Question 1
Give examples of how you use mathematics every day.
Steps only
Examples: check change, measure ingredients, plan a journey and tell the time.
Answer
Possible examples: money, measurement, time and counting.
CLASS 6 · GANITA PRAKASH · CHAPTER 1
Patterns in Mathematics
Spot number and shape patterns. Read short rules and calculations, explore original diagrams, and choose an explanation to understand why a pattern works.
60 worked answers
60 worked answers · short steps first, explanations when you need them
Figure it Out: mathematics around us
Textbook pages 2
Question 2
Give examples of mathematics helping people create useful things.
Steps only
Bridges: lengths, angles and loads.
Computers: numbers and logical rules.
Calendars: repeating time cycles.
Answer
Possible examples include bridges, computers and calendars.
Figure it Out: number sequences
Textbook pages 3
Question 1 · All 1s
Identify the pattern in the all 1s row of Table 1.
Steps only
Every term is 1.
Answer
Every term is 1.
Question 2 · All 1s
Write the next three terms after 1, 1, 1, 1, 1, 1, 1.
Steps only
Every term is 1.
1, 1, 1, 1, 1, 1, 1, 1, 1, 1
Answer
1, 1, 1
Question 1 · Counting numbers
Identify the pattern in the counting numbers row of Table 1.
Steps only
Add 1 each time.
Answer
Add 1 each time.
Question 2 · Counting numbers
Write the next three terms after 1, 2, 3, 4, 5, 6, 7.
Steps only
Add 1 each time.
1, 2, 3, 4, 5, 6, 7, 8, 9, 10
Answer
8, 9, 10
Question 1 · Odd numbers
Identify the pattern in the odd numbers row of Table 1.
Steps only
Start at 1 and add 2 each time.
Answer
Start at 1 and add 2 each time.
Question 2 · Odd numbers
Write the next three terms after 1, 3, 5, 7, 9, 11, 13.
Steps only
Start at 1 and add 2 each time.
1, 3, 5, 7, 9, 11, 13, 15, 17, 19
Answer
15, 17, 19
Question 1 · Even numbers
Identify the pattern in the even numbers row of Table 1.
Steps only
Start at 2 and add 2 each time.
Answer
Start at 2 and add 2 each time.
Question 2 · Even numbers
Write the next three terms after 2, 4, 6, 8, 10, 12, 14.
Steps only
Start at 2 and add 2 each time.
2, 4, 6, 8, 10, 12, 14, 16, 18, 20
Answer
16, 18, 20
Question 1 · Triangular numbers
Identify the pattern in the triangular numbers row of Table 1.
Steps only
Add 2, then 3, then 4, and continue adding the next counting number.
Answer
Add 2, then 3, then 4, and continue adding the next counting number.
Question 2 · Triangular numbers
Write the next three terms after 1, 3, 6, 10, 15, 21, 28.
Steps only
Add 2, then 3, then 4, and continue adding the next counting number.
1, 3, 6, 10, 15, 21, 28, 36, 45, 55
Answer
36, 45, 55
Question 1 · Squares
Identify the pattern in the squares row of Table 1.
Steps only
Multiply each counting number by itself.
Answer
Multiply each counting number by itself.
Question 2 · Squares
Write the next three terms after 1, 4, 9, 16, 25, 36, 49.
Steps only
Multiply each counting number by itself.
1, 4, 9, 16, 25, 36, 49, 64, 81, 100
Answer
64, 81, 100
Question 1 · Cubes
Identify the pattern in the cubes row of Table 1.
Steps only
Multiply each counting number by itself three times.
Answer
Multiply each counting number by itself three times.
Question 2 · Cubes
Write the next three terms after 1, 8, 27, 64, 125, 216.
Steps only
Multiply each counting number by itself three times.
1, 8, 27, 64, 125, 216, 343, 512, 729
Answer
343, 512, 729
Question 1 · Virahanka numbers
Identify the pattern in the virahanka numbers row of Table 1.
