Year 5: Improper fractions and mixed numbers

Key words. An improper fraction has a top number that is bigger than (or equal to) the bottom number, like 74. A mixed number is a whole number and a fraction together, like 134.

Method. Count the equal parts in one whole for the bottom number. Count all the shaded parts for the top number. The full whole ones give the mixed number.

1. Look at how each bar is cut. Every bar is cut into the same number of equal parts. That number is the denominator.
   Each bar has 4 equal parts, so the denominator is 4.
2. Count all the shaded parts. The total number of shaded parts is the numerator.
   1 full bars of 4 parts and 3 more parts: 1 × 4 + 3 = 7
3. Write the improper fraction and the mixed number. There are 1 whole bars shaded and 34 of another.
   74 = 134

Check. 1 × 4 + 3 = 7. ✓

Watch out. Count every shaded part in all the shapes, not just the last one.

Key words. An improper fraction has a top number that is bigger than (or equal to) the bottom number, like 74. A mixed number is a whole number and a fraction together, like 134.

Method. Count the equal parts in one whole for the bottom number. Count all the shaded parts for the top number. The full whole ones give the mixed number.

1. Look at how each circle is cut. Every circle is cut into the same number of equal parts. That number is the denominator.
   Each circle has 5 equal parts, so the denominator is 5.
2. Count all the shaded parts. The total number of shaded parts is the numerator.
   1 full circles of 5 parts and 4 more parts: 1 × 5 + 4 = 9
3. Write the improper fraction and the mixed number. There are 1 whole circles shaded and 45 of another.
   95 = 145

Check. 1 × 5 + 4 = 9. ✓

Watch out. Count every shaded part in all the shapes, not just the last one.

Key words. An improper fraction has a top number that is bigger than (or equal to) the bottom number, like 74. A mixed number is a whole number and a fraction together, like 134.

Method. Count the equal parts in one whole for the bottom number. Count all the shaded parts for the top number. The full whole ones give the mixed number.

1. Look at how each bar is cut. Every bar is cut into the same number of equal parts. That number is the denominator.
   Each bar has 3 equal parts, so the denominator is 3.
2. Count all the shaded parts. The total number of shaded parts is the numerator.
   3 full bars of 3 parts and 2 more parts: 3 × 3 + 2 = 11
3. Write the improper fraction and the mixed number. There are 3 whole bars shaded and 23 of another.
   113 = 323

Check. 3 × 3 + 2 = 11. ✓

Watch out. Count every shaded part in all the shapes, not just the last one.

Key words. An improper fraction has a top number that is bigger than (or equal to) the bottom number, like 74. A mixed number is a whole number and a fraction together, like 134.

Method. An improper fraction has more parts than one whole. Divide the top by the bottom to find how many whole ones it holds.

1. Divide 7 by 4. The fraction line means divide. The answer tells us how many whole ones we can make.
   7 ÷ 4 = 1 remainder 3
2. The whole number is 1. 1 lots of 4 quarters make 1 whole ones: 1 × 4 = 4.
   7 − 4 = 3 quarters are left over.
3. Write the leftover parts as a fraction. The remainder is the numerator. The denominator stays 4.
   1 and 34 = 134

Check. Change back: 1 × 4 + 3 = 7 = 7. ✓

Watch out. The leftover (remainder) goes on top of the same denominator. Do not put it over the whole number.

Key words. An improper fraction has a top number that is bigger than (or equal to) the bottom number, like 74. A mixed number is a whole number and a fraction together, like 134.

Method. An improper fraction has more parts than one whole. Divide the top by the bottom to find how many whole ones it holds.

1. Divide 9 by 5. The fraction line means divide. The answer tells us how many whole ones we can make.
   9 ÷ 5 = 1 remainder 4
2. The whole number is 1. 1 lots of 5 fifths make 1 whole ones: 1 × 5 = 5.
   9 − 5 = 4 fifths are left over.
3. Write the leftover parts as a fraction. The remainder is the numerator. The denominator stays 5.
   1 and 45 = 145

Check. Change back: 1 × 5 + 4 = 9 = 9. ✓

Watch out. The leftover (remainder) goes on top of the same denominator. Do not put it over the whole number.

Key words. An improper fraction has a top number that is bigger than (or equal to) the bottom number, like 74. A mixed number is a whole number and a fraction together, like 134.

