### Summary

Students will use their knowledge of graphing, linear, and quadratic functions to interpret a tortoise - hare scenario.

### Essential Question(s)

How can mathematical expressions represent physical phenomena?

### Snapshot

**Engage**

Students will watch short clip of the tortoise and hare race or any other similar race, such as student-appropriate car chase scenes, or have them read the Aesop fable. Ask students where they find math in the clip or story.

**Explore**

Students will be given equations for the hare and for the tortoise. Students will graph the equations and determine when and where the hare catches up to the tortoise.

**Explain**

Students will solve their problem in an alternate way using their equations after the teacher models the process.

**Extend**

Students will write a story using equations and interpret a different story using graphs.

**Evaluate**

Students will evaluate each other’s stories and share solutions.

### Materials

Video similar to the tortoise and hare race given in resources or the Aesop fable

Tortoise and Hare scenes (one scene per pair of students)

"Police and the Car" graph with equations as a picture or transparency to display from projector, smartboard, or overhead projector

Graph paper

Graphing calculator

### Engage

Show students a short clip of the tortoise and hare race or any other similar race, such as student-appropriate car chase scenes, or have them read the Aesop fable.

Ask students where they find math in the clip or story.

Sample student responses: rates, hare is fast and tortoise is slow, motion, speed, acceleration, distance

### Explore

Give each pair of students a tortoise and hare scene (A, B, C, or D).

Ask students to graph the equations for each animal and write a CER Statement about where the tortoise and the hare will meet or pass.

An sample student response might be: The animals met at 5 min because both equations have a solution of 5 minutes and 20 feet which means that after 5 minutes both animals will have traveled 20 feet but after that point, the turtle will begin to pass the tortoise.

Using a clothesline activity, have students discuss the process they used to answer the question. Clothesline is a strategy that can be used to explore, clarify, and develop knowledge by having students evaluate their thoughts, opinions, and understanding. Divide students into two groups. Have group "A" stand shoulder to shoulder; group "B" will then go stand in front of a student in group "A." Students in group "A" will explain their thoughts/solutions to the student standing in front of them. Group "B" students will then evaluate and discuss the thoughts/solutions that were shared. After two minutes, students in Group "B" rotate to the next person and the process continues.

Pose the following questions to the class:

Did either animal have a head start and how do you know?

What differences do you see in the trajectory of the hare versus the tortoise?

Why is the tortoise linear and the hare quadratic?

What is the difference in the speed of the tortoise and the hare? How does that difference show up in the graph and the equation?

What else can you tell from the constants in the equations?

### Explain

Show students the graph of the "Police and the Car" along with the equations.

Model how to solve the equations simultaneously, setting f(x) = g(x) and solving for x.

Compare your calculated x value with the graph value.

Have students, using their original tortoise and hare scene, determine mathematically (solving the two equations simultaneously) when and where the hare catches the tortoise.

Have two pairs form a larger group and discuss their solutions.

Ask the groups to compare the answer from their graphs to the answer they calculated and discuss any similarities/differences.

### Extend

Engage students in a brainstorming session using think/pair/share by asking students to think about objects that demonstrate linear and quadratic behaviors then have students discuss their answers in pairs then have the pairs share with the whole class.

Ask students to write a story using equations similar to the tortoise and hare story.

Share the story guidelines with students: the story must involve at least two characters and there should be at least two equations. At the end of the story, a question must be asked that can be solved using the information from the story.

After writing the story, each student should come up with a solution and place it on a separate sheet of paper.

### Evaluate

Have students Commit and Toss their stories.

Each student will read and work the story received from the Commit and Toss.

Allow time for students to get with the writer of the story and compare solutions.

Have students complete a three-minute free write over the essential question: how did you use mathematical expressions to express physical phenomena?

### Resources

Tortoise – Hare videohttp://www.youtube.com/watch?v=YubN1MPVN-4

Printed version of Tortoise and Hare (Aesop)http://www.eastoftheweb.com/cgi-bin/version_printable.pl?story_id=TorHar.shtml

C.E.R. Instructional Strategy: K20 Center. (n.d.). Instructional Strategies. Copyright 2015, Board of Regents of the University of Oklahoma. Retrieved from https://learn.k20center.ou.edu/strategy/d9908066f654727934df7bf4f506fc09

Commit and Toss Instructional Strategy: K20 Center. (n.d.). Instructional Strategies. Copyright 2015, Board of Regents of the University of Oklahoma. Retrieved from https://learn.k20center.ou.edu/strategy/d9908066f654727934df7bf4f505b3d0