Year 5: Scaling, rates and combinations

Key words. To partition a number is to split it into parts, such as 47 = 40 + 7. A partial product is the answer to one part of a multiplication. The grid method (or area method) puts the parts in a table and multiplies each one.

Method. Draw a bar model, choose the operation, then work it out.

1. Draw a bar model. Each box is the same size. The longer bar is the same box repeated.
2. Choose the operation. “4 times as many” means 4 equal parts, each the same size as a pen’s amount. Multiply by 4.
   45p × 4
3. Partition the bigger number. Split it into hundreds, tens and ones. Each part is easy to multiply.
   45 = 40 + 5
4. Multiply each part by 4. Use a times table fact, then put the zeros back.
   40 × 4 = 160
   5 × 4 = 20
5. Add the answers. Add them one at a time, starting with the biggest.
   160 + 20 = 180
6. Write the answer. Say whose amount it is.
   180p

Check. Estimate: 45 is about 50. 50 × 4 = 200. Our answer, 180, is close to 200. ✓

Watch out. Multiply every part. Do not forget a part, and do not lose the zeros: 40 × 6 is 240, not 24.

Key words. Short multiplication is a written method for multiplying by a one-digit number. Write the numbers in columns and multiply one column at a time, starting with the ones. If a column makes 10 or more, write the ones digit and carry the tens digit to the next column.

Method. Draw a bar model, choose the operation, then work it out.

1. Draw a bar model. Each box is the same size. The longer bar is the same box repeated.
2. Choose the operation. “6 times as many” means 6 equal parts, each the same size as Mia’s amount. Multiply by 6.
   150 m × 6
3. Write the numbers in columns. Ones go under ones, tens under tens and so on. The one-digit number goes under the ones.

   	H	T	O
   	1	5	0
   ×			6

4. Multiply the ones. Always start with the ones, on the right.
   0 × 6 = 0
   0 is less than 10, so write 0. Nothing to carry.
5. Multiply the tens. Now the next column to the left.
   5 × 6 = 30
   30 is 3 hundreds and 0 tens. Write 0 in the tens column and carry 3.
6. Multiply the hundreds. Multiply, then add the 3 we carried.
   1 × 6 = 6, and 6 + 3 = 9
   9 is less than 10, so write 9. Nothing to carry.
7. Read the answer from the final layout. Check that it has about the right number of digits.

   	H	T	O
   	1	5	0
   ×			6
   	9	0	0
   	3

8. Write the answer. Say whose amount it is.
   900 m

Check. Estimate: 150 is about 200. 200 × 6 = 1,200. Our answer, 900, is a bit less than 1,200. ✓

Watch out. A zero digit still needs a step: 0 × the number is 0, but do not forget to add the carried digit.

Key words. Long multiplication is a written method for multiplying by a number with two or more digits. You multiply by each digit of the second number on its own row, then add the rows. A place holder zero goes at the end of the tens row, because that row is worth tens.

Method. Draw a bar model, choose the operation, then work it out.

1. Draw a bar model. Each box is the same size. The longer bar is the same box repeated.
2. Choose the operation. “12 times as many” means 12 equal parts, each the same size as a bag’s amount. Multiply by 12.
   35 marbles × 12
3. Split the multiplier. Multiplying by 12 means multiplying by 2 and by 10, then adding. Write the numbers in columns.
   12 = 10 + 2, so 35 × 12 = (35 × 2) + (35 × 10)
   

   		3	5
   ×		1	2

4. Row 1: multiply by the 2 (the ones digit). This is short multiplication, starting with the ones. Work out 35 × 2.
   Ones: 5 × 2 = 10. Write 0 and carry 1.
   Tens: 3 × 2 = 6, and 6 + 1 (carried) = 7. Write 7.
   Row 1 = 70
5. Row 2: write a 0 first, then multiply by the 1. The 1 is really 10, so this row is worth tens. The 0 in the ones column is a place holder. Now multiply 35 by 1.
   Write 0 in the ones column.
   Ones: 5 × 1 = 5. Write 5.
   Tens: 3 × 1 = 3. Write 3.
   Row 2 = 35 with a 0 on the end = 350
6. Add the two rows. The sum of the rows is the answer, because 26 = 20 + 6.

