This MCQ module is based on: Shape Sequences and Chapter Exercises
TOPIC 3 OF 41
Shape Sequences and Chapter Exercises
🎓 Class 6
Mathematics
CBSE
Theory
Ch 1 — Patterns in Mathematics
⏱ ~40 min
🌐 Language: [gtranslate]
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This mathematics assessment will be based on: Shape Sequences and Chapter Exercises
Targeting Class 6 level in Number Patterns, with Basic difficulty.
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1.6 Patterns in Shapes — Shape Sequences
Patterns are not limited to numbers — they also appear in shape sequences?. Just as number sequences follow rules, shape sequences follow visual rules where each shape builds on the previous one.
Regular Polygons
A regular polygon? has all sides equal and all angles equal. The sequence of regular polygons starts with a triangle (3 sides), then a quadrilateral (4 sides), pentagon (5 sides), hexagon (6 sides), and so on.
Sequence of Regular Polygons
Stacked Squares
In this sequence, we stack squares on top of each other: first one square, then a 2-high stack, then 3-high, and so on. The number of small squares in each shape gives the counting numbers?: 1, 2, 3, 4, 5, ...
Stacked Squares — 1, 2, 3, 4, 5
Each shape adds one more square → counting numbers: 1, 2, 3, 4, 5, ...
Stacked Triangles
In this sequence, we stack triangles row by row: first one triangle, then a row of 3 (1+3=4), then 1+3+5=9, and so on. The number of small triangles gives the square numbers: 1, 4, 9, 16, 25, ... because each row adds the next odd number of triangles.
Stacked Triangles — 1, 4, 9
Row counts: 1, 3, 5, ... (odd numbers). Total = 1, 4, 9, ... (square numbers!)
Complete Graphs
A complete graph? \(K_n\) connects \(n\) points where every pair of points is joined by a line. The number of lines in \(K_n\) gives the triangular numbers!
Complete Graphs K₂ through K₆
Lines: 1, 3, 6, 10, 15 — the triangular numbers!
| Graph | Points | Lines | Triangular Number |
|---|---|---|---|
| \(K_2\) | 2 | 1 | \(T_1 = 1\) |
| \(K_3\) | 3 | 3 | \(T_2 = 3\) |
| \(K_4\) | 4 | 6 | \(T_3 = 6\) |
| \(K_5\) | 5 | 10 | \(T_4 = 10\) |
| \(K_6\) | 6 | 15 | \(T_5 = 15\) |
💡 Did You Know?
The number of lines in a complete graph \(K_n\) equals the \((n-1)\)-th triangular number: \(\frac{n(n-1)}{2}\). This is because each new point connects to all existing points!
Koch Snowflake
The Koch Snowflake? is a famous fractal shape. Starting with a triangle, you replace the middle third of each side with two new segments forming a bump. Repeating this creates an infinitely complex, beautiful snowflake pattern.
Koch Snowflake — First 3 Stages
At each stage, the number of sides multiplies by 4: 3 → 12 → 48 → 192 → ...
Regular Polygon Explorer
Bloom: L4 AnalyseDrag the slider to see regular polygons from 3 to 20 sides. Notice how the shape approaches a circle!
6
Figure it Out — Shape Sequences (Pages 11–12)
Q1. 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?
Answer:
Number of sides: 3, 4, 5, 6, 7, 8, 9, 10, ... → counting numbers starting from 3.
Number of corners: 3, 4, 5, 6, 7, 8, 9, 10, ... → the same sequence.
Why: In any polygon, the number of sides always equals the number of corners (vertices). Each side connects two vertices, and each vertex is where two sides meet.
Number of sides: 3, 4, 5, 6, 7, 8, 9, 10, ... → counting numbers starting from 3.
Number of corners: 3, 4, 5, 6, 7, 8, 9, 10, ... → the same sequence.
Why: In any polygon, the number of sides always equals the number of corners (vertices). Each side connects two vertices, and each vertex is where two sides meet.
Q2. Count the number of lines in each shape of the sequence of Complete Graphs \(K_2, K_3, K_4, K_5, K_6\). Which number sequence do you get? Can you explain why?
Answer: The number of lines: 1, 3, 6, 10, 15 → triangular numbers.
Why: When you add point \(n\) to the graph, it connects to all \(n-1\) existing points. So the total lines = \(0 + 1 + 2 + 3 + \cdots + (n-1) = \frac{n(n-1)}{2}\), which is the \((n-1)\)-th triangular number.
Why: When you add point \(n\) to the graph, it connects to all \(n-1\) existing points. So the total lines = \(0 + 1 + 2 + 3 + \cdots + (n-1) = \frac{n(n-1)}{2}\), which is the \((n-1)\)-th triangular number.
Q3. How many little squares are there in each shape of the sequence of Stacked Squares? Which number sequence does this give?
Answer: The number of little squares: 1, 2, 3, 4, 5, ... → counting numbers.
Each new shape adds one more square on top, so the n-th shape has exactly \(n\) squares.
