Quick break from report card writing to post about our math this week.

We continue to reinforce the fact that there is often more than one way to a solution, while also discussing the fact that some solutions are more *efficient* than others. Efficiency is a goal, but getting an accurate solution is the bigger goal (and *understanding* the biggest goal of all). For instance, when we received those order forms for the Kingston Frontenacs, we talked about how much money the school would get if 100 tickets were sold. I hope a glance at the student thinking I’ve recorded here gives you a glimpse of the different routes that are possible to this solution. Is there anything inherently superior to seeing that 7 x 5 will get you there as fast as 3.5 x 100?

We’ve been mucking about with fractions for the past few weeks. This is a perennial challenge for many students (and teachers)–I’m not sure why, but it is. I had been working on various tasks with groups at different stages of understanding, but felt like we needed a break from this focussed and fairly abstract stuff. Also, I felt like we needed to return to a task where different math concepts converge. So this week I gave everybody the same task: to **design a schoolyard**. I am delighted with all the learning that flowed from this.

I knew that this was going to present different challenges for different kids, but also that the tasks–and the arithmetic involved–would be solved in various ways. We hadn’t even talked about the concept of **area** (though I knew they should have encountered it in Grade 4). I explained the task and then spent the next two days just letting them work away at it, meeting with different pairs to discuss the concepts they needed to work on.

On the third day, we discussed the first task: calculating the total area of the schoolyard without counting every square. This is *really* a discussion about **multiplication**. By exploring different understandings and strategies, we touched on the concept of *arrays*–the repeating rows and columns. For some students, the important step was identifying that there was a repeated addition of 20 (which could be simplified as 2). For other students, who grasped the multiplication, the important step was seeing a formula emerge or be confirmed: that we can multiply **length times width** to get area. They then got to test and apply this in subsequent steps.

Other concepts that came up included whether *half* has to be rectangle, or whether you can in fact have “half” the schoolyard covered in grass without having all in one continuous space. And one I didn’t anticipate (and at first said “No” to until I thought about the challenge students were giving themselves): *Can our “grass” area also have “woods” on it? In other words can we overlap our “half” and our “quarter” and still fulfill the task? *Hmmm*.* (And, Presley, what will happen to our grass if we fill it with trees?) Two different groups then had to figure out why one design ended up with a lot more “extra” space than the other.

Our fourth day began with a lesson about the relationship between fractions. Fascinating things came up. Students could see that if we divide a shape in half and then half again, we have quarters. They could see that 1/4 is *four equal pieces*. But when I only divided *one* of the halves, leaving only *three *pieces, there was a delicious uncertainty and confusion. Were these pieces they had just called *quarters *” still” quarters”?

The absolutely *vital* concept we had stumbled upon is that the denominator–4–means not *just* that we have four pieces. It fundamentally means that *we would need four pieces of this size to make the whole. *I pulled a quarter (25 cents) out of my pocket.They knew this didn’t mean I had cut a dollar into four pieces, but rather that four of these pieces had the value of a dollar. And what happens when we keep dividing our “halves” in half?

This was *such *a rich conversation. But it reminds me of why fractions can be so challenging: there is a lot of *language* wrapped in fractions and this can seem confusing. “One sixteenth is half of one eighth. You need half as many eighths as sixteenths to make one whole.” *Phew! *These concepts need to be worked and explored again and again. Seems like it’s probably time for some cooking!

I am hopeless at filming and capturing students discussion and work, and we didn’t have time this week to ask them to record themselves. But hopefully these pictures give a glimpse of the creative thinking that occurred:

*Note: **What did I leave out of our “Schoolyard Design” assignment? Well, the *school*. Kyle was quick to point this out; others seemed remarkably untroubled by this omission.*