Year 4: Improper fractions and mixed numbers

Key words. An improper fraction has a top number at least as big as the bottom, like 7/4. A mixed number is a whole number and a fraction, like 1 3/4.

Method. Count how many parts are in one whole for the bottom number. Count all the shaded parts for the top number. Count the full wholes for the mixed number.

1. Count the equal parts in ONE whole. This is the bottom number. All the wholes are cut the same way.
   Each whole is cut into 4 equal parts, so the bottom number is 4.
2. Count ALL the shaded parts. This is the top number. It can be bigger than the bottom number.
   7 parts are shaded, so the top number is 7.   74 is an improper fraction.
3. Count the whole ones and the parts left over. 4 parts make 1 whole.
   1 whole one are full and 3 parts are shaded in the next one.
   74 = 134

Check. 1 × 4 + 3 = 7. ✓

Watch out. The top number counts ALL the shaded parts, in every picture.

Key words. An improper fraction has a top number at least as big as the bottom, like 7/4. A mixed number is a whole number and a fraction, like 1 3/4.

Method. Count how many parts are in one whole for the bottom number. Count all the shaded parts for the top number. Count the full wholes for the mixed number.

1. Count the equal parts in ONE whole. This is the bottom number. All the wholes are cut the same way.
   Each whole is cut into 3 equal parts, so the bottom number is 3.
2. Count ALL the shaded parts. This is the top number. It can be bigger than the bottom number.
   5 parts are shaded, so the top number is 5.   53 is an improper fraction.
3. Count the whole ones and the parts left over. 3 parts make 1 whole.
   1 whole one are full and 2 parts are shaded in the next one.
   53 = 123

Check. 1 × 3 + 2 = 5. ✓

Watch out. The top number counts ALL the shaded parts, in every picture.

Key words. An improper fraction has a top number at least as big as the bottom, like 7/4. A mixed number is a whole number and a fraction, like 1 3/4.

Method. Count how many parts are in one whole for the bottom number. Count all the shaded parts for the top number. Count the full wholes for the mixed number.

1. Count the equal parts in ONE whole. This is the bottom number. All the wholes are cut the same way.
   Each whole is cut into 5 equal parts, so the bottom number is 5.
2. Count ALL the shaded parts. This is the top number. It can be bigger than the bottom number.
   9 parts are shaded, so the top number is 9.   95 is an improper fraction.
3. Count the whole ones and the parts left over. 5 parts make 1 whole.
   1 whole one are full and 4 parts are shaded in the next one.
   95 = 145

Check. 1 × 5 + 4 = 9. ✓

Watch out. The top number counts ALL the shaded parts, in every picture.

Key words. An improper fraction has a top number at least as big as the bottom, like 7/4. A mixed number is a whole number and a fraction, like 1 3/4.

Method. Count how many parts are in one whole for the bottom number. Count all the shaded parts for the top number. Count the full wholes for the mixed number.

1. Count the equal parts in ONE whole. This is the bottom number. All the wholes are cut the same way.
   Each whole is cut into 6 equal parts, so the bottom number is 6.
2. Count ALL the shaded parts. This is the top number. It can be bigger than the bottom number.
   11 parts are shaded, so the top number is 11.   116 is an improper fraction.
3. Count the whole ones and the parts left over. 6 parts make 1 whole.
   1 whole one are full and 5 parts are shaded in the next one.
   116 = 156

Check. 1 × 6 + 5 = 11. ✓

Watch out. The top number counts ALL the shaded parts, in every picture.

Key words. An improper fraction has a top number at least as big as the bottom, like 7/4. A mixed number is a whole number and a fraction, like 1 3/4.

Method. A mixed number is whole ones and a fraction. Change the wholes into parts, add the extra parts, and keep the bottom number.

1. Find how many parts are in the whole ones. Each whole is cut into 4 equal parts.
   1 × 4 = 4 parts
2. Add the extra parts. The fraction part has 3 more parts.
   4 + 3 = 7
3. Write the improper fraction. We count in quarters so the bottom number stays 4.
   134 = 74

Check. Go back: 7 ÷ 4 = 1 remainder 3, which is 134. ✓

Watch out. Do not forget to add the extra parts. And do not change the bottom number.