   		3	5
   ×		1	2
   		7	0	35 × 2
   +	3	5	0	35 × 10
   	4	2	0

   
   70 + 350 = 420
7. Write the answer. Say whose amount it is.
   420 marbles

Check. Estimate: 35 is about 40; 12 is about 10. 40 × 10 = 400. Our answer, 420, is close to 400. ✓

Watch out. Do not forget the 0 place holder on the second row. Without it you are multiplying by 1, not by 10, and the answer is ten times too small.

Key words. Scaling means making an amount bigger or smaller by multiplying. “3 times as long” means multiply by 3. “Half as long” means divide by 2.

Method. Cut the bar into 2 equal parts. Find the size of one part, then that is the answer.

1. Draw a bar model. Long rope’s bar is cut into 2 equal parts. Short rope has 1 of these parts.
2. Find the size of one part. 2 equal parts make 86 m, so one part is 86 m ÷ 2.
   2 × 40 = 80, and 86 − 80 = 6
   2 × 3 = 6
   40 + 3 = 43, so 86 ÷ 2 = 43
3. Write the answer. Say whose amount it is.
   Short rope: 43 m

Check. 1 × 43 = 43, and 2 × 43 = 86. ✓

Watch out. Short rope’s amount must be smaller than Long rope’s amount.

Key words. Scaling means making an amount bigger or smaller by multiplying. “3 times as long” means multiply by 3. “Half as long” means divide by 2.

Method. Cut the bar into 4 equal parts. Find the size of one part, then that is the answer.

1. Draw a bar model. Big bag’s bar is cut into 4 equal parts. Small bag has 1 of these parts.
2. Find the size of one part. 4 equal parts make 120 g, so one part is 120 g ÷ 4.
   4 × 30 = 120, so 120 ÷ 4 = 30
3. Write the answer. Say whose amount it is.
   Small bag: 30 g

Check. 1 × 30 = 30, and 4 × 30 = 120. ✓

Watch out. Small bag’s amount must be smaller than Big bag’s amount.

Key words. Scaling means making an amount bigger or smaller by multiplying. “3 times as long” means multiply by 3. “Half as long” means divide by 2.

Method. Cut the bar into 10 equal parts. Find the size of one part, then that is the answer.

1. Draw a bar model. Road’s bar is cut into 10 equal parts. Path has 1 of these parts.
2. Find the size of one part. 10 equal parts make 450 m, so one part is 450 m ÷ 10.
   10 × 40 = 400, and 450 − 400 = 50
   10 × 5 = 50
   40 + 5 = 45, so 450 ÷ 10 = 45
3. Write the answer. Say whose amount it is.
   Path: 45 m

Check. 1 × 45 = 45, and 10 × 45 = 450. ✓

Watch out. Path’s amount must be smaller than Road’s amount.

Key words. Scaling means making an amount bigger or smaller by multiplying. “3 times as long” means multiply by 3. “Half as long” means divide by 2.

Method. Cut the bar into 4 equal parts. Find the size of one part, then multiply by 3.

1. Draw a bar model. Tank’s bar is cut into 4 equal parts. Jug has 3 of these parts.
2. Find the size of one part. 4 equal parts make 80 litres, so one part is 80 litres ÷ 4.
   4 × 20 = 80, so 80 ÷ 4 = 20
3. Multiply by 3. Jug has 3 parts of 20 litres.
   20 × 3 = 60
4. Write the answer. Say whose amount it is.
   Jug: 60 litres

Check. 3 × 20 = 60, and 4 × 20 = 80. ✓

Watch out. Jug’s amount must be smaller than Tank’s amount.

Key words. Scaling means making an amount bigger or smaller by multiplying. “3 times as long” means multiply by 3. “Half as long” means divide by 2.

Method. Draw a bar model. The big amount is made of equal boxes. Divide it by the number of boxes to find one box.

