Lesson 10-4: Multiplication, Area, and Volume (Day 104)

Before we start the lesson, there's one thing I wish to clarify regarding the traditionalists' debate. For earlier today, I received a comment from Benjamin Leis, the leader of a middle school math club:

http://mymathclub.blogspot.com/

Benjamin Leis:
I just noticed this mention here. Anyway thanks for reading my blog. Its quite bemusing reading oneself being classified on the traditionalist vs. reform spectrum. 

I wanted to clarify a few things. One that post was more for the internet and anyone reading than for my kids. One of the challenges of working with a spread of kids in different grades who are doing anything from Math7 to Geometry this year is finding the right material to focus on. So no one is referring to formulas directly but I was excited enough to put it together just because I hadn't seen something similar elsewhere. My main takeaway I hope others see is how the formulas change between representations and the structure that shows. 

For what its worth, I'd self classify as pragmatic. My main goal is joy and I experiment with various approaches and follow what seems to work with the students. I am really interested in rigor and particularly developing problem solving but I'm still learning myself what that fully means.

Here is my response:

Yes, back in August there was an article that came out about traditionalists, and I was looking for math teachers who blogged in August. That's how I ended up dragging you into the big debate.

On your blog I found things that traditionalists like (rigor and solving complex problems) and things that reformers like (use of Desmos and other software). I suppose that in the end, I'm pragmatic as well. I taught for one year at a charter middle school, where I tried to follow the best of both approaches to teaching. We used a curriculum developed by Illinois State that includes both a traditional textbook and a STEM project guide. Many of my students had very low math skills, as opposed to your math club students who participate in different competitions.

I included both your comment and my response on my blog today to make things clear. I enjoyed reading your latest blog post on your snow day and the triangle in a square proof. Congrats on your team's silver medal and good luck at your state Mathcounts competition!

By the way, this is the latest Leis post that I mentioned above:

http://mymathclub.blogspot.com/2019/02/triangle-in-square-areas.html

He posts an interesting Geometry problem:

Given an equilateral triangle AEF embedded in a square ABCD,  prove the area of CEF = ADF + ABE

Click the link above to see the full proof. His idea is to rearrange the three triangles, one right on top of another. All three triangles have the same base (a side of the equilateral triangle), and then he uses 30-60-90 triangles to show that ADF and ABE have the same height with CEF twice that height. In our text though, we don't reach 30-60-90 triangles until Chapter 14.

Today on her Mathematics Calendar 2019, Theoni Pappas writes:

A tetrahedron is a polyhedron with ____ faces.

Fortunately, we just read about this two weeks ago, in Lesson 9-7 of the U of Chicago text:

"Polyhedra can be classified by the number of faces. Below are pictured a tetrahedron (4 faces) and a hexahedron (6 faces)."

Therefore the desired number of faces is four -- and of course, today's date is the fourth.(Yes, this is much more trivial than the Leis problem above.

Lesson 10-4 of the U of Chicago text is called "Multiplication, Area, and Volume." In the modern Third Edition of the text, multiplication, area, and volume appear in Lesson 10-2.

This is what I wrote last year about today's lesson:

In many ways Lesson 10-4 is more algebraic than geometric. Indeed, this lesson introduces geometric models to justify the FOIL method of multiplying polynomials.

In today's Lesson 10-4, we ask questions such as, how is the volume of a box affected by tripling its length, width, and height? The answer is that it increases by a factor of 27.

One of the review questions asks for the volume of an Uno card. (I wonder whether there are any Wild Draw 4 cards in this deck!) The bonus question asks students to cube a binomial.