Key words. An improper fraction has a top number at least as big as the bottom, like 7/4. A mixed number is a whole number and a fraction, like 1 3/4.

Method. A mixed number is whole ones and a fraction. Change the wholes into parts, add the extra parts, and keep the bottom number.

1. Find how many parts are in the whole ones. Each whole is cut into 5 equal parts.
   2 × 5 = 10 parts
2. Add the extra parts. The fraction part has 1 more part.
   10 + 1 = 11
3. Write the improper fraction. We count in fifths so the bottom number stays 5.
   215 = 115

Check. Go back: 11 ÷ 5 = 2 remainder 1, which is 215. ✓

Watch out. Do not forget to add the extra parts. And do not change the bottom number.

Key words. An improper fraction has a top number at least as big as the bottom, like 7/4. A mixed number is a whole number and a fraction, like 1 3/4.

Method. A mixed number is whole ones and a fraction. Change the wholes into parts, add the extra parts, and keep the bottom number.

1. Find how many parts are in the whole ones. Each whole is cut into 8 equal parts.
   2 × 8 = 16 parts
2. Add the extra parts. The fraction part has 3 more parts.
   16 + 3 = 19
3. Write the improper fraction. We count in eighths so the bottom number stays 8.
   238 = 198

Check. Go back: 19 ÷ 8 = 2 remainder 3, which is 238. ✓

Watch out. Do not forget to add the extra parts. And do not change the bottom number.

Key words. An improper fraction has a top number at least as big as the bottom, like 7/4. A mixed number is a whole number and a fraction, like 1 3/4.

Method. A mixed number is whole ones and a fraction. Change the wholes into parts, add the extra parts, and keep the bottom number.

1. Find how many parts are in the whole ones. Each whole is cut into 4 equal parts.
   3 × 4 = 12 parts
2. Add the extra parts. The fraction part has 1 more part.
   12 + 1 = 13
3. Write the improper fraction. We count in quarters so the bottom number stays 4.
   314 = 134

Check. Go back: 13 ÷ 4 = 3 remainder 1, which is 314. ✓

Watch out. Do not forget to add the extra parts. And do not change the bottom number.

Key words. An improper fraction has a top number at least as big as the bottom, like 7/4. A mixed number is a whole number and a fraction, like 1 3/4.

Method. A mixed number is whole ones and a fraction. Change the wholes into parts, add the extra parts, and keep the bottom number.

1. Find how many parts are in the whole ones. Each whole is cut into 6 equal parts.
   1 × 6 = 6 parts
2. Add the extra parts. The fraction part has 5 more parts.
   6 + 5 = 11
3. Write the improper fraction. We count in sixths so the bottom number stays 6.
   156 = 116

Check. Go back: 11 ÷ 6 = 1 remainder 5, which is 156. ✓

Watch out. Do not forget to add the extra parts. And do not change the bottom number.

Key words. An improper fraction has a top number at least as big as the bottom, like 7/4. A mixed number is a whole number and a fraction, like 1 3/4.

Method. A mixed number is whole ones and a fraction. Change the wholes into parts, add the extra parts, and keep the bottom number.

1. Find how many parts are in the whole ones. Each whole is cut into 7 equal parts.
   4 × 7 = 28 parts
2. Add the extra parts. The fraction part has 2 more parts.
   28 + 2 = 30
3. Write the improper fraction. We count in sevenths so the bottom number stays 7.
   427 = 307

Check. Go back: 30 ÷ 7 = 4 remainder 2, which is 427. ✓

Watch out. Do not forget to add the extra parts. And do not change the bottom number.

Key words. An improper fraction has a top number at least as big as the bottom, like 7/4. A mixed number is a whole number and a fraction, like 1 3/4.

Method. An improper fraction is more than 1 whole. Find how many wholes it contains. The parts left over make the fraction part.

1. Ask: how many wholes are in it? 4 quarters make 1 whole. So we ask how many 4s fit into 9.
   4 × 2 = 8, which is not more than 9.
   4 × 3 = 12, which is more than 9.
   So there are 2 whole ones.
2. Find how many parts are left over. Take the parts in the whole ones away from all the parts.
   9 − 8 = 1
3. Write the mixed number. The whole number is 2. The left over parts make the fraction, still quarters.
   94 = 214

Check. Go back: 2 × 4 + 1 = 9 = 9. ✓

Watch out. The bottom number stays the same. Only the top number and the whole number change.

