AP 6th Class Maths Textbook Solutions – 1. PATTERNS IN MATHEMATICS (2026-27)

1 PATTERNS IN MATHEMATICS

Figure it Out - 1.1 (Page 6)

• 1. Can you think of other examples where mathematics helps us in our everyday lives?
  Mathematics helps us in calculating money and change while shopping, measuring ingredients for cooking recipes, managing daily time and schedules using clocks, calculating travel distance and speed, and measuring height and weight.
• 2. How has mathematics helped propel humanity forward? (You might think of examples involving: carrying out scientific experiments; running our economy and democracy; building bridges, houses or other complex structures; making TVs, mobile phones, computers, bicycles, trains, cars, planes, calendars, clocks, etc.)
  Mathematics is the foundational language of science and technology. It has propelled humanity forward by enabling:
  1. Engineering and Construction: Designing safe bridges, dams, roads, and skyscrapers through geometry and structural calculations.
  2. Technology and Computing: Creating microchips, computers, smartphones, and the internet using binary logic and algorithms.
  3. Space Exploration and Transportation: Calculating satellite trajectories, launching space rockets to the Moon and Mars, and designing airplanes and high-speed trains.
  4. Economy and Governance: Managing national budgets, banking systems, demographic census data, and statistical analysis for public policy.

Figure it Out - 1.2 (Page 10)

• 1. Can you recognise the pattern in each of the sequences in Table 1?
  Yes, each sequence follows a specific rule such as constant terms, adding fixed differences, adding consecutive counting numbers, squaring, cubing, adding the previous two terms, or repeatedly multiplying by a constant number.
• 2. Rewrite each sequence of Table 1 in your notebook, along with the next three numbers in each sequence. After each sequence, write in your own words what is the rule for forming the numbers in the sequence.
  1. All 1's: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
  Rule: Every term is the number 1.
  
  2. Counting numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
  Rule: Start with 1 and add 1 to get each subsequent number.
  
  3. Odd numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19
  Rule: Start with 1 and add 2 to the previous number.
  
  4. Even numbers: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20
  Rule: Start with 2 and add 2 to the previous number.
  
  5. Triangular numbers: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55
  Rule: Start with 1, then add consecutive counting numbers: +2, +3, +4, +5, +6, +7, +8, +9, +10, ...
  
  6. Squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100
  Rule: Multiply each counting number by itself ($n \times n$).
  
  7. Cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729
  Rule: Multiply each counting number by itself three times ($n \times n \times n$).
  
  8. Virahanka / Fibonacci numbers: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89
  Rule: Start with 1, 2; each subsequent term is the sum of the two preceding terms.
  
  9. Powers of 2: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512
  Rule: Start with 1 and multiply by 2 to get the next term.
  
  10. Powers of 3: 1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683
  Rule: Start with 1 and multiply by 3 to get the next term.

Figure it Out - 1.3 (Page 12)

• 1. Copy the pictorial representations of the number sequences in Table 2 in your notebook, and draw the next picture for each sequence!
  - All 1's: 1 dot.
  - Counting numbers: A row of 6 dots.
  - Odd numbers: Two rows of dots with 1 extra dot (total 11 dots arranged as an L-shape/pairs).
  - Even numbers: Two parallel rows of 6 dots each (total 12 dots).
  - Triangular numbers: A triangle with 6 rows containing 1 + 2 + 3 + 4 + 5 + 6 = 21 dots.
  - Squares: A square grid of 6 rows and 6 columns = 36 dots.
  - Cubes: A 3D cube model of dimensions $6 \times 6 \times 6$ = 216 small unit cubes.
• 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?
  - Triangular numbers: Because these numbers of dots can be arranged in the shape of an equilateral triangle.
  - Square numbers: Because these numbers of dots can be arranged in a perfect square grid with equal rows and columns.
  - Cubes: Because these numbers of unit cubes can be stacked to form a perfect solid three-dimensional cube with equal length, breadth, and height.
• 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! Try representing some other numbers pictorially in different ways!
  - 36 as a Triangular number: Form a triangle of 8 rows: $1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36$ dots.
  - 36 as a Square number: Form a square grid of 6 rows and 6 columns: $6 \times 6 = 36$ dots.
  - Other examples: The number 6 can be arranged as a triangle ($1 + 2 + 3 = 6$) and as a rectangle ($2 \times 3 = 6$). The number 1 is simultaneously a triangular number, square number, and cube number.
• 4. What would you call the following sequence of numbers? 1, 7, 19, 37 ... What is the next number in the sequence?
  - They are called hexagonal numbers (or centered hexagonal numbers).
  - Differences:
  $7 - 1 = 6 = 6 \times 1$
  $19 - 7 = 12 = 6 \times 2$
  $37 - 19 = 18 = 6 \times 3$
  - Next difference is $6 \times 4 = 24$.
  - Next number: $37 + 24 =$ 61.
• 5. Can you think of pictorial ways to visualise the sequence of Powers of 2? Powers of 3?
  - Powers of 2: Can be visualised by geometric dimensions: Point (0D, 1 dot), Line segment with 2 endpoints (1D, 2 vertices), Square (2D, 4 vertices), Cube (3D, 8 vertices), Hypercube (4D, 16 vertices); or by repeated binary tree branching where each branch splits into 2.
  - Powers of 3: Can be visualised by a tree branching diagram where every node splits into 3 branches ($1 \rightarrow 3 \rightarrow 9 \rightarrow 27 \rightarrow 81$), or by dividing squares/cubes into 3 equal parts repeatedly.

