Multiple representations of equivalence

A couple of math minutes today – one that laid out beautifully as I’d envisioned it, and one that gave me another reminder of how much I’m still chasing L’s abilities. Let’s start with the “fail,” shall we?

I’ve been keeping a clipboard in the car lately. Sometimes it has coloring pages (primarily from Dinosaur Train, but also color by number or other items I happen upon). Other times, it contains tasks that I estimate she can complete independently and quickly. This first math minute was one of those activities. It is a simple page I found online that is missing a sign (either subtraction or addition) in a series of number sentences. I passed the clipboard to L after she was buckled up in the Whole Foods parking lot. I pulled out of the parking lot, turned right, turned left, and entered the freeway. When I entered the freeway, I realized I hadn’t heard anything from L.

I asked, “So, what does that paper ask you to do?”

L replied, “I’m done.”

She passed me the clipboard. She was done.



Ok. Unexpected! I asked her how she knew what to do. She explained to me (as if speaking to a simpleton, by the way), that one could simply look at the answer and the first number. If the answer was larger, it would be an addition problem; a smaller answer would indicate a subtraction problem.

Fair enough.

Luckily, later this afternoon, I rebounded! I’ve been wanting her to think beyond simple mathematical equations (even multi-digit ones) because she’s pretty comfortable with the processes of addition and subtraction. I’d like her to begin thinking logically, rather than simply procedurally. I found this “balance the scale” page and decided to use it as the basis for this thinking. The goal of this paper would be to have kids identify which two addition sentences have the same answer and place them on either side of the balance.

I decided that I wanted her to explore this concept visually and kinesthetically, as well as have multiple ways to conceptualize the solutions. I pulled out three of our sets of math manipulatives:

All three models gave her the ability to combine terms as well as visually represent equivalence. She never saw the worksheet. I simply asked her to get the 2, 3, 4, and 5 inchimals. I prepared sets of 2, 3, 4, and 5 snap cubes. She got the 2, 3, 4, and 5 bananas. She then worked with her materials to determine that you could make two number sentences which equaled 7 (2+5; 3+4). Once she’d determined that, I asked her to show me on all three sets of materials so that she could physically see the balance points of the monkey and bucket balances, as well as see the same heights of the two stacks of inchimals.

Scene re-creation

Scene re-creation

We proceeded similarly for the next set. By the last two sets, I asked her to explain to me why the solutions made sense. She was able to verbalize that both sides equaled the same total (and give the accurate total). Job well done!

So, one very successful learning opportunity today, as well as another opportunity for mommy to remember that the tasks I anticipate as tricky are often not…