This is the marking copy and it carries every answer. Her sheet is the other file — same numbering, same margin codes, so you can mark straight down the page.
Write the missing number in the number pattern below: 7648, 7798, 7948, __________, 8248
answer 8098
method Find the step first: 7798 − 7648 = 150, and 7948 − 7798 = 150 as well. So the pattern adds 150 each time: 7948 + 150 = 8098. (Check forwards: 8098 + 150 = 8248 ✓)
watch Guessing from the last gap only. Check the step between at least two pairs, then check your answer reaches the number after it.
A road divider is painted in repeating segments: white 2 m, black 5 m, white 2 m, black 5 m, and so on, beginning and ending with white. If the road divider is 149 m long, how many white segments are there?
[white 2m][ black 5m ][white 2m][ black 5m ][white 2m] ... |<----- one repeat = 7 m ----->| the whole divider is 149 m
answer 22
method One repeat is a white and a black together: 2 + 5 = 7 m. In 149 m: 149 ÷ 7 = 21 repeats with 2 m left over. The 21 repeats give 21 white segments, and the last 2 m is exactly one more white segment. So 21 + 1 = 22.
watch Answering 21 and ignoring the 2 m remainder — which is precisely the length of one more white segment, and is why the divider ends white.
Study the pattern of hexagons made from sticks: Pattern 1 is one hexagon (6 sticks), Pattern 2 is two hexagons sharing a side (11 sticks), Pattern 3 is three in a row (16 sticks). In a particular pattern, 206 sticks are used. What is the pattern number?
Pattern 1 Pattern 2 Pattern 3
/ \ / \ / \ / \ / \ / \
| | | | | | | | |
\ / \ / \ / \ / \ / \ /
6 sticks 11 sticks 16 sticksanswer 41
method The first hexagon takes 6 sticks. Every hexagon after that shares a side with the one before, so it only needs 5 more. Sticks = 6 + 5 × (pattern number − 1). Setting that to 206: 206 − 6 = 200 extra sticks, and 200 ÷ 5 = 40 more hexagons, so the pattern number is 40 + 1 = 41.
watch Dividing 206 by 6 because each hexagon "has 6 sides". After the first, each new hexagon adds only 5 sticks — the shared side is already there.
Bob wrote letters in a repeating pattern: Z E S T Z E S T Z E S T Z E … How many letters ‘Z’ and ‘T’ are there altogether if there is a total of 87 letters in the whole series?
Z E S T Z E S T Z E S T Z E ... |<- one repeat = 4 letters ->| 87 letters altogether
answer 43
method One repeat is Z E S T — 4 letters. In 87 letters: 87 ÷ 4 = 21 repeats with 3 left over. The 21 repeats give 21 Zs and 21 Ts. The 3 leftover letters restart the pattern as Z, E, S — one more Z and no more T. So Z = 22 and T = 21, giving 22 + 21 = 43.
watch Splitting the leftovers evenly, or counting a T in the remainder. The leftover always starts from the BEGINNING of the pattern, so it reaches Z before it reaches T.
Sally arranges a pattern with sticks and coins. Pattern 1 uses 6 coins and 7 sticks (13 in all), Pattern 2 uses 9 coins and 12 sticks (21 in all), Pattern 3 uses 12 coins and 17 sticks (29 in all). There is a total of 93 coins and sticks in one of the patterns. What is the Pattern Number?
Pattern | coins | sticks | total
--------+-------+--------+------
1 | 6 | 7 | 13
2 | 9 | 12 | 21
3 | 12 | 17 | 29
4 | ? | ? | ?answer 11
method Look at the totals: 13, 21, 29. Each one is 8 more than the last. So the total is 13 + 8 × (pattern number − 1). Setting that to 93: 93 − 13 = 80, and 80 ÷ 8 = 10 more steps, so the pattern number is 10 + 1 = 11.
watch Dividing 93 by 8 and answering 11 remainder 5 without checking, or forgetting the +1 because Pattern 1 is the starting point, not a step.