Method. An improper fraction has more parts than one whole. Divide the top by the bottom to find how many whole ones it holds.

1. Divide 13 by 3. The fraction line means divide. The answer tells us how many whole ones we can make.
   13 ÷ 3 = 4 remainder 1
2. The whole number is 4. 4 lots of 3 thirds make 4 whole ones: 4 × 3 = 12.
   13 − 12 = 1 thirds are left over.
3. Write the leftover parts as a fraction. The remainder is the numerator. The denominator stays 3.
   4 and 13 = 413

Check. Change back: 4 × 3 + 1 = 13 = 13. ✓

Watch out. The leftover (remainder) goes on top of the same denominator. Do not put it over the whole number.

Key words. An improper fraction has a top number that is bigger than (or equal to) the bottom number, like 74. A mixed number is a whole number and a fraction together, like 134.

Method. An improper fraction has more parts than one whole. Divide the top by the bottom to find how many whole ones it holds.

1. Divide 17 by 6. The fraction line means divide. The answer tells us how many whole ones we can make.
   17 ÷ 6 = 2 remainder 5
2. The whole number is 2. 2 lots of 6 sixths make 2 whole ones: 2 × 6 = 12.
   17 − 12 = 5 sixths are left over.
3. Write the leftover parts as a fraction. The remainder is the numerator. The denominator stays 6.
   2 and 56 = 256

Check. Change back: 2 × 6 + 5 = 17 = 17. ✓

Watch out. The leftover (remainder) goes on top of the same denominator. Do not put it over the whole number.

Key words. An improper fraction has a top number that is bigger than (or equal to) the bottom number, like 74. A mixed number is a whole number and a fraction together, like 134.

Method. An improper fraction has more parts than one whole. Divide the top by the bottom to find how many whole ones it holds.

1. Divide 22 by 8. The fraction line means divide. The answer tells us how many whole ones we can make.
   22 ÷ 8 = 2 remainder 6
2. The whole number is 2. 2 lots of 8 eighths make 2 whole ones: 2 × 8 = 16.
   22 − 16 = 6 eighths are left over.
3. Write the leftover parts as a fraction. The remainder is the numerator. The denominator stays 8.
   2 and 68 = 268
4. Simplify the fraction part. 6 and 8 can both be divided by 2.
   68 = 34, so 268 = 234

Check. Change back: 2 × 8 + 6 = 22 = 22. ✓

Watch out. The leftover (remainder) goes on top of the same denominator. Do not put it over the whole number.

Key words. An improper fraction has a top number that is bigger than (or equal to) the bottom number, like 74. A mixed number is a whole number and a fraction together, like 134.

Method. An improper fraction has more parts than one whole. Divide the top by the bottom to find how many whole ones it holds.

1. Divide 12 by 4. The fraction line means divide. The answer tells us how many whole ones we can make.
   12 ÷ 4 = 3 remainder 0
2. The whole number is 3. 3 lots of 4 quarters make 3 whole ones: 3 × 4 = 12.
   Nothing is left over.
3. Write the answer. It is a whole number.
   3

Check. Change back: 3 × 4 + 0 = 12 = 12. ✓

Watch out. The leftover (remainder) goes on top of the same denominator. Do not put it over the whole number.

Key words. An improper fraction has a top number that is bigger than (or equal to) the bottom number, like 74. A mixed number is a whole number and a fraction together, like 134.

Method. Count how many parts there are altogether. A whole number is worth the denominator in parts.

1. Change the 1 whole into thirds. One whole is 33, so 1 wholes are 1 lots of 3.
   1 × 3 = 3, so 1 = 33
2. Add the 2 extra thirds. The fraction part is already in the same size parts.
   3 + 2 = 5
3. Write the improper fraction. The denominator stays the same.
   123 = 53

Check. Change back: 5 ÷ 3 = 1 remainder 2, so 53 = 123. ✓

Watch out. Multiply the whole number by the denominator, then add the numerator. Do not just write the digits next to each other.

Key words. An improper fraction has a top number that is bigger than (or equal to) the bottom number, like 74. A mixed number is a whole number and a fraction together, like 134.

Method. Count how many parts there are altogether. A whole number is worth the denominator in parts.

