Year 6 Algebra: check-up and SATs-style questions

Key words. A formula is a rule written with letters, such as P = 2a + 2b. To substitute means to put a number in place of a letter.

Method. Put the number in place of the letter, multiply, then add or take away.

1. Write the formula. The letter stands for a number we know, so we put it in.
   5n + 3
2. Put 8 in place of n. Multiply first. Multiplication comes before addition and subtraction.
   5 × 8 = 40
3. Add 3. This is the last step of the formula.
   40 + 3 = 43
4. Write the answer.
   43

Check. Work it out again: 5 × 8 = 40, then 40 + 3 = 43. ✓

Watch out. Multiply before you add or take away. Do not do the sum from left to right.

Key words. A formula is a rule written with letters, such as P = 2a + 2b. To substitute means to put a number in place of a letter.

Method. Put the numbers in place of the letters, then work it out in the right order.

1. Write the formula. It tells us which numbers to use.
   P = 2a + 2b
2. Put in a = 11 and b = 3. Each letter is replaced by its number.
   P = 2 × 11 + 2 × 3
3. Multiply first. Multiplication comes before addition.
   22 + 6
4. Add. This gives P.
   22 + 6 = 28
5. Write the answer.
   28

Check. 11 + 3 + 11 + 3 = 28. ✓

Watch out. Multiply each letter by 2 before you add.

Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.

Method. Undo the add or take away first, then undo the multiplication or division. Check at the end.

1. Write the equation. Both sides are equal.
   4n + 5 = 33
2. Undo the add 5: take 5 from both sides. Taking 5 away from both sides keeps the equation true.
   4n = 28
3. Undo the × 4: divide both sides by 4. Dividing both sides by 4 leaves the letter on its own.
   n = 28 ÷ 4 = 7
4. Check: put the answer back into the equation. If both sides match, the answer is right.
   4 × 7 + 5 = 33 ✓
5. Write the answer.
   n = 7

Check. 4 × 7 + 5 = 33. ✓

Watch out. Work backwards, doing the opposite operation. Do the last thing first.

Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.

Method. Undo the add or take away first, then undo the multiplication or division. Check at the end.

1. Write the equation. Both sides are equal.
   3x - 6 = 21
2. Undo the take away 6: add 6 to both sides. Adding the same number to both sides keeps the equation true.
   3x = 21 + 6 = 27
3. Undo the × 3: divide both sides by 3. Dividing both sides by 3 leaves the letter on its own.
   x = 27 ÷ 3 = 9
4. Check: put the answer back into the equation. If both sides match, the answer is right.
   3 × 9 - 6 = 21 ✓
5. Write the answer.
   x = 9

Check. 3 × 9 - 6 = 21. ✓

Watch out. Work backwards, doing the opposite operation. Do the last thing first.

Key words. A term is one number in a sequence. The term-to-term rule says how to get from one term to the next. In a linear sequence the same amount is added (or taken away) each time.

Method. The rule is to add 5 each time. Use it to find the next terms.

1. Find the difference between two terms next to each other. If the difference is always the same, the sequence is linear.
   12 - 7 = 5
2. Check the other pairs of terms. They must have the same difference, or the rule is not linear.
   12 - 7 = 5, 17 - 12 = 5
3. Apply the rule: add 5. This gives the next term.
   17 + 5 = 22
4. Do the same again for the term after that. The same rule works every time.
   22 + 5 = 27
5. Write the answer.
   22, 27

Check. Start from the last term: 17 + 5 = 22. ✓

Watch out. Check that the difference is the same for every pair of neighbouring terms before you use it.

Key words. A term is one number in a sequence. The term-to-term rule says how to get from one term to the next. In a linear sequence the same amount is added (or taken away) each time.

Method. Find the difference and say whether the terms go up or down.

