Problem Set 53 – Square and Triangular NumbersClass 5 Maths · Chapter 15 Patterns · Problem Set 53 – Maharashtra Board

Summary: Problem Set 53 – Square and Triangular Numbers

If a number of dots can be arranged in a square — the same number in every row and column — it is a square number: 2 × 2 = 4, 3 × 3 = 9, 4 × 4 = 16 …

A square number is a number multiplied by itself. 50 is not one, because 7 × 7 = 49 and 8 × 8 = 64 jump right over it.

Adding counting numbers from 1 gives 1 + 2 = 3, 1 + 2 + 3 = 6, then 10, 15, 21… These dots form triangles with equal sides, so they are triangular numbers.

To get the next triangular number, add the next counting number: 21 + 7 = 28. Some numbers, like 36, are both square and triangular.

Floor tiles make patterns that cover a surface with no gaps and no overlaps.

Key points to remember

• Square number = a number × itself
• Square numbers: 4, 9, 16, 25, 36, 49, 64, 81, 100
• Triangular number = 1 + 2 + 3 + … up to some number
• Triangular numbers: 3, 6, 10, 15, 21, 28, 36, 45, 55
• Next triangular = previous + next counting number
• 36 is both square (6 × 6) and triangular (1 + 2 + … + 8)
• Square and triangular numbers never end
• Tiles fit with no gaps and no overlaps

Problem Set 53 – Square and Triangular Numbers – Notes

1. Square numbers

• Arrange dots in rows and columns with the same number in each: 5 rows of 5 = 25 dots.
• So square numbers come from multiplying a number by itself: 4, 9, 16, 25, 36, 49, 64, 81, 100.
• In this chapter 4 (2 × 2) is counted as the first square number, 9 as the second, and so on.

2. Is it a square number?

• Try multiplying numbers by themselves until you reach or pass the number.
• 60: 7 × 7 = 49, 8 × 8 = 64. 60 lies between, so it is NOT a square number.
• 81: 9 × 9 = 81, so it IS a square number.

3. Triangular numbers

• Add counting numbers starting from 1: 1 + 2 = 3, 1 + 2 + 3 = 6, 1 + 2 + 3 + 4 = 10.
• Draw them as dots: 1 dot on top, 2 below it, 3 below those… they make triangles with equal sides.
• Here 3 is the first triangular number, 6 the second, 10 the third, then 15, 21, 28, 36, 45, 55, 66.

4. Patterns between them

• Square numbers grow by odd numbers: 4 + 5 = 9, 9 + 7 = 16, 16 + 9 = 25.
• Triangular numbers grow by the next counting number: 10 + 5 = 15, 15 + 6 = 21.
• 36 is in both lists. You can see these numbers in stacked pipes, rows of bowls and a chessboard (8 × 8 = 64 squares).

5. Patterns in floor tiles

• Tiles are laid so that there is no gap between them and none lies on top of another.
• Squares, rectangles, triangles with equal sides and six-sided (hexagon) tiles can all cover a floor this way.
• Colouring half the tiles differently makes beautiful borders for saris, shawls and rangolis.

Formulas at a glance

Key terms

Problem Set 53 – Square and Triangular Numbers – Flashcards

Read the question, answer it in your head, then tap the card to check. 10 cards.

Problem Set 53 – Square and Triangular Numbers – MCQ Quiz

10 questions. Tap an option — you will see at once if it is right and why.

How to study Problem Set 53 – Square and Triangular Numbers

⏱ About 25 minutes

1. Draw (10 min)
   Draw dots for 4, 9, 16 as squares and 3, 6, 10 as triangles.
2. List (10 min)
   Write square numbers up to 100 and triangular numbers up to 66 and look for the growth patterns.
3. Practise (5 min)
   Do the 4 'Try it yourself' questions, take the quiz and play 🎮 Play.