1. Draw a bar model. Mia’s bar is 3 equal boxes. Tom’s bar is 1 box. The long bar is worth 72.
2. Choose the operation. 3 equal boxes make 72, so one box is 72 shared into 3 equal parts. Use the opposite of multiplying.
   3 × ? = 72
3. Work out one box. Use multiplication and partitioning to find what, times 3, makes 72.
   3 × 20 = 60, and 72 − 60 = 12
   3 × 4 = 12
   20 + 4 = 24, so 72 ÷ 3 = 24
4. Write the answer. Say whose amount it is.
   Tom: 24

Check. 3 × 24 = 72. ✓

Watch out. Tom has the smaller amount, so the answer must be smaller than 72.

Key words. Scaling means making an amount bigger or smaller by multiplying. “3 times as long” means multiply by 3. “Half as long” means divide by 2.

Method. Draw a bar model. The big amount is made of equal boxes. Divide it by the number of boxes to find one box.

1. Draw a bar model. film’s bar is 6 equal boxes. advert’s bar is 1 box. The long bar is worth 90.
2. Choose the operation. 6 equal boxes make 90, so one box is 90 shared into 6 equal parts. Use the opposite of multiplying.
   6 × ? = 90
3. Work out one box. Use multiplication and partitioning to find what, times 6, makes 90.
   6 × 10 = 60, and 90 − 60 = 30
   6 × 5 = 30
   10 + 5 = 15, so 90 ÷ 6 = 15
4. Write the answer. Say whose amount it is.
   advert: 15

Check. 6 × 15 = 90. ✓

Watch out. advert has the smaller amount, so the answer must be smaller than 90.

Key words. A rate compares two amounts, such as £3 for 5 apples or 12 km each hour. A unit rate is the amount for just one (one apple, one hour).

Method. Write the amounts in a table. Find the number you multiply by to get from one amount to the one you want. Multiply the other amount by the same number.

1. Write what you know in a table. A ratio table keeps matching amounts side by side.

   Pens	4	12
   Cost	£6	?

2. Find the multiplier. How many times bigger is 12 than 4? Count in 4s or use a times table.
   4 × 3 = 12, so the multiplier is 3
3. Multiply the other amount by the same number. If you buy 3 times as much, you pay 3 times as much.
   £6 × 3
4. Partition the bigger number. Split it into hundreds, tens and ones. Each part is easy to multiply.
   6 = 6
5. Multiply each part by 3. Use a times table fact, then put the zeros back.
   6 × 3 = 18
6. Add the answers. Add them one at a time, starting with the biggest.
   18
7. Write the answer. Include the unit.
   £18

Check. Work back: £18 is 3 times the first amount, and £6 × 3 = £18. ✓

Watch out. Multiply BOTH amounts by the same number so that the rate stays the same.

Key words. A rate compares two amounts, such as £3 for 5 apples or 12 km each hour. A unit rate is the amount for just one (one apple, one hour).

Method. Write the amounts in a table. Find the number you multiply by to get from one amount to the one you want. Multiply the other amount by the same number.

1. Write what you know in a table. A ratio table keeps matching amounts side by side.

   Cakes	6	24
   Cost	£9	?

2. Find the multiplier. How many times bigger is 24 than 6? Count in 6s or use a times table.
   6 × 4 = 24, so the multiplier is 4
3. Multiply the other amount by the same number. If you buy 4 times as much, you pay 4 times as much.
   £9 × 4
4. Partition the bigger number. Split it into hundreds, tens and ones. Each part is easy to multiply.
   9 = 9
5. Multiply each part by 4. Use a times table fact, then put the zeros back.
   9 × 4 = 36
6. Add the answers. Add them one at a time, starting with the biggest.
   36
7. Write the answer. Include the unit.
   £36

Check. Work back: £36 is 4 times the first amount, and £9 × 4 = £36. ✓

Watch out. Multiply BOTH amounts by the same number so that the rate stays the same.

Key words. A rate compares two amounts, such as £3 for 5 apples or 12 km each hour. A unit rate is the amount for just one (one apple, one hour).

