Diagonals of Rectangles
Width
Height
Dimensions
6 × 4
Squares
8
How many squares will the diagonal pass through for a rectangle?
Recorded Results
Getting Started
1
Try a few rectangles and record each result. How many squares does the diagonal pass through for
3 × 2? For 5 × 3? For 7 × 4?2
What happens with squares (same width and height)? Try
1×1, 2×2, 3×3, 4×4. Can you spot the pattern?3
Keep the width fixed at 6 and change the height from 1 to 10. Record each result. What pattern do you notice?
Spotting Patterns
4
Keep the difference between width and height the same. Try
3×1, 4×2, 5×3, 6×4 (difference is always 2). What do you notice about the number of squares?5
What happens when the height is double the width? Try
2×4, 3×6, 4×8, 5×10.6
What about when the height is triple the width? Try
2×6, 3×9, 4×12. Compare this to the doubling results.7
Try rectangles where width and height are consecutive numbers:
1×2, 2×3, 3×4, 4×5. Can you predict the next one?Common Factors
8
Compare
2×3 and 4×6. The second rectangle is double the first. What happens to the number of squares? Now try 6×9 (triple). What’s going on?
Hint: try calculating the HCF (highest common factor) of the width and height for each rectangle.
9
Look at all the results where the number of squares doesn’t seem to fit a simple pattern. What do the width and height of these rectangles have in common?
10
Try
6×4 and 3×2. The diagonal passes through 8 squares and 4 squares. What is the HCF of 6 and 4? Of 3 and 2? Can you see a connection?11
Look at the diagram carefully when the HCF is greater than 1. Where does the diagonal cross a grid intersection point? What has this got to do with common factors?
Finding the Rule
12
For rectangles where the HCF of width and height is 1 (called “coprime”), can you find a formula for the number of squares?
Hint: look at how the number of squares relates to width + height.
13
Now generalise: can you come up with a formula that works for ALL rectangles, including ones where the width and height share a common factor?
Hint: the formula involves width, height, and their HCF.
14
Test your formula on
12×8. Does it predict the correct answer? Try 15×10.15
Can you explain why your formula works? Think about when the diagonal crosses a vertical grid line, a horizontal grid line, or both at the same time.
Extension
16
What is the maximum number of squares a diagonal can pass through for a rectangle with perimeter 20?
17
For which rectangles does the diagonal pass through exactly half the total number of unit squares? Is this ever possible?
18
Now imagine a 3D cuboid — a diagonal going from one corner to the opposite corner through a block of unit cubes. How many cubes would the diagonal pass through for a
2×3×4 cuboid? Can you extend your formula to 3D?
Hint: in 3D, you need to consider HCF of pairs AND the HCF of all three.
How to Use
The Investigation
Draw a rectangle on squared paper with whole-number width and height. Draw a diagonal from one corner to the opposite corner. How many squares does the diagonal pass through?
Using This Tool
1. Set the width and height using the sliders or number inputs.
2. The highlighted squares are the ones the diagonal passes through.
3. Click Record to save each result to the table below.
4. Use Hide Count, Hide Line, and Hide Shading to challenge students to predict before revealing.
5. Try Predict mode for a random rectangle challenge!
2. The highlighted squares are the ones the diagonal passes through.
3. Click Record to save each result to the table below.
4. Use Hide Count, Hide Line, and Hide Shading to challenge students to predict before revealing.
5. Try Predict mode for a random rectangle challenge!
The Big Question
Can you find a formula that predicts the number of squares for any rectangle? There is one — and it’s beautifully simple.