CBSE Class 6 Mathematics Patterns in Mathematics Assignment Set 01

Very Short Answer Questions

Question 1. Mathematics is, in large part, a search for two things. Name them.
Answer:
Mathematics searches for patterns and for the explanations of why those patterns exist.

Teacher's Note:
1. Finding the pattern is only half the job.
2. The 'why' is the other half.

Question 2. Name the branch of Mathematics that studies patterns in whole numbers.
Answer:
Patterns in the whole numbers 0, 1, 2, 3, … are studied in number theory.

Teacher's Note:
1. Number theory: numbers.
2. Geometry: shapes.

Question 3. Write the next term of each of these sequences.
(i) 2, 4, 6, 8, 10, …
(ii) 1, 2, 4, 8, 16, …
Answer:
(i) Add 2 each time: the next even number is 12.
(ii) Double each time: 16 × 2 = 32.

Teacher's Note:
1. First find the rule, then apply it once.
2. Adding and doubling are different rules.

Question 4. Why are the numbers 1, 3, 6, 10, 15, … called triangular numbers?
Answer:
Because that many dots can be arranged as a perfect triangle, with 1, 2, 3, 4, … dots in the rows.

Teacher's Note:
1. Draw the first three triangles of dots.
2. Each new triangle adds one longer row.

Question 5. Why are the numbers 1, 8, 27, 64, 125, … called cube numbers?
Answer:
Because that many small cubes can be stacked to build a perfect cube, for example 2 × 2 × 2 = 8 and 3 × 3 × 3 = 27.

Teacher's Note:
1. A cube number is a number multiplied by itself three times.
2. Think of building blocks.

Question 6. State the rule used to form the Virahānka numbers 1, 2, 3, 5, 8, 13, …
Answer:
After the first two terms 1 and 2, each number is the sum of the two numbers just before it, for example 5 + 8 = 13.

Teacher's Note:
1. Two terms are needed to make the next one.
2. Check the rule on two places in the list.

Question 7. Write the next number in the sequence of powers of 3: 1, 3, 9, 27, 81, …
Answer:
Each term is 3 times the one before, so the next is 81 × 3 = 243.

Teacher's Note:
1. Powers of 3 multiply by 3.
2. Do not add 3.

Question 8. Name the branch of Mathematics that studies patterns in shapes.
Answer:
Patterns in shapes are studied in geometry.

Teacher's Note:
1. Geometry covers 1D, 2D, 3D and more.
2. Pair it with number theory.

Question 9. How many sides does a regular decagon have?
Answer:
A regular decagon has 10 sides (and also 10 corners).

Teacher's Note:
1. Deca means ten.
2. Sides and corners are always equal in number.

Question 10. What does the word “regular” tell us about a regular polygon?
Answer:
All its sides are of equal length and all its corners look the same (the angles are equal).

Teacher's Note:
1. Two conditions: equal sides and equal angles.
2. A square is a regular quadrilateral.

Question 11. Find the value of 1 + 3 + 5 + 7 + 9 and name the kind of number you get.
Answer:
1 + 3 + 5 + 7 + 9 = 25, a square number (5 × 5).

Teacher's Note:
1. Five odd numbers give 5 × 5.
2. Odd numbers add up to squares.

Question 12. Name the shape sequence of Table 3 in which every pair of dots is joined by a line.
Answer:
It is the sequence of Complete Graphs K₂, K₃, K₄, K₅, K₆, …

Teacher's Note:
1. The small number tells how many dots.
2. Every pair of dots is joined.

Question 13. Which number sequence do you get when you start adding up the All 1’s sequence as 1, 1 + 1, 1 + 1 + 1, …?
Answer:
You get the counting numbers 1, 2, 3, 4, 5, …, since each new 1 raises the total by exactly one.

Teacher's Note:
1. Each total counts how many 1’s were added.
2. Running totals are the key idea.

Question 14. Which number sequence do you get when you start adding up the counting numbers as 1, 1 + 2, 1 + 2 + 3, …?
Answer:
You get the triangular numbers 1, 3, 6, 10, 15, …, since each new row of dots makes a bigger triangle.

Teacher's Note:
1. Each new counting number is a new row of dots.
2. Link the sum to the picture.

Question 15. Find the value of 1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1 and name the kind of number you get.
Answer:
Counting up to 6 and back down gives 6 × 6 = 36, a square number.

Teacher's Note:
1. The top number decides the square.
2. Check by adding: 21 + 15 = 36.

Question 16. Add the two triangular numbers 21 and 28 that come one after the other. What kind of number do you get?
Answer:
21 + 28 = 49 = 7 × 7, a square number.

Teacher's Note:
1. Neighbouring triangular numbers always add to a square.
2. Two triangles fit into one square.