Method. Write the amounts in a table. Find the number you multiply by to get from one amount to the one you want. Multiply the other amount by the same number.

1. Write what you know in a table. A ratio table keeps matching amounts side by side.

   Paint (litres)	5	25
   Area covered (m²)	40	?

2. Find the multiplier. How many times bigger is 25 than 5? Count in 5s or use a times table.
   5 × 5 = 25, so the multiplier is 5
3. Multiply the other amount by the same number. If you buy 5 times as much, you pay 5 times as much.
   40 × 5
4. Partition the bigger number. Split it into hundreds, tens and ones. Each part is easy to multiply.
   40 = 40
5. Multiply each part by 5. Use a times table fact, then put the zeros back.
   40 × 5 = 200
6. Add the answers. Add them one at a time, starting with the biggest.
   200
7. Write the answer. Include the unit.
   200

Check. Work back: 200 is 5 times the first amount, and 40 × 5 = 200. ✓

Watch out. Multiply BOTH amounts by the same number so that the rate stays the same.

Key words. A rate compares two amounts, such as £3 for 5 apples or 12 km each hour. A unit rate is the amount for just one (one apple, one hour).

Method. Write the amounts in a table. Find the number you multiply by to get from one amount to the one you want. Multiply the other amount by the same number.

1. Write what you know in a table. A ratio table keeps matching amounts side by side.

   Apples	5	?
   Cost	£3	£21

2. Find the multiplier. How many times bigger is £21 than £3?
   3 × 7 = 21, so the multiplier is 7
3. Multiply the other amount by the same number. If you spend 7 times as much, you get 7 times as much.
   5 × 7
4. Partition the bigger number. Split it into hundreds, tens and ones. Each part is easy to multiply.
   5 = 5
5. Multiply each part by 7. Use a times table fact, then put the zeros back.
   5 × 7 = 35
6. Add the answers. Add them one at a time, starting with the biggest.
   35
7. Write the answer. Include the unit.
   35

Check. Work back: 35 is 7 times the first amount, and 5 × 7 = 35. ✓

Watch out. Multiply BOTH amounts by the same number so that the rate stays the same.

Key words. A rate compares two amounts, such as £3 for 5 apples or 12 km each hour. A unit rate is the amount for just one (one apple, one hour).

Method. Write the amounts in a table. Find the number you multiply by to get from one amount to the one you want. Multiply the other amount by the same number.

1. Write what you know in a table. A ratio table keeps matching amounts side by side.

   Bananas	3	?
   Cost	£4	£24

2. Find the multiplier. How many times bigger is £24 than £4?
   4 × 6 = 24, so the multiplier is 6
3. Multiply the other amount by the same number. If you spend 6 times as much, you get 6 times as much.
   3 × 6
4. Partition the bigger number. Split it into hundreds, tens and ones. Each part is easy to multiply.
   3 = 3
5. Multiply each part by 6. Use a times table fact, then put the zeros back.
   3 × 6 = 18
6. Add the answers. Add them one at a time, starting with the biggest.
   18
7. Write the answer. Include the unit.
   18

Check. Work back: 18 is 6 times the first amount, and 3 × 6 = 18. ✓

Watch out. Multiply BOTH amounts by the same number so that the rate stays the same.

Key words. A rate compares two amounts, such as £3 for 5 apples or 12 km each hour. A unit rate is the amount for just one (one apple, one hour).

Method. The rate says how far per hour. Multiply it by the number of hours.

1. Write the rate in a table. It goes 12 km every hour.

   Time (hours)	1	5
   Distance (km)	12	?

2. Choose the operation. In 5 hours there are 5 lots of 12 km, so multiply.
   12 km × 5
3. Partition the bigger number. Split it into hundreds, tens and ones. Each part is easy to multiply.
   12 = 10 + 2
4. Multiply each part by 5. Use a times table fact, then put the zeros back.
   10 × 5 = 50
   2 × 5 = 10
5. Add the answers. Add them one at a time, starting with the biggest.
   50 + 10 = 60
6. Write the answer. Include the unit.
   60 km

Check. Estimate: 12 is about 10. 10 × 5 = 50. Our answer, 60, is close to 50. ✓

Watch out. The unit of the answer is km, not km per hour.

