Fraction Flags

When I was 10 years old I did a project on flags. It was one of my many happy school experiences and ever since then I've always loved flags. Whenever I go to a different part of the world I always try to get a local flag. I visited Vermont a couple of summers ago and managed to track down and buy this wonderful Green Mountain Boys flag. One of the things I like about flags is that they are positively brimming with maths. For example, with the Green Mountain Boys flag, what is the ratio of length to width? What fraction of the flag is green?
I get the sense that my vexillophilia is something that is shared with a lot of students. We took advantage of this recently in a Grade 3/4 split class. We wanted to see what students knew (or didn't know) about fractions. In particular, we wanted to see what they knew about equivalent fractions.
As a Minds On activity, we played this Flags Recognition Game (left) and we were quite impressed as to how well the kids did; between them they covered a lot of cultures and heritages and they were proud to share this.
We then followed this up with this Fraction Flags Game where we asked the kids how we could colour a flag according to some given fractions. For example, we asked them how we could colour a flag that is half gold, a quarter white and a quarter green. We could display these suggestions quickly on the interactive white board as shown below:

As the students explained their solutions we heard the beginnings of some ideas about equivalence: it was a perfect time to get them to design their own flags based on a 6 by 4 template similar to the flags above. This is the task we gave them:

Design a flag for this school or for your family. Explain what fraction of the flag each colour covers.

We gave them some guidelines as to what makes a good flag as shown below:

We did this so that they wouldn't design flags that made it very difficult to see what fraction each colour covers. The great thing about this activity is that it forced out some common fraction misconceptions that we could address using the students' work.
The first misconception is that equivalent fractions are congruent. In other words, some students think that if two fraction pieces look different, then they cannot be equivalent. Looking at these flags, though, allowed us to challenge this notion:

 In the example on the left, most students would agree that each colour covers a quarter but would be less sure about the example on the right. By comparing the area of each piece though we were able to challenge their misconception and get them to realise that equivalent fractions aren't necessarily congruent. After this student had completed his flag, we asked him to draw another with the colours covering the same fractions:

The second misconception is that fraction pieces have to be joined together. For example, some students would look at this flag...

...and not see that half of the flag is blue. They might say that two-quarters of the flag is blue but believe that it can only be a half if the two blue sections are together. This misconception prevents them from seeing equivalence.
This student (below) clearly explained how she could pick up one of the coloured pieces and 'match' it with a piece of the same colour and so each colour covers a quarter of the flag.

And this student is already comfortable with the idea of equivalence as seen in his answer:

 

This student clearly explained how both of his flags were half blue and half green. If a student is unsure about this we could show them both and ask "Which is more blue?" This forces them to think about bringing the blues together and hence seeing how 12/24 (or 3/6) is equivalent to a half.

So by the end, we certainly got students to realise that equivalent fractions don't necessarily look the same and that the fractions pieces don't have to be joined together. We would need to back up these ideas by considering fractions that are sets of abjects (and not just fractions as area as in these examples).
And just for the record, my favourite flag is the West Riding of Yorkshire Flag (below). What math questions can you see in it?

Using Tangrams

I have been growing more and more fond of tangrams recently. I used to think they were just for making pictures that might (or might not) look like a man on a sledge (right).
Now I use them for exploring geometric properties and also for some proportional reasoning problems. Recently I went into a Grade 3 class and, to get them used to the tangrams, we began by telling them story of how tangrams were invented (as told in Virginia Pilegard's book The Warlord's Puzzle.) Basically, a special tile that was made for a VIP fell and broke into seven pieces. Many people tried to fit the pieces back together to make a square but with no success. Then came along a little boy who solved the puzzle and achieved fame and fortune (Hollywood should really make a film about this).
I then asked the students to take the two large triangles and ask them what shapes they can make with these. Usually, it doesn't take too long before students come up with a square, a parallelogram and a triangle (as shown below).

But after a while we got these other shapes which begged the question: What are the names of these shapes?

It is so important that students can construct irregular polygons to help them truly appreciate geometric properties.

Our next puzzle was straightforward to present to them:

If the small triangle costs 10 cents, how much does the whole tangram cost?

Occasionally, we get some students say "70 cents" as they think each piece is 10 cents; we reply to this by asking why the smallest piece would cost the same as the largest piece. The reasoning and proving, reflecting, representing and communicating that came out of this problem was great to see as seen in the video below:

I have varied this problem for older students by asking them to find the price of each piece if the whole puzzle cost $4.00. Also you could ask the students what fraction of the whole tangram each piece is.
What I love about tangrams is that, given the right questions, students really do use them to help model their thinking. They also get students thinking about how shapes can be decomposed and recomposed. This is an essential skill that will come in useful later when they work with areas of irregular shapes.
And next time you make a toasted sandwich, how about you tangramise it?!