AP 6th Maths "Patterns in Mathematics" Unit Answers 2026-27
📘 Lesson 1: Patterns in Mathematics — Answers (Pages 2–36)
Class 6 Maths | Simple answers for young learners
📄 Page 6 — Figure it Out 1.1 APTEACHERS.IN
1 Can you think of other examples where mathematics helps us in our everyday lives?
Answer: Yes! Maths helps us while cooking (measuring ingredients), shopping (counting money and change), telling time, playing games (keeping score), and building things like houses and furniture.
2 How has mathematics helped propel humanity forward?
Answer: Mathematics helped us build bridges and houses, make TVs, mobile phones and computers, run our economy, and even send rockets to the Moon and Mars. Understanding number patterns in the stars helped scientists discover the laws of gravity.
📄 Page 10 — Figure it Out 1.2
1 Can you recognise the pattern in each of the sequences in Table 1?
Answer: Yes. Each sequence follows its own simple rule — some add a fixed number, some multiply, and some depend on shape patterns (see Q2 below).
2 Rewrite each sequence with the next three numbers and the rule.
| Sequence | Next 3 Terms | Rule |
|---|---|---|
| All 1's | 1, 1, 1 | Every term is 1 |
| Counting numbers | 8, 9, 10 | Add 1 each time |
| Odd numbers | 15, 17, 19 | Add 2 each time |
| Even numbers | 16, 18, 20 | Add 2 each time |
| Triangular numbers | 36, 45, 55 | Add the next counting number |
| Squares | 64, 81, 100 | n × n |
| Cubes | 343, 512, 729 | n × n × n |
| Virahanka/Fibonacci | 34, 55, 89 | Add the previous two terms |
| Powers of 2 | 128, 256, 512 | Multiply by 2 each time |
| Powers of 3 | 2187, 6561, 19683 | Multiply by 3 each time |
📄 Page 12 — Figure it Out 1.3
1 Copy the pictorial representations from Table 2 and draw the next picture for each.
Answer: Draw one more step following the same rule — e.g. next Counting-number picture has 6 dots, next Odd-number picture has 11 dots, next Square picture is a 6×6 dot grid, next Cube picture is a 6×6×6 cube.
2 Why are 1,3,6,10,15 called triangular numbers, 1,4,9,16,25 called square numbers, and 1,8,27,64,125 called cubes?
Answer: Because their dots can be arranged exactly into those shapes — triangular numbers form a triangle, square numbers form a perfect square grid, and cubes form a solid 3-D cube shape.
3 36 is both a triangular and a square number — show this with pictures.
Answer: 36 dots can form a triangle with 8 rows (1+2+3+4+5+6+7+8 = 36) and also a 6×6 square grid (6×6 = 36). Same 36 dots, two different shapes!
4 What would you call the sequence 1, 7, 19, 37...? What is the next number?
Answer: These are hexagonal numbers. The gaps increase by 6 each time (6, 12, 18, 24...), so the next number is 37 + 24 = 61.
5 Can you think of pictorial ways to visualise Powers of 2 and Powers of 3?
Answer: Powers of 2 can be shown as a point, then a line, then a square, then a cube, doubling in dimension each time. Powers of 3 can be shown as a growing 3×3×3 style grid that triples in size at each step.
📄 Pages 14–16 — In-text Questions (Odd Numbers & Squares)
• Why does adding up odd numbers always give square numbers? Will it happen forever?
Answer: Yes, it always happens. A picture of dots arranged in L-shaped bands around a corner shows each new odd number fitting perfectly around the last square to make the next bigger square.
• What is the sum of the first 10 odd numbers? The first 100 odd numbers?
Answer: Sum of first 10 odd numbers = 10 × 10 = 100. Sum of first 100 odd numbers = 100 × 100 = 10,000.
📄 Page 18 — Figure it Out 1.4 (Q1–6)
1 Why does adding counting numbers up and down give square numbers?
Answer: Arranging the up-and-down sum as a dot picture also fills a perfect square grid, so it always equals n × n.
2 What is 1+2+3+...+99+100+99+...+3+2+1?
Answer: It equals 100 × 100 = 10,000.
3 Which sequence do you get adding the All 1's sequence up? And up-and-down?
Answer: Adding up gives Counting numbers (1,2,3,4...). Adding up-and-down gives Odd numbers (1,3,5,7...).
4 Which sequence do you get adding Counting numbers up? Smaller picture explanation?
Answer: You get Triangular numbers (1,3,6,10...). A simple staircase of dots — one dot, then two, then three — shows this clearly.
5 What happens when you add pairs of consecutive triangular numbers (1+3, 3+6, 6+10...)?
Answer: You get Square numbers (4, 9, 16, 25...). Two triangles (one upside down) fit together to make a square.
6 Add up Powers of 2 (1, 1+2, 1+2+4, 1+2+4+8...). Now add 1 to each. What do you get?
Answer: Sums are 1, 3, 7, 15, 31... Adding 1 gives 2, 4, 8, 16, 32 — the Powers of 2 again, one step ahead!
📄 Page 20 — Figure it Out 1.4 (Q7–9) Patterns in Mathematics
7 Multiply triangular numbers by 6 and add 1. Which sequence do you get?
Answer: 1×6+1=7, 3×6+1=19, 6×6+1=37... These are Hexagonal numbers — six triangles arranged around one central dot.
