Two Squared
Two Squared
If you double the sides of a square, the area becomes four times the size. It is quadrupled.
We can try the same thing with a rectangle and a rhombus.
How do the four smaller ones fit into the larger one?
We can then try with equilateral triangles:
And "L" shapes:
What has to be done to make these fit?
We could try with other shapes like hexagons.
These have to be cut and rearranged.
What is the least number of cuts needed to fit four hexagons into one larger hexagon with sides double the length?
Why do this problem?
This problem helps learners to come to terms with the concept of area and also to explore the properties of many straight-sided shapes. It requires some quite difficult visualisation when more complex shapes are used. However, the beginning is very accessible so everyone can become involved quickly.
Possible approach
Some of the later shapes require considerable manipulation to fit in the four smaller shapes. The hexagons are likely to prove the first real difficulty and children will reach different solutions. They may need to count little triangles on the isometric paper to be convinced that they have actually fitted four hexagons into the larger one. Possibly the simplest solution is to cut one hexagon into three rhombi and fit all together like this:
If many learners are having difficulties, the diagram above could be drawn on the board. Isometric paper can be very useful at this point.
At the end, with the whole group together, some solutions and sketches could be discussed. You may want to draw out the fact that doubling the sides of the shape has resulted in four times the area, not double the area. With very visual work such as this some children show unexpected talent and others, who usually succeed at once, have blocks. This means that it is specially helpful if later the children have a few minutes to jot down their feelings.
How did you feel if you got stuck?
What or who helped you?
How did you feel when you cracked the problem?
Key questions
Possible extension
Learners could carry on using this sheet which contains further shapes to explore. They could also draw some shapes of their own on squared or isometric dotty paper. Alternatively, or as well, they could go on to making the bigger shapes three times the size of the small ones.
Possible support
Some children would benefit from having a go with differently sized squares and rectangles until they have built up their confidence.You may also like
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