1. Change the 2 wholes into quarters. One whole is 44, so 2 wholes are 2 lots of 4.
   2 × 4 = 8, so 2 = 84
2. Add the 3 extra quarters. The fraction part is already in the same size parts.
   8 + 3 = 11
3. Write the improper fraction. The denominator stays the same.
   234 = 114

Check. Change back: 11 ÷ 4 = 2 remainder 3, so 114 = 234. ✓

Watch out. Multiply the whole number by the denominator, then add the numerator. Do not just write the digits next to each other.

Key words. An improper fraction has a top number that is bigger than (or equal to) the bottom number, like 74. A mixed number is a whole number and a fraction together, like 134.

Method. Count how many parts there are altogether. A whole number is worth the denominator in parts.

1. Change the 3 wholes into fifths. One whole is 55, so 3 wholes are 3 lots of 5.
   3 × 5 = 15, so 3 = 155
2. Add the 1 extra fifths. The fraction part is already in the same size parts.
   15 + 1 = 16
3. Write the improper fraction. The denominator stays the same.
   315 = 165

Check. Change back: 16 ÷ 5 = 3 remainder 1, so 165 = 315. ✓

Watch out. Multiply the whole number by the denominator, then add the numerator. Do not just write the digits next to each other.

Key words. An improper fraction has a top number that is bigger than (or equal to) the bottom number, like 74. A mixed number is a whole number and a fraction together, like 134.

Method. Count how many parts there are altogether. A whole number is worth the denominator in parts.

1. Change the 4 wholes into eighths. One whole is 88, so 4 wholes are 4 lots of 8.
   4 × 8 = 32, so 4 = 328
2. Add the 5 extra eighths. The fraction part is already in the same size parts.
   32 + 5 = 37
3. Write the improper fraction. The denominator stays the same.
   458 = 378

Check. Change back: 37 ÷ 8 = 4 remainder 5, so 378 = 458. ✓

Watch out. Multiply the whole number by the denominator, then add the numerator. Do not just write the digits next to each other.

Key words. An improper fraction has a top number that is bigger than (or equal to) the bottom number, like 74. A mixed number is a whole number and a fraction together, like 134.

Method. Count how many parts there are altogether. A whole number is worth the denominator in parts.

1. Change the 2 wholes into tenths. One whole is 1010, so 2 wholes are 2 lots of 10.
   2 × 10 = 20, so 2 = 2010
2. Add the 7 extra tenths. The fraction part is already in the same size parts.
   20 + 7 = 27
3. Write the improper fraction. The denominator stays the same.
   2710 = 2710

Check. Change back: 27 ÷ 10 = 2 remainder 7, so 2710 = 2710. ✓

Watch out. Multiply the whole number by the denominator, then add the numerator. Do not just write the digits next to each other.

Key words. An improper fraction has a top number that is bigger than (or equal to) the bottom number, like 74. A mixed number is a whole number and a fraction together, like 134.

Method. Count how many parts there are altogether. A whole number is worth the denominator in parts.

1. Change the 5 wholes into halves. One whole is 22, so 5 wholes are 5 lots of 2.
   5 × 2 = 10, so 5 = 102
2. Add the 1 extra halves. The fraction part is already in the same size parts.
   10 + 1 = 11
3. Write the improper fraction. The denominator stays the same.
   512 = 112

Check. Change back: 11 ÷ 2 = 5 remainder 1, so 112 = 512. ✓

Watch out. Multiply the whole number by the denominator, then add the numerator. Do not just write the digits next to each other.

Key words. An improper fraction has a top number that is bigger than (or equal to) the bottom number, like 74. A mixed number is a whole number and a fraction together, like 134.

Method. Count the equal steps from the nearest whole number (or from 0).

1. Find the whole numbers either side. The point is between two whole numbers.
   It is between 1 and 2.
2. Count the small steps. Each step is 14. There are 4 equal steps between each pair of whole numbers.
   From 1 it is 1 step along, so it is 1 quarter more than 1.
3. Write it as a mixed number and as an improper fraction. 1 wholes are 4 quarters. Add 1 more.
   114 = 54

Check. 1 × 4 + 1 = 5. ✓

Watch out. Count the steps between the numbers, not the tick marks. There is one more tick mark than there are steps.

Key words. An improper fraction has a top number that is bigger than (or equal to) the bottom number, like 74. A mixed number is a whole number and a fraction together, like 134.

Method. Count the equal steps from the nearest whole number (or from 0).