Steps only
After the first two terms, add the previous two terms.
Answer
After the first two terms, add the previous two terms.
Question 2 · Virahanka numbers
Write the next three terms after 1, 2, 3, 5, 8, 13, 21.
Steps only
After the first two terms, add the previous two terms.
1, 2, 3, 5, 8, 13, 21, 34, 55, 89
Answer
34, 55, 89
Question 1 · Powers of 2
Identify the pattern in the powers of 2 row of Table 1.
Steps only
Start at 1 and multiply by 2 each time.
Answer
Start at 1 and multiply by 2 each time.
Question 2 · Powers of 2
Write the next three terms after 1, 2, 4, 8, 16, 32, 64.
Steps only
Start at 1 and multiply by 2 each time.
1, 2, 4, 8, 16, 32, 64, 128, 256, 512
Answer
128, 256, 512
Question 1 · Powers of 3
Identify the pattern in the powers of 3 row of Table 1.
Steps only
Start at 1 and multiply by 3 each time.
Answer
Start at 1 and multiply by 3 each time.
Question 2 · Powers of 3
Write the next three terms after 1, 3, 9, 27, 81, 243, 729.
Steps only
Start at 1 and multiply by 3 each time.
1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683
Answer
2187, 6561, 19683
Figure it Out: visual number patterns
Textbook pages 5
Question 1 · All 1s
Draw the next picture in the all 1s row of Table 2.
Steps only
Next count = 1
Use 1 dots.
1 dots
Answer
The next picture represents 1.
Question 1 · Counting numbers
Draw the next picture in the counting numbers row of Table 2.
Steps only
Next count = 6
Use 6 dots.
6 dots
Answer
The next picture represents 6.
Question 1 · Odd numbers
Draw the next picture in the odd numbers row of Table 2.
Steps only
Next count = 11
Use 11 dots.
11 dots
Answer
The next picture represents 11.
Question 1 · Even numbers
Draw the next picture in the even numbers row of Table 2.
Steps only
Next count = 12
Use 12 dots.
12 dots
Answer
The next picture represents 12.
Question 1 · Triangular numbers
Draw the next picture in the triangular numbers row of Table 2.
Steps only
Next count = 21
Use 6 rows.
21 dots
Answer
The next picture represents 21.
Question 1 · Squares
Draw the next picture in the squares row of Table 2.
Steps only
Next count = 36
Use 6 units along each side.
36 dots
Answer
The next picture represents 36.
Question 1 · Cubes
Draw the next picture in the cubes row of Table 2.
Steps only
Next count = 216
Use 6 units along each side.
6 × 6 × 6: 216 small cubes
Answer
The next picture represents 216.
Question 2
Why are the numbers called triangular numbers, squares and cubes?
Steps only
Triangles: rows of 1, 2, 3, … dots.
Squares: equal numbers of rows and columns.
Cubes: equal numbers of units in length, breadth and height.
6 dots
9 dots
3 × 3 × 3: 27 small cubes
Answer
Their names describe the shapes that their units form.
Question 3
Show that 36 is both triangular and square.
Steps only
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36
6 × 6 = 36
36 dots
36 dots
Answer
36 can form an eight-row triangle or a 6 by 6 square.
Question 4
Name the pattern 1, 7, 19, 37 and find its next term.
Steps only
Differences: 6, 12, 18; next difference = 24
37 + 24 = 61
1 dots
7 dots
19 dots
37 dots
61 dots
Answer
Centred hexagonal numbers; next term 61.
Question 5
Give a pictorial rule for powers of 2 and powers of 3.
Steps only
Double the number of dots each time: 1, 2, 4, 8, …
Triple the number of dots each time: 1, 3, 9, …
1 dots
2 dots
4 dots
8 dots
1 dots
3 dots
9 dots
Answer
Use doubling for powers of 2 and tripling for powers of 3.