1. Find the difference between two terms next to each other. This is the amount we add or take away.
   27 - 30 = -3
2. Check that every pair has the same difference. Then the rule is the same all the way along.
   3, 3, 3 (sizes of the jumps)
3. Write the rule. Say whether we add or subtract, and by how much.
   subtract 3

Check. Start at 30 and apply the rule: 30 - 3 = 27. ✓

Watch out. Say "add 4" or "subtract 4", not "plus 4" or "4 more" with no operation.

Key words. The nth term is a rule that gives any term. The letter n is the position of the term: the first term has n = 1, the second term has n = 2.

Method. Find the difference: it is the number in front of n. Then find the number that makes the first term right.

1. Find the difference between two terms next to each other. It is the same every time, because the sequence is linear.
   10 - 4 = 6
2. Start the rule with 6n, because the difference is 6. The number in front of n is the difference.
   6n + ?
3. Find the number that is added: put n = 1 in the rule. The first term is 4, so the rule must give 4 when n = 1.
   6 × 1 + ? = 4, so ? = 4 - 6 = -2
4. Check with the second term. Put n = 2 in the rule: the answer must be 10.
   6 × 2 - 2 = 10. ✓
5. Write the nth term.
   6n - 2

Check. Put n = 2 in 6n - 2: this gives the second term of the sequence. ✓

Watch out. The number in front of n is the difference, but the rule also needs the number that is added (or taken away).

Key words. A term is one number in a sequence. The term-to-term rule says how to get from one term to the next. In a linear sequence the same amount is added (or taken away) each time.

Method. Work out how far the number is from the first term. If that is a whole number of jumps, the number is in the sequence.

1. Take away the first term from 62. We want to know how far the term is from the start.
   62 - 5 = 57
2. Divide by the difference, 4. The sequence goes up by the same amount each time.
   57 ÷ 4 = 57/4
3. It is not a whole number (or it is below the first term), so the number is not in the sequence. Every term is the first term plus a whole number of jumps.
   57/4 is not a whole number
4. Write the answer.
   No

Check. Terms are 5, 9, 13 ... and never reach 62. ✓

Watch out. The number must be exactly one of the terms. A number between two terms is not in the sequence.

Key words. A pair is two numbers that together make an equation true. Positive whole numbers are 1, 2, 3, 4 and so on (no zero, fractions or decimals).

Method. Work through the possible values of one letter, in order, and work out the other letter each time.

1. Write the equation. We are looking for pairs that make it true.
   a + b = 7
2. Start with a = 1 and work out b. Keep only the values that give a whole number. Keep going, one value at a time, so that no pair is missed.

   a	b	Check
   1	6	1 + 6 = 7
   2	5	2 + 5 = 7
   3	4	3 + 4 = 7
   4	3	4 + 3 = 7
   5	2	5 + 2 = 7
   6	1	6 + 1 = 7

3. Stop when there are no more pairs with positive numbers. Once a number would be below 1, there is nothing left to find.
   the last pair is 6 and 1
4. Count the pairs. Each row of the table is one pair.
   6 pairs
5. Write the answer.
   6

Check. Every row was checked in the equation. The table has 6 rows and no pair is repeated. ✓

Watch out. Start from 1, not from 0, for positive whole numbers. Make sure you have not counted a pair twice.

Key words. A combination is one way of choosing an amount of each thing. To enumerate means to list every possibility in order, so that none is missed.

Method. List every choice in order, check each one, then count the ways.

1. Write the total and the value of each item. One 2p coin is worth 2p. One 5p coin is worth 5p.
   15p from 2p and 5p items
2. Choose how many 5p coins, starting at 0, and work out how much is left. The rest of the total must be made from 2p coins.
   the rest must be a whole number of the other item
3. Keep the list in order, so that no way is missed or repeated. Trying each choice in turn gives a complete list.

   Number of 2p coins	Number of 5p coins	Check
   0	3	0 × 2 + 3 × 5 = 15
   5	1	5 × 2 + 1 × 5 = 15

4. Count the ways. Each row is one way.
   2 ways
5. Write the answer.
   2

Check. Each row adds to 15p, and no row is repeated. ✓

Watch out. Check that every row uses the right number of items. Do not forget the rows where one item is not used, if the question allows it.

