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. 

Morning Math

I'm really enjoying math in the morning.

There's something fresh and new and hopeful about mornings lately and, even though I'm not doing anything ground breaking, I'm really enjoying how connected everything seems to be these days, mathematically speaking. 

In mid-October I mapped out a basic math plan when it was clear the girl (now 7.5) needed and wanted more and different kinds of math challenges.  I decided to call the plan 'algebra' because, from what I've read, algebra combines a number of skills and concepts that we have to start learning anyhow at the primary level.  And, because the girl has often balked when I introduce new math stuff, calling it algebra motivates her to give it a try; 'algebra' is a big kid skill, and she really, really wants to be big.  

So, in the morning it's been fun to open my math folders and give my lovely child her choice of math activities.

Solve for x (conceptualizing equality and sameness, sums and differences) or math card games (3 digit mental sums and differences)?

Christmas themed beginner Sudoku puzzles or Factor Dominoes?

Growing patterns or Bean Soup (fractions, multiplication, division)?



















 "What's division, Mama?"

"You know, like when you wanted to see if you were halfway through your reader.  There were 96 pages and you figured out in your head that half of 90 was 45 and then you..."

"Oh, yeah."

Later that morning at the science museum she built this:

"Hey Mama, look!  I got the water to cover the whole area!"

"Cool!  That's an example of division, too.  The water is being distributed evenly across the table."

So, here I am, trying to connect our morning math time to the bigger math picture that I'm constructing in my brain.  Like algebra, as I've mentioned.  What is algebra???  I did some algebra when I was in high school, but failed literally and horribly (although I aced geometry).  But why let that stop me?  After some research I decided that the concepts of balance and sameness, solving for unknown quantities, and growing patterns were all pretty darn interesting and relevant whatever we called it and away we went.

We began by building and analyzing growing patterns from some pattern starters I found.  Here's one she came up with on her own:

And, when we do the occasional worksheet (the one below is skip counting/multiplication) I look for ways to extend the activity forward, even if I'm not completely sure I'm right.  The sheet below was for figuring out numbers of feathers on different numbers of hats, legs on rabbits, and petals on flowers.  She got the skip counting patterns easily, so I described the data another way by saying:

"The number of hats you have is multiplied by three feathers per hat which gives you the total number of feathers," while writing h x 3 = f.  The next two examples she talked herself through the whole thing and wrote it down.   But maybe it would have been better to say, "The total number of feathers you have is equal to the total number of hats multiplied by 3"??  Writing all this out has me thinking of another way I could have done that, but I think sometimes its okay for me to muddle through stuff like this.

And, if skip counting is coming easily, why not challenge her to fill in the chart backwards instead? Ooooh, that was cause for consternation, but she had been saying "This is easy, this is easy, " all morning.  I know for sure she needs more experience with subtraction, I know being able to invert a procedure or concept is an important skill, and I also know that if something is too challenging she'll just give up.  So, here was an opportunity for deepening her experience with a skip counting chart and providing just the right amount of challenge at the same time. 

Filling in the charts backwards really made her think.  That's what she wanted, right?

Anyhow, it's nice to be entering a stage with her where we can sit down and 'do math' outright instead of trying to leave it around the house for her to discover, although I'm keeping that strategy in my kip for now!  More than anything, it's been lovely doing math, in the morning, with the winter sunlight streaming in, with a child who is finally (finally!) allowing herself to be interested and excited about exploring this mysterious and wonderful subject.