Key words. A rate compares two amounts, such as £3 for 5 apples or 12 km each hour. A unit rate is the amount for just one (one apple, one hour).

Method. The rate says how far per hour. Multiply it by the number of hours.

1. Write the rate in a table. It goes 45 km every hour.

   Time (hours)	1	4
   Distance (km)	45	?

2. Choose the operation. In 4 hours there are 4 lots of 45 km, so multiply.
   45 km × 4
3. Partition the bigger number. Split it into hundreds, tens and ones. Each part is easy to multiply.
   45 = 40 + 5
4. Multiply each part by 4. Use a times table fact, then put the zeros back.
   40 × 4 = 160
   5 × 4 = 20
5. Add the answers. Add them one at a time, starting with the biggest.
   160 + 20 = 180
6. Write the answer. Include the unit.
   180 km

Check. Estimate: 45 is about 50. 50 × 4 = 200. Our answer, 180, is close to 200. ✓

Watch out. The unit of the answer is km, not km per hour.

Key words. A rate compares two amounts, such as £3 for 5 apples or 12 km each hour. A unit rate is the amount for just one (one apple, one hour).

Method. The rate says how far per minute. Multiply it by the number of minutes.

1. Write the rate in a table. It goes 28 bolts every minute.

   Time (minutes)	1	15
   Distance (bolts)	28	?

2. Choose the operation. In 15 minutes there are 15 lots of 28 bolts, so multiply.
   28 bolts × 15
3. Split the multiplier. Multiplying by 15 means multiplying by 5 and by 10, then adding. Write the numbers in columns.
   15 = 10 + 5, so 28 × 15 = (28 × 5) + (28 × 10)
   

   		2	8
   ×		1	5

4. Row 1: multiply by the 5 (the ones digit). This is short multiplication, starting with the ones. Work out 28 × 5.
   Ones: 8 × 5 = 40. Write 0 and carry 4.
   Tens: 2 × 5 = 10, and 10 + 4 (carried) = 14. Write 4 and carry 1.
   Hundreds: nothing left to multiply. Write the carried 1.
   Row 1 = 140
5. Row 2: write a 0 first, then multiply by the 1. The 1 is really 10, so this row is worth tens. The 0 in the ones column is a place holder. Now multiply 28 by 1.
   Write 0 in the ones column.
   Ones: 8 × 1 = 8. Write 8.
   Tens: 2 × 1 = 2. Write 2.
   Row 2 = 28 with a 0 on the end = 280
6. Add the two rows. The sum of the rows is the answer, because 26 = 20 + 6.

   		2	8
   ×		1	5
   	1	4	0	28 × 5
   +	2	8	0	28 × 10
   	4	2	0

   
   140 + 280 = 420
7. Write the answer. Include the unit.
   420 bolts

Check. Estimate: 28 is about 30; 15 is about 20. 30 × 20 = 600. Our answer, 420, is a bit less than 600. ✓

Watch out. The unit of the answer is bolts, not bolts per minute.

Key words. A rate compares two amounts, such as £3 for 5 apples or 12 km each hour. A unit rate is the amount for just one (one apple, one hour).

Method. Multiply the cost of one by the number you buy. Work in pence so there is no decimal point.

1. Write the cost of one. The rate is 120p for each kg.
   1 kg costs 120p
2. Choose the operation. 7 kg cost 7 lots of the cost of one, so multiply.
   120p × 7
3. Write the numbers in columns. Ones go under ones, tens under tens and so on. The one-digit number goes under the ones.