Key words. An improper fraction has a top number at least as big as the bottom, like 7/4. A mixed number is a whole number and a fraction, like 1 3/4.

Method. An improper fraction is more than 1 whole. Find how many wholes it contains. The parts left over make the fraction part.

1. Ask: how many wholes are in it? 3 thirds make 1 whole. So we ask how many 3s fit into 7.
   3 × 2 = 6, which is not more than 7.
   3 × 3 = 9, which is more than 7.
   So there are 2 whole ones.
2. Find how many parts are left over. Take the parts in the whole ones away from all the parts.
   7 − 6 = 1
3. Write the mixed number. The whole number is 2. The left over parts make the fraction, still thirds.
   73 = 213

Check. Go back: 2 × 3 + 1 = 7 = 7. ✓

Watch out. The bottom number stays the same. Only the top number and the whole number change.

Key words. An improper fraction has a top number at least as big as the bottom, like 7/4. A mixed number is a whole number and a fraction, like 1 3/4.

Method. An improper fraction is more than 1 whole. Find how many wholes it contains. The parts left over make the fraction part.

1. Ask: how many wholes are in it? 5 fifths make 1 whole. So we ask how many 5s fit into 11.
   5 × 2 = 10, which is not more than 11.
   5 × 3 = 15, which is more than 11.
   So there are 2 whole ones.
2. Find how many parts are left over. Take the parts in the whole ones away from all the parts.
   11 − 10 = 1
3. Write the mixed number. The whole number is 2. The left over parts make the fraction, still fifths.
   115 = 215

Check. Go back: 2 × 5 + 1 = 11 = 11. ✓

Watch out. The bottom number stays the same. Only the top number and the whole number change.

Key words. An improper fraction has a top number at least as big as the bottom, like 7/4. A mixed number is a whole number and a fraction, like 1 3/4.

Method. An improper fraction is more than 1 whole. Find how many wholes it contains. The parts left over make the fraction part.

1. Ask: how many wholes are in it? 6 sixths make 1 whole. So we ask how many 6s fit into 13.
   6 × 2 = 12, which is not more than 13.
   6 × 3 = 18, which is more than 13.
   So there are 2 whole ones.
2. Find how many parts are left over. Take the parts in the whole ones away from all the parts.
   13 − 12 = 1
3. Write the mixed number. The whole number is 2. The left over parts make the fraction, still sixths.
   136 = 216

Check. Go back: 2 × 6 + 1 = 13 = 13. ✓

Watch out. The bottom number stays the same. Only the top number and the whole number change.

Key words. An improper fraction has a top number at least as big as the bottom, like 7/4. A mixed number is a whole number and a fraction, like 1 3/4.

Method. An improper fraction is more than 1 whole. Find how many wholes it contains. The parts left over make the fraction part.

1. Ask: how many wholes are in it? 8 eighths make 1 whole. So we ask how many 8s fit into 17.
   8 × 2 = 16, which is not more than 17.
   8 × 3 = 24, which is more than 17.
   So there are 2 whole ones.
2. Find how many parts are left over. Take the parts in the whole ones away from all the parts.
   17 − 16 = 1
3. Write the mixed number. The whole number is 2. The left over parts make the fraction, still eighths.
   178 = 218

Check. Go back: 2 × 8 + 1 = 17 = 17. ✓

Watch out. The bottom number stays the same. Only the top number and the whole number change.

Key words. An improper fraction has a top number at least as big as the bottom, like 7/4. A mixed number is a whole number and a fraction, like 1 3/4.

Method. An improper fraction is more than 1 whole. Find how many wholes it contains. The parts left over make the fraction part.

1. Ask: how many wholes are in it? 3 thirds make 1 whole. So we ask how many 3s fit into 10.
   3 × 3 = 9, which is not more than 10.
   3 × 4 = 12, which is more than 10.
   So there are 3 whole ones.
2. Find how many parts are left over. Take the parts in the whole ones away from all the parts.
   10 − 9 = 1
3. Write the mixed number. The whole number is 3. The left over parts make the fraction, still thirds.
   103 = 313

Check. Go back: 3 × 3 + 1 = 10 = 10. ✓

Watch out. The bottom number stays the same. Only the top number and the whole number change.

