Claim Challenge Convince

The last two posts have been about discussion that's come up in class about the fraction talk squares that two students, Beatriz and Maria, made.

They both have some interesting triangles, where the students are really not sure of the fraction. It turned out even better than I hoped because one of the students made a claim about this:

The claim went up on the Mathematical Claims Board, and we returned to it twice. (I've blogged this on the class blog too.)

I asked on Twitter:

#MTBoS, looking for a simple proof that a ▲ splits into 2 =area ▲s at a midpoint https://t.co/L9LCpmk51d @mathhombre pic.twitter.com/YgED586st7

— Simon Gregg (@Simon_Gregg) March 29, 2016

And got some helpful replies, like this one from Daniel Scher:

Lovely challenge & tool: Find as many ways as you can to ÷ a ▲ into 4 equal-area ▲s @dpscher https://t.co/plthIK7Im5 pic.twitter.com/Uex1V3wrYz

— Simon Gregg (@Simon_Gregg) March 29, 2016

Some people referred me to the area = 1/2 height x base rule, but this is something my class doesn't know and don't need to know yet. (It seems to me further on than the thing we're trying to prove.)

John Golden suggested on Twitter that we see how you can get same-area triangles (if you start with the same "base"):

> 
 

We also had a look at this:

This convinced a lot of the students that Maryam's claim was true.

But Miguel was still not convinced. He said:
"Technology isn't perfect. Is that really a proof?"
Yes, he really did say that! Christmas came early to 4G.

Dan Finkel suggested on Twitter that we change our triangle into a parallelogram and it will make it easier to see the truth:


We had a really good discussion about this, and quite a few students tried to persuade Miguel. There were a number of really good arguments to show how the two parts of the triangle in fact have equal areas. Hannah, for instance, suggested further splitting the pink parallelogram and the green parallelogram with their other diagonals. A lot of people thought this helped to see it.

After this, I asked the students to write about what had happened with this claim, to write what they thought about it all. Mostly great results:

Looking at Beatriz's Fraction Talk Square

So, we made some fraction talk squares.

I thought I'd document today's fraction talk lesson in a bit more detail, as we did a longer, 45-minute lesson on this rather than our usual ten minutes.

We started off working with a partner, looking at Beatriz's design:

One pair was measuring.

Some pairs made a little triangle

I circulated asking for explanations, and chipping in here and there. For instance:

Then we came together and shared what we'd decided:

We were divided in our opinion about whether the top right triangles were the same fraction or not (this was what I was hoping for), and so I moved on to Maria's square. This time, straightforward for most of the square, but what about those two on the left? Opinion was even more divided.

Then Maryam made her claim: "As long as it's the midpoint, it doesn't matter how you split the triangle; it will always be half." Yay! That will go on the claims board, and it's even more what I was hoping might happen!

 To help people consider her claim, I drew this on the whiteboard:

Some students said it wasn't clear because of my drawing. OK then, I said, how about I draw it on Geogebra? 

I used the midpoint tool. Maryam was still persuaded the two half-triangles were the same.

This time most people thought they were different. I said perhaps we'd need to come back to this. K. said that we needed to call Albert Einstein back from the grave to tell us. I was doubtful about both the possibility and necessity of that. 

I was planning for them tomorrow to have a look at this image, made on an online geoboard, looking generally at all the fractions, and later focusing on the yellow triangle:

But, now Maryam has framed her claim differently to how I had, perhaps this diagram would be a better one to anatomise:

I think maybe we can see that whichever way we cut up the yellow triangle, we cut it in half.

Watch this space...

Make a Fraction Talk Square

Nat Banting got me working out how to screencast with the Smartboard (and spotlit it too). The sound's still not good, but I'm hoping I'll work that out.

One of my first recordings was this, during a fraction talk.

@FractionTalks But I liked Beatriz's question: "Are we going to be able to make some of these?" pic.twitter.com/8tR2HxhBLk

— Simon Gregg (@Simon_Gregg) March 21, 2016

I was glad I followed up Beatriz's question and we made them with Geogebra:

@FractionTalks In fact, I liked Beatriz's suggestion so much, we did make some of our own #fractiontalks! pic.twitter.com/Fi6xxU1SuZ

— Simon Gregg (@Simon_Gregg) March 21, 2016

It was a great way into Geogebra apart from anything else - simply using Midpoint and Segment tools (Click on the tools second and third from the left.)

We only used the two tools and yet the students had a lot of freedom - they produced some interesting squares. We talked about two of them, that involved halving in particular.

The other two involve an interesting feature of triangles.

I'm wondering if someone will say that the triangles on the left of Maria's square are the same area? Or the small ones on the top right of Beatriz's? And where do we go from there?

Code to embed the Geogebra app:

<iframe height="556px" scrolling="no" src="https://www.geogebra.org/material/iframe/id/WUtYR9Mf/width/1007/height/556/border/888888/rc/false/ai/false/sdz/true/smb/false/stb/true/stbh/true/ld/false/sri/true/at/auto" style="border: 0px;" width="1007px"> </iframe>

Five regular routines

As my teaching evolves to start with students' thinking and talking, I'm usually beginning my hour long maths lessons with routines that encourage these things. Here are five favourites:

Quick Dot Images

See this by Steve Wyborney. I show an image quickly, then show it a second time; the idea is that almost all students should be able to have seen how many dots there are. But the interesting thing is not how many, but how. Students really like sharing their ways. And the message is: this is something you can do your way; you didn't need to be told how to see this. And - your way is worth sharing.

Students like to do the annotating themselves.
Often I do it: to speed things up, and to model revoicing.

Which One Doesn’t Belong 

Christopher Danielson got the ball rolling with this. Since then, Mary Bourassa has done a great job of collecting WODBs together at wodb.ca . I've blogged about them a while back, and recently.

The idea is that you can justify the choice of any of the four items as the odd one out.

Estimate 

Students write three numbers on a blank number line on their whiteboard, one too big, one too small and as close as you can (Graham Fletcher has some; these can be extended into whole lesson activities which are great modelling activities - and then of course there’s our regular, estimation180).

Estimation is such a key skill that we've also devoted some blogging time to it, with each student making their own estimation challenge.

See, think, wonder 

Last year it was notice and wonder, but I like the differentiation of notice into see and think, and feel it's worth distinguishing the two. (Both are brilliant though!) I'm reading Making Thinking Visible, and this is one of their routines.

The point is just to present an image and see what people see, preferably recording it all. 

This one developed into a really interesting debate

I've also adapted this to be what equations do you see, often presenting an image of Cuisenaire rods:

Counting Circles 

I've blogged about this before. The idea is to count from different starting points, with different jumps. We do it round in a circle. Then we stop, and think about what the number three or four jumps on would be. Students share how they know.

NEW!
Fraction Talks

I haven't used these yet, but this is a great idea, and Nat Banting is developing a really useful site with lots of images: fractiontalks.com - watch this space!