Class 6 Ganita Prakash Chapter 1 Patterns in Mathematics Solutions

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Class 6 Ganita Prakash · Chapter 1
Patterns in Mathematics

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🗝️ Key ideas of this chapter

• Mathematics is the search for patterns and for the reasons why they happen.
• Number sequences: counting, odd, even, triangular (1, 3, 6, 10 …), squares (1, 4, 9 …), cubes (1, 8, 27 …), Virahānka (1, 2, 3, 5, 8 …), powers of 2 and 3.
• 1 + 3 + 5 + … (first n odd numbers) = n × n · 1 + 2 + … + n + … + 2 + 1 = n × n · 1 + 2 + 3 + … = triangular numbers.
• Two neighbouring triangular numbers add to a square · 6 × triangular + 1 = hexagonal · adding hexagonal numbers gives cubes.
• Shape sequences: regular polygons (3, 4, 5 … sides), complete graphs (1, 3, 6, 10 … lines), stacked squares and triangles (squares), Koch snowflake (3, 12, 48 … segments).

📖 Textbook Page 2

📖 Page 2Figure it Out · Q1 (Math Talk)

Can you think of other examples where mathematics helps us in our everyday lives?

1Shopping: we add prices, check the bill and count the change.

2Cooking: we measure cups of rice and water, and double a recipe for more people.

3Time: we read clocks and calendars, and plan how long homework or a journey will take.

4Travel: we read distances, compare bus fares and work out the shortest route.

5Games and sports: we keep scores, count runs, and see patterns in moves.

6Saving money: we add up our savings every week and plan what we can buy.

✅ Answer: Mathematics helps in shopping, cooking, telling time, travelling, sports, saving money, measuring, and many more daily activities.

💡 Tip: This is a ‘Math Talk’ question — any correct example from your own life is a good answer.

📖 Page 2Figure it Out · Q2 (Math Talk)

How has mathematics helped propel (push) humanity forward?

1Science: experiments are measured and their results are studied using numbers and graphs.

2Economy and democracy: banks, budgets, taxes, and counting votes all use mathematics.

3Buildings and bridges: engineers calculate lengths, angles and loads so that structures stay strong and safe.

4Machines: TVs, mobile phones, computers, bicycles, trains, cars and planes are designed using mathematics.

5Calendars and clocks: patterns in the motion of the Sun and Moon helped people measure days, months and years.

6Space: patterns in the motion of planets led to the theory of gravitation and to sending rockets to the Moon and Mars.

✅ Answer: Mathematics is used in science, money, voting, building, machines, calendars, clocks and space travel — it has helped humans make progress in every field.

📖 Textbook Page 3

📖 Page 3Figure it Out · Q1 (Math Talk)

Can you recognise the pattern in each of the sequences in Table 1?

1Look at how each number is made from the one before it (add? multiply?), or from its position (1st, 2nd, 3rd …).

2The table below gives the rule for each sequence.

Sequence	Numbers	Rule (pattern)
All 1’s	1, 1, 1, 1, 1, 1, 1, …	Every number is 1.
Counting numbers	1, 2, 3, 4, 5, 6, 7, …	Add 1 each time.
Odd numbers	1, 3, 5, 7, 9, 11, 13, …	Start at 1 and add 2 each time.
Even numbers	2, 4, 6, 8, 10, 12, 14, …	Start at 2 and add 2 each time.
Triangular numbers	1, 3, 6, 10, 15, 21, 28, …	Add 2, then 3, then 4, … (add one more each time).
Squares	1, 4, 9, 16, 25, 36, 49, …	1 × 1, 2 × 2, 3 × 3, … (a number times itself).
Cubes	1, 8, 27, 64, 125, 216, …	1 × 1 × 1, 2 × 2 × 2, 3 × 3 × 3, …
Virahānka numbers	1, 2, 3, 5, 8, 13, 21, …	Each number = sum of the two numbers before it.
Powers of 2	1, 2, 4, 8, 16, 32, 64, …	Multiply by 2 each time (doubling).
Powers of 3	1, 3, 9, 27, 81, 243, 729, …	Multiply by 3 each time.

