Example 1
How many squares of any size are in a 3 by 3 grid?
- 1 by 1: 9.
- 2 by 2: 4 (the top-left corner of one can be in 2 rows and 2 columns).
- 3 by 3: 1. Total 9 + 4 + 1 = 14.
Answer: 14
🦘 Kangaroo Grade 3 · session 6 of 24 · 60 minutes · ⭐ Plus
Counting squares, triangles and shapes
How many squares are on a chessboard? Not 64! Count shapes by size, never one by one, and the big questions become short sums.
A complete lesson · 60 minutes
Teach it, see it, practise it, all on this page.
Open this page with your learner and work down it, or press Teach this lesson to go through it one step at a time on a big screen. Every section has something to do.
- Warm-up4 min
- Squares8 min
- Rectangles9 min
- Triangles7 min
- Pairs7 min
- Activity 17 min
- Activity 27 min
- Activity 36 min
- Quiz4 min
- Set the project1 min
By the end you can
- Count all the squares in a grid size by size
- Count rectangles and segments in a strip
- Count triangles in fan figures
- Count handshakes and lines without missing any
Before you start
- Adding small numbers quickly
- Square numbers (1, 4, 9, 16, …) help
Have ready
- Squared paper
- Coloured pencils
Grade 3 · ages 8 to 9 · contest level 3–4
Warm-up · think first
How many squares are on a chessboard? Most people say 64. Is that all of them?
Show the answer
No. There are 204 squares: 64 small ones and also 2 by 2, 3 by 3, … up to the whole 8 by 8 board. You will be able to count them by the end of the lesson.
Teacher tip: Take a few guesses and write them down. Come back to them at the end of part 1.
Part 1 of 4 · learn, see, practise
Squares in a grid
Count squares one size at a time. In a 3 by 3 grid there are 9 small squares, 4 squares of size 2 by 2 and 1 big 3 by 3 square: 14 altogether.
In an n by n grid there are n × n small squares, then (n − 1) × (n − 1) squares of the next size, and so on down to 1.
Example 2
How many squares are in a grid 2 squares tall and 4 squares wide?
- 1 by 1: 8.
- 2 by 2: 3 (they fit in 3 positions along the row).
- 8 + 3 = 11.
Answer: 11
Square finder
Choose the grid size and the square size. Every square of that size is outlined in turn and counted.
- Count all the squares in a 4 by 4 grid.
- Show the 2 by 2 squares in a 5 by 5 grid.
- Find the grid that has exactly 55 squares.
Practise it
Start with the warm-up and go as far as you can. The stretch questions are meant to be hard.
Warm-up
Warm-up
Core
Core
Stretch 🔥
Stretch 🔥
Answers (for the teacher)
- 16
- 5
- 9
- 30
- 26
- 204
⭐ Part of Pure Faith Plus
The rest of this lesson is in Plus
Already have Plus? Log in and the whole lesson opens.
You have seen the plan, the warm-up and part 1. With Plus you also get:
- Part 2: Rectangles in a strip, with practice at three levels
- Part 3: Triangles in figures, with practice at three levels
- Part 4: Count without missing any, with practice at three levels
- Class activity 1: Grid square hunt
- Class activity 2: Rectangle count
- Class activity 3: Tournament planner
- A 8-question quiz
- After-class project: Shape hunt poster
- Homework and a printable practice sheet with answers
- Teach mode and the minute-by-minute teacher's plan
The first 2 lessons of every course, every game and all maths practice stay free.
That is ₹208 a month for every lesson of every course. Prefer a teacher? Ask about live classes.
FAQ
Counting squares, triangles and shapes: questions
How many squares are on a chessboard?
204: 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1, counting every size from 1 by 1 to 8 by 8.
How do you count triangles in a figure?
Find triangles that share a corner and count the pieces of the opposite side like a strip: 3 pieces give 3 + 2 + 1 = 6.
Why are counting questions hard in Math Kangaroo?
It is easy to miss the bigger shapes. Counting one size at a time, and writing each size down, fixes that.
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