1. Find the whole numbers either side. The point is between two whole numbers.
   It is between 2 and 3.
2. Count the small steps. Each step is 14. There are 4 equal steps between each pair of whole numbers.
   From 2 it is 1 step along, so it is 1 quarter more than 2.
3. Write it as a mixed number and as an improper fraction. 2 wholes are 8 quarters. Add 1 more.
   214 = 94

Check. 2 × 4 + 1 = 9. ✓

Watch out. Count the steps between the numbers, not the tick marks. There is one more tick mark than there are steps.

Key words. An improper fraction has a top number that is bigger than (or equal to) the bottom number, like 74. A mixed number is a whole number and a fraction together, like 134.

Method. Count the equal steps from the nearest whole number (or from 0).

1. Find the whole numbers either side. The point is between two whole numbers.
   It is between 2 and 3.
2. Count the small steps. Each step is 14. There are 4 equal steps between each pair of whole numbers.
   From 2 it is 3 steps along, so it is 3 quarters more than 2.
3. Write it as a mixed number and as an improper fraction. 2 wholes are 8 quarters. Add 3 more.
   234 = 114

Check. 2 × 4 + 3 = 11. ✓

Watch out. Count the steps between the numbers, not the tick marks. There is one more tick mark than there are steps.

Key words. An improper fraction has a top number that is bigger than (or equal to) the bottom number, like 74. A mixed number is a whole number and a fraction together, like 134.

Method. Count the equal steps from the nearest whole number (or from 0).

1. Find the whole numbers either side. The point is between two whole numbers.
   It is between 1 and 2.
2. Count the small steps. Each step is 15. There are 5 equal steps between each pair of whole numbers.
   From 1 it is 2 steps along, so it is 2 fifths more than 1.
3. Write it as a mixed number and as an improper fraction. 1 wholes are 5 fifths. Add 2 more.
   125 = 75

Check. 1 × 5 + 2 = 7. ✓

Watch out. Count the steps between the numbers, not the tick marks. There is one more tick mark than there are steps.

Key words. An improper fraction has a top number that is bigger than (or equal to) the bottom number, like 74. A mixed number is a whole number and a fraction together, like 134.

Method. Count the equal steps from the nearest whole number (or from 0).

1. Find the whole numbers either side. The point is between two whole numbers.
   It is between 2 and 3.
2. Count the small steps. Each step is 15. There are 5 equal steps between each pair of whole numbers.
   From 2 it is 1 step along, so it is 1 fifth more than 2.
3. Write it as a mixed number and as an improper fraction. 2 wholes are 10 fifths. Add 1 more.
   215 = 115

Check. 2 × 5 + 1 = 11. ✓

Watch out. Count the steps between the numbers, not the tick marks. There is one more tick mark than there are steps.

Key words. An improper fraction has a top number that is bigger than (or equal to) the bottom number, like 74. A mixed number is a whole number and a fraction together, like 134.

Method. Count the equal steps from the nearest whole number (or from 0).

1. Find the whole numbers either side. The point is between two whole numbers.
   It is between 2 and 3.
2. Count the small steps. Each step is 15. There are 5 equal steps between each pair of whole numbers.
   From 2 it is 4 steps along, so it is 4 fifths more than 2.
3. Write it as a mixed number and as an improper fraction. 2 wholes are 10 fifths. Add 4 more.
   245 = 145

Check. 2 × 5 + 4 = 14. ✓

Watch out. Count the steps between the numbers, not the tick marks. There is one more tick mark than there are steps.

Key words. An improper fraction has a top number that is bigger than (or equal to) the bottom number, like 74. A mixed number is a whole number and a fraction together, like 134.

Method. An improper fraction has more parts than one whole. Divide the top by the bottom to find how many whole ones it holds.

1. Divide 17 by 5. The fraction line means divide. The answer tells us how many whole ones we can make.
   17 ÷ 5 = 3 remainder 2
2. The whole number is 3. 3 lots of 5 fifths make 3 whole ones: 3 × 5 = 15.
   17 − 15 = 2 fifths are left over.
3. Write the leftover parts as a fraction. The remainder is the numerator. The denominator stays 5.
   3 and 25 = 325

Check. Change back: 3 × 5 + 2 = 17 = 17. ✓

Watch out. The leftover (remainder) goes on top of the same denominator. Do not put it over the whole number.