Key words. A formula is a rule written with letters, such as P = 2a + 2b. To substitute means to put a number in place of a letter.

Method. Put the number in place of the letter, multiply, then add or take away.

1. Write the formula. The letter stands for a number we know, so we put it in.
   5n - 3
2. Put 7 in place of n. Multiply first. Multiplication comes before addition and subtraction.
   5 × 7 = 35
3. Take away 3. This is the last step of the formula.
   35 - 3 = 32
4. Write the answer.
   32

Check. Work it out again: 5 × 7 = 35, then 35 - 3 = 32. ✓

Watch out. Multiply before you add or take away. Do not do the sum from left to right.

Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.

Method. Undo the add or take away first, then undo the multiplication or division. Check at the end.

1. Write the equation. Both sides are equal.
   4x + 6 = 30
2. Undo the add 6: take 6 from both sides. Taking 6 away from both sides keeps the equation true.
   4x = 24
3. Undo the × 4: divide both sides by 4. Dividing both sides by 4 leaves the letter on its own.
   x = 24 ÷ 4 = 6
4. Check: put the answer back into the equation. If both sides match, the answer is right.
   4 × 6 + 6 = 30 ✓
5. Write the answer.
   x = 6

Check. 4 × 6 + 6 = 30. ✓

Watch out. Work backwards, doing the opposite operation. Do the last thing first.

Key words. A term is one number in a sequence. The term-to-term rule says how to get from one term to the next. In a linear sequence the same amount is added (or taken away) each time.

Method. The rule is to subtract 4 each time. Use it to find the next terms.

1. Find the difference between two terms next to each other. If the difference is always the same, the sequence is linear.
   9 - 13 = -4
2. Check the other pairs of terms. They must have the same difference, or the rule is not linear.
   9 - 13 = -4, 5 - 9 = -4, 1 - 5 = -4
3. Apply the rule: subtract 4. This gives the next term.
   1 - 4 = -3
4. Write the answer.
   -3

Check. Start from the last term: 1 - 4 = -3. ✓

Watch out. Check that the difference is the same for every pair of neighbouring terms before you use it.

Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.

Method. Undo the add or take away first, then undo the multiplication or division. Check at the end.

1. Write the equation. Both sides are equal.
   4m + 10 = 42
2. Undo the add 10: take 10 from both sides. Taking 10 away from both sides keeps the equation true.
   4m = 32
3. Undo the × 4: divide both sides by 4. Dividing both sides by 4 leaves the letter on its own.
   m = 32 ÷ 4 = 8
4. Check: put the answer back into the equation. If both sides match, the answer is right.
   4 × 8 + 10 = 42 ✓
5. Write the answer.
   m = 8

Check. 4 × 8 + 10 = 42. ✓

Watch out. Work backwards, doing the opposite operation. Do the last thing first.

Key words. A formula is a rule written with letters, such as P = 2a + 2b. To substitute means to put a number in place of a letter.

Method. Put the number in place of the letter, multiply, then add or take away.

1. Write the formula. The letter stands for a number we know, so we put it in.
   5n + 4
2. Put 40 in place of n. Multiply first. Multiplication comes before addition and subtraction.
   5 × 40 = 200
3. Add 4. This is the last step of the formula.
   200 + 4 = 204
4. Write the answer.
   204

Check. Work it out again: 5 × 40 = 200, then 200 + 4 = 204. ✓

Watch out. Multiply before you add or take away. Do not do the sum from left to right.

Key words. A pair is two numbers that together make an equation true. Positive whole numbers are 1, 2, 3, 4 and so on (no zero, fractions or decimals).

Method. Work through the possible values of one letter, in order, and work out the other letter each time.

