A professor of mine once said, “Don’t memorise formulas, memorise arguments.” An argument in math is like a story: it takes us from an unsuspecting beginning into a journey of challenge, surprise, and insight. What we take home at the end is, perhaps, a formula. The meaning of the formula is that it reminds us of the argument, and allows us to take what we learned from the journey and apply it to new situations.
Why should we memorise formulas for the areas of rectangles, triangles, trapezoids, and parallelograms when the same fundamental argument is behind them all? Instead of memorising, take the journey.
This week’s puzzles are all about area, but from an angle you may not have seen before. Formulas will get you only so far. You’ll need arguments.
Puzzle 1
What area of each of the shapes below is shaded?
Puzzle 2
Let’s say the red square has area 1. In other words, we’ll use the red square as kind of ruler and measure the other squares against it.
Part 1. Show that the small blue square has area 2.
Part 2. Find the are of the purple square.
Part 3. Of the following numbers, one of them cannot be the area of a square with all its corners on a grid, no matter how large the grid is. Which one is it?
17, 18, 19, 20
Click here to read the answer.
Research: Is there any kind of pattern in the possible areas a grid square can have?
Puzzle 3
What are the areas of the three triangles below?
I’ll just leave you with the answers for this one. Try and work out how you get there...
The areas of the triangles are, from least to greatest:
9/8, 9/4, and 27/8.
Or, to see the pattern more clearly: 9/8, 18/8, 27/8. That’s 1 1/8, 2 2/8 and 3 3/8.