1
Hidden rectangles in a puzzle
Discover Mathematics P3: Term 1, Lessons 31–40
GoalCount every rectangle in a puzzle by counting by size, so that none is missed.
New words
rectangle · a shape with 4 sides and 4 square cornerssquare · a rectangle with 4 equal sidescount by size · count the small ones first, then the bigger ones
Explanation
A puzzle can hide many rectangles, because small rectangles join to make bigger ones. A square is also a rectangle. To count them all without missing any, count by size. Look at a strip of 3 boxes. First count the rectangles made of 1 box: there are 3. Then count the rectangles made of 2 boxes side by side: there are 2. Then count the rectangle made of all 3 boxes: there is 1. The total is 3 + 2 + 1 = 6. Colouring each rectangle you count, or writing a number in it, helps you keep track.
Worked examples
Here is a square cut into 4 small squares. Count by size: 4 small squares, 2 rectangles lying down (the top row and the bottom row), 2 rectangles standing up (the left column and the right column) and 1 big square. The total is 4 + 2 + 2 + 1 = 9.
A strip has 4 boxes. Rectangles of 1 box: 4. Rectangles of 2 boxes: 3. Rectangles of 3 boxes: 2. The rectangle of 4 boxes: 1. The total is 4 + 3 + 2 + 1 = 10.
This grid has 2 rows and 3 columns. Count by size, using rows × columns. 1 × 1: 6. 1 × 2: 4. 1 × 3: 2. 2 × 1: 3. 2 × 2: 2. 2 × 3: 1. The total is 6 + 4 + 2 + 3 + 2 + 1 = 18 rectangles.
Class activity
Rectangle hunt. Draw a strip of 5 boxes on squared paper. Colour the rectangles of 1 box red, those of 2 boxes blue and so on, and write how many you find of each size. Add them. Then try a strip of 6 boxes and look for a pattern in the totals.
Practice
1
Count all the rectangles in this strip.
5 + 4 + 3 + 2 + 1 = 15 rectangles.
2
How many squares of any size are in a 2 by 2 grid of squares?
There are 4 small squares and 1 big square, so 5 squares.
3
Mona counted the rectangles in a strip of 3 boxes and got 5. What did she probably miss? What is the right total?
She probably missed the long rectangle made of all 3 boxes. The right total is 3 + 2 + 1 = 6.
4
A grid has 3 rows and 2 columns. How many rectangles does it have? Think of the grid with 2 rows and 3 columns.
18 rectangles. Turning the grid on its side does not change the number of rectangles.
5
Write the addition that counts all the rectangles in a strip of 6 boxes.
6 + 5 + 4 + 3 + 2 + 1 = 21 rectangles.