Design and discovery in mathematics

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

Remember to click on these to see them better. As I await delivery of my new glasses, I am noticing this more. ūüôā

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:

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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.


Math Movies

I think I teach math a little differently every year. ¬†(I know I do). ¬†We began the year looking at patterns in numbers, and then spent some time looking at the important idea of “proportionality”–this is what we were exploring in If the World Were a Village (here).¬†¬†We’re moving¬†into an extended look at larger numbers, place value and operational problem solving (that means figuring stuff out with adding and subtraction and multiplication and division). ¬†If I am doing my job right (an open question) we will kind of spiral back to each concept throughout the year–so proportionality will be revisited¬†through place value, but also through fractions and through graphing.

My goal is to bring the idea of inquiry into our math experience, balanced with instruction and practice in¬†arithmetichow to perform the different operations, as well as greater “automaticity” with mental math. ¬†Most importantly, I want all students to¬†believe they can be good at math (and that it can be fun)!

As a kind of pre-assessment, but also as a way of challenging the various kids in various ways, I tried something a bit different last¬†week. ¬†The focus: ¬†10,000. ¬†I tried to come up with five different tasks aimed at testing their understanding of how larger numbers are constructed. ¬†Bless their hearts, their engagement with my weird questions was universally fabulous! ¬†In each, they had to figure out how different “units” could be used to reach 10,000: ¬†in some cases this was tens, hundreds and thousands; in another it was 500; in others the numbers were less round and they had to do some significant figuring (and re-figuring).

Then I asked them to make a film that explained what they did and why.  Today we had a little film festival (complete with popcorn:  is this a physical or a chemical change??) so everybody could see what everybody else had been up to.  Check these out!

(Warning:  you may have to adjust the volume as you go).