   	H	T	O
   	1	2	0
   ×			7

4. Multiply the ones. Always start with the ones, on the right.
   0 × 7 = 0
   0 is less than 10, so write 0. Nothing to carry.
5. Multiply the tens. Now the next column to the left.
   2 × 7 = 14
   14 is 1 hundred and 4 tens. Write 4 in the tens column and carry 1.
6. Multiply the hundreds. Multiply, then add the 1 we carried.
   1 × 7 = 7, and 7 + 1 = 8
   8 is less than 10, so write 8. Nothing to carry.
7. Read the answer from the final layout. Check that it has about the right number of digits.

   	H	T	O
   	1	2	0
   ×			7
   	8	4	0
   	1

8. Change pence to pounds. 100p = £1, so divide by 100 (the digits move 2 places to the right).
   840p = £8.40
9. Write the answer. Include the unit.
   £8.40

Check. Estimate: 120 is about 100. 100 × 7 = 700. Our answer, 840, is close to 700. ✓

Watch out. Pence and pounds are different units: 250p is £2.50, not £250.

Key words. A rate compares two amounts, such as £3 for 5 apples or 12 km each hour. A unit rate is the amount for just one (one apple, one hour).

Method. Multiply the cost of one by the number you buy. Work in pence so there is no decimal point.

1. Write the cost of one. The rate is 89p for each kg.
   1 kg costs 89p
2. Choose the operation. 5 kg cost 5 lots of the cost of one, so multiply.
   89p × 5
3. Partition the bigger number. Split it into hundreds, tens and ones. Each part is easy to multiply.
   89 = 80 + 9
4. Multiply each part by 5. Use a times table fact, then put the zeros back.
   80 × 5 = 400
   9 × 5 = 45
5. Add the answers. Add them one at a time, starting with the biggest.
   400 + 45 = 445
6. Change pence to pounds. 100p = £1, so divide by 100 (the digits move 2 places to the right).
   445p = £4.45
7. Write the answer. Include the unit.
   £4.45

Check. Estimate: 89 is about 90. 90 × 5 = 450. Our answer, 445, is close to 450. ✓

Watch out. Pence and pounds are different units: 250p is £2.50, not £250.

Key words. A rate compares two amounts, such as £3 for 5 apples or 12 km each hour. A unit rate is the amount for just one (one apple, one hour).

Method. Multiply the cost of one by the number you buy. Work in pence so there is no decimal point.

1. Write the cost of one. The rate is 650p for each kg.
   1 kg costs 650p
2. Choose the operation. 4 kg cost 4 lots of the cost of one, so multiply.
   650p × 4
3. Write the numbers in columns. Ones go under ones, tens under tens and so on. The one-digit number goes under the ones.

   	Th	H	T	O
   		6	5	0
   ×				4

4. Multiply the ones. Always start with the ones, on the right.
   0 × 4 = 0
   0 is less than 10, so write 0. Nothing to carry.
5. Multiply the tens. Now the next column to the left.
   5 × 4 = 20
   20 is 2 hundreds and 0 tens. Write 0 in the tens column and carry 2.
6. Multiply the hundreds. Multiply, then add the 2 we carried.
   6 × 4 = 24, and 24 + 2 = 26
   26 is 2 thousands and 6 hundreds. Write 6 in the hundreds column and carry 2.
7. Write the carried digit. There is nothing left to multiply in the thousands column, but we carried 2 into it.
   Write 2 in the thousands column.
8. Read the answer from the final layout. Check that it has about the right number of digits.

   	Th	H	T	O
   		6	5	0
   ×				4
   	2	6	0	0
   	2	2

9. Change pence to pounds. 100p = £1, so divide by 100 (the digits move 2 places to the right).
   2,600p = £26.00
10. Write the answer. Include the unit.
    £26.00

Check. Estimate: 650 is about 700. 700 × 4 = 2,800. Our answer, 2,600, is close to 2,800. ✓

Watch out. Pence and pounds are different units: 250p is £2.50, not £250.

Key words. A combination is one choice from each group, such as one top and one hat. If there are 3 tops and 5 hats, each top can go with 5 hats, so there are 3 × 5 combinations.

Method. Each choice from the first group can go with every choice from the second group. Multiply the number of choices.

