Unit 3, Lesson 1
Understanding Proportional Relationships
Grade 8
1.1
Warm-up
two graphs. both graphs, horizontal axis, scale 0 to 18, by 2's. vertical axis, scale 0 to 6, by 1's. graph on left horizontal axis labeled distance traveled in centimeters, vertical axis labeled elapsed time in seconds. 2 lines on graph, labeled g and f. line g passes through origin and 6 comma 3. line f passes through origin and 8 comma 2. graph on the right, axes and lines not labeled. line passes through origin and 3 comma 1.
What do you notice? What do you wonder?
1.2
Activity
A ladybug and ant move at constant speeds. The diagrams with tick marks show their positions at different times, as measured by the front of each bug’s head. Each tick mark represents 1 centimeter.
4 number lines, 30 evenly spaced tick marks. first line, 0 seconds, lady bug and ant both at 5th tick mark, labeled start. second line, 2 seconds, lady bug at 9th tick mark and ant at 11th tick mark. thrid line, 4 seconds, lady bug at 13th tick mark and ant at 17th tick mark. fourth line, 6 seconds, lady bug at 17th tick mark and ant at 23rd tick mark.
graph. horizontal axis, distance traveled in centimeters, 20 units. vertical axis, elapsed time in seconds, 6 units. 2 lines graphed, labeled u and v. line u passes through origin and 2nd line to the right and 1st line up. line v passes through origin and 3rd line to the right and 1st line up.
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Lines and 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.
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How long does it take the ladybug to travel 12 centimeters? The ant?
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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.
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Mark and label the point on line and the point on line that represent the time and position of each bug after traveling 1 centimeter.
Student Lesson Summary
Graphing is a way to help make sense of relationships.
But the graph of a line on a coordinate plane without labels or a scale isn’t very helpful. Without labels, we can’t tell what the graph is about or what units are being used. Without an appropriate scale, we can’t tell any specific values.
Here are the same graphs, but now with labels and a scale:
Notice how adding labels lets us know that the relationship compares time and distance and helps to understand both the speed and pace of two different items. When adding labels to axes, be sure to include units, such as minutes and miles.
Notice how adding a scale makes it possible to identify specific points and values. When adding a scale to an axis, be sure that the space between each grid line represents the same amount.
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