So the missing number is 2.

Key words. An improper fraction has a top number that is bigger than (or equal to) the bottom number, like 74. A mixed number is a whole number and a fraction together, like 134.

Method. An improper fraction has more parts than one whole. Divide the top by the bottom to find how many whole ones it holds.

1. Divide 23 by 4. The fraction line means divide. The answer tells us how many whole ones we can make.
   23 ÷ 4 = 5 remainder 3
2. The whole number is 5. 5 lots of 4 quarters make 5 whole ones: 5 × 4 = 20.
   23 − 20 = 3 quarters are left over.
3. Write the leftover parts as a fraction. The remainder is the numerator. The denominator stays 4.
   5 and 34 = 534

Check. Change back: 5 × 4 + 3 = 23 = 23. ✓

Watch out. The leftover (remainder) goes on top of the same denominator. Do not put it over the whole number.

So the missing number is 5.

Key words. An improper fraction has a top number that is bigger than (or equal to) the bottom number, like 74. A mixed number is a whole number and a fraction together, like 134.

Method. Count how many parts there are altogether. A whole number is worth the denominator in parts.

1. Change the 2 wholes into eighths. One whole is 88, so 2 wholes are 2 lots of 8.
   2 × 8 = 16, so 2 = 168
2. Add the 3 extra eighths. The fraction part is already in the same size parts.
   16 + 3 = 19
3. Write the improper fraction. The denominator stays the same.
   238 = 198

Check. Change back: 19 ÷ 8 = 2 remainder 3, so 198 = 238. ✓

Watch out. Multiply the whole number by the denominator, then add the numerator. Do not just write the digits next to each other.

So the missing number is 19.

Key words. An improper fraction has a top number that is bigger than (or equal to) the bottom number, like 74. A mixed number is a whole number and a fraction together, like 134.

Method. Count how many parts there are altogether. A whole number is worth the denominator in parts.

1. Change the 4 wholes into sixths. One whole is 66, so 4 wholes are 4 lots of 6.
   4 × 6 = 24, so 4 = 246
2. Add the 1 extra sixths. The fraction part is already in the same size parts.
   24 + 1 = 25
3. Write the improper fraction. The denominator stays the same.
   416 = 256

Check. Change back: 25 ÷ 6 = 4 remainder 1, so 256 = 416. ✓

Watch out. Multiply the whole number by the denominator, then add the numerator. Do not just write the digits next to each other.

So the missing number is 25.

Key words. To add or subtract fractions, the denominators (bottom numbers) must be the same. A common denominator is a bottom number that both denominators go into.

Method. Make sure the denominators are the same. Then add the numerators and keep the denominator.

1. Check the denominators. To add fractions, the parts must be the same size.
   Both denominators are 5, so we can add straight away.
2. Add the numerators. We are adding parts of the same size, so we add the top numbers. The bottom number stays the same.
   35 + 45 = 3 + 45 = 75
3. The answer is 1 or more, so change it to a mixed number or a whole number. 75 means 7 ÷ 5.
   75 = 125
4. Write the answer.
   35 + 45 = 125

Check. Take away: 125 − 45 = 35. ✓

Watch out. Add the numerators only. The denominator does not get added: 1/4 + 1/4 is 2/4, not 2/8.

Key words. To add or subtract fractions, the denominators (bottom numbers) must be the same. A common denominator is a bottom number that both denominators go into.

Method. Make sure the denominators are the same. Then add the numerators and keep the denominator.

1. Check the denominators. To add fractions, the parts must be the same size.
   Both denominators are 4, so we can add straight away.
2. Add the numerators. We are adding parts of the same size, so we add the top numbers. The bottom number stays the same.
   34 + 34 = 3 + 34 = 64
3. The answer is 1 or more, so change it to a mixed number or a whole number. 64 means 6 ÷ 4.
   64 = 124
4. Simplify the fraction part. 2 and 4 can both be divided by 2.
   24 = 2 ÷ 24 ÷ 2 = 12
   So the answer is 112.
5. Write the answer.
   34 + 34 = 112

Check. Take away: 112 − 34 = 34. ✓

Watch out. Add the numerators only. The denominator does not get added: 1/4 + 1/4 is 2/4, not 2/8.

Key words. To add or subtract fractions, the denominators (bottom numbers) must be the same. A common denominator is a bottom number that both denominators go into.

