Year 6 Maths reasoning (SATs) Algebra and sequences: check-up and practice test questions

Method. Find the difference between terms, then apply it again.

1. Find the rule. Each term is the one before with add 7.
   16 - 9 = 7
2. Apply the rule to the last term. Take the last term, 30, and add 7.
   30 + 7 = 37
3. Write the answer.
   37

Check. 37 - 30 = 7. ✓

Watch out. Check the difference between every pair of terms, not just the first two.

Method. Find the difference between the known terms, then count to the gap.

1. Find the difference. The terms go subtract 4 each time.
   difference = -4
2. Work out the missing term. Count from a known term to the gap.
   12 - 12 = 0
3. Write the answer.
   0

Check. 0 is in the same pattern: the difference is -4. ✓

Watch out. The term is missing from the list, so count the steps carefully.

Method. Start at the first term and add the difference for each step after it.

1. Find the first term and the difference. The first term is 7, and the difference is 5.
   7 and 5
2. Count the steps to the nth term. The 15th term is 14 steps from the first term.
   15 - 1 = 14 steps
3. Work out the term. Start at 7 and add 5 for each step.
   7 + 70 = 77
4. Write the answer.
   77

Check. 7 + (15 - 1) x 5 = 77. ✓

Watch out. The 1st term has 0 steps added, so the 10th term has 9 steps.

Method. The number in front of n is the difference. Then work out what to add or subtract.

1. Find the number that goes with n. The term goes up by 2 each time, so the formula has 2n.
   2n
2. Check the first term. When n = 1, 2 x 1 + (the rest) must equal 3.
   2 + 1 = 3
3. Write the formula. Use the number that makes the first term correct.
   2n + 1
4. Write the answer.
   2n + 1

Check. When n = 2, the term is 5. ✓

Watch out. Check your formula with n = 1 and n = 2.

Method. Check whether the difference from the first term is a multiple of the difference.

1. Find the difference from the first term. 31 - 4 = 27.
   27
2. Divide by the difference. 27 / 3 = 9, a whole number.
   9
3. Decide. So 31 is term number 10.
   term 10
4. Write the answer.
   yes

Check. The terms are 4 + 3 x (whole number). ✓

Watch out. A number is in the sequence only if it can be reached by adding the difference repeatedly.

Method. Replace n with the number, then multiply before you add.

1. Write the formula. Total = 5 x n + 8.
   T = 5n + 8
2. Substitute the value. Put n = 6 into the formula.
   5 x 6 + 8
3. Work out. Multiply first, then add.
   30 + 8 = 38
4. Write the answer.
   38

Check. 30 + 8 = 38. ✓

Watch out. Multiply before you add. 3n + 2 with n = 4 is 14, not 20.

Method. Use inverse operations: subtract, then divide.

1. Undo the addition. Take 3 from both sides.
   4x = 39 - 3 = 36
2. Undo the multiplication. Divide both sides by 4.
   x = 36 / 4 = 9
3. Write the answer.
   9

Check. 4 x 9 + 3 = 39. ✓

Watch out. Undo the addition or subtraction first, then the multiplication.

Method. Write an equation with n, then solve it by undoing the operations.

1. Write the equation. Let the number be n. Multiply by 3, add 7, and get 40.
   3n + 7 = 40
2. Subtract the added number. Take 7 from 40.
   3n = 33
3. Divide by the multiplier. Divide 33 by 3.
   n = 11
4. Write the answer.
   11

Check. 3 x 11 + 7 = 40. ✓

Watch out. Undo the operations in reverse order.

Method. List the pairs systematically, starting with the smallest value of a.

1. List the pairs with a smaller than b. Start with a = 1 and count up, so a is always less than b.
   1 + 13, 2 + 12, 3 + 11, 4 + 10, 5 + 9, 6 + 8
2. Count the pairs. Each pair is one answer.
   6 pairs
3. Write the answer.
   6

Check. Each pair adds to 14. ✓

Watch out. Do not count the pair where a = b, and do not count a pair twice.

Method. Subtract the known number from the total.

1. Write the number sentence. 24 + b = 60.
   24 + b = 60
2. Find the missing number. Take 24 from 60.
   b = 60 - 24
3. Write the answer.
   36

Check. 24 + 36 = 60. ✓

Method. Find the difference between terms, then apply it again.

