Finding Area By Combining Shapes

I have been visiting Aoki-Chuo school in Kawaguchi, Japan this week as part of the "World Tour of Maths".

It has been an amazing experience. I have been treated like a rock star. The staff and students have been so friendly and helpful. What a great school!

Cut To The Chase

Anyway, one of the lessons I was watching involved Year 6 working out the area of shapes by looking at the shapes they are made up of.

Now, we are not talking about two triangles make a square here.

We are looking at finding an area like this:

Problem Solving the Japanese Way


One of the things I was seeing in Japanese classrooms was the way the students were encouraged, even expected, to find multiple ways to solve a problem.

Here are four ways that one student demonstrated that found the area of the shape:

Method 1

He works out that a quarter of a circle overlapped on another quarter circle makes the required overlapped shape.

So he calculates that two quarter circles have a combined area of 157cm².

If you subtract the area of the square, you will have the overlapping shape remaining.

157cm² - 100cm² = 57cm².

Method 2

In this one, he works out the area of the entire circle, subtracts this from the area of the bigger square and gets the area of the four corner pieces. This he divides by 4 to get the area of one corner piece. Then he multiplies by 2 and subtracts this total from the area of the smaller square to get the required solution.

Method 3

This time he uses a triangle and sees that the difference between the area of the triangle and the area of the quarter circle will give him half of his required shape. 

Method 4

His final method is to subtract a quarter circle from the square to get the outside corner piece. This he doubles and then subtracts from the square to get the internal shape.

What next?

Well, after they had time to work independently on their solutions, the teacher selected a few students to come and present their ideas to the class. The presentation is a very important part of the lesson and the children take it very seriously. The get a few minutes to draw their solution on some A3 paper and then stand up and talk about it. At the end they say something like, "This is what I have found to be true." and the class responds, "I agree." Then they ask for any questions, which they answer. Then they are thanked by the teacher and get a round of applause from the class.

After all the selected children had presented their solutions, the teacher left their diagrams on the board and asked the class to find similarities and differences between the various methods. This was important because the final step was to make a generalisation, a statement that could be used to help solve similar problems in the future. It is like the class summarising their learning for the lesson.

So what?

I was blown away. In the space of 45 minutes the class answered one question.

It wasn't 57 different questions from the textbook. It was one question.

But the depth of learning was very impressive. 

And at the end of the lesson, the purpose was manifestly clear - it was all about thinking.

Quality not quantity.