## Saturday, 27 April 2013

### MAT: Tile and Error

You know that we like using 'x'
To do algebra is not that complex!

A small square will be one by one
The area, the area is how it's done!
Unknown length shows 'x' without a doubt.

-Avril Lavigne ParaB Conic, "Y-Squared Tile"

Blogging about PD (Professional Development) here. I should do more of this. That way, I'll remember it better, and it could be of more use to others. I sort of did it as part of DITLife, but here it wasn't part of a school day, it was a Saturday morning. The "Ottawa Valley Spring Math Forum", as presented by Claire Bonner, Rebecca Black, Robin McAteer and Anne Holness.

Topic: From Patterns to Algebra.

In other words - at least for me - the bridge from middle school (Using: Input x 3) to high school (Using: 3x). Immediate diversion to write about what's in the parentheses there... since when does "take the number and triple it" become "triple of some unknown that looks like a multiplication symbol"?? Answer: Nowhere in the curriculum. Just, you know, one of those adjustments we expect Grade 9s to make.

Along the same lines, we tell students to always do the parentheses first, before exponents. Then we start writing parentheses as multiplication, and wonder why they have trouble with order of operations. Yeah.

Anyway, here's some of my "take aways" from today:

1) Students - and people - have a tendency to look down the table for patterns (+5,+5,+5,+5) rather than across (value times five). Putting things in order does not help them make the transition.

2) A mistake textbooks make is giving a lovely pattern of squares, then immediately saying "create a table". That changes the context. One should be able to come up with a "rule" even WITHOUT a table. Along the same lines, you don't want to introduce 'x' and 'y' too early, or you're probably doing a disservice. They're very abstract - patterns are concrete.

3) Pausing to ask a student "Why? Explain yourself!" is important not just for that student. Granted, it forces that student to try and articulate their intuition. Good. But it can also allow you to see whether they're seeing what you think, and perhaps more importantly, give OTHER students additional time to think about things.

4) Why DO we kill mixed numbers as soon as students enter high school? If you have "6 times 2 (1/2)", you don't have to do "6 times (5/2)", you can do "6 times 2" PLUS "6 times 1/2". Boom. Distributive law.

5) Patterns show equivalent expressions. Consider "3x+7" and "3(x+3)-2". Equivalent? Well, put down 3 "x-tiles" and 7 "unit tiles". Now put down three groups of "x-tiles" & 3 "unit tiles". Include two negative units. Zero pairs. Boom, same thing. No algebra.

6) Cutting up the "x-tiles" according to values of x makes more sense than swapping in unit chips for x. Which I'm not articulating well, but we sort of need elasticized tiles to represent x. Ones that students won't fire at each other.

7) Possible group activity for solving equations: "You're the right side (3x-4). I'm the left side (2x). We're equal. So of what value are our 'x' tiles to make it work?" (No stealing my tiles! But you can give me zero pairs, aka 'Thanks for nothing!'.)

8) Sequences, which start counting at '1'... I've blogged about that issue before. But it occurred today that RECURSIVE sequences DO run down the table. You need to know entry 98 to figure out entry 99. Inefficient, but there's the connection between recursive addition (of a constant) modeling a linear pattern.

And I'm doing recursion this coming week. I have no idea how much use that connection will be, but here's hoping I remember to incorporate it.

And that's whyyyyyy I smile, it's been a while,
Since every day and everything has felt this riiiight,
And now. YOU turn it all around...

-ParaB  Avril Lavigne, "Smile"