Study the pattern. What is the missing number in the box? star × star = 49; moon + moon + moon + moon = 100; star × moon = ?
star × star = 49 moon + moon + moon + moon = 100 star × moon = ?
answer 175
method From the first line, a number times itself is 49, so the star is 7. From the second, four moons make 100, so the moon is 100 ÷ 4 = 25. Then star × moon = 7 × 25 = 175.
watch Reading the second line as 4 × moon = 100 and then halving instead of dividing by 4, or taking the star as 24.5 (half of 49) instead of the number that multiplies by ITSELF.
A length of sticky tape is made up of repeated designs: a 5 cm striped band, then a 3 cm checked band, then a 4 cm grid band, repeating. The sticky tape is 80 cm long. How many checked bands are there altogether?
| striped | checked | grid | striped | checked | grid | ... |<- 5 cm ->|<- 3 cm ->|<-4cm->| |<-------- one repeat = 12 cm -------->| the whole tape is 80 cm
answer 7
method One full repeat is 5 + 3 + 4 = 12 cm. In 80 cm: 80 ÷ 12 = 6 repeats with 8 cm left over. Those 6 repeats give 6 checked bands. The leftover 8 cm starts the next repeat: 5 cm of stripes, then 3 cm of checks — exactly enough for one more complete checked band. So 6 + 1 = 7.
watch Answering 6 and ignoring the leftover. Always ask what the remaining centimetres are long enough to reach.
Some circles and triangles are arranged in a repeating pattern: triangle, circle, circle, circle, triangle, circle, circle, circle, triangle, … How many circles are there if there are 122 shapes?
/\ O O O /\ O O O /\ ... |<- one repeat = 4 shapes ->| (1 triangle and 3 circles)
answer 91
method One repeat is a triangle and 3 circles — 4 shapes. In 122 shapes: 122 ÷ 4 = 30 repeats with 2 shapes left over. The 30 repeats give 30 × 3 = 90 circles. The 2 leftover shapes continue the pattern: a triangle, then a circle. So 90 + 1 = 91 circles.
watch Answering 90 and forgetting the leftovers, or counting the leftover 2 shapes as 2 circles. The pattern always restarts with a TRIANGLE.
Study the staircase pattern: Figure 1 has 2 squares along the bottom and 1 above the right-hand one (3 squares). Figure 2 has 3 along the bottom, then 2, then 1 (6 squares). Figure 3 has 4, then 3, then 2, then 1 (10 squares). Find the total number of squares needed to form Figure 50.
Figure 1 Figure 2 Figure 3
[]
[] [] [][]
[][] [][] [][][]
[][][] [][][][]
3 6 10 squaresanswer 1326
method Each figure is a staircase. Figure 1 is 2 + 1 = 3, Figure 2 is 3 + 2 + 1 = 6, Figure 3 is 4 + 3 + 2 + 1 = 10. So Figure n counts down from (n + 1) to 1. Figure 50 is 51 + 50 + 49 + … + 1. Pair the ends: 51 + 1 = 52, 50 + 2 = 52, and so on — there are 51 numbers, giving 51 × 52 ÷ 2 = 1326.
watch Counting down from 50 instead of 51. Figure 1's bottom row has 2 squares, not 1, so the bottom row of Figure 50 has 51.
Jamie decorated a square classroom of side 6 m 40 cm. She tied balloons in a repeating pattern on a string and hung the string around the classroom once. In every 80 cm of string the pattern holds 4 small balloons (with big balloons between them). How many small balloons did she use to decorate the 4 sides of the classroom?
(BIG) o o (BIG) o o (BIG) |<--------- 80 cm --------->| o = small balloon · 4 small balloons in every 80 cm
answer 128
method Find how much string is needed: the classroom is a square of side 6 m 40 cm = 640 cm, so once round is 4 × 640 = 2560 cm. The pattern repeats every 80 cm, so it repeats 2560 ÷ 80 = 32 times. Each repeat carries 4 small balloons, so she used 32 × 4 = 128 small balloons.
watch Working out the balloons for one side and forgetting to multiply by 4, or counting the big balloons too. The question asks only for the SMALL ones.