How We Teach is the Message

Math Talk: ¼ of a Square (Part 2)

How might we use a Math Talk to help students connect fractions and a host of two-dimensional geometry concepts? How might this same Math Talk take learners into “a broad mathematical terrain of interconnected concepts, procedures, representations, and explanations”? (Kazemi & Hintz, 2014) How might a Why? Let’s Justify discussion routine overlaid on an extension Math Talk help deepen our understanding of why a particular geometric sketch is accurate? Let’s explore these ideas further…!

[Part 1 of this blog post discussed the original Math Talk and an Open Strategy Share discussion structure.]

Here is a “spin-off” Math Talk that emerged from the Open Strategy Share discussed in Part 1.

The group was presented with this sketch from one of their peers and asked to consider whether this image represented ¼ shaded. Note that the learner used a dot to mark the intended midpoint of the bottom of the square.

Triangle 1

Does the image above represent ¼ of the square shaded? How do you know?

We discussed this representation in pairs and small groups, then we shared some of our thinking whole group. For this particular Math Talk, we were using a Why? Let’s Justify discussion structure. Instead of trying to generate multiple solution pathways, we were considering one particular (potential) solution and working on justifying why this was or was not a solution.

Kazemi & Hintz (2014) summarize the goal of this structure as “to generate justifications for why a particular mathematical strategy works” (p. 3). Kazemi & Hintz continue: “During a Why? Let’s Justify discussion, the talk narrows upon a general claim in order to closely examine the mathematics and generate a discussion for it.” (p. 74).

The following justifications emerged from the group:

B suggested that the shaded triangle did represent ¼ of the square, since the base was half of the square’s side length and the height of the triangle was equal to the side of the square.

This verbal description resonated with some learners in the group, but others were unconvinced.

S suggested that so long as the bottom vertices of the triangle remain unchanged the top vertex of the triangle could actually lie anywhere along the top side of the square — and the triangle would always have ¼ of the area of the square. We sketched out some examples of S’s idea:

Triangle 2

S contended that all three of the triangles shown above represent ¼ of the square shaded in. The group agreed that the yellow triangle (a right triangle with its top vertex at the midpoint of the upper side of the square) definitely represented ¼ of the square, but some learners remained unconvinced about the other two examples.

A suggested that we draw in an altitude for each of the triangles we had sketched as a group. Our diagrams now looked like this:

Triangle 3

Adding in this additional structure (the altitudes) revealed something that wasn’t yet visible for all learners — namely, S’s contention that all three triangles had the same base and height, so must have the same area.

[Side note: This is a great example of what Jill Gough and Jennifer Wilson describe as “adding in an auxiliary line…to make what isn’t pictured visible” from their learning progression for SMP 7: I can look for and make use of structure. Check out their fabulous and thoughtful work on their blogs linked above.]

Consider how this Math Talk feels different than the Open Strategy Share version shared in Part 1 of this post. The Why? Let’s Justify discussion structure took us into another part of the mathematics — not merely considering a wealth of different approaches, but deeply considering one approach and focusing on creating convincing explanations to justify the mathematics used in that strategy.

In discussing the Why? Let’s Justify structure, Kazemi & Hintz (2014) contend: “Certain types of mathematical ideas lend themselves to a discussion that generates a justification for that idea” (p. 74). Not every problem or prompt is innately geared towards this particular discussion structure.

So what made this particular mathematical idea (specifically that the green triangle shown above does or does not represent ¼ of the square) a good fit for a Why? Let’s Justify discussion structure?

First, this particular representation did not look like ¼ of the square at first glance. A deeper analysis of the shape was required in order to determine whether or not it represented ¼ of the square–and deeper still to justify why.

Second, during the Open Strategy Share for the preceding Math Talk, most of the group’s sketches of ¼-shaded involved only congruent shapes. The approach used in this Why? Let’s Justify discussion did not subdivide the square into congruent shapes/pieces. Learners would have to rely on different reasoning (beyond subdividing a shape into congruent pieces and shading ¼ of the congruent pieces) to justify whether or not this specific triangle represented ¼ of the square.

Third, this particular triangle exposed that we (as learners) lose some of our intuition about triangles when they are obtuse triangles. We don’t often “see” the altitude of an obtuse triangle in the way that we see (and reason with) the heights of acute and right triangles. This drawing exposed some of that loss of intuition and made us dig deeper into why something was true.

For these three reasons (and perhaps others?), this prompt providing a meaningful task for learners to engage in a Why? Let’s Justify discussion routine.

Side Note: Kazemi & Hintz (2014) challenge that “During the discussion, we need to listen for procedural descriptions that students might give and be sure not to have the conversation stop there. When we press beyond procedural explanations into explanations that include reasoning, we are supporting students in justifying their ideas” (p. 56).

This paragraph still convicts me. Did I press far enough in the justification discussion outlined above? Why does any triangle with a base half the width of the square and a height of the square have an area of ¼ of the square? Maybe I should have pushed our thinking a little bit further here…

How does the Why? Let’s Justify discussion structure compare to the Open Strategy Share structure? In what ways does it open up a different part of the mathematics? What additional considerations might a teacher attend to when planning a Math Talk like this?

(See Elham Kazemi & Allison Hintz’s book Intentional Talk: How to Structure and Lead Productive Mathematical Discussions for more information on the Open Strategy Share and Why? Let’s Justify discussion structures.)

Michelle Rinehart

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