CBSE Class 6 Mathematics Patterns in Mathematics Worksheet Set 01

Short Answer Questions

Question 1. Mathematics is often described as the search for patterns and for the explanations of why those patterns exist. Explain this statement in your own words and give one example from everyday life in which finding a pattern helps us.
Answer:
Mathematics does two jobs. First, it looks for patterns, things that repeat or grow in a regular way.
Second, and just as important, it asks why the pattern must happen. Knowing the reason lets us use the pattern in new places.
Example: the days of the week repeat every 7 days. Because of this pattern we can make a calendar and tell today which day a birthday will fall on next year.
In simple words: Maths finds what repeats and explains why it repeats, so we can use it.

Teacher's Note:
1. Give both parts: the pattern and the reason.
2. Any honest daily-life example earns credit.

Question 2. Write the next three numbers in each sequence and state the rule you used.
(i) 1, 3, 5, 7, 9, 11, …
(ii) 2, 4, 6, 8, 10, 12, …
Answer:
(i) 13, 15, 17. These are the odd numbers; each number is 2 more than the one before it.
(ii) 14, 16, 18. These are the even numbers; each number is 2 more than the one before it (each is a counting number doubled).
In simple words: Keep adding 2 in both sequences.

Teacher's Note:
1. Same rule, different starting number.
2. Name the sequence as well as the rule.

Question 3. The Virahānka numbers are 1, 2, 3, 5, 8, 13, 21, … Write the rule for forming this sequence and find the next two numbers after 21.
Answer:
Rule: the first two numbers are 1 and 2; after that each number is the sum of the two numbers just before it.
Check: 2 + 3 = 5, 3 + 5 = 8, 5 + 8 = 13, 8 + 13 = 21.
Next: 13 + 21 = 34 and 21 + 34 = 55.
In simple words: Add the last two numbers to get the next one: 34, then 55.

Teacher's Note:
1. Always use the two most recent terms.
2. Show at least one check of the rule.

Question 4. Using dots, explain why the numbers 1, 3, 6, 10, 15, … are called triangular numbers and why the numbers 1, 8, 27, 64, … are called cubes.
Answer:
Triangular numbers: that many dots fit neatly into a triangle: a row of 1; then rows of 1 and 2; then 1, 2 and 3; and so on. Each new picture gets one extra row at the bottom.
Cubes: that many small cubes stack into a bigger cube: 1 × 1 × 1 = 1, 2 × 2 × 2 = 8, 3 × 3 × 3 = 27, 4 × 4 × 4 = 64.
In simple words: The names come from the shapes the objects make: a triangle or a cube.

Teacher's Note:
1. A quick sketch helps the explanation.
2. Cubes need three equal factors.

Question 5. Look at the sequence 1, 7, 19, 37, 61, …
(i) Name this sequence.
(ii) Write the next number and show how you got it.
Answer:
(i) These are the hexagonal numbers, drawn as rings of dots around a centre dot.
(ii) The jumps are 7 − 1 = 6, 19 − 7 = 12, 37 − 19 = 18, 61 − 37 = 24.
The next jump is 30, so the next number is 61 + 30 = 91.
In simple words: The jumps grow by 6 each time, so add 30 to 61 to get 91.

Teacher's Note:
1. Look at the differences.
2. Check: 6 × 15 + 1 = 91.

Question 6. The polygons shown in the chapter are called regular polygons.
(i) What does the word ‘regular’ tell you about such a polygon?
(ii) Name the regular polygons that have 7, 9 and 10 sides.
Answer:
(i) All the sides are of equal length and all the corners look exactly alike (the angles are equal).
(ii) 7 sides: heptagon; 9 sides: nonagon; 10 sides: decagon.
In simple words: Regular means equal sides and equal corners; 7, 9, 10 sides are heptagon, nonagon, decagon.

Teacher's Note:
1. Learn the polygon names from 3 to 10.
2. Both conditions are needed for 'regular'.

Question 7. Show that 1 + 3 + 5 + 7 + 9 + 11 + 13 = 49, and write the rule that connects the sum of the first few odd numbers with the square numbers.
Answer:
Adding one at a time: 1 + 3 = 4, 4 + 5 = 9, 9 + 7 = 16, 16 + 9 = 25, 25 + 11 = 36, 36 + 13 = 49.
So 1 + 3 + 5 + 7 + 9 + 11 + 13 = 49.
Rule: the sum of the first n odd numbers is n × n. Here 7 odd numbers were added and 7 × 7 = 49.
In simple words: Every running total is a square, and seven odd numbers give 7 × 7 = 49.

Teacher's Note:
1. The running totals 4, 9, 16, … are all squares.
2. Count the terms to find n.

