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Illustrative Math
Grade 8
Lesson 1: Understanding Proportional Relationships
Let’s study some graphs.
Illustrative Math Unit 8.3, Lesson 1 (printable worksheets)
Lesson 1 Summary
The following diagram shows how to graph a proportional relationship from a story.

Lesson 1.1 Notice and Wonder: Two Graphs
What do you notice? What do you wonder?
Lesson 1.2 Moving Through Representations
A ladybug and ant move at constant speeds. The diagrams with tick marks show their positions at different times. Each tick mark represents 1 centimeter.
- Lines u and v also show the positions of the two bugs. Which line shows the ladybug’s movement?
Which line shows the ant’s movement? Explain your reasoning.
- How long does it take the ladybug to travel 12 cm? The ant?
- Scale the vertical and horizontal axes by labeling each grid line with a number. You will need to use the time and distance information shown in the tick-mark diagrams.
- Mark and label the point on line u and the point on line v that represent the time and position of each bug after traveling 1 cm.
Are you ready for more?
- How fast is each bug traveling?
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Show Answer
The ladybug is traveling at 2 cm/s.
The ant is traveling at 3 cm/s.
- Will there ever be a time when the purple bug (ant) is twice as far away from the start as the red bug (ladybug)? Explain or show your reasoning.
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Show Answer
At anytime t, the ladybug would be 2t cm from the start and the ant would be 3t from the start.
It is not possible to find a time t such that
2 · 2t = 3t
Lesson 1.3 Moving Twice as Fast
Refer to the tick-mark diagrams and graph in the earlier activity when needed.
- Imagine a bug that is moving twice as fast as the ladybug. On each tick-mark diagram, mark the position of this bug.
- Plot this bug’s positions on the coordinate axes with lines u and v, and connect them with a line.
- Write an equation for each of the three lines.
Lesson 1 Practice Problems
- Priya jogs at a constant speed. The relationship between her distance and time is shown on the graph. Diego bikes at a constant speed twice as fast as Priya. Sketch a graph showing the relationship between Diego’s distance and time.
- A you-pick blueberry farm offers 6 lbs of blueberries for $16.50.
Sketch a graph of the relationship between cost and pounds of blueberries.
- A line contains the points (-4,1) and (4,6). Decide whether or not each of these points is also on the line:
- The points (2,-4),(x,y), A, and B all lie on the line. Find an equation relating x and y.
The Open Up Resources math curriculum is free to download from the Open Up Resources website and is also available from Illustrative Mathematics.
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