Math Talk (Page 16)

• By drawing a similar picture, can you say what is the sum of the first 10 odd numbers?
  The sum of the first 10 odd numbers is equal to $10^2$ (a $10 \times 10$ square grid of dots):
  $1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 = 10 \times 10 =$ 100.
• Now by imagining a similar picture, or by drawing it partially, as needed, can you say what is the sum of the first 100 odd numbers?
  The sum of the first 100 odd numbers equals $100^2$ (a $100 \times 100$ square grid of dots):
  $\text{Sum} = 100 \times 100 =$ 10,000.

Figure it Out - 1.4 (Page 18)

• 1. Can you find a similar pictorial explanation for why adding counting numbers up and down, i.e., 1, 1+2+1, 1+2+3+2+1, ... gives square numbers?
  Consider an $n \times n$ square dot grid rotated diagonally. Counting dots along the diagonal columns gives the sum $1 + 2 + 3 + \dots + n + \dots + 3 + 2 + 1$. Since the entire grid contains $n \times n = n^2$ dots, adding counting numbers up and down always equals a square number.
• 2. By imagining a large version of your picture, or drawing it partially, as needed, can you see what will be the value of 1 + 2 + 3 + ... + 99 + 100 + 99 + ... + 3 + 2 + 1?
  Here the peak number is 100. According to the pattern:
  $\text{Sum} = 100^2 = 100 \times 100 =$ 10,000.
• 3. Which sequence do you get when you start to add the All 1's sequence up? What sequence do you get when you add the All 1's sequence up and down?
  - Adding up: $1, 1+1=2, 1+1+1=3, 1+1+1+1=4, \dots$ gives the Counting numbers (Natural numbers) sequence: $1, 2, 3, 4, 5, \dots$
  - Adding up and down: $1, 1+1+1=3, 1+1+1+1+1=5, \dots$ gives the Odd numbers sequence: $1, 3, 5, 7, 9, \dots$
• 4. Which sequence do you get when you start to add the counting numbers up? Can you give a smaller pictorial explanation?
  - Adding counting numbers up gives: $1, 1+2=3, 1+2+3=6, 1+2+3+4=10, 1+2+3+4+5=15, \dots$
  This is the Triangular numbers sequence.
  - Pictorial Explanation: In a dot triangle, row 1 has 1 dot, row 2 has 2 dots, row 3 has 3 dots, and so on. Stacking these rows creates a triangular arrangement representing the cumulative sum.
• 5. What happens when you add up pairs of consecutive triangular numbers? That is, take 1 + 3, 3 + 6, 6 + 10, 10 + 15, ... Which sequence do you get? Why? Can you explain it with a picture?
  - $1 + 3 = 4 = 2^2$
  - $3 + 6 = 9 = 3^2$
  - $6 + 10 = 16 = 4^2$
  - $10 + 15 = 25 = 5^2$
  We get the sequence of Square numbers (Squares).
  Explanation: Two consecutive triangular shapes of sizes $T_{n-1}$ and $T_n$ can be fitted together along their diagonals to form a perfect $n \times n$ square grid.
• 6. What happens when you start to add up powers of 2 starting with 1, i.e., take 1, 1 + 2, 1 + 2 + 4, 1 + 2 + 4 + 8, ...? Now add 1 to each of these numbers - what numbers do you get? Why does this happen?
  - Sums: $1 = 1$, $1+2 = 3$, $1+2+4 = 7$, $1+2+4+8 = 15$, $1+2+4+8+16 = 31, \dots$
  - Adding 1 to each sum gives: $1+1=2$, $3+1=4$, $7+1=8$, $15+1=16$, $31+1=32, \dots$
  We get the Powers of 2 ($2^1, 2^2, 2^3, 2^4, 2^5, \dots$).
  Why: The sum of powers of 2 from $2^0$ to $2^n$ always equals $2^{n+1} - 1$. Adding 1 yields the exact next power of 2: $2^{n+1}$.