Question 8. In the Koch Snowflake sequence the numbers of line segments are 3, 12, 48, 192, … Describe this pattern in words and write the fifth number of the sequence.
Answer:
Each number is 4 times the one before it, starting from 3: 3 × 4 = 12, 12 × 4 = 48, 48 × 4 = 192.
So the sequence is 3 times the powers of 4.
The fifth number is 192 × 4 = 768.
In simple words: Multiply by 4 each time; the fifth number is 768.

Teacher's Note:
1. Each segment becomes 4 segments.
2. Do the multiplication carefully.

Question 9. Use the rule “the sum of the first n odd numbers is n × n” to answer the following.
(i) Find the sum of the first 25 odd numbers.
(ii) How many odd numbers, starting from 1, must be added to get a total of 144?
Answer:
(i) 25 × 25 = 625.
(ii) We need n × n = 144. Since 12 × 12 = 144, we must add 12 odd numbers, that is, all the odd numbers from 1 to 23.
In simple words: Square 25 to get 625; 144 is 12 × 12, so 12 odd numbers are needed.

Teacher's Note:
1. Part (ii) uses the rule backwards.
2. The 12th odd number is 2 × 12 − 1 = 23.

Question 10. Find the value of 1 + 2 + 3 + ⋯ + 14 + 15 + 14 + ⋯ + 3 + 2 + 1 and give a reason for your answer.
Answer:
The counting numbers are added up to 15 and then back down to 1.
Such an up-and-down sum is always a square: 1 = 1, 1 + 2 + 1 = 4, 1 + 2 + 3 + 2 + 1 = 9, and so on.
Here the top number is 15, so the value is 15 × 15 = 225.
In simple words: Up to 15 and back down gives 15 × 15 = 225.

Teacher's Note:
1. Show the small cases to support the rule.
2. Check: 120 + 105 = 225.

Question 11. Add the pairs of consecutive triangular numbers 15 + 21 and 21 + 28. Which sequence do the two answers belong to? Explain why this happens.
Answer:
15 + 21 = 36 and 21 + 28 = 49. Both are square numbers.
Reason: a triangular number is a triangle of dots. Turn the smaller triangle upside down and place it beside the bigger one; together they fit exactly into a square of dots.
So two consecutive triangular numbers always add up to a square number.
In simple words: Two neighbouring dot triangles fit together into a square.

Teacher's Note:
1. A sketch with 3 + 6 = 9 makes the idea clear.
2. 36 = 6 × 6 and 49 = 7 × 7.

Question 12. A student adds powers of 2 as follows: 1 + 2 + 4 + 8 + 16 + 32.
(i) Find the total.
(ii) Add 1 to this total and say what you notice about the answer.
Answer:
(i) 1 + 2 + 4 + 8 + 16 + 32 = 63.
(ii) 63 + 1 = 64, which is the very next power of 2 after 32.
Adding up powers of 2 always gives one less than the next power of 2.
In simple words: The total is 63, just one short of the next power of 2, 64.

Teacher's Note:
1. Try smaller cases: 1 + 2 = 3 = 4 − 1.
2. State the general pattern.

Question 13. Differentiate between triangular numbers and square numbers. Also name one number smaller than 50 that belongs to both sequences.
Answer:
Triangular numbers (1, 3, 6, 10, 15, …) count dots arranged in a triangle; each is found by adding the next counting number: add 2, then 3, then 4, …
Square numbers (1, 4, 9, 16, 25, …) count dots arranged in a square; each is a counting number multiplied by itself.
36 belongs to both: 36 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 and 36 = 6 × 6.
In simple words: Triangular numbers make triangles, squares make squares, and 36 makes both.

Teacher's Note:
1. 1 also belongs to both, but 36 is the more interesting answer.
2. Give the dot shape for each.

Question 14. Multiply the 5th triangular number by 6 and add 1.
(i) Which number do you get?
(ii) To which sequence does it belong, and which term of that sequence is it?
Answer:
(i) The 5th triangular number is 15, and 6 × 15 + 1 = 90 + 1 = 91.
(ii) 91 belongs to the hexagonal numbers 1, 7, 19, 37, 61, 91, … It is the 6th term.
In simple words: 6 × 15 + 1 = 91, the 6th hexagonal number.

Teacher's Note:
1. List the triangular numbers to find the 5th.
2. Using the nth triangular number gives the (n + 1)th hexagonal number.

