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We mathematicians often like to talk about introducing proof early. When discussing these ideas with academics, they have sometimes countered with the idea that mathematical proof isn’t so important for beginning mathematicians– but the principles underlying it – assumptions, reasoning, and justifying – are.
One major strength of geometry and shape is that it offers an accessible way in to abstract reasoning for even young students. How? Here’s an example:
Can you draw a shape in each cell in the table? Does it matter what the central shape is? Are there some shapes for which you can/cannot complete the whole table? What about just some cells in the table? Convince yourself….
Complete the table when your starting shape in the central cell is each of the following:
What if you could place one of the cards in any cell? Can you complete the challenge then? Convince me!
This can be simplified for younger students by scaffolding (e.g. start with the central cell and ask them to spend some time trying to find a shape with the same numerical area as perimeter) or by modifying (e.g. using pegboards to explore making shapes with different areas and perimeters before they start).
It can also be extended by asking pupils to draw their own Carroll diagram with the shapes in and asking another pupil to write the labels in – or asking them to draw graphs instead of shapes. How will they define the perimeter and the area in this case?
SOMETHING FOR YOU TO TRY:
KS1: Put the four shapes above into two groups. How have you decided which ones go together? Can you group them again, a different way?
KS2: Print out the shapes and use some string to order the four shapes above by perimeter. Then draw another shape that has a bigger perimeter than all of them. How can you convince me?
KS3: Complete the activity as shown above. Present your findings to the rest of the class.
KS4: Using squared paper, or using a dynamic geometry package, draw nine different triangles that fit into the nine cells of the Carroll diagram above. Can it be done? Explain your reasoning.
KS5: The perimeter of a given triangle, whose sides are in the ratio of 3:4:5, is greater by 4m. than that of a given square. The area of the square is greater by 25sq.m. than that of the triangle. Find the lengths of the sides of the square and of the triangle. (adapted from Underground Maths)