Page 20

• 7. What happens when you multiply the triangular numbers by 6 and add 1? Which sequence do you get? Can you explain it with a picture?
  - For $T_1 = 1$: $6(1) + 1 = 7$
  - For $T_2 = 3$: $6(3) + 1 = 19$
  - For $T_3 = 6$: $6(6) + 1 = 37$
  - For $T_4 = 10$: $6(10) + 1 = 61$
  We get the sequence of Hexagonal numbers (Centered hexagonal numbers): 1, 7, 19, 37, 61, ...
  Explanation: A regular hexagon is made of 6 identical triangular sectors arranged around 1 central dot. Hence, total dots = $6 \times (\text{Triangular Number}) + 1$.
• 8. What happens when you start to add up hexagonal numbers, i.e., take 1, 1 + 7, 1 + 7 + 19, 1 + 7 + 19 + 37, ...? Which sequence do you get? Can you explain it using a picture of a cube?
  - $1 = 1 = 1^3$
  - $1 + 7 = 8 = 2^3$
  - $1 + 7 + 19 = 27 = 3^3$
  - $1 + 7 + 19 + 37 = 64 = 4^3$
  - $1 + 7 + 19 + 37 + 61 = 125 = 5^3$
  We get the sequence of Cubes ($1, 8, 27, 64, 125, \dots$).
  Explanation: Looking at a 3D cube of side $n$ from one corner, peeling away the outer shell of unit cubes leaves a smaller cube of side $(n-1)$. The outer layer contains exactly $n^3 - (n-1)^3 = 3n^2 - 3n + 1$ unit cubes, which corresponds to the $n^{\text{th}}$ centered hexagonal number.
• 9. Find your own patterns or relations in and among the sequences in Table 1. Can you explain why they happen with a picture or otherwise?
  Relation: Consecutive Square Differences are Odd Numbers:
  $2^2 - 1^2 = 4 - 1 = 3$
  $3^2 - 2^2 = 9 - 4 = 5$
  $4^2 - 3^2 = 16 - 9 = 7$
  $5^2 - 4^2 = 25 - 16 = 9$
  Reason: Expanding an $n \times n$ square grid of dots to an $(n+1) \times (n+1)$ grid requires adding an L-shaped border of $2n+1$ dots, which is always an odd number.

Figure it Out - 1.5 (Page 24)

• 1. Can you recognise the pattern in each of the sequences in Table 3?
  Yes: (i) Regular Polygons increase by 1 side at each step. (ii) Complete Graphs add 1 vertex connected to all existing vertices. (iii) Stacked Squares grow by adding rows and columns forming an $n \times n$ grid. (iv) Stacked Triangles add rows of triangles forming larger triangles. (v) Koch Snowflake repeatedly replaces each straight segment with an equilateral bump.
• 2. Try and redraw each sequence in Table 3 in your notebook. Can you draw the next shape in each sequence? Why or why not? After each sequence, describe in your own words what is the rule or pattern for forming the shapes in the sequence.
  1. Regular Polygons: Next shape is a regular Hendecagon (11-sided polygon). Rule: Increase the number of equal sides and equal angles by 1.
  
  2. Complete Graphs: Next shape is $K_7$ (7 vertices). Rule: Add a 7th vertex and connect it with straight lines to all other 6 vertices.
  
  3. Stacked Squares: Next shape is a $6 \times 6$ square grid (36 unit squares). Rule: Add one row and one column of unit squares.
  
  4. Stacked Triangles: Next shape is a large triangle containing 6 rows (total $6^2 = 36$ small triangles). Rule: Add a new bottom row containing the next odd number of unit triangles.
  
  5. Koch Snowflake: Next shape is the 5th iteration. Rule: Divide every straight segment into three equal parts and build an outward equilateral triangle on the middle third. (It is challenging to draw by hand because the segments become very small).

