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.

If you know that a pair of numbers has a certain sum or difference, which such pair of numbers has the greatest or the least possible product? How can you prove this?

30 minutes

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.

30 minutes

10 minutes