Lesson 1

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understanding proportional relationships

Warm Up

Lesson

Summary

Cool

Down

Reflections

Student

Task

Learning

Targets

Practice Problems

Lesson Video

Unit 5

linear relationships

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1.1 Warm Up - notice and wonder: two graphs

What do you notice? What do you wonder?

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Learning Targets

• I can graph a proportional relationship from a story.
• I can use the constant of proportionality to compare the pace of different animals.
• The purpose of this lesson is to get students thinking about what makes a “good” graph by first considering what are the components of a graph.

Success Criteria

• Today I am learning how to think about what makes a “good” graph by first considering what are the components of a graph.

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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. Each bug’s position is measured at the front of its head.

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1.2 moving through representations

1. 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.
2. How long does it take the ladybug to travel 12 cm? The ant?
3. 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.
4. 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.

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Are you Ready for more?

1. How fast is each bug traveling?

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• 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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1.3 moving twice as fast

Imagine a bug that is moving twice as fast as the ladybug. On each tick-mark diagram, mark the position of this bug.

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1.3 moving twice as fast

Plot this bug’s positions on the coordinate axes with lines u and v, and connect them with a line.

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Write an equation for each of the three lines.

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1.3 lesson Synthesis

What would the graph of a bug going 3 times faster than the ant look like?

What would an equation showing the relationship between the bugs' distance and time look like?

If we wanted to scale the graph so we could see how long it takes the ladybug to travel 50 cm, what numbers could we use on the vertical axis?

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Lesson Summary

Graphing is a way to help us make sense of relationships. But the graph of a line on a coordinate axes without scale or labels isn’t very helpful. For example, let’s say we know that on longer bike rides Kiran can ride 4 miles every 16 minutes and Mai can ride 4 miles every 12 minutes. Here are the graphs of these relationships:

Without labels we can’t even tell which line is Kiran and which is Mai! Without labels and a scale on the axes, we can’t use these graphs to answer questions like:

1. Which graph goes with which rider?
2. Who rides faster?
3. If Kiran and Mai start a bike trip at the same time, how far are they after 24 minutes?
4. How long will it take each of them to reach the end of the 12 mile bike path?

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Lesson Summary

Here are the same graphs, but now with labels and scale. Kiran can ride 4 miles every 16 minutes and Mai can ride 4 miles every 12 minutes.

• Which graph goes with which rider?

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• Who rides faster?

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• If Kiran and Mai start a bike trip at the same time, how far are they after 24 minutes?

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• How long will it take each of them to reach the end of the 12 mile bike path?

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1.4 Cool Down: Turtle race

This graph represents the positions of two turtles in a race.

1. On the same axes, draw a line for a third turtle that is going half as fast as the turtle described by line g.

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2. Explain how your line shows that the turtle is going half as fast.

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Reflections

• Can you graph a proportional relationship from story?

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• Can you use the constant of proportionality to compare the pace of different animals?

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Practice Problems

Must do: All problems

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Lesson Video

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