Year 6 Arithmetic (SATs): Systematic tables and combinations
Part of the Year 6 Arithmetic (SATs) algebra and formulae chapter. Chapter overview · whole chapter on one page
Systematic tables and combinations
Rule
To list all the combinations of two variables, go through one variable in order (0, 1, 2, ...) and find the other variable each time. A table keeps the list in order, so no possibility is missed or repeated.
Key ideas
- Go through one variable in order, and find the other each time.
- A systematic table shows every possibility once.
- Check each pair in the original equation.
Method. Go through the number of 5p coins in order, and find the 2p coins each time.
- Try 0 five-pence coins. Then 2p coins make 10p, so 5 coins.(5, 0)
- Try 1 five-pence coin. Then 5p is left, which is not a multiple of 2.no
- Try 2 five-pence coins. Then no money is left, so 0 two-pence coins.(0, 2)
- Write the answer. 2 ways
Check. There are 2 ways. ✓
Method. Make a table: try y = 0, 1, 2, 3 and find x.
- y = 0. x = 6 - 0 = 6.(6, 0)
- y = 1. x = 6 - 2 = 4.(4, 1)
- y = 2. x = 6 - 4 = 2.(2, 2)
- y = 3. x = 6 - 6 = 0.(0, 3)
- Write the answer. (6, 0), (4, 1), (2, 2), (0, 3)
Check. Each pair gives 6 when put into x + 2y. ✓
Method. Each shirt can go with each pair of trousers, so multiply.
- List the choices. Shirts: 3. Trousers: 2.3 and 2
- Multiply. There are 3 × 2 = 6 outfits.3 × 2 = 6
- Check with a table. Each of the 3 shirts can be paired with 2 trousers.6 pairs
- Write the answer. 6
Check. The table has 6 cells. ✓
Exercise 6E Practice: systematic tables and combinations
Non-calculatorMethod. Use a table of the 5p coins, then find the 2p coins.
- Try the 5p coins. The 5p coins can be 0, 1 or 2.0, 1, 2
- Find the 2p coins. (6, 0), (1, 2).(6 and 0), (1 and 2)
- Count. There are 2 ways.2
- Write the answer. 2 ways
Check. 6 × 2 = 12 and 1 × 2 + 2 × 5 = 12. ✓
Method. Multiply the choices.
- Multiply. 4 × 3 = 12.12
- Write the answer. 12
Check. Each of 4 shirts with each of 3 shorts. ✓
Method. Find y for each x.
- x = 0. y = 8 - 0 = 8.(0, 8)
- x = 1. y = 7.(1, 7)
- x = 2 and 3. y = 6 and 5.(2, 6), (3, 5)
- Write the answer. (0, 8), (1, 7), (2, 6), (3, 5)
Check. Each pair adds to 8. ✓
Method. Use a table of the cakes, then find the sweets.
- Try 0 cakes. 12 ÷ 2 = 6 sweets.(0, 6)
- Try 1 cake. 8 left, so 4 sweets.(1, 4)
- Try 2 cakes. 4 left, so 2 sweets.(2, 2)
- Try 3 cakes. No money left, so 0 sweets.(3, 0)
- Write the answer. 4 ways
Check. Each way adds to 12p. ✓
Method. Try each number of adults, then find the children.
- Adults 0, 1, 2, 3. Each adult costs £4, so there are 3 adults at most.0, 1, 2, 3
- Find the children. 0 adults: 6 children, 1 adults: 4 children, 2 adults: 2 children, 3 adults: 0 children(0, 6), (1, 4), (2, 2), (3, 0)
- Write the answer. (0, 6), (1, 4), (2, 2), (3, 0)
Check. Each pair pays exactly £12. ✓
Method. List them in a systematic way: start with 1, then 2, then 3.
- Start with 1 as the tens digit. 12 and 13.12, 13
- Start with 2 as the tens digit. 21 and 23.21, 23
- Start with 3 as the tens digit. 31 and 32.31, 32
- Write the answer. 6
Check. 6 numbers in total. ✓
Method. Try each x, then find y.
- x = 1, 2, 3, 4. y = 4, 3, 2, 1.(1, 4), (2, 3), (3, 2), (4, 1)
- Write the answer. 4
Check. Each pair adds to 5. ✓
Method. Test each choice against the question.
- Read what the question asks. Decide what you need to find or check.
- Test the choice. 2 + 2 × 4 = 2 + 8 = 10, which is not 8, so (2, 4) is not a solution.(2, 4) is right
- Reject the other choices. (8, 0): 8 + 0 = 8. (4, 2): 4 + 4 = 8. (6, 1): 6 + 2 = 8. All these work.(8, 0) and (4, 2) and (6, 1) are not right
- Write the answer. (2, 4)
Check. (2, 4) is right. ✓
Method. Go through the values in order.
- List in order. Start with the smallest value and go up.in order
- Check. Each row gives one possibility; the table shows them all.all shown
- Write the answer. It lists each value in order, so each possibility appears once and none is missed.
Check. The table lists every possibility once. ✓