Method. Make sure the denominators are the same. Then add the numerators and keep the denominator.

1. Check the denominators. To add fractions, the parts must be the same size.
   Both denominators are 6, so we can add straight away.
2. Add the numerators. We are adding parts of the same size, so we add the top numbers. The bottom number stays the same.
   56 + 56 = 5 + 56 = 106
3. The answer is 1 or more, so change it to a mixed number or a whole number. 106 means 10 ÷ 6.
   106 = 146
4. Simplify the fraction part. 4 and 6 can both be divided by 2.
   46 = 4 ÷ 26 ÷ 2 = 23
   So the answer is 123.
5. Write the answer.
   56 + 56 = 123

Check. Take away: 123 − 56 = 56. ✓

Watch out. Add the numerators only. The denominator does not get added: 1/4 + 1/4 is 2/4, not 2/8.

Key words. To add or subtract fractions, the denominators (bottom numbers) must be the same. A common denominator is a bottom number that both denominators go into.

Method. Make sure the denominators are the same. Then add the numerators and keep the denominator.

1. Check the denominators. To add fractions, the parts must be the same size.
   Both denominators are 10, so we can add straight away.
2. Add the numerators. We are adding parts of the same size, so we add the top numbers. The bottom number stays the same.
   710 + 910 = 7 + 910 = 1610
3. The answer is 1 or more, so change it to a mixed number or a whole number. 1610 means 16 ÷ 10.
   1610 = 1610
4. Simplify the fraction part. 6 and 10 can both be divided by 2.
   610 = 6 ÷ 210 ÷ 2 = 35
   So the answer is 135.
5. Write the answer.
   710 + 910 = 135

Check. Take away: 135 − 910 = 710. ✓

Watch out. Add the numerators only. The denominator does not get added: 1/4 + 1/4 is 2/4, not 2/8.

Key words. To compare fractions is to say which is bigger. < means “is less than” and > means “is greater than”. A common denominator is a bottom number that both denominators go into.

Method. The denominators are the same, so compare the numerators.

1. Write the mixed numbers as improper fractions. This makes every number a fraction, so we can compare them.
   315 = 3 × 5 + 15 = 165
2. Look at the denominators. If the bottom numbers are the same, the parts are the same size.
   Both are 5, so we can compare the top numbers.
3. Compare the top numbers. With the same bottom number, the fraction with the bigger top number is bigger.
   17 > 16
4. Write the answer. Use the original fractions in your answer.
   175 > 315

Check. Look at the changed fractions again: 175 and 165 have the same denominator, so the bigger numerator is the bigger fraction. ✓

Watch out. A bigger denominator does not mean a bigger fraction: 1/8 is smaller than 1/4.

Key words. To compare fractions is to say which is bigger. < means “is less than” and > means “is greater than”. A common denominator is a bottom number that both denominators go into.

Method. The denominators are the same, so compare the numerators.

1. Write the mixed numbers as improper fractions. This makes every number a fraction, so we can compare them.
   234 = 2 × 4 + 34 = 114
2. Look at the denominators. If the bottom numbers are the same, the parts are the same size.
   Both are 4, so we can compare the top numbers.
3. Compare the top numbers. With the same bottom number, the fraction with the bigger top number is bigger.
   11 > 10
4. Write the answer. Use the original fractions in your answer.
   234 > 104

Check. Look at the changed fractions again: 114 and 104 have the same denominator, so the bigger numerator is the bigger fraction. ✓

Watch out. A bigger denominator does not mean a bigger fraction: 1/8 is smaller than 1/4.

Key words. To compare fractions is to say which is bigger. < means “is less than” and > means “is greater than”. A common denominator is a bottom number that both denominators go into.

Method. The denominators are the same, so compare the numerators.

1. Write the mixed numbers as improper fractions. This makes every number a fraction, so we can compare them.
   113 = 1 × 3 + 13 = 43
2. Look at the denominators. If the bottom numbers are the same, the parts are the same size.
   Both are 3, so we can compare the top numbers.
3. Compare the top numbers. With the same bottom number, the fraction with the bigger top number is bigger.
   5 > 4
4. Write the answer. Use the original fractions in your answer.
   53 > 113

Check. Look at the changed fractions again: 53 and 43 have the same denominator, so the bigger numerator is the bigger fraction. ✓

Watch out. A bigger denominator does not mean a bigger fraction: 1/8 is smaller than 1/4.

Key words. The line in a fraction means divide. So 34 means 3 ÷ 4.

Method. The fraction line means divide. So a ÷ b is the fraction a over b. If the top is bigger, change it to a mixed number.

1. Write the division as a fraction. A fraction line means divide, so the top number is the number being shared.
   3 ÷ 4 = 34
2. Think of sharing. Cut each of the 3 wholes into 4 equal parts. Each person gets 1 part of each whole.
   3 parts of size 14 make 34.
3. Write the answer.
   34

Check. Multiply back: 34 × 4 = 3. ✓

Watch out. Put the number you are dividing (the one being shared) on the top.

Key words. The line in a fraction means divide. So 34 means 3 ÷ 4.

Method. The fraction line means divide. So a ÷ b is the fraction a over b. If the top is bigger, change it to a mixed number.

1. Write the division as a fraction. A fraction line means divide, so the top number is the number being shared.
   5 ÷ 8 = 58
2. Think of sharing. Cut each of the 5 wholes into 8 equal parts. Each person gets 1 part of each whole.
   5 parts of size 18 make 58.
3. Write the answer.
   58

Check. Multiply back: 58 × 8 = 5. ✓

Watch out. Put the number you are dividing (the one being shared) on the top.

Key words. The line in a fraction means divide. So 34 means 3 ÷ 4.

Method. The fraction line means divide. So a ÷ b is the fraction a over b. If the top is bigger, change it to a mixed number.

1. Write the division as a fraction. A fraction line means divide, so the top number is the number being shared.
   7 ÷ 4 = 74
2. Change it to a mixed number. 7 wholes shared between 4: each gets 1 whole, and 3 wholes are left to share.
   7 ÷ 4 = 1 remainder 3
3. Write the left-over part as a fraction. The 3 left over are shared between 4, so each gets 34.
   1 and 34 = 134
4. Write the answer.
   134

Check. Multiply back: 134 × 4 = 7. ✓

Watch out. Put the number you are dividing (the one being shared) on the top.

Key words. The line in a fraction means divide. So 34 means 3 ÷ 4.

Method. The fraction line means divide. So a ÷ b is the fraction a over b. If the top is bigger, change it to a mixed number.

1. Write the division as a fraction. A fraction line means divide, so the top number is the number being shared.
   11 ÷ 3 = 113
2. Change it to a mixed number. 11 wholes shared between 3: each gets 3 wholes, and 2 wholes are left to share.
   11 ÷ 3 = 3 remainder 2
3. Write the left-over part as a fraction. The 2 left over are shared between 3, so each gets 23.
   3 and 23 = 323
4. Write the answer.
   323

Check. Multiply back: 323 × 3 = 11. ✓

Watch out. Put the number you are dividing (the one being shared) on the top.

Key words. Always means true every time. Never means false every time. Sometimes means true for some numbers and false for others.

Method. Test the statement with some numbers. If it is always true, never true, or true for some numbers only, that gives the answer.

1. Try some examples. Testing numbers helps us see if the statement is true every time.
   75 is more than 1 whole (7 > 5).
2. Look for an example that does not work. One example that breaks the rule is a counter-example.
   55 is an improper fraction but it is equal to 1, not more than 1.
3. Decide: always, sometimes or never? An improper fraction is 1 or more. It is more than 1 unless the top and the bottom are equal.
   Sometimes

Check. We found a fraction that is more than 1 and one that is not. ✓

Watch out. To show a statement is “always” true you need a reason. A few examples are not enough to prove it, but one counter-example shows it is not always true.

Key words. Always means true every time. Never means false every time. Sometimes means true for some numbers and false for others.

Method. Test the statement with some numbers. If it is always true, never true, or true for some numbers only, that gives the answer.

1. Try some examples. Testing numbers helps us see if the statement is true every time.
   112 is more than 1
   234 is more than 2
   5110 is more than 5
2. Decide: always, sometimes or never? A mixed number is a whole number (1 or more) plus a fraction, so it is always bigger than its whole number part, and so more than 1.
   Always

Check. Every example is more than 1. ✓

Watch out. To show a statement is “always” true you need a reason. A few examples are not enough to prove it, but one counter-example shows it is not always true.