Figure it Out - 1.6 (Page 24)

• 1. Count the number of sides in each shape in the sequence of Regular Polygons. Which number sequence do you get? What about the number of corners in each shape in the sequence of Regular Polygons? Do you get the same number sequence? Can you explain why this happens?
  - Number of sides: 3, 4, 5, 6, 7, 8, 9, 10, ... (Counting numbers starting from 3).
  - Number of corners (vertices): 3, 4, 5, 6, 7, 8, 9, 10, ...
  - Same sequence? Yes, we get the exact same sequence.
  - Why: In any closed polygon, every pair of adjacent sides meets at exactly one corner (vertex), making the number of sides always equal to the number of vertices.
• 2. Count the number of lines in each shape in the sequence of Complete Graphs. [From table 3 (ii)] Which number sequence do you get? Can you explain why?
  - $K_2$: 1 line
  - $K_3$: 3 lines
  - $K_4$: 6 lines
  - $K_5$: 10 lines
  - $K_6$: 15 lines
  We get the sequence of Triangular numbers: 1, 3, 6, 10, 15, ...
  Why: Adding the $n^{\text{th}}$ vertex connects it to all preceding $(n-1)$ vertices, adding $(n-1)$ new edges: $\text{Total lines} = 1 + 2 + 3 + \dots + (n-1) = \frac{n(n-1)}{2}$.

Page 26

• 3. How many little squares [From table 3 (iii)] are there in each shape of the sequence of Stacked Squares? Which number sequence does this give? Can you explain why?
  - Grid 1: $1 \times 1 = 1$
  - Grid 2: $2 \times 2 = 4$
  - Grid 3: $3 \times 3 = 9$
  - Grid 4: $4 \times 4 = 16$
  - Grid 5: $5 \times 5 = 25$
  We get the sequence of Square numbers (Squares): 1, 4, 9, 16, 25, ...
  Why: An $n \times n$ stacked square consists of $n$ rows with $n$ unit squares each, giving a total of $n \times n = n^2$ little squares.
• 4. How many little triangles [From table 3 (iv)] are there in each shape of the sequence of Stacked Triangles? Which number sequence does this give? Can you explain why? (Hint: In each shape in the sequence, how many triangles are there in each row?)
  - Shape 1 (1 row): 1 triangle
  - Shape 2 (2 rows): $1 + 3 = 4$ triangles
  - Shape 3 (3 rows): $1 + 3 + 5 = 9$ triangles
  - Shape 4 (4 rows): $1 + 3 + 5 + 7 = 16$ triangles
  - Shape 5 (5 rows): $1 + 3 + 5 + 7 + 9 = 25$ triangles
  We get the sequence of Square numbers (Squares): 1, 4, 9, 16, 25, ...
  Why: Each subsequent row adds consecutive odd numbers of small triangles ($1, 3, 5, 7, 9, \dots$). The sum of the first $n$ odd numbers is always $n^2$.
• 5. To get from one shape to the next shape in the Koch Snowflake sequence, one replaces each line segment '-' by a 'speed bump'. How many total line segments are there in each shape of the Koch Snowflake? What is the corresponding number sequence?
  - Initial triangle: 3 segments
  - 1st iteration: $3 \times 4 = 12$ segments
  - 2nd iteration: $12 \times 4 = 48$ segments
  - 3rd iteration: $48 \times 4 = 192$ segments
  - 4th iteration: $192 \times 4 = 768$ segments
  The corresponding number sequence is 3, 12, 48, 192, 768, ... (given by $3 \times 4^n$).

Page 28

• Can you give me some examples for Symmetrical objects in the Nature?
  1. Bilateral Symmetry: Butterfly wings, human face, crab, banyan/peepal leaf.
  2. Radial Symmetry: Starfish, sunflower, daisy, jellyfish, cross-section of an orange.

CHAPTER MASTERY (Page 34)

• 1. Which of the following is Virahanka/Fibonacci sequence?
  A) 1, 1, 3, 5, 6, ---

  B) 1, 2, 4, 8, ---

  C) 1, 2, 3, 5, 8, ---

  D) 1, 3, 9, 27, ---
  Answer: C) 1, 2, 3, 5, 8, ---
• 2. Match the following:

  A	Ans	B
  (i) square numbers	( b )	a) 1, 3, 6, 10, ---
  (ii) cube numbers	( c )	b) 1, 4, 9, 16, ---
  (iii) Triangular numbers	( a )	c) 1, 8, 27, 64, ---

  Answer: B) (i)-b, (ii)-c, (iii)-a
• 3. What is the next number in the sequence: 1, 7, 19, 37, ---
  A) 61

  B) 63

  C) 59

  D) 47
  Answer: A) 61
• 4. Represent the third hexagonal number (19) through a diagram.
  The 3rd hexagonal number (19) is drawn as a centered hexagon of dots consisting of:
  - 1 dot at the center.
  - Ring 1 around the center: 6 dots.
  - Ring 2 (outer boundary): 12 dots.
  $\text{Total dots} = 1 + 6 + 12 = 19$.
• 5. 36 is an example for both triangular and square number.
• 6. Assertion (A): 1, 4, 9, 16, --- are square numbers.
  Reason (R): By adding the counting numbers up and down we get square numbers.
  A) Both (A) and (R) are correct and (R) is the correct explanation of (A)

  B) Both (A) and (R) are correct and (R) is not the correct explanation of (A)

  C) (A) is correct and (R) is false.

