standard_description: Prove the Laws of Sines and Cosines and use them to solve problems.
problem_activity: Warm-up
In this activity, students encounter two distance-time ratios in which one quantity (distance) has the same value and the other quantity (time) has different values. Students interpret what the ratios mean in context, i.e., in terms of the speeds of two runners. There are several ways to reason about this with or without double number lines. Students may argue that since both runners ran for the same distance but Mai ran a shorter amount of time, she ran at a greater speed.
Students may also say that if Mai could run 3 miles in 24 minutes, at that speed, she would run more than 3 miles in 30 minutes. Since Jada only ran 3 miles in 30 minutes, Mai ran faster. The double number line that corresponds to these arguments may be as shown below.
[IMG]
[IMG]
As students work, monitor for the different ways they reason about the situation and identify a few students to share different approaches later.
Launch
Explain to students that a treadmill is an exercise machine for walking or running. Explain that while the runner does not actually go anywhere on a treadmill, a computer inside the treadmill keeps track of the distance traveled as if she were running outside.
If desired, this video shows a person starting a treadmill and walking at a constant speed for a few seconds.
Video
Treadmill
available at
https://player.vimeo.com/video/304136549
.
Students work on all parts of the activity silently and individually, then share their explanation with a partner.
Student Facing
Mai and Jada each ran on a treadmill. The treadmill display shows the distance, in miles, each person ran and the amount of time it took them, in minutes and seconds.
Here is Mai’s treadmill display:
[IMG]
Here is Jada’s treadmill display:
[IMG]
What is the same about their workouts? What is different about their workouts?
If each person ran at a constant speed the entire time, who was running faster? Explain your reasoning.
Student Response
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Anticipated Misconceptions
Because a person running on a treadmill does not actually go anywhere, it may be challenging to think about a distance covered. If this comes up, suggest that students think about running the given distances outside on a straight, flat road at a constant speed.
Activity Synthesis
Select a few students to share their reasoning about the speeds of the runners. If no students use a double number line to make an argument, illustrate one of their explanations using a double number line.
Remind students that even though a double number line is not always necessary, it can be a helpful tool to support arguments about ratios in different contexts.


aligns: no