✅ Answer: See the rule for each sequence in the table.

📖 Page 3Figure it Out · Q2 (a)

Write the next three numbers of the sequence 1, 1, 1, 1, 1, 1, 1, … (All 1’s) and write its rule.

1Rule: Every number is 1.

2Use the rule on the last number: Every number is 1, so the next three are 1, 1, 1.

✅ Answer: 1, 1, 1. Rule: Every number is 1.

📖 Page 3Figure it Out · Q2 (b)

Write the next three numbers of the sequence 1, 2, 3, 4, 5, 6, 7, … (Counting numbers) and write its rule.

1Rule: Add 1 each time.

2Use the rule on the last number: 7 + 1 = 8, 8 + 1 = 9, 9 + 1 = 10.

✅ Answer: 8, 9, 10. Rule: Add 1 each time.

📖 Page 3Figure it Out · Q2 (c)

Write the next three numbers of the sequence 1, 3, 5, 7, 9, 11, 13, … (Odd numbers) and write its rule.

1Rule: Start at 1 and add 2 each time.

2Use the rule on the last number: 13 + 2 = 15, 15 + 2 = 17, 17 + 2 = 19.

✅ Answer: 15, 17, 19. Rule: Start at 1 and add 2 each time.

📖 Page 3Figure it Out · Q2 (d)

Write the next three numbers of the sequence 2, 4, 6, 8, 10, 12, 14, … (Even numbers) and write its rule.

1Rule: Start at 2 and add 2 each time.

2Use the rule on the last number: 14 + 2 = 16, 16 + 2 = 18, 18 + 2 = 20.

✅ Answer: 16, 18, 20. Rule: Start at 2 and add 2 each time.

📖 Page 3Figure it Out · Q2 (e)

Write the next three numbers of the sequence 1, 3, 6, 10, 15, 21, 28, … (Triangular numbers) and write its rule.

1Rule: Add 2, then 3, then 4, … (add one more each time).

2Use the rule on the last number: 28 + 8 = 36, 36 + 9 = 45, 45 + 10 = 55.

✅ Answer: 36, 45, 55. Rule: Add 2, then 3, then 4, … (add one more each time).

⚠️ Common mistake: Writing 30, 32, 34 by adding 2 each time — the triangular numbers do not grow by the same amount.

💡 Tip: Triangular numbers: the gaps are 2, 3, 4, 5, 6, 7 — so the next gaps are 8, 9, 10.

📖 Page 3Figure it Out · Q2 (f)

Write the next three numbers of the sequence 1, 4, 9, 16, 25, 36, 49, … (Squares) and write its rule.

1Rule: 1 × 1, 2 × 2, 3 × 3, … (a number times itself).

2Use the rule on the last number: 8 × 8 = 64, 9 × 9 = 81, 10 × 10 = 100.

✅ Answer: 64, 81, 100. Rule: 1 × 1, 2 × 2, 3 × 3, … (a number times itself).

📖 Page 3Figure it Out · Q2 (g)

Write the next three numbers of the sequence 1, 8, 27, 64, 125, 216, … (Cubes) and write its rule.

1Rule: 1 × 1 × 1, 2 × 2 × 2, 3 × 3 × 3, …

2Use the rule on the last number: 7 × 7 × 7 = 343, 8 × 8 × 8 = 512, 9 × 9 × 9 = 729.

✅ Answer: 343, 512, 729. Rule: 1 × 1 × 1, 2 × 2 × 2, 3 × 3 × 3, …

⚠️ Common mistake: Writing 6 × 3 = 18 for 6 × 6 × 6. A cube is a number multiplied by itself THREE times.

📖 Page 3Figure it Out · Q2 (h)

Write the next three numbers of the sequence 1, 2, 3, 5, 8, 13, 21, … (Virahānka numbers) and write its rule.

1Rule: Each number = sum of the two numbers before it.