  D) (A) is false and (R) is correct.
  Answer: B) Both (A) and (R) are correct and (R) is not the correct explanation of (A)
• 7. What is a pattern?
  A pattern is an arrangement or sequence of numbers, shapes, colours, or designs that repeats or grows in a regular, predictable way according to a specific mathematical rule.
• 8. Which mathematical sequence is often seen in natural patterns like sunflowers and pinecones?
  The Virahanka / Fibonacci sequence ($1, 2, 3, 5, 8, 13, 21, \dots$).

CHAPTER MASTERY (Page 36)

• 9. How is the Fibonacci sequence connected to spirals in nature?
  In natural spirals (such as snail shells, sunflower seed arrangements, and pinecone scales), the adjacent seed counts or spiral arcs grow in proportions given by consecutive Fibonacci numbers ($1, 2, 3, 5, 8, 13, \dots$), producing the logarithmic golden spiral.
• 10. Why do honeycombs have a hexagonal pattern?
  Regular hexagons tile a plane perfectly without any gaps or overlaps while having the minimum perimeter for a given area. This allows bees to store the maximum amount of honey using the least amount of beeswax and energy.
• 11. Explain how symmetry helps in nature.
  1. Balance and Movement: Bilateral symmetry helps animals and birds maintain physical balance, run efficiently, and fly smoothly.
  2. Pollination and Survival: Radial symmetry in flowers attracts pollinating insects like bees and butterflies from all directions.
• 12. Find the next three terms in the pattern: 2, 4, 8, 16, ___
  Each term is multiplied by 2 (Powers of 2).
  - $16 \times 2 = 32$
  - $32 \times 2 = 64$
  - $64 \times 2 = 128$
  Next three terms: 32, 64, 128.
• 13. The arrangement of petals in a flower follows Fibonacci numbers. If a flower has 13 petals, predict the number of petals in the next in this pattern.
  The Fibonacci sequence is $1, 2, 3, 5, 8, 13, 21, 34, \dots$
  The term following 13 is $8 + 13 =$ 21 petals.
• 14. Observe these numbers: 1, 4, 9, 16, 25... Identify the pattern and explain the rule.
  - Pattern: Sequence of Square numbers ($1^2, 2^2, 3^2, 4^2, 5^2, \dots$).
  - Rule: The $n^{\text{th}}$ term is obtained by multiplying the counting number $n$ by itself ($T_n = n \times n = n^2$), or by adding consecutive odd numbers ($1, 1+3=4, 4+5=9, 9+7=16, 16+9=25$).
• 15. Compare a fern leaf and a snowflake. What common pattern do they share?
  Both share a Fractal pattern (self-similarity across different scales). In both a fern and a snowflake, smaller branches repeat the exact geometric shape of the larger overall structure.
• 16. Do you think patterns in nature can help in technological design? Give an example.
  Yes (Biomimicry).
  - Example: Engineers designed the aerodynamic nose of the Japanese Bullet Train (Shinkansen) modeled after the beak of the kingfisher bird to reduce noise and air resistance when exiting tunnels. Honeycomb structures are also used in aircraft wings for high strength and low weight.
• 17. Explain the efficiency of nature of spirals in space usage?
  Spirals based on the golden ratio and Fibonacci sequence allow compact packing of seeds and florets without leaving empty gaps or overcrowding the center, ensuring maximum exposure to sunlight and efficient space utilization.
• 18. Design a pattern using shapes that could represent growth in nature. Describe your rule.
  - Design: Concentric circles or nested squares expanding outward.
  - Rule: Start with a unit circle of radius 1 cm. At each step, increase the radius by adding the next Fibonacci number: $r_1=1\text{ cm}, r_2=2\text{ cm}, r_3=3\text{ cm}, r_4=5\text{ cm}, r_5=8\text{ cm}$. This simulates natural growth patterns seen in tree rings and ripples in water.