Question 17. Multiply the triangular number 21 by 6 and then add 1. To which number sequence does your answer belong?
Answer:
6 × 21 + 1 = 126 + 1 = 127.
It belongs to the hexagonal numbers 1, 7, 19, 37, 61, 91, 127, …

Teacher's Note:
1. 6 × triangular + 1 is a hexagonal number.
2. Multiply first, then add 1.

Question 18. How many little squares are there in the 6th shape of the Stacked Squares sequence?
Answer:
The 6th shape is a 6 × 6 arrangement, so it has 36 little squares.

Teacher's Note:
1. nth shape has n × n squares.
2. Draw a small grid if unsure.

Question 19. How many lines are there in the complete graph K₆?
Answer:
K₆ has 6 dots: \( \frac{6 \times 5}{2} \) = 15 lines, the 5th triangular number.

Teacher's Note:
1. Each line joins two dots, so divide by 2.
2. Line counts are triangular numbers.

Question 20. The number 36 can be drawn both as a triangle of dots and as a square of dots. What does this tell us about numbers?
Answer:
The same number can be pictured in more than one way and can play different roles depending on the situation.

Teacher's Note:
1. 36 is triangular and square.
2. Such double roles often hint at hidden links.

Question 21. Write the number that comes just after 61 in the sequence of hexagonal numbers.
Answer:
6 × 15 + 1 = 91 (also 61 + 30 = 91).

Teacher's Note:
1. Use the triangular number 15.
2. Gaps grow by 6: 24, then 30.

Question 22. Describe in one or two lines a picture that can be used to show the sequence of powers of 2.
Answer:
Take a whole shape as 1, cut it into 2 equal parts, then cut each part into 2 again, and so on.
The numbers of parts are 1, 2, 4, 8, 16, …

Teacher's Note:
1. Each cut doubles the number of pieces.
2. Folding paper works the same way.

Question 23. How is one shape of the Koch Snowflake sequence changed to get the next shape, and how many line segments does the 2nd shape have?
Answer:
Every line segment is replaced by a “speed bump” of 4 smaller segments.
So the 2nd shape has 3 × 4 = 12 line segments.

Teacher's Note:
1. One segment becomes four.
2. Start from the 3 sides of a triangle.

Question 24. Give one example of how understanding a mathematical pattern has helped humanity move forward.
Answer:
Patterns in the motion of stars and planets led to the theory of gravitation, which lets us launch satellites and send rockets to the Moon and Mars.

Teacher's Note:
1. Another example: patterns in genomes help doctors cure diseases.
2. One clear example is enough.

Question 25. Without adding the numbers one by one, find the sum of the first 25 odd numbers. Give the reason for your method in one line.
Answer:
Sum = 25 × 25 = 625.
Reason: a square grid splits into L-shaped layers of 1, 3, 5, … dots, so the first 25 odd numbers fill a 25 × 25 square.

Teacher's Note:
1. The n × n rule saves time.
2. The L-shaped picture is the reason.

Question 26. Find the value of 1 + 2 + 3 + ⋯ + 59 + 60 + 59 + ⋯ + 3 + 2 + 1.
Answer:
Up to 60 and back down gives a square: 60 × 60 = 3600.

Teacher's Note:
1. The top number is 60.
2. Up-and-down sums are squares.

Question 27. Find the sum of the first six hexagonal numbers 1 + 7 + 19 + 37 + 61 + 91 and name the sequence to which the answer belongs.
Answer:
1 + 7 + 19 + 37 + 61 + 91 = 216 = 6 × 6 × 6, a cube number.
Sums of hexagonal numbers always give cubes.

Teacher's Note:
1. Six hexagonal numbers give 6 × 6 × 6.
2. Add step by step to check.

Question 28. How many little triangles are there in the 9th shape of the Stacked Triangles sequence? Write the reason in one line.
Answer:
9 × 9 = 81 little triangles.
Reason: its rows hold 1, 3, 5, …, 17 triangles, and odd numbers add up to a square.

Teacher's Note:
1. Stacked triangles give squares, not triangular numbers.
2. The last row has 2 × 9 − 1 = 17.

Question 29. How many line segments are there in the 5th shape of the Koch Snowflake sequence?
Answer:
The counts are 3, 12, 48, 192, 768, so the 5th shape has 768 segments (3 × 4 × 4 × 4 × 4).

Teacher's Note:
1. Multiply by 4 four times.
2. Write all five terms.

Question 30. Explain in one or two lines why counting the sides and counting the corners of the regular polygons gives the same number sequence.
Answer:
A polygon is a closed figure: every side ends where the next side begins.
So each side matches exactly one corner, and both counts give 3, 4, 5, 6, …

Teacher's Note:
1. Pair each side with its end corner.
2. This holds for any polygon, regular or not.