2Use the rule on the last number: 13 + 21 = 34, 21 + 34 = 55, 34 + 55 = 89.

✅ Answer: 34, 55, 89. Rule: Each number = sum of the two numbers before it.

💡 Tip: These numbers are named after the Indian scholar Virahānka, who found them while studying rhythms of poetry.

📖 Page 3Figure it Out · Q2 (i)

Write the next three numbers of the sequence 1, 2, 4, 8, 16, 32, 64, … (Powers of 2) and write its rule.

1Rule: Multiply by 2 each time (doubling).

2Use the rule on the last number: 64 × 2 = 128, 128 × 2 = 256, 256 × 2 = 512.

✅ Answer: 128, 256, 512. Rule: Multiply by 2 each time (doubling).

📖 Page 3Figure it Out · Q2 (j)

Write the next three numbers of the sequence 1, 3, 9, 27, 81, 243, 729, … (Powers of 3) and write its rule.

1Rule: Multiply by 3 each time.

2Use the rule on the last number: 729 × 3 = 2187, 2187 × 3 = 6561, 6561 × 3 = 19683.

✅ Answer: 2187, 6561, 19683. Rule: Multiply by 3 each time.

📖 Textbook Page 4

📖 Page 4Table 2 · Understand the pictures

Look at Table 2 (pictures of number sequences). How does each picture show its number?

Each number is shown as a picture of dots (or small cubes). The picture grows by a rule.

1All 1’s: always one dot.

2Counting numbers: a row with 1, 2, 3, … dots.

3Odd numbers: two rows, the bottom row has one dot more than the top row (1, 3, 5, …).

4Even numbers: two equal rows (2, 4, 6, …).

5Triangular numbers: rows of 1, 2, 3, … dots that make a triangle.

6Squares: equal rows and columns making a square (2 × 2, 3 × 3, …).

7Cubes: small cubes stacked to make a bigger cube (2 × 2 × 2, 3 × 3 × 3, …).

✅ Answer: Each sequence has its own picture rule — rows, triangles, squares or cubes — and the picture grows by the same rule each time.

📖 Textbook Page 5

📖 Page 5Figure it Out · Q1 (Math Talk)

Copy the pictorial representations of the number sequences in Table 2 and draw the next picture for each sequence.

1All 1’s: one dot again.

2Counting numbers: a row of 6 dots.

3Odd numbers: 5 dots on top and 6 dots below = 11.

4Even numbers: two rows of 6 = 12.

5Triangular numbers: rows of 1, 2, 3, 4, 5, 6 = 21.

6Squares: 6 rows of 6 = 36.

7Cubes: a 6 × 6 × 6 cube = 216 small cubes.

The next picture of each sequence is in the dotted gold box.

✅ Answer: Next pictures: 1, 6, 11, 12, 21, 36 and 216 (shown in gold).

📖 Page 5Figure it Out · Q2

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?

1Triangular numbers: that many dots can be arranged in a triangle — rows of 1, 2, 3, 4, … dots.

2Square numbers: that many dots can be arranged in a square with equal rows and columns (1 × 1, 2 × 2, 3 × 3, …).

3Cubes: that many small cubes can be stacked to make a bigger cube (1 × 1 × 1, 2 × 2 × 2, 3 × 3 × 3, …).

The shape gives the name.

✅ Answer: Because their dots (or small cubes) can be arranged exactly in the shape of a triangle, a square and a cube.

📖 Page 5Figure it Out · Q3

36 is both a triangular number and a square number. Make pictures showing this. Try showing some other numbers in different ways.

1Triangle: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36, so 36 dots fit in 8 rows.

2Square: 6 × 6 = 36, so 36 dots fit in 6 rows of 6.

3Other examples: 6 is a triangle (1 + 2 + 3) and a rectangle (2 × 3); 12 is a rectangle 3 × 4 and also 2 × 6; 10 is a triangle (1 + 2 + 3 + 4) and also two rows of 5.

The same 36 dots can be arranged as a triangle and as a square.