Each new shape adds one more square on top, so the n-th shape has exactly \(n\) squares.
Q4. How many little triangles are there in each shape of the sequence of Stacked Triangles? Which number sequence does this give?
Answer: The number of little triangles: 1, 4, 9, 16, 25, ... → square numbers.
Why: In the n-th shape, the rows have 1, 3, 5, ..., (2n−1) triangles. The total = sum of first \(n\) odd numbers = \(n^2\).
Why: In the n-th shape, the rows have 1, 3, 5, ..., (2n−1) triangles. The total = sum of first \(n\) odd numbers = \(n^2\).
Q5. To get from one shape to the next in the Koch Snowflake, one replaces each line segment '___' with a 'speed bump' shape '_/\_'. How many line segments are there in each shape of the Koch Snowflake? What sequence does this give?
Answer: Number of line segments: 3, 12, 48, 192, 768, ...
Each stage multiplies by 4 (each segment becomes 4 new segments).
This is the sequence: \(3 \times 4^0,\; 3 \times 4^1,\; 3 \times 4^2,\; 3 \times 4^3, \ldots = 3, 12, 48, 192, \ldots\)
It is 3 times the powers of 4.
Each stage multiplies by 4 (each segment becomes 4 new segments).
This is the sequence: \(3 \times 4^0,\; 3 \times 4^1,\; 3 \times 4^2,\; 3 \times 4^3, \ldots = 3, 12, 48, 192, \ldots\)
It is 3 times the powers of 4.
Chapter Summary
📖 Key Takeaways
- Mathematics is fundamentally the search for patterns and their explanations
- Number sequences follow definite rules: counting, odd, even, triangular, square, cube, Virahanka, powers of 2
- Many sequences have beautiful visual representations (dots forming triangles, squares, etc.)
- Sequences are related to each other: e.g., sum of odd numbers = square numbers
- Shape sequences like regular polygons, stacked triangles, complete graphs, and Koch snowflakes also follow patterns
- Shape sequences often connect back to number sequences (stacked triangles → square numbers, complete graphs → triangular numbers)
🔍 Activity 1.3 — Complete Graphs and Handshakes
Bloom: L3 Apply
Materials needed: Paper, pencil, ruler
🤔 PREDICT FIRST: If 6 people are at a party and each person shakes hands with every other person exactly once, how many handshakes happen in total? Is this related to any sequence you know?
- Draw 3 dots (representing people) and connect every pair. Count the lines. (Answer: 3)
- Draw 4 dots and connect every pair. Count the lines. (Answer: 6)
- Draw 5 dots and connect every pair. Count the lines. (Answer: 10)
- Draw 6 dots and connect every pair. Count the lines. (Answer: 15)
- Write the sequence: 1, 3, 6, 10, 15. What sequence is this?
✅ Observation & Explanation
The number of handshakes follows the triangular numbers: 1, 3, 6, 10, 15, ...For 6 people: 15 handshakes = \(T_5 = \frac{6 \times 5}{2} = 15\).
This is exactly the complete graph \(K_6\)! Each person is a point, and each handshake is a line connecting two points.
📋
Competency-Based Questions
Scenario: A school art club is creating a mural based on mathematical patterns. They plan to include regular polygons from triangle to decagon, stacked triangles showing square numbers, and a Koch snowflake fractal. The teacher asks students to find the number patterns hidden in each design.
Q1. If the Koch snowflake starts with 3 line segments, how many line segments will it have after 5 stages?
L3 Apply
Answer: (C) 3072 — After \(n\) stages: \(3 \times 4^n\). So after 5 stages: \(3 \times 4^5 = 3 \times 1024 = 3072\).
Q2. A student claims "Complete graphs and handshake problems are the same thing mathematically." Evaluate this claim with an example.
L5 Evaluate
Model Answer: The claim is correct. In both cases, we count the number of unique pairs from \(n\) items. In a complete graph \(K_n\), each pair of points is connected by a line. In a handshake problem with \(n\) people, each pair of people shakes hands once. Both give \(\frac{n(n-1)}{2}\) — the triangular numbers. Example: 5 people → \(\frac{5 \times 4}{2} = 10\) handshakes = lines in \(K_5\).
Q3. In the stacked triangles sequence, the 4th shape has 16 small triangles. How many small triangles does the 10th shape have? Explain your reasoning.
L4 Analyse
Answer: The n-th shape has \(n^2\) small triangles. So the 10th shape has \(10^2 = \mathbf{100}\) small triangles. This is because each row \(k\) has \(2k - 1\) triangles (an odd number), and the sum of the first \(n\) odd numbers equals \(n^2\).
HOT Q. Design your own shape sequence where the number of elements in each shape follows the triangular numbers (1, 3, 6, 10, 15, ...). Describe the rule clearly.
L6 Create
Hint: Think about stacking a different shape (e.g., dots, hexagons, circles) where each new row adds one more element than the previous row. For instance: row 1 has 1 dot, row 2 has 2 dots, row 3 has 3 dots — totals give 1, 3, 6, 10, ... which are triangular numbers! Or use the handshake/complete graph idea.