Question 15. The numbers of lines in the complete graphs K₂, K₃, K₄, K₅, K₆ are 1, 3, 6, 10, 15.
(i) Which number sequence is this?
(ii) How many lines will K₈ have? Give a reason for your answer.
Answer:
(i) These are the triangular numbers.
(ii) K₇ has 15 + 6 = 21 lines. To make K₈, the 8th dot is joined to each of the 7 dots already there, adding 7 lines.
So K₈ has 21 + 7 = 28 lines.
In simple words: Each new dot adds as many lines as there are old dots, so K₈ has 28 lines.

Teacher's Note:
1. Check: 8 × 7 ÷ 2 = 28.
2. Build up step by step from K₆.

Question 16. In the sequence of Stacked Triangles, the shapes are made up of 1, 4, 9, 16, … little triangles.
(i) How many little triangles are there in the 6th shape?
(ii) Explain your answer by counting the little triangles row by row.
Answer:
(i) The 6th shape has 36 little triangles.
(ii) From the top, the rows hold 1, 3, 5, 7, 9 and 11 little triangles; each row has 2 more than the row above.
Total = 1 + 3 + 5 + 7 + 9 + 11 = 36 = 6 × 6, since odd numbers add up to a square.
In simple words: Six rows of 1, 3, 5, 7, 9, 11 triangles add to 36.

Teacher's Note:
1. The bottom row of the nth shape has 2n − 1 triangles.
2. Link back to odd numbers adding to squares.

Question 17. The sum of the first 100 odd numbers is 10,000, and the value of 1 + 2 + 3 + ⋯ + 99 + 100 + 99 + ⋯ + 2 + 1 is also 10,000. Explain why these two sums, which look so different, give the same answer.
Answer:
Both sums count the dots of the same 100 × 100 square grid, so both are 10,000.
First way: cut the grid into L-shaped bands, one inside the other. The bands hold 1, 3, 5, 7, … dots, so the first 100 odd numbers fill the square.
Second way: cut the same grid along slanting lines. The slanting lines hold 1, 2, 3, …, 99, 100, 99, …, 2, 1 dots.
Two ways of cutting the same picture must give the same total.
In simple words: They are two different ways of counting the dots of one square.

Teacher's Note:
1. Draw a small 4 × 4 grid to show both cuttings.
2. Same picture, same total.

Question 18. (i) Find the 20th triangular number by adding the counting numbers up to 20.
(ii) Multiply this triangular number by 6 and add 1. To which sequence does your answer belong?
Answer:
(i) 1 + 2 + 3 + ⋯ + 20. Pair the numbers: 1 + 20 = 21, 2 + 19 = 21, and so on, giving 10 pairs of 21.
So the 20th triangular number is 10 × 21 = 210.
(ii) 6 × 210 + 1 = 1260 + 1 = 1261. Six times a triangular number plus 1 is always a hexagonal number, so 1261 belongs to the hexagonal numbers 1, 7, 19, 37, 61, 91, …
In simple words: Pairing gives 210, and 6 × 210 + 1 = 1261 is a hexagonal number.

Teacher's Note:
1. Pairing first and last makes long sums easy.
2. 1261 is the 21st hexagonal number.

Question 19. In the Koch Snowflake, every line segment of one shape is replaced by four smaller segments to make the next shape.
(i) Explain why the number of line segments gets multiplied by 4 at every step.
(ii) Find the number of line segments in the 6th shape of the sequence.
Answer:
(i) At each step every segment is removed and 4 smaller segments are drawn in its place. So each old segment turns into 4 new ones, and the total is multiplied by 4.
(ii) Start with 3 and multiply by 4 again and again: 3, 12, 48, 192, 768, 3072.
So the 6th shape has 3072 line segments (3 × 4 × 4 × 4 × 4 × 4).
In simple words: Each segment becomes four, so multiply by 4 each time to reach 3072.

Teacher's Note:
1. One-to-four replacement means × 4.
2. Count the steps: five multiplications for the 6th shape.

Question 20. Anita keeps adding hexagonal numbers one after another: 1, 1 + 7, 1 + 7 + 19, … She stops the moment her total becomes exactly 1000.
(i) How many hexagonal numbers did she add in all?
(ii) Which hexagonal number was the last one she added?
Answer:
(i) Adding hexagonal numbers gives the cubes: 1, 8, 27, 64, … Since 1000 = 10 × 10 × 10, she added 10 hexagonal numbers.
(ii) After 9 hexagonal numbers the total was 9 × 9 × 9 = 729.
So the last number added was 1000 − 729 = 271.
Check: the 9th triangular number is 45, and 6 × 45 + 1 = 271.
In simple words: 1000 is the 10th cube, so she added 10 numbers, the last being 1000 − 729 = 271.

Teacher's Note:
1. Running totals of hexagonal numbers are cubes.
2. Subtract consecutive cubes to get a hexagonal number.