Many numbers can be drawn in more than one way.

✅ Answer: 36 = 1 + 2 + … + 8 (triangle) = 6 × 6 (square). The same number can be drawn in different ways.

💡 Tip: 1 is also both triangular and square. The next number that is both is 1225 (= 35 × 35).

📖 Page 5Figure it Out · Q4

What would you call the sequence 1, 7, 19, 37, …? Draw these and find the next number.

What do we call this sequence of numbers?

1They are called hexagonal numbers — the dots make a hexagon (6 sides).

2Find the differences: 7 − 1 = 6, 19 − 7 = 12, 37 − 19 = 18.

3The differences are 6, 12, 18 — they go up by 6. The next difference is 24.

4Next number = 37 + 24 = 61. (In the picture, a new ring of 24 dots goes around the hexagon of 37.)

Hexagonal numbers: each new picture adds a ring of dots around the old hexagon.

✅ Answer: Hexagonal numbers. The next number is 61.

⚠️ Common mistake: Writing 55 by adding 18 again. The gap grows by 6 each time (6, 12, 18, 24).

📖 Page 5Figure it Out · Q5

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

1Powers of 2 (1, 2, 4, 8, 16, …): each picture is two copies of the picture before it, put side by side.

2Powers of 3 (1, 3, 9, 27, 81, …): each picture is three copies of the picture before it.

3Another way for powers of 2: point (1 dot), line segment (2 dots), square (4 corners), cube (8 corners), two joined cubes (16 corners).

Doubling and tripling pictures for powers of 2 and powers of 3.

Another way: point (1), segment (2), square (4), cube (8), and two joined cubes (16). Each shape is two copies of the previous one, joined.

✅ Answer: Make each new picture from 2 copies (powers of 2) or 3 copies (powers of 3) of the previous picture.

📖 Textbook Page 6

📖 Page 6Think · Adding odd numbers

What happens when we keep adding odd numbers: 1, 1 + 3, 1 + 3 + 5, …? Does it happen forever? Why?

11 = 1, 1 + 3 = 4, 1 + 3 + 5 = 9, 1 + 3 + 5 + 7 = 16, 1 + 3 + 5 + 7 + 9 = 25, 1 + 3 + 5 + 7 + 9 + 11 = 36.

2The answers 1, 4, 9, 16, 25, 36 are the square numbers.

3Why: take a square of dots and cut it into L-shapes. The L-shapes hold 1, 3, 5, 7, … dots (see the picture).

4Each new L adds 2 more dots than the last L (one more on each arm), so the L’s are always the odd numbers.

5Such a picture can be drawn for a square of any size, so the pattern goes on forever.

Each L-shape holds the next odd number of dots: 1 + 3 + 5 + 7 + 9 + 11 = 36 = 6 × 6.

✅ Answer: Adding the odd numbers starting from 1 always gives a square number: the first n odd numbers add to n × n.

📖 Textbook Page 7

📖 Page 7Think

Using a similar picture, what is the sum of the first 10 odd numbers?

1The first 10 odd numbers are 1, 3, 5, 7, 9, 11, 13, 15, 17, 19.

2They fill a square of 10 rows and 10 columns.

3Sum = 10 × 10 = 100.

4Check: 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 = 100 ✔

✅ Answer: 100

💡 Tip: Count the odd numbers (here 10), then multiply that count by itself.

📖 Page 7Think

Now imagine a similar picture. What is the sum of the first 100 odd numbers?

1The first 100 odd numbers fill a square of 100 rows and 100 columns.

2Sum = 100 × 100 = 10,000.

3(The 100th odd number is 199, so this is 1 + 3 + 5 + … + 199.)

✅ Answer: 10,000

⚠️ Common mistake: Writing 100 × 2 = 200. The sum is the number of odd numbers times ITSELF: 100 × 100.

📖 Page 7Adding up and down

Look at the pattern 1 = 1, 1 + 2 + 1 = 4, 1 + 2 + 3 + 2 + 1 = 9, … What do you notice? Find 1 + 2 + 3 + 4 + 5 + 6 + 7 + 6 + 5 + 4 + 3 + 2 + 1.

