Patterns and rules worksheet

Part A: Working out and filling in

1. 1. 8

   Count the 🍓 in one group of 4, then use 15 ÷ 4 to see how many whole groups there are and how many are left.

   Groups of 4, each with 2 🍓. 15 ÷ 4 = 3 r 3: 3 groups hold 6, and the 3 left over hold 2, so 8 in all.

2. 2. 45

   Look at how the dots are laid out: how many rows, and how many in each, in patterns 1, 2 and 3? Then picture pattern 9.

   The counts go up by 2, 3, … and each increase is 1 more. Quicker to look at the layout: pattern 9 has 9 rows, which add up to 9 + 8 + … + 1 = 45 dots.

3. 3. 121

   Look at the gaps from 49 to 64 and from 64 to 81. What pattern do the gaps follow?

   The gaps are 15, 17, 19, each 2 more, so the next gap is 21: 100 + 21 = 121. They are square numbers: 121 = 11 × 11.

4. 4. 27, 47

   From 17 to 37 is two steps. Find how much it goes up in all, then split that into two equal steps.

   From 17 to 37 is two steps adding 20, so each step adds 10: 17, 27, 37, 47, 57.

5. 5. 19

   Go from left to right and do only the first step to 8. What does it become?

   First step: 8 becomes 24. Second step: 24 becomes 19.

6. 6. 34

   Read the rule in the box and do it to 25.

   Put 25 into the machine and do what the box says: 34.

7. 7. 8, 8

   What is the gap between neighbours? That is what n is multiplied by. Then use the 1st number, 16, to find what is added.

   It adds 8 each time, so 8 × n; 8 × 1 = 8 needs 8 more to make 16: 8 × n + 8.

8. 8. 29

   Work backwards: 91 came from × 3 then + 4, so undo the + 4 first.

   Work backwards: 91 − 4 = 87, 87 ÷ 3 = 29: the 29th.

9. 9. 8

   List them in order: what is the smallest △ can be, and the largest? Remember ○ must be from 1 to 9 too.

   List them in order: △ is at least 1 and at most 8, each with one ○, so 8 pairs.

10. 10. 2

    First find when it is equal: what makes □ + 18 = 21? Then think whether □ must be bigger or smaller.

    First find when it is equal: □ + 18 = 21 when □ = 3. To be less than 21, □ must be less than 3: at most 2.

Part B: Choose, or true or false

1. 11. B. 🌸

   Find the group that repeats and how many are in it, then see which place in the group comes after the 7th.

   It repeats in groups of 2. After the 7th comes the 8th, which by the pattern is 🌸.

2. 12. A

   See how many sticks each new pattern adds and work out how many pattern 5 needs; then count each picture's sticks and check its shape.

   Each pattern adds 2 sticks, so pattern 5 needs 11, in the same shape.

Part C: Word problems

Marking: 2 for the working, 1 for the answer, 1 for the sentence.

1. 13. A. Friday

   The week repeats every 7 days. Find the remainder of 29 ÷ 7; whole weeks don't change the day.

   The week repeats every 7 days. 29 ÷ 7 = 4 r 1: after 4 weeks it is Thursday again; count on 1 from there to reach Friday.

2. 14. 10 houses

   How many more sticks does each house add? Work out what n houses use, then see how many 54 sticks can make.

   Each house adds 5 sticks: n houses use 5 × n + 1. 10 houses use 5 × 10 + 1 = 51; one more would need 56, more than 54.

3. 15. 29 tiles

   Write down the gaps between 1, 4 and 9 and see their pattern; or draw it to see where the new tiles go.

   1, 4, 9, 16, … go up by 3, 5, 7, …: from a side of 14 to 15 you add a row, a column and the corner: 14 + 14 + 1 = 29.

4. 16. 81

   Each time 1 more goes in, how much more comes out? Compare 27 and 33.

   Each 1 more in gives 6 more out: × 6. 3 × 6 is 9 short: + 9. 12 × 6 + 9 = 81.

5. 17. 23 ways

   The red bag holds at least 3. What is the most it can hold, keeping at least 3 for the blue bag?

   The red bag can hold 3, …, 25, and each has exactly one blue amount to match: 23 ways.

6. 18. 18 people

   More than 25 doesn't include exactly 25. What is the fewest people that is more than 25?

   “More than 25” doesn't include 25, so at least 26 are needed: 26 − 8 = 18.