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NCERT Solutions · Maths Ch 1
Patterns in Mathematics
25 questions · 4 exercises
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Ex 1.1
Ex 1.2
Ex 1.3
Ex 1.4
NCERT answers with full exam-style working, step by step. Tap a question to open its solution.
1
EXERCISE 1.1 · 2 QUESTIONS
What is Mathematics?
Q1
Can you think of other examples where mathematics helps us in our everyday lives?
(2 marks)
Q2
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.)
(2 marks)
2
EXERCISE 1.2 · 7 QUESTIONS
Number Sequences
Q3
Can you recognise the pattern in each of the sequences in Table 1? (All 1's; Counting numbers; Odd numbers; Even numbers; Triangular numbers 1, 3, 6, 10, 15, 21, 28; Squares 1, 4, 9, 16, 25, 36, 49; Cubes 1, 8, 27, 64, 125, 216; Virahānka numbers 1, 2, 3, 5, 8, 13, 21; Powers of 2; Powers of 3)
(3 marks)
Q4
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 the rule for forming the numbers in the sequence.
(3 marks)
Q5
Copy the pictorial representations of the number sequences in Table 2 in your notebook, and draw the next picture for each sequence.
(3 marks)
Q6
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?
(1 mark)
Q7
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!
(3 marks)
Q8
What would you call the sequence 1, 7, 19, 37? Draw these in your notebook. What is the next number in the sequence?
(3 marks)
Q9
Can you think of pictorial ways to visualise the sequence of powers of 2? Powers of 3?
(2 marks)
3
EXERCISE 1.3 · 10 QUESTIONS
Relations among Number Sequences
Q10
By drawing a similar picture, what is the sum of the first 10 odd numbers? What is the sum of the first 100 odd numbers?
(2 marks)
Q11
Can you find a pictorial explanation for why adding counting numbers up and down, i.e., 1, 1 + 2 + 1, 1 + 2 + 3 + 2 + 1, … gives square numbers?
(3 marks)
Q12
By imagining a large version of your picture, or drawing it partially, what will be the value of 1 + 2 + 3 + … + 99 + 100 + 99 + … + 3 + 2 + 1?
(2 marks)
Q13
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?
(2 marks)
Q14
Which sequence do you get when you start to add the counting numbers up? Can you give a smaller pictorial explanation?
(3 marks)
Q15
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?
(2 marks)
Q16
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?
(3 marks)
Q17
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?
(3 marks)
Q18
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?
(2 marks)
Q19
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?
(2 marks)
4
EXERCISE 1.4 · 6 QUESTIONS
Patterns in Shapes
Q20
Can you recognise the pattern in each of the sequences in Table 3 (shape sequences: Regular Polygons, Complete Graphs, Stacked Squares, Stacked Triangles, Koch Snowflake)? Redraw each sequence and describe the rule for forming the shapes.
(3 marks)
Q21
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? Do you get the same number sequence? Why?
(2 marks)
Q22
Count the number of lines in each shape in the sequence of Complete Graphs. Which number sequence do you get? Can you explain why?
(2 marks)
Q23
How many little squares are there in each shape of the sequence of Stacked Squares? Which number sequence does this give? Can you explain why?
(2 marks)
Q24
How many little triangles 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?)
(3 marks)
Q25
To get from one shape to the next 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?
(2 marks)