Odd-number sums
Textbook pages 7
Question First 10 odd numbers
Find the sum of the first 10 odd positive integers.
Steps only
Sum = 10 × 10 = 100
Answer
100
Question First 100 odd numbers
Find the sum of the first 100 odd positive integers.
Steps only
Sum = 100 × 100 = 10000
Answer
10000
Figure it Out: relations between sequences
Textbook pages 8–9
Question 1
Use a picture to explain 1; 1 + 2 + 1; 1 + 2 + 3 + 2 + 1; … .
Steps only
1 = 1²
1 + 2 + 1 = 2²
1 + 2 + 3 + 2 + 1 = 3²
4 dots
9 dots
16 dots
Answer
These are square numbers.
Question 2
Evaluate 1 + 2 + … + 99 + 100 + 99 + … + 2 + 1.
Steps only
Sum = 100² = 10,000
Answer
10,000
Question 3 · Adding ones
What sequence comes from cumulative sums of 1, 1, 1, …?
Steps only
1; 1 + 1 = 2; 1 + 1 + 1 = 3; …
Answer
Counting numbers: 1, 2, 3, 4, … .
Question 3 · Up and down
What sequence comes from adding the all-ones sequence up and down?
Steps only
1; 1 + 1 + 1 = 3; 1 + 1 + 1 + 1 + 1 = 5; …
Answer
Odd numbers: 1, 3, 5, 7, … .
Question 4
What sequence comes from adding 1, then 1 + 2, then 1 + 2 + 3, and so on?
Steps only
1, 3, 6, 10, 15, …
1 dots
3 dots
6 dots
10 dots
15 dots
Answer
Triangular numbers.
Question 5
Add consecutive triangular numbers: 1 + 3, 3 + 6, 6 + 10, … .
Steps only
1 + 3 = 4
3 + 6 = 9
6 + 10 = 16
10 + 15 = 25
6 dots
10 dots
16 dots
Answer
Square numbers: 4, 9, 16, 25, … .
Question 6
Find cumulative sums of powers of 2, then add 1 to each sum.
Steps only
Sums: 1, 3, 7, 15, 31, …
Add 1: 2, 4, 8, 16, 32, …
Answer
Adding 1 gives the next powers of 2.
Question 7
Multiply each triangular number by 6 and add 1. What pattern results?
Steps only
6 × 1 + 1 = 7
6 × 3 + 1 = 19
6 × 6 + 1 = 37
6 × 10 + 1 = 61
7 dots
19 dots
37 dots
61 dots
Answer
Centred hexagonal numbers: 7, 19, 37, 61, 91, … .
Question 8
Find 1; 1 + 7; 1 + 7 + 19; 1 + 7 + 19 + 37; … .
Steps only
1 = 1³
1 + 7 = 8 = 2³
1 + 7 + 19 = 27 = 3³
1 + 7 + 19 + 37 = 64 = 4³
1 × 1 × 1: 1 small cubes
2 × 2 × 2: 8 small cubes
3 × 3 × 3: 27 small cubes
4 × 4 × 4: 64 small cubes
Answer
Cube numbers: 1, 8, 27, 64, 125, … .
Question 9
Give your own relation between sequences in Table 1.
Steps only
Add 1 to each odd number.
1 + 1 = 2; 3 + 1 = 4; 5 + 1 = 6; …
Answer
One example: odd numbers plus 1 give even numbers.
Figure it Out: shape sequences
Textbook pages 10–11
Question 1 · Regular polygons
Describe the rule for the regular polygons sequence in Table 3.
Steps only
Add one side and one vertex.
8 sides and 8 vertices
9 sides and 9 vertices
10 sides and 10 vertices
Answer
Add one side and one vertex.
Question 2 · Regular polygons
How do you draw the next picture in the regular polygons sequence?
Steps only
Add one side and one vertex.
11 sides: an undecagon.
