##
* *Reflection: Intervention and Extension
Push Ups - Practice with an Arithmetic Sequence - Section 3: Discussion

I find I need to do some scaffolding around writing explicit equations from the recursive equation that can start in this lesson. For some of my students, it is difficult to see a starting value in addition to the constant difference and how those go together to write the explicit equation. Rather than focus on the starting value, I like to focus on the constant difference and show students that as long as you're repeatedly adding the same number in the out column of a table, that number has to go in front of the coefficient in the explicit equation (we can talk about repeated addition as multiplication here). From there, I ask students how they can adjust their equation by adding or subtracting a number to make the values in their table work. I realize that this focuses more on strategy and procedural fluency rather than building conceptual understanding, but for some students I think this is a better starting point and their understanding can build from here.

*Scaffolding the Explicit Equation*

*Intervention and Extension: Scaffolding the Explicit Equation*

# Push Ups - Practice with an Arithmetic Sequence

Lesson 3 of 10

## 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.

*60 minutes*

#### Opening

*5 min*

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.

#### Resources

*expand content*

#### Investigation

*30 min*

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.

*expand content*

#### Discussion

*20 min*

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.

#### Resources

*expand content*

#### Closing

*5 min*

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?

*expand content*

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- UNIT 1: Introduction to Algebra: Focus on Problem Solving
- UNIT 2: Multiple Representations: Situations, Tables, Graphs, and Equations
- UNIT 3: Systems of Equations and Inequalities
- UNIT 4: Quadratics!
- UNIT 5: Data and Statistics
- UNIT 6: Arithmetic & Geometric Sequences
- UNIT 7: Functions

- LESSON 1: Exploring an Arithmetic Sequence - Dots
- LESSON 2: Exploring a Geometric Sequence - Different Dots
- LESSON 3: Push Ups - Practice with an Arithmetic Sequence
- LESSON 4: Chain Emails - Practice with a Geometric Sequence
- LESSON 5: Decreasing Arithmetic Sequences
- LESSON 6: Decreasing Geometric Sequences
- LESSON 7: Arithmetic Means
- LESSON 8: Geometric Means
- LESSON 9: Arithmetic & Geometric Sequences: Putting It All Together
- LESSON 10: Sequences Assessment