8 Add up hexagonal numbers (1, 1+7, 1+7+19...). Which sequence?
Answer: You get Cubes (1, 8, 27, 64...). Each hexagonal "layer" added builds up one more layer of a solid cube.
9 Find your own pattern among the sequences in Table 1.
Answer (example): Cubes can be made by adding consecutive odd numbers: 1=1³, 3+5=8=2³, 7+9+11=27=3³.
📄 Page 24 — Figure it Out 1.5 & 1.6 (Q1–2)
Figure it Out 1.5
1 Can you recognise the pattern in each sequence of Table 3?
Answer: Regular Polygons gain one side each time; Complete Graphs gain one point (connected to all others); Stacked Squares/Triangles grow one row bigger; Koch Snowflake gets a small "bump" added to every edge.
2 Redraw each sequence and draw the next shape. Describe the rule.
Answer: Next Polygon = 11 sides; next Complete Graph = K7 (7 points); next Stacked Square = 6×6 grid; next Stacked Triangle = one more row added; next Koch Snowflake = every edge gets a smaller bump.
Figure it Out 1.6
1 Count sides and corners in the Regular Polygons sequence. Same number sequence?
Answer: Both sides and corners follow 3,4,5,6,7,8,9,10 — the Counting numbers starting at 3. This happens because every polygon has equal numbers of sides and corners.
2 Count lines in the Complete Graphs sequence (K2–K6). Which sequence?
Answer: Lines: 1, 3, 6, 10, 15 — the Triangular numbers. Each new point connects to all previous points, adding one more line-group each time.
📄 Page 26 — Figure it Out 1.6 (Q3–5) Patterns in Mathematics
3 How many little squares in each Stacked Squares shape? Which sequence?
Answer: 1, 4, 9, 16, 25 — the Square numbers (an n×n grid has n² small squares).
4 How many little triangles in each Stacked Triangles shape? Which sequence?
Answer: 1, 4, 9, 16, 25 — the Square numbers again! Each row has an odd number of small triangles (1,3,5,7...), and adding consecutive odd numbers gives square numbers.
5 How many line segments in each Koch Snowflake shape? What sequence?
Answer: 3, 12, 48, 192... — that is 3 × Powers of 4. Each segment turns into 4 smaller ones every step.
📄 Page 28 — In-text Question (Symmetry)
• Can you give examples of symmetrical objects in nature?
Answer: Butterflies, leaves, flowers, and starfish are all symmetrical — their two halves (or all sides) match each other.
📄 Page 34 — Chapter Mastery (Q1–8) Patterns in Mathematics
1 Which is the Virahanka/Fibonacci sequence?
A)1,1,3,5,6 B)1,2,4,8 C)1,2,3,5 D)1,3,9,27
Answer: C) 1, 2, 3, 5
2 Match: (i) square (ii) cube (iii) triangular numbers with a)1,3,6,10 b)1,4,9,16 c)1,8,27,64
Answer: B) (i)-b, (ii)-c, (iii)-a
3 Next number in 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.
Answer: Draw 1 central dot surrounded by two rings of dots (6 dots in the first ring, 12 in the second) making 19 dots total in a hexagon shape.
5 ______ is an example of both a triangular and a square number.
Answer: 36
6 Assertion(A): 1,4,9,16 are square numbers. Reason(R): Adding counting numbers up and down gives square numbers.
Answer: A) Both (A) and (R) are correct, and (R) is the correct explanation of (A)
7 What is a pattern?
Answer: A pattern is an arrangement of numbers, shapes, or objects that repeats or changes in a regular, predictable way.
8 Which sequence is often seen in sunflowers and pinecones?
Answer: The Fibonacci (Virahanka) sequence
📄 Page 36 — Chapter Mastery (Q9–18) & Summary APTEACHERS.IN
9 How is the Fibonacci sequence connected to spirals in nature?
Answer: Each new Fibonacci number grows from the two before it, and this growth naturally curves into the spiral shapes seen in shells and sunflowers.
10 Why do honeycombs have a hexagonal pattern?
Answer: Hexagons fit together with no gaps and use the least wax to store the most honey — the most efficient shape.
11 Explain how symmetry helps in nature.
Answer: Symmetry helps living things stay balanced, move well, and grow evenly — like butterflies flying straight or starfish moving in any direction.
12 Find the next 3 terms: 2, 4, 8, 16, __, __, __
Answer: 32, 64, 128 (each term doubles)
13 Petals follow Fibonacci numbers. If a flower has 13 petals, predict the next.
Answer: 21 petals (since 8+13=21)
14 Observe 1, 4, 9, 16, 25... Identify the pattern and rule.
Answer: These are Square numbers. Rule: each term is a number multiplied by itself (n × n).
15 Compare a fern leaf and a snowflake. What common pattern do they share?
Answer: Both show a fractal pattern — the same small shape repeats again at smaller and smaller sizes.
16 Do patterns in nature help in technological design? Give an example.
Answer: Yes — solar panels are arranged in spiral (Fibonacci) patterns to capture maximum sunlight, inspired by sunflowers.
17 Explain the efficiency of spirals in space usage.
Answer: A spiral arrangement lets seeds or petals pack tightly into a small space without wasting room or overlapping.
18 Design a pattern using shapes that could represent growth in nature. Describe your rule.
Answer (example): Draw a small triangle, then a bigger triangle made of a new row of small triangles added below, growing bigger each time — like a plant sprouting new branches. Rule: add one more row every step.
0 comments:
Give Your valuable suggestions and comments