## Reflection: Connection to Prior Knowledge Finding Roots of All Sorts - Section 2: Finishing Up, Q&A, and Work Collection for Can You Factor It?

When students learn how to factor quadratic expressions for the first time, they are drawing on an array of both procedural and conceptual prior knowledge.  There's plenty I want students to understand, but the initial hook is to frame this as a puzzle: what are two numbers that have this sum and that product?

As I've written above, some students can make sense of this task pretty quickly, so that by the end of the first week of Unit 6, they're ready to investigate and begin to use the discriminant on their own terms.

Other students need a little more help developing their understanding of factoring.  It would be counterproductive to hammer on the discriminant before these students know enough about factoring to appreciate the usefulness of the discriminant in the first place.  Students must really get the idea that some quadratic expressions are factorable and others are not.  For anyone who needs help getting to that point, we revisit two related approaches from earlier in the year: guess and check, and making lists.

In one of my sections today, we ended up making a list like this, which students have seen before.  By engaging in this sort of list-making, students recognize that there's actually a pretty limited number of possibilities, and therefore a limited number of different products we can make from a pair of positive integers whose sum is 17.  I asked students if this list is complete, and how they know.  Then I asked if them to consider a pair of numbers whose sum is 17 and whose product is 65.  The list helps to make it clear that if such numbers exists, they will not be integers.  If the numbers we're looking for are not integers, we recognize, then this problem just got a little more complicated.  The moment kids understand that distinction is exciting, for them and for me.  And once we get there, well, then we're ready to see what the discriminant can do for us!

Connection to Prior Knowledge: Scaffolding with Prior Knowledge

# Finding Roots of All Sorts

Lesson 5 of 21

## Big Idea: In this fast-paced lesson, students are introduced to as many ideas as they can handle, while also being given space to make their own sense of those ideas.

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43 minutes

### James Dunseith

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