11 sides and 11 vertices
Answer
11 sides: an undecagon.
Question 1 · Complete graphs
Describe the rule for the complete graphs sequence in Table 3.
Steps only
Add a vertex and connect it to every existing vertex.
4 vertices: 6 lines
5 vertices: 10 lines
6 vertices: 15 lines
Answer
Add a vertex and connect it to every existing vertex.
Question 2 · Complete graphs
How do you draw the next picture in the complete graphs sequence?
Steps only
Add a vertex and connect it to every existing vertex.
7 vertices with 21 connecting lines.
7 vertices: 21 lines
Answer
7 vertices with 21 connecting lines.
Question 1 · Stacked squares
Describe the rule for the stacked squares sequence in Table 3.
Steps only
Increase the number of rows and columns by one.
9 small squares
16 small squares
25 small squares
Answer
Increase the number of rows and columns by one.
Question 2 · Stacked squares
How do you draw the next picture in the stacked squares sequence?
Steps only
Increase the number of rows and columns by one.
A 6 by 6 arrangement: 36 small squares.
36 small squares
Answer
A 6 by 6 arrangement: 36 small squares.
Question 1 · Stacked triangles
Describe the rule for the stacked triangles sequence in Table 3.
Steps only
Add the next row containing the next odd number of small triangles.
9 small triangles
16 small triangles
25 small triangles
Answer
Add the next row containing the next odd number of small triangles.
Question 2 · Stacked triangles
How do you draw the next picture in the stacked triangles sequence?
Steps only
Add the next row containing the next odd number of small triangles.
6 rows containing 36 small triangles.
36 small triangles
Answer
6 rows containing 36 small triangles.
Question 1 · Koch snowflake
Describe the rule for the koch snowflake sequence in Table 3.
Steps only
Replace each segment by four shorter segments forming an outward bump.
Stage 1: 12 segments
Stage 2: 48 segments
Stage 3: 192 segments
Answer
Replace each segment by four shorter segments forming an outward bump.
Question 2 · Koch snowflake
How do you draw the next picture in the koch snowflake sequence?
Steps only
Replace each segment by four shorter segments forming an outward bump.
The next stage has 768 segments.
Stage 4: 768 segments
Answer
The next stage has 768 segments.
Figure it Out: count parts of shapes
Textbook pages 11–12
Question 1
Count sides and vertices in the regular polygon sequence.
Steps only
Sides: 3, 4, 5, 6, 7, …
Vertices: 3, 4, 5, 6, 7, …
3 sides and 3 vertices
4 sides and 4 vertices
5 sides and 5 vertices
6 sides and 6 vertices
Answer
The two counts are equal: counting numbers starting at 3.
Question 2
Count lines in complete graphs with 2, 3, 4, 5 and 6 vertices.
Steps only
1, 3, 6, 10, 15
A new vertex connects to every previous vertex.
2 vertices: 1 lines
3 vertices: 3 lines
4 vertices: 6 lines
5 vertices: 10 lines
6 vertices: 15 lines
Answer
Triangular numbers: 1, 3, 6, 10, 15.
Question 3
Count the small squares in the stacked-square sequence.
Steps only
1², 2², 3², 4², 5²
= 1, 4, 9, 16, 25
1 small squares
4 small squares
9 small squares
16 small squares
25 small squares
Answer
Square numbers.
Question 4
Count the small triangles in the stacked-triangle sequence.
Steps only
Row counts: 1, 3, 5, 7, …
Totals: 1, 4, 9, 16, 25, …
1 small triangles
4 small triangles
9 small triangles
16 small triangles
25 small triangles
Answer
Square numbers.
Question 5
Count the line segments after successive Koch-snowflake replacements.
Steps only
3, 12, 48, 192, 768, …
Multiply by 4 at each replacement.
Stage 0: 3 segments
Stage 1: 12 segments
Stage 2: 48 segments
Stage 3: 192 segments
Answer
3 times powers of 4.