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# Sums, Differences and Products

Lesson 5 of 9

## Objective: SWBAT describe the relationship between the products of pairs of numbers that have fixed sums and differences and one of the numbers. SWBAT use multiple representations to solve optimization problems involving quadratic functions.

#### Warm-Up

*30 min*

Today’s warm-up is beginning to push students to a more abstract understanding of quadratic functions. The first problem on the warm-up is meant to ensure that all students have gotten the basic key idea of the past two lessons, and it is beginning to push students to write the function in different ways. This is open-ended on purpose to see what students come up with and to give them the chance to start asking questions about how to rewrite these functions.

The sum and difference problems are repeated, with the same two questions. Students who struggle to get started can be encouraged to data tables and graphs to get started trying to find functions to represent the situation. The questions are presented the same way for both problems, even though the situations are different (one has a maximum and the other has a minimum), so that students will have to figure this out as they explore the relationships.

Problems (4) and (5) are designed to push students to think about these relationships out of context (**MP2**) and hopefully students will make connections between these problems and the previous problems. The big idea is that when solving an optimization problem involving a quadratic function the vertex is the solution, so finding vertices of parabolas is useful. This can also illustrate how different forms of quadratic functions can be useful in different ways.

Ask students, “Why is finding the vertices useful?” and “How does writing the function in different ways help use get information about the function?” The idea is for students to make meaning of these more abstract calculations by relating these problems to the optimization problems from the past few days.

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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
- UNIT 7: Polynomial Functions
- UNIT 8: Exponential Functions
- UNIT 9: Ferris Wheels
- UNIT 10: Circles
- UNIT 11: Radical Functions
- UNIT 12: Cubic Functions

- LESSON 1: Rectangles with Fixed Perimeters
- LESSON 2: Equivalent Expressions Rectangles with Fixed Perimeters
- LESSON 3: More Rectangles with Fixed Perimeters
- LESSON 4: Multiple Representations Fixed Perimeter Problems
- LESSON 5: Sums, Differences and Products
- LESSON 6: Equivalent Expressions Generalization Constant Sums and Differences
- LESSON 7: Multiple Representations Sums, Differences and Products
- LESSON 8: Review Session and Workshop
- LESSON 9: Fixed Perimeter and Constant Sums/Differences Assessment