Page No. 2 – Q1. Can you think of other examples where mathematics helps us in our everyday lives?
Mathematics helps us in budgeting and shopping (calculating discounts, comparing prices), cooking and baking (measurements and proportions), time management (planning schedules, calculating travel time), and recognising patterns in weather, traffic, or daily routines.
Page No. 2 – Q2. How has mathematics helped propel humanity forward?
Mathematics has propelled humanity forward through economic development, eradication of diseases, and increase in living standards. It is used in scientific experiments, building structures, running economies, and creating technology like TVs, phones, computers, trains, and planes.
Page No. 3 – Q1. Can you recognize the pattern in each of the sequences in Table 1?
Yes. 1,1,1… is all 1’s; 1,2,3… are counting numbers; 1,3,5… are odd numbers; 2,4,6… are even numbers; 1,3,6,10… are triangular numbers; 1,4,9… are squares; 1,8,27… are cubes; 1,2,3,5,8… are Virahanka (Fibonacci) numbers.
Page No. 3 – Q2. Rewrite each sequence along with the next three numbers and write the rule.
- All 1’s: next three are 1, 1, 1 — every number is 1.
- Counting numbers: next three are 8, 9, 10 — add 1 to the previous number.
- Odd numbers: next three are 15, 17, 19 — add 2 to the previous number.
- Even numbers: next three are 16, 18, 20 — add 2 to the previous number.
- Triangular numbers: next three are 36, 45, 55 — add the next natural number.
- Squares: next three are 64, 81, 100 — square the next natural number.
- Cubes: next three are 343, 512, 729 — cube the next natural number.
- Virahanka numbers: next three are 34, 55, 89 — add the last two numbers.
- Powers of 2: next three are 128, 256, 512 — multiply the last number by 2.
- Powers of 3: next three are 2187, 6561, 19683 — multiply the last number by 3.
Page No. 5–6 – Q1. Copy the pictorial representations and draw the next picture for each sequence.
The next picture for each sequence continues the pattern by adding one more row or layer to the existing shape as per the respective sequence rule.
Page No. 5–6 – Q2(a). Why are 1, 3, 6, 10, 15… called triangular numbers?
They are called triangular numbers because that many dots can be arranged to form a triangle.
Q2(b). Why are 1, 4, 9, 16, 25… called square numbers?
They are called square numbers because that many dots can be arranged to form a perfect square.
Q2(c). Why are 1, 8, 27, 64, 125… called cubes?
They are called cubes because that many unit cubes can be arranged to form a larger cube.
Page No. 5–6 – Q3. Show that 36 is both a triangular number and a square number.
36 dots can be arranged in a triangular shape (rows of 1+2+3+4+5+6+7+8 = 36) and also in a 6×6 square grid. Another example is 9, which can be arranged as a 3×3 square or in a cross shape.
Page No. 5–6 – Q4. What are hexagonal numbers? What is the next number?
The sequence is 1, 7, 19, 37, 61… These are called hexagonal numbers. The next number in the sequence is 61.
Page No. 5–6 – Q5. Can you think of pictorial ways to visualise Powers of 2 and Powers of 3?
Powers of 2 can be visualised by repeatedly doubling a set of dots or squares (1 → 2 → 4 → 8…). Powers of 3 can be visualised similarly by tripling each time (1 → 3 → 9 → 27…).
Page No. 7 – Q1. What is the sum of the first 10 odd numbers?
1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 = 100 (= 10²)
Page No. 7 – Q2. What is the sum of the first 100 odd numbers?
Adding 100 consecutive odd numbers gives 100² = 10,000
Page No. 8–9 – Q1. Can you find a pictorial explanation for why adding counting numbers up and down gives square numbers?
1 = 1²; 1+2+1 = 4 = 2²; 1+2+3+2+1 = 9 = 3²; 1+2+3+4+3+2+1 = 16 = 4², and so on. Each forms a symmetric square pattern around a central line.
Page No. 8–9 – Q2. What is the value of 1 + 2 + 3 + … + 100 + … + 3 + 2 + 1?
= 100² = 10,000
Page No. 8–9 – Q3. Which sequence do you get when you add the All 1’s sequence up and down?
Adding up gives counting numbers. Adding up and down gives odd numbers.
Page No. 8–9 – Q4. Which sequence do you get when you add the counting numbers up?
You get the triangular numbers: 1, 3, 6, 10, 15…
Page No. 8–9 – Q5. What happens when you add pairs of consecutive triangular numbers?
1+3=4, 3+6=9, 6+10=16, 10+15=25 — you get the sequence of perfect squares.
Page No. 8–9 – Q6. What happens when you add powers of 2 starting with 1, and then add 1 to each result?
Sums: 1, 3, 7, 15, 31… Adding 1 gives: 2, 4, 8, 16, 32 — the powers of 2. This happens because adding 1 completes the next power of 2.
Page No. 8–9 – Q7. What happens when you multiply triangular numbers by 6 and add 1?
1×6+1=7, 3×6+1=19, 6×6+1=37, 10×6+1=61, 15×6+1=91 — you get the sequence 7, 19, 37, 61, 91…
Page No. 8–9 – Q8. What happens when you add up hexagonal numbers cumulatively?
1=1, 1+7=8, 1+7+19=27, 1+7+19+37=64 — you get the sequence of cubic numbers (1³, 2³, 3³, 4³…).
Page No. 8–9 – Q9. Find your own patterns among the sequences in Table 1.
Squares are formed by adding consecutive odd numbers: 1=1², 1+3=2², 1+3+5=3², 1+3+5+7=4², and so on.
Page No. 11 – Q1. Can you recognise the pattern in each sequence in Table 3?
Yes, each sequence in Table 3 follows a regular geometric pattern where each shape is built upon the previous one by adding sides, corners, or layers.
Page No. 11 – Q2. Redraw each sequence and describe the rule.
Regular Polygons increase by one side each time (triangle → square → pentagon…). Complete graphs add one new vertex connected to all others. Stacked squares and triangles grow by adding one more layer each time.
Page No. 11–12 – Q1. Count sides and corners of Regular Polygons. Which number sequence?
Both the number of sides and corners follow the sequence 3, 4, 5, 6, 7… (counting numbers starting from 3). This is because in any regular polygon, the number of sides equals the number of corners.
Page No. 11–12 – Q2. Count lines in complete graphs. Which number sequence?
The number of lines is 1, 3, 6, 10, 15… — the triangular numbers. This is because the number of edges in a complete graph with n vertices = n(n−1)/2.
Page No. 11–12 – Q3. How many little squares in Stacked Squares? Which sequence?
There are 1, 4, 9, 16, 25… little squares — the sequence of perfect squares. Each stacked square adds rows symmetrically: 1; 1+2+1=4; 1+2+3+2+1=9, and so on.
Page No. 11–12 – Q4. How many little triangles in Stacked Triangles? Which sequence?
There are 1, 4, 9, 16… little triangles — again the sequence of perfect squares, because each row adds triangles in increasing order forming a perfect square total.
Page No. 11–12 – Q5. How many line segments are in each Koch Snowflake shape? Which sequence?
The number of line segments is 3, 12, 48, 192, 768… — this is 3 times the powers of 4 (3×4⁰, 3×4¹, 3×4², …), since each segment is replaced by 4 new segments at every step.