# Push Ups - Practice with an Arithmetic Sequence

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## Objective

SWBAT identify and represent an arithmetic sequence. SWBAT represent an arithmetic sequence using a table, graph, recursive equation, and explicit equation.

#### Big Idea

Scott is getting buff! How many push ups will he do on day n? Students explore an arithmetic sequence and solidify their understanding of how to represent it.

## Opening

5 minutes

The purpose of today's lesson is for students to solidify their understanding of arithmetic sequences. This task is similar to the one in the first lesson of this unit, Exploring an Arithmetic Sequence - Dots, so I try to help students as little as possible in this lesson and let them try to come up with the different representations on their own.

We begin class by reading through Scott's Workout together and then I let students get to work in small groups.

## Investigation

30 minutes

Students will spend the bulk of today's class working together to construct representations that describe this arithmetic sequence.  Rather than guiding them through this process, I will try to refer them back, whenever possible, to their growing dots work. This should be a good way for them to relate the two problems and build their understanding, especially around writing recursive and explicit equations.

Issues I watch for:

• Are students connecting their lines on their graphs? This could lead to a good discussion/introduction about discrete versus linear functions.

• One difference between this problem and the original Dots problem is that this pattern starts at Day 1, rather than 0. It's interesting to see how different students handle this issue and I'll be looking for different examples to share out. This could lead to some students struggling with writing the explicit equation for this problem.

## Discussion

20 minutes

The discussion section of this lesson follows the Scott’s Workout. Some key points I want to highlight in the discussion are:

• I want students to see where the repeated addition of 2 shows up in all of these representations.  I will push for students to make this connection, especially between the recursive and explicit equations.

• I introduce students to the function notations for starting values of a recursive function. This is a new step for them at the this point.

• I want students to be able to identify this pattern as arithmetic by just looking at the recursive function.

• We spend a fair amount of time on student explanations of how they derived the explicit equation.

## Closing

5 minutes

I finish today's class with an exit ticket that asks students to respond to the following prompt:

• Compare your recursive and explicit equations. What information do you need to write each one?