SWBAT write equivalent expressions in generalized form to solve the constant sums and differences problems.

Equivalent expressions give us different information about a situation. Students rewrite functions in different forms and find generalized solutions to save time later.

30 minutes

30 minutes

The investigation from yesterday continues--encourage students to choose a level that challenges them in thinking about these new problems. I give students as much time as possible today to answer the questions on this assessment. The questions are written in a way that allows multiple possible answers--there are still right and wrong answers, but there are many different ways to describe a relationship, which hopefully urges students to think more deeply about all of the questions.

10 minutes

**Instructor's Note:** You can fit this section into the lesson anywhere that feels convenient--it can be right after the warm-up, or you can use this as the closing.

Some students noticed that their partners wrote different different equations than they did. This is because, if you choose 10 as of your numbers, both 40 and -20 have a difference of 30 from this number. When students found different equations from each other, they assumed one of them was wrong. The purpose of this quick task is for them to realize that both of them can arrive at the same correct answer.

I frame this task just by telling students that these are two solutions I saw yesterday--and I want to know if either of them are correct. The point of this discussion is to create some argument or disagreement between students. Eventually, somebody will start to realize that perhaps these are both accurate solutions.

5 minutes

The end of this lesson is a good opportunity to focus on the "Big Ideas." As much as possible when I use a differentiated lesson like this, I want to emphasize the fact that there are key ideas that all students should be grappling with.

In this lesson, some of the key ideas are:

1) Some verbal descriptions lead to linear relationships and others lead to quadratic relationships. What are the key properties of each?

2) When you look at numbers with a constant sum, this creates a linear relationship with a negative slope. When you look at numbers with a constant difference, this creates a linear relationship with a positive slope. Why is this? How do these relationships show up in the parabolas when you look at the products of the numbers?

These questions are both worth having students discuss and write about. I like to use the same questions for lesson closings that I will ask them to write about as part of their assessment, so that they get several chances to think about these questions. Also, I have noticed that the students who learn the content best are the ones who can articulate answers to these types of questions clearly, because they have a big picture understanding. So I like to ask them to do a Think-Pair-Share with these two questions. Alternatively, they can do a "Think-Pair-Write" if you want to get some writing from them on their way out of class. The specific format doesn't matter, as long as students are held accountable to actually thinking about the questions.