Key words. An improper fraction has a top number at least as big as the bottom, like 7/4. A mixed number is a whole number and a fraction, like 1 3/4.

Method. Count the steps between 0 and 1 for the bottom number. Count the steps from 0 to the point for the top number.

1. Count the equal steps between 0 and 1. This is the bottom number. All the steps are the same size.
   There are 4 steps from 0 to 1. Each step is 14.
2. Count the steps from 0 to the point. This is the top number.
   The point is 5 steps from 0, so it is 54.
3. Write the answer. 4 steps make 1 whole.
   54 = 114

Check. 5 steps of 14 is 5 quarters. ✓

Watch out. Count the spaces (steps), not the tick marks.

Key words. An improper fraction has a top number at least as big as the bottom, like 7/4. A mixed number is a whole number and a fraction, like 1 3/4.

Method. Count the steps between 0 and 1 for the bottom number. Count the steps from 0 to the point for the top number.

1. Count the equal steps between 0 and 1. This is the bottom number. All the steps are the same size.
   There are 4 steps from 0 to 1. Each step is 14.
2. Count the steps from 0 to the point. This is the top number.
   The point is 7 steps from 0, so it is 74.
3. Write the answer. 4 steps make 1 whole.
   74 = 134

Check. 7 steps of 14 is 7 quarters. ✓

Watch out. Count the spaces (steps), not the tick marks.

Key words. An improper fraction has a top number at least as big as the bottom, like 7/4. A mixed number is a whole number and a fraction, like 1 3/4.

Method. Count the steps between 0 and 1 for the bottom number. Count the steps from 0 to the point for the top number.

1. Count the equal steps between 0 and 1. This is the bottom number. All the steps are the same size.
   There are 4 steps from 0 to 1. Each step is 14.
2. Count the steps from 0 to the point. This is the top number.
   The point is 9 steps from 0, so it is 94.
3. Write the answer. 4 steps make 1 whole.
   94 = 214

Check. 9 steps of 14 is 9 quarters. ✓

Watch out. Count the spaces (steps), not the tick marks.

Key words. An improper fraction has a top number at least as big as the bottom, like 7/4. A mixed number is a whole number and a fraction, like 1 3/4.

Method. Count the steps between 0 and 1 for the bottom number. Count the steps from 0 to the point for the top number.

1. Count the equal steps between 0 and 1. This is the bottom number. All the steps are the same size.
   There are 4 steps from 0 to 1. Each step is 14.
2. Count the steps from 0 to the point. This is the top number.
   The point is 11 steps from 0, so it is 114.
3. Write the answer. 4 steps make 1 whole.
   114 = 234

Check. 11 steps of 14 is 11 quarters. ✓

Watch out. Count the spaces (steps), not the tick marks.

Key words. An improper fraction has a top number at least as big as the bottom, like 7/4. A mixed number is a whole number and a fraction, like 1 3/4.

Method. Find the jump. Then count on in quarters. Remember that 4 quarters make 1 whole.

1. Find the jump between two numbers you know. The numbers go up by the same amount each time.
   34 to 1: the jump is 14
2. Count on in quarters. Remember that 4 quarters make 1 whole, so 44 is written as 1.
   34,  1,  114,  124,  134,  2,  214
3. Write the missing numbers.
   114,  2

Check. Count on again from the first number and see that the jump is always 14. ✓

Watch out. When the top number reaches the bottom number you have a whole: 44 is written as 1. After the whole, the fractions carry on: 1 14.

Key words. An improper fraction has a top number at least as big as the bottom, like 7/4. A mixed number is a whole number and a fraction, like 1 3/4.

Method. Find the jump. Then count on in thirds. Remember that 3 thirds make 1 whole.

1. Find the jump between two numbers you know. The numbers go up by the same amount each time.
   2 to 213: the jump is 13
2. Count on in thirds. Remember that 3 thirds make 1 whole, so 33 is written as 1.
   113,  123,  2,  213,  223,  3
3. Write the missing numbers.
   123,  223

Check. Count on again from the first number and see that the jump is always 13. ✓

Watch out. When the top number reaches the bottom number you have a whole: 33 is written as 1. After the whole, the fractions carry on: 1 13.