1The answers 1, 4, 9, 16, 25, 36 are square numbers.

2When we go up to n and come back down, the answer is n × n.

3Here we go up to 7, so the answer is 7 × 7 = 49.

4Check: 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28 and 6 + 5 + 4 + 3 + 2 + 1 = 21; 28 + 21 = 49 ✔

✅ Answer: Adding counting numbers up and then down gives square numbers. 1 + 2 + … + 7 + … + 2 + 1 = 49.

📖 Textbook Page 8

📖 Page 8Figure it Out · Q1

Find a pictorial explanation for why adding counting numbers up and down (1, 1 + 2 + 1, 1 + 2 + 3 + 2 + 1, …) gives square numbers.

A square of dots from the book’s question.

1Draw a square of dots, for example 6 rows of 6.

2Read the dots along slanting lines (diagonals), starting from the top-left corner.

3The slanting lines have 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1 dots — up to 6 and back down.

4All these dots make the whole square, so 1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1 = 6 × 6 = 36.

5This works for a square of any size.

Read the square along the slanting lines: 1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1 = 6 × 6 = 36.

✅ Answer: The slanting lines of an n × n square of dots hold 1, 2, …, n, …, 2, 1 dots, so the up-and-down sum = n × n.

📖 Page 8Figure it Out · Q2

Imagining a large version of your picture, find 1 + 2 + 3 + … + 99 + 100 + 99 + … + 3 + 2 + 1.

1We go up to 100 and back down.

2This is the up-and-down sum for a square of 100 rows and 100 columns.

3Value = 100 × 100 = 10,000.

✅ Answer: 10,000

📖 Page 8Figure it Out · Q3

Which sequence do you get when you add the All 1’s sequence up? What do you get when you add the All 1’s up and down?

1Adding up: 1, 1 + 1 = 2, 1 + 1 + 1 = 3, 1 + 1 + 1 + 1 = 4, … → 1, 2, 3, 4, … (counting numbers).

2Up and down: 1; 1 + 1 + 1 = 3; 1 + 1 + 1 + 1 + 1 = 5; 1 + 1 + 1 + 1 + 1 + 1 + 1 = 7; … (going up to the 2nd, 3rd, 4th place and back down).

3This gives 1, 3, 5, 7, … (odd numbers).

✅ Answer: Adding up gives the counting numbers 1, 2, 3, 4, …; adding up and down gives the odd numbers 1, 3, 5, 7, …

⚠️ Common mistake: Thinking ‘up and down’ of All 1’s gives squares. Each step adds two more 1’s (one going up, one coming down), so we get odd numbers.

📖 Page 8Figure it Out · Q4

Which sequence do you get when you start to add the counting numbers up? Can you give a smaller pictorial explanation?

11 = 1, 1 + 2 = 3, 1 + 2 + 3 = 6, 1 + 2 + 3 + 4 = 10, 1 + 2 + 3 + 4 + 5 = 15, …

2These are the triangular numbers.

3Picture: put 1 dot in the first row, 2 dots in the second row, 3 in the third, … The rows make a triangle (or a staircase).

1 + 2 + 3 + 4 + 5 = 15: the rows make a triangle (staircase), so the sums are triangular numbers.

✅ Answer: The triangular numbers 1, 3, 6, 10, 15, 21, …

📖 Page 8Figure it Out · Q5

What happens when you add pairs of consecutive triangular numbers: 1 + 3, 3 + 6, 6 + 10, 10 + 15, …? Which sequence do you get? Why?

11 + 3 = 4, 3 + 6 = 9, 6 + 10 = 16, 10 + 15 = 25, 15 + 21 = 36, …

2These are the square numbers.

3Why: cut a square of dots along a staircase. One part is a triangle with rows 1, 2, 3, 4, 5 (15 dots) and the other part has rows 1, 2, 3, 4 (10 dots).