⚖️ Assertion–Reason Questions
Options:
(A) Both A and R are true, and R is the correct explanation of A.
(B) Both A and R are true, but R is NOT the correct explanation of A.
(C) A is true, but R is false.
(D) A is false, but R is true.
(A) Both A and R are true, and R is the correct explanation of A.
(B) Both A and R are true, but R is NOT the correct explanation of A.
(C) A is true, but R is false.
(D) A is false, but R is true.
Assertion (A): The number of line segments in the Koch Snowflake at stage 3 is 192.
Reason (R): At each stage of the Koch Snowflake, every line segment is replaced by 4 new segments.
Answer: (A) — Both are true. Stage 3: \(3 \times 4^3 = 3 \times 64 = 192\). R correctly explains the multiplication-by-4 rule.
Assertion (A): A regular polygon with 100 sides looks almost like a circle.
Reason (R): As the number of sides of a regular polygon increases, each side becomes shorter and the shape approaches a circle.
Answer: (A) — Both are true, and R is the correct explanation. With 100 equal sides, each side is very short and the polygon closely approximates a circle. In fact, a circle can be thought of as a polygon with infinitely many sides!
Frequently Asked Questions
What are shape sequences in NCERT Class 6 Maths?
Shape sequences are visual patterns where geometric shapes change or grow according to a specific rule. In NCERT Class 6 Ganita Prakash Chapter 1, students explore patterns like triangular dot arrangements, growing squares, and Koch snowflake iterations where each step adds new elements to the shape.
How do you find the next term in a shape sequence?
To find the next term in a shape sequence, observe how the shape changes from one step to the next. Count the number of elements added, identify the direction of growth, and look for symmetry. Apply the discovered rule to predict the next shape. NCERT Class 6 Chapter 1 provides numerous practice exercises.
What is the Koch snowflake pattern in Class 6 Maths?
The Koch snowflake pattern starts with a triangle and adds smaller triangles to the middle of each side in every step. This creates an increasingly complex snowflake shape. NCERT Class 6 Ganita Prakash introduces a simplified version to help students understand shape growth patterns.
How to solve number pattern exercises in Class 6?
To solve number pattern exercises, first calculate differences between consecutive terms. If differences are constant, use arithmetic progression logic. If differences themselves form a pattern, look for quadratic or other relationships. Write the rule as a formula and verify with given terms. Chapter 1 exercises provide extensive practice.
What types of questions come in Class 6 Maths Chapter 1 exercises?
Class 6 Maths Chapter 1 exercises include finding the next terms in number sequences, identifying the rule behind a pattern, completing shape sequences, relating different sequences to each other, and applying patterns to real-world contexts. Both numerical and visual pattern questions are included.
Frequently Asked Questions — Patterns in Mathematics
What is Shape Sequences and Chapter Exercises in NCERT Class 6 Mathematics?
Shape Sequences and Chapter Exercises is a key concept covered in NCERT Class 6 Mathematics, Chapter 1: Patterns in Mathematics. This lesson builds the student's foundation in the chapter by explaining the core ideas with worked examples, definitions, and step-by-step methods aligned to the CBSE curriculum.
How do I solve problems on Shape Sequences and Chapter Exercises step by step?
To solve problems on Shape Sequences and Chapter Exercises, follow the NCERT method: identify the given quantities, choose the relevant formula or theorem, substitute values carefully, and simplify. Class 6 exercises gradually increase in difficulty — start with solved NCERT examples before attempting exercise questions, and always verify your answer by substitution or diagram.
What are the most important formulas for Chapter 1: Patterns in Mathematics?
The essential formulas of Chapter 1 (Patterns in Mathematics) are listed in the chapter summary and highlighted throughout the lesson in formula boxes. Memorise them and practise at least 2–3 problems per formula. CBSE board exams frequently test direct application as well as combined use of multiple formulas from this chapter.
Is Shape Sequences and Chapter Exercises important for the Class 6 board exam?
Shape Sequences and Chapter Exercises is part of the NCERT Class 6 Mathematics syllabus and appears in CBSE board exams. Questions typically include short-answer, long-answer, and competency-based items. Review the NCERT examples, exercise questions, and previous-year board problems on this topic to prepare confidently.
What mistakes should students avoid in Shape Sequences and Chapter Exercises?
Common mistakes in Shape Sequences and Chapter Exercises include skipping steps, misapplying formulas, sign errors, and losing track of units. Write each step clearly, double-check algebraic manipulations, and re-read the question after solving to verify that your answer matches what was asked.
Where can I find more NCERT practice questions on Shape Sequences and Chapter Exercises?
End-of-chapter NCERT exercises for Shape Sequences and Chapter Exercises cover all difficulty levels tested in CBSE exams. After completing them, try the examples again without looking at the solutions, attempt the NCERT Exemplar questions for Chapter 1, and solve at least one previous-year board paper to consolidate your understanding.
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