standard_description: Prove the Laws of Sines and Cosines and use them to solve problems.
problem_activity: Optional activity
In the previous activity, students were given a map with a scale and the amount of time it took to get from one place to another. They used this to estimate the speed of the trip. In this activity, students work with the speed and a map with scale to find the amount of time a trip will take.
The main strategy to expect is to measure the distance between the two locations on the map and use the scale to convert this to the distance between the actual cities. Then students can calculate how long it will take at 15 mph.
Launch
Tell students that they will now use a scale drawing (a map) to solve a different problem about travel, this time focusing on how long it will take. Ask students what is the farthest they have ever biked. How long did it take? Do they know someone who has biked farther or for longer? If so, how far and how long?
Keep students in the same groups. Give students 4–5 minutes of quiet work time followed by partner and whole-class discussion.
Student Facing
A cyclist rides at a constant speed of 15 miles per hour. At this speed, about how long would it take the cyclist to ride from Garden City to Dodge City, Kansas?
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Student Response
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Student Facing
Are you ready for more?
Jada finds a map that says, “Note: This map is not to scale.” What do you think this means? Why is this information important?
Student Response
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Anticipated Misconceptions
The road from Garden City to Dodge City has many twists and bends. Students may not be sure how to treat these. Tell them to make their best estimate. Measuring many small segments of the road will have the advantage that those short segments are straight but it is time consuming. A good estimate will be sufficient here.
Activity Synthesis
First, have students compare answers with a partner and discuss their reasoning until they reach an agreement.
Next, invite students to share how they estimated the distance between the two cities (and how long it takes the cyclist to travel this distance). Ask students to consider the different distances students estimated the trip to be. What are some reasons for the differences? Possible explanations include:
Measurement error
The road is not straight and so needs to be approximated
For students who lay out the scale over and over again to cover the distance, it is difficult to estimate the fraction of the scale at the last step
Because of these different sources of inaccuracy, reporting the distance as 50 miles is reasonable; reporting it as 52 miles would require a lot of time and measurements; and reporting it as 51.6 miles is not reasonable with the given scale and map.
Speaking, Listening: MLR7 Compare and Connect.
As students work to determine the duration of the trip from Garden City to Dodge City, look for students with different strategies for estimating the distance between the two cities. As students investigate each other’s work, ask students to share what worked well in a particular approach. During this discussion, listen for any comments that make the estimation of the distance more precise. Then encourage students to make connections between the various uses of constant speed to calculate the duration of the trip. Amplify language students use to make sense of the cyclist’s constant speed and how it could be represented in the map. This will support constructive conversations as students compare strategies for calculating the duration of a trip and make connections between the quantity and visual representations of constant speed on the map.
Design Principle(s): Cultivate conversation; Maximize meta-awareness


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