4So 10 + 15 = 5 × 5 = 25. Two triangles that fit together always make a square.

A 5 × 5 square cut along a staircase gives two triangles: 10 + 15 = 25.

✅ Answer: Square numbers 4, 9, 16, 25, 36, … Two neighbouring triangular numbers fit together to make a square.

📖 Page 8Figure it Out · Q6

Add up powers of 2: 1, 1 + 2, 1 + 2 + 4, 1 + 2 + 4 + 8, … Now add 1 to each. What numbers do you get? Why?

1Sums: 1, 1 + 2 = 3, 1 + 2 + 4 = 7, 1 + 2 + 4 + 8 = 15, 1 + 2 + 4 + 8 + 16 = 31, …

2Add 1 to each: 2, 4, 8, 16, 32, … — these are powers of 2 again!

3Why: 1 + 1 = 2; then 2 + 2 = 4; then 4 + 4 = 8; then 8 + 8 = 16 … Each time the extra 1 makes the total equal to the NEXT power of 2.

4So 1 + 2 + 4 + … + (a power of 2) is always one less than the next power of 2.

The extra 1 joins the 1 to make 2, then 2 + 2 = 4, 4 + 4 = 8, 8 + 8 = 16.

✅ Answer: The sums are 1, 3, 7, 15, 31, … and adding 1 gives 2, 4, 8, 16, 32, … (powers of 2).

📖 Textbook Page 9

📖 Page 9Figure it Out · Q7

What happens when you multiply the triangular numbers by 6 and add 1? Which sequence do you get? Explain with a picture.

16 × 1 + 1 = 7, 6 × 3 + 1 = 19, 6 × 6 + 1 = 37, 6 × 10 + 1 = 61, 6 × 15 + 1 = 91.

27, 19, 37, 61, 91 are hexagonal numbers (and 1 = 6 × 0 + 1 is the first one).

3Picture: a hexagon of dots = 1 centre dot + 6 equal triangles of dots around it.

4Each of the 6 triangles has a triangular number of dots, so the total is 6 × (triangular number) + 1.

37 = 6 × 6 + 1: six triangles of 6 dots each (one colour each) plus the centre dot.

✅ Answer: We get the hexagonal numbers 7, 19, 37, 61, 91, …

📖 Page 9Figure it Out · Q8

What happens when you add up hexagonal numbers: 1, 1 + 7, 1 + 7 + 19, 1 + 7 + 19 + 37, …? Explain using a picture of a cube.

11 = 1, 1 + 7 = 8, 1 + 7 + 19 = 27, 1 + 7 + 19 + 37 = 64.

2These are the cubes 1 × 1 × 1, 2 × 2 × 2, 3 × 3 × 3, 4 × 4 × 4.

3Why: look at a cube from one corner. The three faces you can see make a hexagon shape.

4For a 3 × 3 × 3 cube, those three faces hold 9 + 6 + 4 = 19 small cubes. Remove them and a 2 × 2 × 2 cube (8) is left; its faces hold 7, and then 1 is left.

5So 27 = 19 + 7 + 1. In the same way any cube is built from layers of hexagonal numbers.

The three faces seen from a corner of a 3-cube show 9 + 6 + 4 = 19 small cubes (a hexagonal number). 27 − 8 = 19.

Look at a cube from one corner: the three faces you see show a hexagon of small cubes. Peel off that outer layer (7, 19, 37 …) and a smaller cube is left.

✅ Answer: We get the cubes 1, 8, 27, 64, 125, …

📖 Page 9Figure it Out · Q9

Find your own patterns or relations in and among the sequences in Table 1. Can you explain why they happen?

1Even = odd + 1: 2 = 1 + 1, 4 = 3 + 1, 6 = 5 + 1 … (each even number is the next number after an odd number).

2Triangular numbers × 2 = rectangles: 1 × 2 = 2, 3 × 2 = 6 = 2 × 3, 6 × 2 = 12 = 3 × 4, 10 × 2 = 20 = 4 × 5 (two triangles make a rectangle).

