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# The Painted Cube Problem

Lesson 1 of 9

## Objective: SWBAT analyze patterns in data tables to determine whether the tables show linear or quadratic relationships and describe these same types of patterns in the context of a real-world problem.

#### Warm-Up

*30 min*

The two problems in this warm-up are designed to prepare students for the day’s lesson and to get them thinking about the big idea of this unit: * the relationship between the behavior of a polynomial function and the function rule*.

The first problem is about data tables. Students may or may not remember previous work on linear and quadratic data tables. The Resource Poster will remind them if they have forgotten. It is worth discussing again why a constant difference creates a linear function and why increasing differences create non-linear functions.

The key idea about these polynomial data tables is that we can examine the differences between consecutive outputs, and the differences between those differences, and the differences between those differences and so on. The number of levels of differences that it takes us to get a sequence of constant differences tells us the degree of the polynomial. This can be proved to be true using calculus (the Power Rule). If you think about the differences as working kind of like derivatives, if you keep finding higher order derivatives of any polynomial function, eventually the degree will be 0, because the Power Rule reduces the degree each time you differentiate. Obviously students will not understand this now, but it is an interesting idea to plant in their minds for later on.

The scaffold for students who need extra support is the Poster to remind students of their previous knowledge about linear and quadratic patterns in data tables. The extension is for students to find equations for as many of the data tables as they can. Additionally, the 2^{nd} and 3^{rd} pages of the warm-up ask students to go beyond the basic expectations.

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#### Closing

*10 min*

The amount of instruction you provide at the end of this lesson really depends on how much progress students make towards solving the problem today. If students complete the data tables for the Painted Cube problem, they will be able to do more analysis of the tables and you can take time at the end of the lesson to discuss how to determine the rules for these data tables. Many of my students had only barely begun to figure out how to complete one row of the table at a time, which is fine for the first day.

I plan to give students some time to think about the exit ticket questions and write their initial thoughts. After they have done this, we will discuss the connection between the first warm-up problem and the Painted Cube problem and give students a little bit more time to predict what type of equation will show up in each column of the data table. This is just a prediction for now, so ask students to justify their thinking but it doesn’t matter if they have the correct predictions at this time.

#### Resources

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- UNIT 1: Linear and Nonlinear Functions
- UNIT 2: Piecewise Functions
- UNIT 3: Absolute Value Functions and More Piecewise Functions
- UNIT 4: Introduction to Quadratic Functions through Applications
- UNIT 5: More Abstract Work with Quadratic Functions
- UNIT 6: Rational Functions
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- UNIT 8: Exponential Functions
- UNIT 9: Ferris Wheels
- UNIT 10: Circles
- UNIT 11: Radical Functions
- UNIT 12: Cubic Functions

- LESSON 1: The Painted Cube Problem
- LESSON 2: The Painted Cube Part 2 and End Behavior
- LESSON 3: Surface Area and Volume Functions
- LESSON 4: Writing Rules for Polynomial Functions using Data Tables
- LESSON 5: Sketching Graphs of Polynomial Functions
- LESSON 6: Compare and Contrast Graphs of Polynomial Functions
- LESSON 7: Relationship between the Degree and the Number of X-intercepts of a Polynomial
- LESSON 8: Writing Equations for Polynomial Graphs
- LESSON 9: Graphing Polynomial Transformations