The Tortoise and the Hare Piece-wise Functions

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Students will be able to write and graph piece-wise functions.

Big Idea

Will the tortoise win again? Students will not only find out but write a mathematical story about it.

Warm up and Homework Check

10 minutes

I include Warm ups with a Rubric as part of my daily routine. My goal is to allow students to work on Math Practice 3 each day. Grouping students into homogeneous pairs provides an opportunity for appropriately differentiated math conversations. The Video Narrative explains this lesson’s Warm Up- Tortoise and Hare Piecewise Functions which asks students to discuss the continuity of functions.

I also use this time to correct and record the previous day's Homework.  

A New Version of the Race, Piecewise Style

8 minutes

In the fifth day using the tortoise and the hare, my students will look at another version of the race using piecewise functions.  This will be their first exposure to these types of functions.  I begin by giving them a coordinate plane showing the new distance functions for each animal.  My students first make sense of the graph by writing statements describing the actions of each animal during the race (Math Practice 4).  We will share these as a class.  This can be a great place for more visual students to shine.  I then give them the terminology, piecewise functions, to represent these types of graphs.  

Writing a Piecewise Function

30 minutes

Before writing piecewise functions for each animal, students need to see how to properly write one.  I have provided them with a Graph of a simple piecewise function.  We write the first half together and they write the second half.  It is important to focus on the equality portion of the signs.  A good question here will be to ask why both end points aren’t filled in.  

The rest of the class will be spent writing piecewise functions for each of the animals (Math Practice 2).  I use the Note Card Activity for this one.  One place students may struggle is in writing the equations for each portion.  A reminder of finding slope may be sufficient.  Another issue students may face is with the y-intercept.  Often this is the starting place for a real-world function.  The challenge there is that the right hand portion of the function doesn’t start at the y-intercept.  I have students test a point from the line into the function to ensure they wrote a proper function. 

Exit Ticket

2 minutes

I use an exit ticket each day as a quick formative assessment to judge the success of the lesson.  

Today’s Exit Ticket asks student to graph a piecewise function.


The first three problems in this Assignment ask the students to write piece-wise functions from real life scenarios.  These help students strengthen their conceptual understanding of piece-wise functions.  The last two problems ask students to graph piece-wise functions.