38 × triangular number + 1 = odd squares: 8 × 1 + 1 = 9, 8 × 3 + 1 = 25, 8 × 6 + 1 = 49, 8 × 10 + 1 = 81.

4Adding cubes gives squares of triangular numbers: 1 = 1 × 1, 1 + 8 = 9 = 3 × 3, 1 + 8 + 27 = 36 = 6 × 6, 1 + 8 + 27 + 64 = 100 = 10 × 10.

5Squares differ by odd numbers: 4 − 1 = 3, 9 − 4 = 5, 16 − 9 = 7 (the L-shapes!).

6Virahānka numbers: every third one is even — 2, 8, 34, 144 …

✅ Answer: Many answers are possible — for example, 1 + 8 + 27 + 64 = 100 = 10 × 10 and 8 × (triangular number) + 1 = an odd square.

💡 Tip: Always check a pattern with more numbers before you believe it, and then try to explain WHY it happens.

📖 Textbook Page 11

📖 Page 11Figure it Out · Q1 (Math Talk)

Can you recognise the pattern in each of the shape sequences in Table 3?

Regular polygons: 3, 4, 5, 6, … equal sides.

Complete graphs: every dot is joined to every other dot.

Stacked squares: n rows of n little squares.

Stacked triangles: the rows have 1, 3, 5, 7, … little triangles.

Koch snowflake: every segment is replaced by 4 smaller segments each time.

1Regular polygons: each shape has one more side than the one before (3, 4, 5, … sides). All sides are equal and all corners look the same.

2Complete graphs: K2, K3, K4, … have 2, 3, 4, … dots, and every dot is joined to every other dot by a line.

3Stacked squares: 1 × 1, 2 × 2, 3 × 3, … little squares — one more row and one more column each time.

4Stacked triangles: a big triangle made of little triangles; each new shape adds one more row at the bottom.

5Koch snowflake: start with a triangle; each time, every straight segment is replaced by a ‘speed bump’ of 4 smaller segments.

✅ Answer: Each shape sequence grows by a fixed rule — see the rule for each in the steps.

📖 Page 11Figure it Out · Q2 (Math Talk)

Redraw each sequence in Table 3. Can you draw the next shape in each sequence? Describe the rule for each.

1Regular polygons: the next shape is a regular 11-sided polygon (hendecagon).

2Complete graphs: next is K7 — 7 dots, each joined to the other 6, so 21 lines.

3Stacked squares: next is a 6 × 6 square made of 36 little squares.

4Stacked triangles: next has 6 rows (1, 3, 5, 7, 9, 11 little triangles) = 36 little triangles.

5Koch snowflake: the next shape is hard to draw because the segments become very tiny — it has 768 segments!

Regular polygons: 3, 4, 5, 6, … equal sides.

Complete graphs: every dot is joined to every other dot.

Stacked squares: n rows of n little squares.

Stacked triangles: the rows have 1, 3, 5, 7, … little triangles.

Koch snowflake: every segment is replaced by 4 smaller segments each time.

✅ Answer: The next shapes are an 11-gon, K7 (21 lines), a 6 × 6 square (36), a 6-row triangle (36) and a Koch snowflake with 768 segments.

📖 Page 11Figure it Out · Q1 (1.6)

Count the number of sides in each shape of the Regular Polygons. Which number sequence do you get? What about the number of corners? Why?

1Sides: triangle 3, quadrilateral 4, pentagon 5, hexagon 6, heptagon 7, octagon 8, nonagon 9, decagon 10.

2So the sides give the counting numbers starting from 3: 3, 4, 5, 6, 7, 8, 9, 10, …

3Corners: also 3, 4, 5, 6, … — the same sequence.

4Why: walk around the shape. After every side you reach a corner, and every corner is followed by a side. Going once around, we meet them one after the other, so their numbers are equal.

Regular polygons: 3, 4, 5, 6, … equal sides (number of sides written inside).

