## Reflection: Intervention and Extension Quadratic Functions in Three Forms - Section 1: Opener: Making Sums of 12, and Some Beautiful Things About Numbers

While set up to be a pretty simple task that builds some foundations for completing the square, this opener provides plenty of opportunities for deeper investigation.

For example, I provide enough space in the chart to list all pairs on non-negative integers whose sum is 12.  But there is nothing in the instructions to say that we must stop there.  If any students ask about including non-integers, that's fabulous!  I tell them to continue the chart with as many pairs of numbers as they'd like, and to tell me what they notice.  Some students will continue the chart past the bottom row, "6 and 6," and they see that pairs of numbers start to repeat.  Combine that observation with an observation of how the product reaches a maximum value of 36, and kids have a glimpse at the idea that a parabola will "turn around."

There's also the whole issue of negative integers.  Once we allow for those, this chart could have an infinite number of unique rows.  What happens in those rows?  Looking ahead, we'll see that a parabola has a minimum or maximum in one vertical direction, but extends infinitely in the other.

For teachers (and your particularly intrepid students), I'll leave it to you to explore what's going on here.  If you've never played with it before try to generalize how any product of two integers can be written as the difference of two squares.

When I teach this lesson, sometimes the opener is short and sweet, and then we move on.  But sometimes kids want to do more, and I'll dig as deep as students are able here, taking the opportunity to touch again on the difference of two squares, and giving some students the extension task of trying to generalize.  Any time it comes naturally, I want students to spend a few moments in the sandbox.

How Deep to You Want to Dig?
Intervention and Extension: How Deep to You Want to Dig?

# Quadratic Functions in Three Forms

Lesson 6 of 21

## Big Idea: An brief adventure in number theory provides some background knowledge for completing the square, then students get to practice manipulating quadratic expressions in different forms.

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### James Dunseith

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