1. Find the rule. Each term is the one before with add 6.
   10 - 4 = 6
2. Apply the rule to the last term. Take the last term, 22, and add 6.
   22 + 6 = 28
3. Write the answer.
   28

Check. 28 - 22 = 6. ✓

Watch out. Check the difference between every pair of terms, not just the first two.

Method. Start at the first term and add the difference for each step after it.

1. Find the first term and the difference. The first term is 10, and the difference is 3.
   10 and 3
2. Count the steps to the nth term. The 12th term is 11 steps from the first term.
   12 - 1 = 11 steps
3. Work out the term. Start at 10 and add 3 for each step.
   10 + 33 = 43
4. Write the answer.
   43

Check. 10 + (12 - 1) x 3 = 43. ✓

Watch out. The 1st term has 0 steps added, so the 10th term has 9 steps.

Method. The number in front of n is the difference. Then work out what to add or subtract.

1. Find the number that goes with n. The term goes up by 4 each time, so the formula has 4n.
   4n
2. Check the first term. When n = 1, 4 x 1 + (the rest) must equal 5.
   4 + 1 = 5
3. Write the formula. Use the number that makes the first term correct.
   4n + 1
4. Write the answer.
   4n + 1

Check. When n = 2, the term is 9. ✓

Watch out. Check your formula with n = 1 and n = 2.

Method. Replace n with the number, then multiply before you add.

1. Write the formula. Total = 3 x n + 9.
   T = 3n + 9
2. Substitute the value. Put n = 8 into the formula.
   3 x 8 + 9
3. Work out. Multiply first, then add.
   24 + 9 = 33
4. Write the answer.
   33

Check. 24 + 9 = 33. ✓

Watch out. Multiply before you add. 3n + 2 with n = 4 is 14, not 20.

Method. Use inverse operations: subtract, then divide.

1. Undo the addition. Take 2 from both sides.
   5x = 37 - 2 = 35
2. Undo the multiplication. Divide both sides by 5.
   x = 35 / 5 = 7
3. Write the answer.
   7

Check. 5 x 7 + 2 = 37. ✓

Watch out. Undo the addition or subtraction first, then the multiplication.

Method. Write an equation with n, then solve it by undoing the operations.

1. Write the equation. Let the number be n. Multiply by 4, add 6, and get 46.
   4n + 6 = 46
2. Subtract the added number. Take 6 from 46.
   4n = 40
3. Divide by the multiplier. Divide 40 by 4.
   n = 10
4. Write the answer.
   10

Check. 4 x 10 + 6 = 46. ✓

Watch out. Undo the operations in reverse order.

Method. List the pairs systematically, starting with the smallest value of a.

1. List the pairs with a smaller than b. Start with a = 1 and count up, so a is always less than b.
   1 + 15, 2 + 14, 3 + 13, 4 + 12, 5 + 11, 6 + 10, 7 + 9
2. Count the pairs. Each pair is one answer.
   7 pairs
3. Write the answer.
   7

Check. Each pair adds to 16. ✓

Watch out. Do not count the pair where a = b, and do not count a pair twice.

Method. Check whether the difference from the first term is a multiple of the difference.

1. Find the difference from the first term. 30 - 2 = 28.
   28
2. Divide by the difference. 28 / 7 = 4, a whole number.
   4
3. Decide. So 30 is term number 5.
   term 5
4. Write the answer.
   yes

Check. The terms are 2 + 7 x (whole number). ✓

Watch out. A number is in the sequence only if it can be reached by adding the difference repeatedly.

Method. Check whether 30 - 1 is a multiple of 4.

1. Subtract 1. 30 - 1 = 29.
   29
2. Check the multiple of 4. 29 / 4 is not a whole number, so 30 is not a term.
   not whole
3. Write the answer.
   Because 4n + 1 is always one more than a multiple of 4, and 30 is 2 more than 28, so it is not a term.

Check. 29 / 4 is not whole. ✓

Method. Find the difference, then count.

1. Difference. 35 - 40 = -5, so subtract 5 each time.
   -5
2. Missing term. 35 - 5 = 30.
   30
3. Write the answer.
   30

Check. 30 - 5 = 25. ✓