
  2. The points should fall on a line. Use a ruler to graph this line.
  3. Compute the slope of the line using several different triangles. Does it matter which triangle you use to compute the slope? Why or why not?
  4. The equation of the line in the experiment has two numbers and two variables. What physical quantities do the two numbers represent? What does represent and what does represent?

Are you ready for more?

A situation is represented by the equation y = 5 + 1/2 x.

  1. Invent a story for this situation.
  2. Graph the equation.
  3. What do the and the 5 represent in your situation?
  4. Where do you see the and 5 on the graph?

Lesson 7.3 Calculate the Slope

  1. For each graph, record:
  2. Describe a procedure for finding the slope between any two points on a line.
  3. Write an expression for the slope of the line in the graph using the letters u,v,s, and t.

Lesson 7 Practice Problems

  1. Here are recipes for two different banana cakes. Information for the first recipe is shown in the table.
    a. If you used 4 cups of sugar, how much flour does each recipe need?
    b. What is the constant of proportionality for each situation and what does it mean?
  2. Show that the two figures are similar by identifying a sequence of translations, rotations, reflections, and dilations that takes the larger figure to the smaller one.
  3. Create a graph that shows three linear relationships with different y-intercepts using the following slopes, and write an equation for each line.
  4. The graph shows the height in inches, h, of a bamboo plant t months after it has been planted.
    a. Write an equation that describes the relationship between h and t. b. After how many months will the bamboo plant be 66 inches tall? Explain or show your reasoning.

The Open Up Resources math curriculum is free to download from the Open Up Resources website and is also available from Illustrative Mathematics.

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.
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