Now that I’m retired from the classroom, summer tends to be my busy time. I just taught the grades 6-9 version of my Visual Algebra workshop. There were quite a few familiar faces among the participants. Some had seen me present at the Asilomar (California Math Council) conference, others at NCTM, yet others at a Math Teachers’ Circle. Some had even taken a version of the same workshop in the past, but felt they needed a refresher. It was great working with teachers who more or less knew what was in store.

As is usually the case, participants contributed ideas that were new to me, even though I had taught this material many times before to both students and teachers. In this post, I will share some of what I learned, even though those discoveries might not be of interest to people who are not already familiar with my curriculum creations.

### Function Diagrams

In grades 6-9, I mostly use function diagrams to represent basic operations, and linear functions. These two worksheets are excellent discussion starters in pre-algebra and Algebra 1. I regret that I did not use function diagrams (and cartesian graphs) to explore signed number arithmetic in the workshop. I’ll try to squeeze that into the Boston iteration of the workshop in August. (You can still sign up for that one!)

Function diagrams, of course, have many uses beyond the basics, which I usually share in my Algebra 2 / Precalculus workshops. This did not stop Eben from experimenting by making his own diagrams, and admiring the beauty of the diagram for y = 1/x:

(While Eben made his version of the diagram with a pencil, on graph paper, I made this one in a few seconds using this applet. For a fun animated version of y=1/x, go here.)

Eben also discovered an interesting phenomenon in the diagram for y = x^{2}. See below for the diagram, showing only the in-out lines for integer x’s from -3 to 4. *Notice the pairs of parallels*. I had never thought about those. It turns out each pair originates in two values of x that add up to 1. Why should that be? See if you can figure it out. I will post the answer in the comments next week if no one else does.

There’s a lot more information about function diagrams on my website.

### Advanced Lab Gear

The most basic (and most important) use of algebra manipulatives is the geometric representation of the distributive law — the so-called area model. Back in the late 1980’s, I designed the Lab Gear manipulatives by incorporating the best ideas of my predecessors (particularly Peter Rasmussen and Mary Laycock) into a new design that would also expand the usefulness of the tool beyond the basics. In my Visual Algebra workshop I present most of the uses of the Lab Gear, usually ending with some of the trickier applications: a geometrically correct representation of minus, a powerful introduction to completing the square, and a strategy to simplify algebraic fractions. Let me demonstrate this last one with an example: (2x+4)/(4x+2). Setting the blocks up like this reveals a common dimension, which is the common factor:

So “top and bottom” can be divided by 2. Julie noticed that the resulting fraction can be seen by looking at the blocks from the side. For example, the numerator, unsimplified is:

But looking at it *from the side*, you see that the simplified numerator is x+2. That’s a fun shortcut! The simplified fraction, thus, is (x+2)/(2x+1)

### Also…

The one criticism from otherwise extremely positive evaluations was that I was too slow in stopping off-topic side conversations. I apologize for that. This makes me think I should write up some notes on teaching teachers: how is it the same as and different from teaching kids? Maybe in a future post.

— Henri

Great! I too had never noticed this before. The key is recognizing the meaning of parallel arrows/segments on a mapping diagram: the difference in the source,h, equals the difference in the target, f(x+h)-f(x). For the quadratic function f(x)=x^2 this gives a quadratic equation in h, leading to the solution h=0 or h=1-2x. So x + (x+h) =1. 🙂

It would be interesting to explore this with other functions yielding similar visual niceties of mapping diagrams.

Thanks for sharing. Martin

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Great! I too had never noticed this before. The key is recognizing the meaning of parallel arrows/segments on a mapping diagram: the difference in the source,h, equals the difference in the target, f(x+h)-f(x). For the quadratic function f(x)=x^2 this gives a quadratic equation in h, leading to the solution h=0 or h=1-2x. So x + (x+h) =1. 🙂

It would be interesting to explore this with other functions yielding similar visual niceties of mapping diagrams.

Thanks for sharing. Martin

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Yep!

Conversely, if (x+h)+x=1, then ∆y=(x+h)^2–x^2=((x+h)–x)((x+h)+x)=(x+h)–x=h=∆x. So the average rate of change between x and x+h is 1, and thus the in-out lines are parallel.

I learned about function diagrams decades ago from Martin. He calls them mapping diagrams, and has a huge website about them: https://flashman.neocities.org

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