Trying to find math inside everything else

Posts tagged ‘illustrative mathematics’

A Way to Foster Productive Struggle?

My school has been trying to better create conditions for productive struggle in our classes, because a lot of students have taken a very receiving stance. So early in our Area and Volume unit, I decided to use this task from Illustrative Mathematics.

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The task is a 7th grade task, and so involved nothing new for my high school geometry students – just area and perimeter/circumference. But the task has a lot of parts, not all of which are obvious from looking at it. So I gave them task, and then I was “less helpful.” In fact, I barely spoke during the lesson, only quietly clarifying things, but reflecting their proximity questions back towards themselves and their other group members.

Almost every group that attempted the task solved the problem on their own. (I followed up with an extension where they designed their own stained class on the coordinate plane and found the price using the same pricing, for those who finished quickly.) I had a group of three girls who don’t usually feel very confident in my class feel like rock stars after figuring the whole thing out themselves.

A few days ago, I saw this tweet:

I thought it really applied here. While the content was still related to what we were learning in high school geometry, the opportunity to solve a complex task with little scaffolding was really helped by using a task from an earlier grade. I recommend it.

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Quadrilateral Congruence

Stressful as it is, I am loving teaching new courses. When I first start teaching, I felt like I was learning new stuff all the time, stuff about algebra (and how it connects to other courses) that I didn’t know I didn’t know, and now it keeps happening with geometry, especially with the more transformational tinge CC geometry has.

One of the things that struck me was, last week, when I used this Illustrative Mathematics task as a follow-up to my lesson about the diagonals of quadrilaterals. I feel like the understanding I had internalized that you can prove triangles congruent with less information because they are rigid structures, but quadrilaterals are not, so there are no quadrilateral congruence theorems. But I realized that’s not true.

Last time, we constructed all of the special quadrilaterals by taking a triangle and applying a rigid motion transformation. That meant that every special quadrilateral can be split into two congruent triangles. Therefore, if you had enough information to prove one pair of triangles is congruent, you could prove the whole quadrilaterals are congruent.

Parallelogram SSSS

So if we’re looking at SSSS in terms of the triangles, we really only know two sides of the triangles. Since that’s not information to prove the triangles congruent, then it’s not enough for the parallelograms. But SAS is enough for the triangles, so it’s enough for the parallelograms.

Isosceles Trapezoid SSA

Here’s a non-parallelogram example. Here are two isosceles trapezoids with the same diagonals, same legs, and the same angle between the diagonals and one of the bases, but the trapezoids are not congruent. But that’s because, when you look at the triangles, we have Angle-Side-Side, which we all know is not a congruence theorem. If, instead, we had had SSS (a leg, a base, and a diagonal), then they would be congruent.