Wednesday, January 2, 2013

All-in-One? Using Shapes to Explore Number & Algebra Concepts

I'll start right out by saying that I'm pretty sure this was not the right activity at the right time for my 7.5 year old darling girl, but I did learn a lot about where her mathematical thinking is right now, which is always helpful.

In the last one and a half years my inquiry into elementary math education has kept pace with her math learning.  We've discovered so much together and it's been an incredible learning process for both of us.  Lately, though, it seems like the big picture concepts have clicked for me but as I try to move forward myself I end up rushing her.  The following activity is a case in point, but I still think it has merit for some child, somewhere!  Here's how it played out:

Back in November I bought a travel set of attribute blocks.  We haven't done much with them yet, but after looking through the little activity booklet that came with it I found an activity that piqued my interest.  It had to do with figuring out how many blocks of a certain shape (or combination of shapes) you would need to have to get a total number of sides.  It looked vaguely algebraic to me but was presented as a mental math activity.  So, I thought I'd create my own version of the activity on paper to make it a little easier to follow and to visually reinforce the differences between shapes.

This is the first worksheet I made.  In the first example I labeled the triangle with a 1 (meaning one group of three sides) and she had to figure out how many more hexagons made it add up to a total of 15 sides.  The second problem also had one shape already labeled, but in the final two problems I left it to her to figure out how many of both shapes.  You can see little pencil marks around the shapes at the bottom where she counted the sides one by one and then made notes for herself.

She was not completely happy with this activity (and was in a bad mood, distracted by whether the word 'futzy' was an insult or not).  Grumpy or not I think it really stretched her capacity in a good way, well enough for me to try again.  In the second iteration I asked her to write the total number of sides under each shape which I think really helped.  It was easier for her this time around.  You also might notice that I rephrased the question a little.

Then the holidays interceded with math learning.  Over that time, though, I did some thinking about how perhaps this kind of activity could be used to reinforce the concepts of multiples and the commutative property.  For example, 4 three-sided shapes (triangles) have the same number of edges as 3 four-sided shapes (squares or rectangles).  I also wanted to continue to stretch her idea of what the equal sign means; not necessarily a result, but a relationship -- various expressions of the same idea.

Here is the third activity.  In it I intentionally grew the numbers from 6 to 12 to 24 to 48. 

This time her strategy right out of the gates was skip counting whole groups of sides to work toward her answer instead of counting individual edges.  This means to me that somehow in the last three weeks her brain has begun to 'group' with more facility. I think this because, in the same time period, she has also experienced a huge jump in her reading abilities -- from having to sound out familiar words as if they were new every time to simply looking at a word and knowing what it says.  The math concept of 'grouping' and the reading concept of 'chunking' are essentially the same skill -- smaller items grouped into a larger whole. I saw that click into gear today with my daughter as she went to skip counting unbidden.

Anyhow, she moved through this last activity fairly quickly until the last two problems.  After it was finally over she proclaimed, "That was hard!  I hated it!  Forty-eight is such a big number!!"

That proclamation was revealing to me -- at this point in the game she's got facility with multiples of 0,1, 2, 3, 4, 5, 10 and 11.  The larger numbers are still a lot of work in terms of multiplication.  Being able to decompose a number like 48 was just too much at this moment in time. Ultimately, I think it's a call to put on my own brakes and, instead of trying to rush us forward, really dig into the mysteries of number composition and decomposition.  I know numbers are my weak point, so this will be good for me personally as well. 

Epilogue: After drafting this post this afternoon and then leaving to let it sit for a while I ran across the multiplication card game called Snap it Up which I found a month or so ago while at Goodwill (read about moreof my thrifted math here!).  I decided to give it a try and what do you know?  It was fun for both of us!  One interesting observation was that when I said 'what's x times y'  she'd give me a blank look but when I said 'what are two fives...' or 'how many tens make eighty' she totally got it.  I love it when the math stars align for us like this.  It happens a lot, actually, but I am grateful each and every time. 


  1. I never thought about the way grouping skills in math and reading are related, but it makes sense! And now I realize that it's exactly what seems to be going on with my son's reading and arithmetic skills. Thank you for this insight, Malke!

  2. I woke up this morning with another piece of 'proof' about my little theory -- over the holidays she was beading jewelry to sell in her store and every piece was created with a linear pattern unit made with no prompting from me. It's a brain thing, I tell ya, but I also know that it's partly the work we've done on patterns and groupings clicking in when the brain is ready.

    And then we were doing some work this morning on skip counting in the Beast Academy 3A book and she made the same observation about big numbers that she had about 48 (as mentioned above, in the post). I think what she means is that multiplication makes numbers get big QUICKLY! It's a great observation, that! I also started reading/working through Greg Tang's MATHterpieces book -- different ways to combine groups of numbers to make the same sum. It's a perfect resource for my goal to look more deeply into number composition/division.

  3. I posted your url to the group that had a link to this article:

  4. I did a shape - algebra lesson last week with tak-tiles (have you come across them?) - not so number-related, but seems to get brilliantly and simply to the root of algebra. See:

  5. I had not heard of tak tiles, but they look so interesting!!! I loved the graphic you made, but it's going to take a while to wrap my head around it... :-)

  6. Tak-tiles are hard to get hold of now. My colleague and friend Isobel pointed out that pattern blocks are also good for that kind of basic algebra too - and much easier to get hold of. Have you used them? I like to get the kids to make different patterns of dodecagons with them:

  7. So, is the idea to have three starting 'areas' (shapes) and make the design using only those three, then figuring out the area(?) as expressed algebraically? I'll have to go back to the tak tiles post and look more closely.

    As you probably know by now, I've just figured out factoring -- not sure if I'm ready to tackle algebraic expressions yet! (Although we have been doing some of that -- working on paper with the 'balance' concept, solving for x, challenging the idea of the equal sign, making growing patterns...but I've still got a lot of anxiety around it to work through ;-)

  8. Well, you've got so many brilliant ideas, you could afford to leave aside anything that brings back bad memories!

    The basic ideas are that 1. We can make letters stand for areas, rather than finding a number; 2. We can sometimes simplify. And 3. Mathematics as a tool for making beauty.


Thanks for reading. I would love to hear your thoughts and comments!


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