✅ Answer: Sides: 3, 4, 5, 6, … (counting numbers from 3). Corners: the same sequence, because each side ends at a corner and sides and corners take turns.

📖 Page 11Figure it Out · Q2 (1.6)

Count the number of lines in each shape of the Complete Graphs. Which number sequence do you get? Can you explain why?

1K2: 1 line, K3: 3 lines, K4: 6 lines, K5: 10 lines, K6: 15 lines.

21, 3, 6, 10, 15 are the triangular numbers.

3Why: when we add a new dot, it must be joined to every old dot.

4K3 adds 2 lines to K2, K4 adds 3 lines, K5 adds 4 lines, K6 adds 5 lines.

5So the lines are 1, 1 + 2, 1 + 2 + 3, 1 + 2 + 3 + 4, … — adding counting numbers gives triangular numbers.

Complete graphs: every dot is joined to every other dot.

✅ Answer: 1, 3, 6, 10, 15, … — the triangular numbers.

⚠️ Common mistake: Counting each line twice (once from each end). One line joins TWO dots — count it only once.

📖 Textbook Page 12

📖 Page 12Figure it Out · Q3

How many little squares are there in each shape of the Stacked Squares? Which number sequence does this give? Why?

Stacked squares: n rows of n little squares.

1The shapes have 1, 4, 9, 16, 25 little squares.

2This is the sequence of square numbers.

3Why: the n-th shape has n rows with n little squares in each row, so it has n × n little squares.

✅ Answer: 1, 4, 9, 16, 25, … — the square numbers.

📖 Page 12Figure it Out · Q4

How many little triangles are there in each shape of the Stacked Triangles? Which number sequence does this give? Why? (Hint: count the triangles in each row.)

Stacked triangles: the rows have 1, 3, 5, 7, … little triangles.

1Rows from the top have 1, 3, 5, 7, … little triangles (pointing up and down).

2Shape 1: 1; shape 2: 1 + 3 = 4; shape 3: 1 + 3 + 5 = 9; shape 4: 1 + 3 + 5 + 7 = 16; shape 5: 25.

3These are the square numbers.

4Why: each row adds the next odd number, and adding odd numbers gives square numbers.

Rows have 1, 3, 5, 7 little triangles: 1 + 3 + 5 + 7 = 16 = 4 × 4 (gold point up, blue point down).

✅ Answer: 1, 4, 9, 16, 25, … — the square numbers.

⚠️ Common mistake: Counting only the triangles that point up (1, 3, 6, 10). Count the upside-down ones too.

📖 Page 12Figure it Out · Q5

In the Koch Snowflake, each line segment is replaced by a ‘speed bump’. How many line segments are there in each shape? What is the number sequence?

Each straight segment becomes a ‘speed bump’ of 4 smaller segments.

1Shape 1 (triangle): 3 segments.

2Each segment becomes 4 smaller segments, so we multiply by 4 each time.

3Shape 2: 3 × 4 = 12; shape 3: 12 × 4 = 48; shape 4: 48 × 4 = 192; shape 5: 192 × 4 = 768.

4The sequence is 3, 12, 48, 192, 768, … = 3 times the powers of 4 (1, 4, 16, 64, 256, …).

Koch snowflake: every segment is replaced by 4 smaller segments each time.

✅ Answer: 3, 12, 48, 192, 768, … (3 × powers of 4).

🌍 Where do we see patterns?

• Nature: petals of flowers, honeycomb hexagons, spirals of a sunflower (Virahānka numbers!).
• Daily life: calendars, tiles on the floor, rangoli and kolam designs, bricks in a wall.
• Technology: computers count in powers of 2 (2, 4, 8, 16 … GB).

🚀 Link to higher classes

• Class 7: Number Play and algebra — writing patterns using letters (2n, n × n).
• Class 8: A Square and A Cube — squares, cubes and their patterns in detail.

⬆ Top 🏠 Class 6 Hub

🔑 keytoenjoylearningmaths.blogspot.com · Solutions written in our own words, based on NCERT Ganita Prakash Class 6 (Grade 6)