1. List the choices. There are 4 tops and 6 skirts.
   Tops: top 1, top 2, top 3, top 4
   Skirts: skirt A, skirt B, skirt C, skirt D, skirt E, skirt F
2. Draw a table of every pair. Each row is one top. Each box is one pair. The boxes are numbered.

   tops \ skirts	skirt A	skirt B	skirt C	skirt D	skirt E	skirt F
   top 1	1	2	3	4	5	6
   top 2	7	8	9	10	11	12
   top 3	13	14	15	16	17	18
   top 4	19	20	21	22	23	24

3. Count the pairs by multiplying. Every one of the 4 rows has 6 pairs.
   4 × 6 = 24 outfits

Check. Count the numbered boxes: the last one is 24. ✓

Watch out. Do not add. 4 + 6 = 10 would only count each item once, not every pair.

Key words. A combination is one choice from each group, such as one top and one hat. If there are 3 tops and 5 hats, each top can go with 5 hats, so there are 3 × 5 combinations.

Method. Each choice from the first group can go with every choice from the second group. Multiply the number of choices.

1. List the choices. There are 3 sizes and 5 flavours.
   Sizes: small, medium, large
   Flavours: strawberry, banana, chocolate, mango, vanilla
2. Draw a table of every pair. Each row is one size. Each box is one pair. The boxes are numbered.

   sizes \ flavours	strawberry	banana	chocolate	mango	vanilla
   small	1	2	3	4	5
   medium	6	7	8	9	10
   large	11	12	13	14	15

3. Count the pairs by multiplying. Every one of the 3 rows has 5 pairs.
   3 × 5 = 15 milkshakes

Check. Count the numbered boxes: the last one is 15. ✓

Watch out. Do not add. 3 + 5 = 8 would only count each item once, not every pair.

Key words. A combination is one choice from each group, such as one top and one hat. If there are 3 tops and 5 hats, each top can go with 5 hats, so there are 3 × 5 combinations.

Method. Multiply the number of choices in every group.

1. Write the number of choices in each group. There are three groups, so we have three numbers.
   3 starters
   4 main courses
   2 desserts
2. Combine the first two groups. Each of the 3 starters goes with each of the 4 main courses.
   3 × 4 = 12
3. Now combine with the third group. Each of those 12 combinations goes with each of the 2 desserts.
   12 × 2 = 24 meals

Check. Try another order: 4 × 2 = 8, then 8 × 3 = 24. ✓

Watch out. With three groups you multiply all three numbers, not just two.

Key words. A counter-example is one example that shows a statement is false.

Method. Try some examples. If the statement is true for every example you can find, ask why. If you find one that is false, it cannot be “always”.

1. Try some numbers. Test the statement with some examples.
   Try 12: 12 × 3 = 36, then a third of 36 is 12.
   Try 20: 20 × 3 = 60, then a third of 60 is 20.
2. Explain why. A reason shows what is happening for every number, not just the ones we tried.
   Making something 3 times as big and then finding a third of it are opposite (inverse) operations, so you always get back to where you started.
3. Decide.
   Always

Check. Both examples returned to the starting number. ✓

Watch out. “Always” means every time. “Sometimes” needs one example that works and one that does not. “Never” means no example works.

Key words. A counter-example is one example that shows a statement is false.

Method. Try some examples. If the statement is true for every example you can find, ask why. If you find one that is false, it cannot be “always”.

1. Try some numbers. Test the statement with some examples.
   4 tops and 6 skirts: 24 outfits, which is more than 6.
   1 top and 6 skirts: 1 × 6 = 6 outfits, which is not more than 6.
2. Look for one that does not work. One example that breaks the rule is called a counter-example.
   1 × 6 = 6 is not bigger than 6.
3. Explain why. A reason shows what is happening for every number, not just the ones we tried.
   If both groups have more than one item, multiplying gives more than either group. But if one group has just 1 item, the number of combinations is equal to the size of the other group.
4. Decide.
   Sometimes

Check. We found an example each way. ✓

Watch out. “Always” means every time. “Sometimes” needs one example that works and one that does not. “Never” means no example works.