1. Write the equation. We are looking for pairs that make it true.
   a + b = 16, both even
2. Start with a = 1 and work out b. Keep only the values that give a whole number. Keep going, one value at a time, so that no pair is missed.

   a	b	Check
   2	14	2 + 14 = 16, both even
   4	12	4 + 12 = 16, both even
   6	10	6 + 10 = 16, both even
   8	8	8 + 8 = 16, both even
   10	6	10 + 6 = 16, both even
   12	4	12 + 4 = 16, both even
   14	2	14 + 2 = 16, both even

3. Stop when there are no more pairs with positive numbers. Once a number would be below 1, there is nothing left to find.
   the last pair is 14 and 2
4. Count the pairs. Each row of the table is one pair.
   7 pairs
5. Write the answer.
   7

Check. Every row was checked in the equation. The table has 7 rows and no pair is repeated. ✓

Watch out. Start from 1, not from 0, for positive whole numbers. Make sure you have not counted a pair twice.

Key words. The nth term is a rule that gives any term. The letter n is the position of the term: the first term has n = 1, the second term has n = 2.

Method. Solve the equation for n. If n is a whole number that is 1 or more, the number is a term.

1. Write the equation. We ask: for which n is the term equal to 150?
   7n + 1 = 150
2. Take 1 away from both sides. This leaves the part with n.
   7n = 149
3. Divide both sides by 7. This gives n, the position of the term.
   n = 149 ÷ 7 = 149/7
4. Decide. The position must be a whole number, 1 or more.
   n = 149/7, which is not a whole number, so 150 is not a term
5. Write the answer.
   No

Check. Try the nearest whole numbers: 7 × 21 + 1 and 7 × 22 + 1 are not 150. ✓

Watch out. The position n must be a whole number. A number between two terms is never a term.

Key words. A combination is one way of choosing an amount of each thing. To enumerate means to list every possibility in order, so that none is missed.

Method. List every choice in order, check each one, then count the ways.

1. Write the total and the value of each item. One 3p coin is worth 3p. One 5p coin is worth 5p.
   37p from 3p and 5p items
2. Choose how many 5p coins, starting at 0, and work out how much is left. The rest of the total must be made from 3p coins.
   the rest must be a whole number of the other item
3. Keep the list in order, so that no way is missed or repeated. Trying each choice in turn gives a complete list.

   Number of 3p coins	Number of 5p coins	Check
   4	5	4 × 3 + 5 × 5 = 37
   9	2	9 × 3 + 2 × 5 = 37

4. Count the ways. Each row is one way.
   2 ways
5. Write the answer.
   2

Check. Each row adds to 37p, and no row is repeated. ✓

Watch out. Check that every row uses the right number of items. Do not forget the rows where one item is not used, if the question allows it.

Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.

Method. Undo the add or take away first, then undo the multiplication or division. Check at the end.

1. Write the equation. Both sides are equal.
   6n - 2 = 118
2. Undo the take away 2: add 2 to both sides. Adding the same number to both sides keeps the equation true.
   6n = 118 + 2 = 120
3. Undo the × 6: divide both sides by 6. Dividing both sides by 6 leaves the letter on its own.
   n = 120 ÷ 6 = 20
4. Check: put the answer back into the equation. If both sides match, the answer is right.
   6 × 20 - 2 = 118 ✓
5. Write the answer.
   n = 20

Check. 6 × 20 - 2 = 118. ✓

Watch out. Work backwards, doing the opposite operation. Do the last thing first.

Key words. A pair is two numbers that together make an equation true. Positive whole numbers are 1, 2, 3, 4 and so on (no zero, fractions or decimals).

Method. Add the two equations to find a, then find b.

1. Write the two equations. Both are true for the same a and b.
   a + b = 18 and a - b = 4
2. Add the two equations. The b terms cancel: +b and -b.
   2a = 22
3. Divide by 2. This gives a.
   a = 22 ÷ 2 = 11
4. Find b. Put a = 11 into a + b = 18.
   11 + b = 18, so b = 7
5. Check both equations. Both must be true.
   11 + 7 = 18 and 11 - 7 = 4
6. Write the answer. The question asks for a.
   11

Check. 11 + 7 = 18 and 11 - 7 = 4. ✓

Watch out. Check both equations with both numbers.