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Chapter 1: Patterns In Mathematics

    What is Mathematics?

    Patterns in Numbers

    Visualising Number Sequences

    Relations among Number Sequences

    Patterns in Shapes

    Relation to Number Sequences

    Summary

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Chapter 1: Patterns In Mathematics > Visualising Number Sequences

Visualising Number Sequences

Many number sequences can be visualised using pictures. Visualising mathematical objects through pictures or diagrams can be a very fruitful way to understand mathematical patterns and concepts. Let us represent the first seven sequences in Table 1 using the following pictures.

n1n2n3n4n5Sequence Type
All 1’s
Counting numbers
Odd numbers
Even numbers
Triangular numbers
Squares
Cubes

🎯 Sequence Master 🎯

Display Type:

Welcome to Sequence Master!

Select a sequence type above to see how it works. Watch the patterns appear with animation and learn visually!

Select a sequence to begin...
Loading question...

🔺 Triangle Arrangement Challenge

Drag the 10 circles to form a triangular pattern
💡 Hint: Arrange them like bowling pins (1 on top, then 2, then 3, then 4)
🎯 Keep arranging...
🎉
Excellent Work!
You've mastered the triangle arrangement!
Great spatial reasoning skills! 🌟

Figure it Out

(1) Place the picture for each sequence in the correct box.

Cubes
Odd Numbers
Counting Numbers
Squares
Triangular Numbers
All 1's
Even Numbers

Visualising Square Numbers

12

square-1

22

square-2

32

square-3

square-4

square-5

square-6

Visualising Triangular Numbers

1

triangle-1

3

triangle-2

6

triangle-3

10

triangle-4

15

triangle-5

21

triangle-6
x=

If we use polygons with 6 sides, we get the sequence of Hexagonal numbers.

(2) Why are 1, 3, 6, 10, 15, … called triangular numbers? Why are 1, 4, 9, 16, 25, … called square numbers or squares? Why are 1, 8, 27, 64, 125, … called cubes?

Instruction

Triangular numbers are called so because they can be arranged in the shape of an triangle. Each number represents the number of dots that can form a triangle when arranged in rows.
Thus, 1 (a single dot), (a triangle with rows: 1 + ), (a triangle with rows: 1 + + ), (a triangle with rows: 1 + 2 + + ), and so on.
Square numbers are called so because they can be arranged in the shape of a .
= 1 × 1 (a 1×1 square), = 2 × 2 (a 2×2 square), = 3 × 3 (a 3×3 square), and so on.
Cube numbers are called so because they can be arranged to form a in dimensions.
= 1 × 1 × 1 (a 1×1×1 cube), = 2 × 2 × 2 (a 2×2×2 cube), = 3 × 3 × 3 (a 3×3×3 cube), and so on.

(3) You will have noticed that 36 is both a triangular number and a square number! That is, 36 dots can be arranged perfectly both in a triangle and in a square. Make pictures in your notebook illustrating this!

This shows that the same number can be represented differently, and play different roles, depending on the context. Try representing some other numbers pictorially in different ways!

Instruction

36 when represented as a square number has number of rows with each row having number of dots.
36 when represented as a triangular number has number of rows with the number of dots for each consecutive row being: 1, , , , , , , .

(4) What would you call the following sequence of numbers?

That’s right, they are called hexagonal numbers! Draw these in your notebook. What is the next number in the sequence?

(5) Can you think of pictorial ways to visualise the sequence of Powers of 2? Powers of 3?

Here is one possible way of thinking about Powers of 2: