Students will extend their use of patterning to connect different representations of linear relations. More specifically, students will be exposed to rates of change, calculating the slope of a linear relation, building linear equations, and solving linear equations.

The purpose of this Task is:

• to help students develop an understanding of how various representations of linear relations are connected;
• create a linear equation given two points; and,
• solve linear equations.

There is a version 2 of this task that shows students a clear initial value. If you’d prefer that version look to the resources on this page.

The mathematical ideas we are intentionally drawing out involve connecting various representations of linear relations, building linear equations from two or more points, and solving linear equations.

This will be achieved by first having students watch how the cost of renting a scooter is related to the time rented. You can learn more about the actual scooters and the pricing structure at lyft.com.

Show students this video:

Ask students to engage in a notice and wonder protocol. ANYTHING and EVERYTHING that comes to mind is fair game.

Here’s some of the “everything and anything” students noticed and wondered on chart paper:

• I noticed that they were riding scooters;
• I noticed that they weren’t wearing helmets;
• I noticed the map;
• I noticed that the cost changes;
• I wonder where that was?
• I wonder how much the scooters cost?
• I wonder what the range means?

Now we can focus in on the big question for this task:

How much does it cost to ride the scooter the entire length of the outlined route?

We can now ask students to make an estimate using their reasoning skills. Asking students to think about quantities that would be too high and too low to be realistic is always helpful for students to set a reasonable range.

Students will also be uncomfortable here because the length of time of the trip or how the scooter charges customers has not been revealed yet. We encourage you to hold off on revealing these answers because it will build anticipation. This anticipation is what helps develop a high level of curiosity and almost naturally encourages students to want to start formulating a plan.

At this point, we want to give students the opportunity to improve their estimates by engaging in developing a problem solving strategy.

Ask students:

If we are going to improve our estimates, what information will we need?

Take a minute to think independently and then discuss your thoughts with an elbow partner.

After students have been given some time to think independently, instruct the students to turn and talk with elbow partners.

As the conversation gains momentum in the room, ask students to begin sharing out to the class.

While it might feel like you are cutting discussions short, waiting until the discussions end will mean less students will be eager to share with the whole group.

As students voice the information they wish to see, ask them:

What would you do with that information if we were able to share that with you?

Listen closely here as their responses give you important formative assessment details about their prior knowledge and their thinking into how they might begin solving this problem.

After students have shared their “information wishlist” and what they would use the information for, you can reveal this image of three (3) snapshots showing the cost at different times during the trip. 

We have intentionally included three (3) points along the journey so that students can verify that the relationship between the cost and the time is linear.

Key Teacher Move:

If students explore the information given in the previous image, some will soon notice that not only does the cost of renting a scooter depend on the time the scooter has been rented for, but that there is also an initial value (i.e.: a flat fee) making this a partial variation (not directly proportional).

As students are discussing the implications of the cost and time at three (3) points along the journey, you can ask them to update their estimates with the following information:

How much did the trip cost if we know the duration was 15 minutes and 9 seconds?

Along with the previous prompt, you can share this image outlining the trip and the total time of the trip.

*Note that the cost of the trip is missing here.

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After consolidating learning using student generated solution strategies and by extending their thinking intentionally, we can share what the actual cost of the trip was:

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Provide students an opportunity to reflect on their learning by offering these consolidation prompts to be completed independently.

Consolidation Prompt #1:

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Consolidation Prompt #2:

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We suggest collecting these reflections as an additional opportunity to engage in the formative assessment process to inform next steps for individual students as well as how the whole class will proceed.

Dive into some extensions with your students for purposeful practice around this particular concept.

Extend #1:

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Extend #2:

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Extend #3:

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