Key words. An improper fraction has a top number at least as big as the bottom, like 7/4. A mixed number is a whole number and a fraction, like 1 3/4.

Method. Find the jump. Then count back in fifths. Remember that 5 fifths make 1 whole.

1. Find the jump between two numbers you know. The numbers go down by the same amount each time.
   225 to 2: the jump is 25
2. Count back in fifths. Remember that 1 whole is 55.
   225,  2,  135,  115,  45,  25
3. Write the missing numbers.
   135,  45

Check. Count back again from the first number and see that the jump is always 25. ✓

Watch out. Be careful when you pass a whole number going backwards: 1 is the same as 55.

Key words. The numerator is the top number. The denominator is the bottom number.

Method. Fractions with the same bottom number are made of equal pieces. Add the top numbers and keep the bottom number the same.

1. Check the bottom numbers. They are all 5. That means every piece is the same size (fifths), so we can add them.
   45 + 35
2. Add the top numbers. The top numbers tell us how many pieces we have. We are counting pieces of the same size.
   4 + 3 = 7
3. Keep the bottom number the same. The pieces are still fifths. Adding does not change the size of the pieces.
   45 + 35 = 75
4. Write the answer as a mixed number. 7 fifths is more than 1 whole. 5 fifths make 1 whole, so we have 1 whole and 2 fifths left over.
   75 = 125

Check. Take away one of the fractions from the answer: 7 − 3 = 4. That is the top of the other fraction. ✓

Watch out. Do not add the bottom numbers. The pieces stay fifths, so the bottom number stays 5.

Key words. The numerator is the top number. The denominator is the bottom number.

Method. Fractions with the same bottom number are made of equal pieces. Add the top numbers and keep the bottom number the same.

1. Check the bottom numbers. They are all 7. That means every piece is the same size (sevenths), so we can add them.
   57 + 47
2. Add the top numbers. The top numbers tell us how many pieces we have. We are counting pieces of the same size.
   5 + 4 = 9
3. Keep the bottom number the same. The pieces are still sevenths. Adding does not change the size of the pieces.
   57 + 47 = 97
4. Write the answer as a mixed number. 9 sevenths is more than 1 whole. 7 sevenths make 1 whole, so we have 1 whole and 2 sevenths left over.
   97 = 127

Check. Take away one of the fractions from the answer: 9 − 4 = 5. That is the top of the other fraction. ✓

Watch out. Do not add the bottom numbers. The pieces stay sevenths, so the bottom number stays 7.

Key words. The numerator is the top number. The denominator is the bottom number.

Method. Fractions with the same bottom number are made of equal pieces. Add the top numbers and keep the bottom number the same.

1. Check the bottom numbers. They are all 8. That means every piece is the same size (eighths), so we can add them.
   58 + 68
2. Add the top numbers. The top numbers tell us how many pieces we have. We are counting pieces of the same size.
   5 + 6 = 11
3. Keep the bottom number the same. The pieces are still eighths. Adding does not change the size of the pieces.
   58 + 68 = 118
4. Write the answer as a mixed number. 11 eighths is more than 1 whole. 8 eighths make 1 whole, so we have 1 whole and 3 eighths left over.
   118 = 138

Check. Take away one of the fractions from the answer: 11 − 6 = 5. That is the top of the other fraction. ✓

Watch out. Do not add the bottom numbers. The pieces stay eighths, so the bottom number stays 8.

Key words. The numerator is the top number. The denominator is the bottom number.

Method. Fractions with the same bottom number are made of equal pieces. Add the top numbers and keep the bottom number the same.

1. Check the bottom numbers. They are all 9. That means every piece is the same size (ninths), so we can add them.
   79 + 69
2. Add the top numbers. The top numbers tell us how many pieces we have. We are counting pieces of the same size.
   7 + 6 = 13
3. Keep the bottom number the same. The pieces are still ninths. Adding does not change the size of the pieces.
   79 + 69 = 139
4. Write the answer as a mixed number. 13 ninths is more than 1 whole. 9 ninths make 1 whole, so we have 1 whole and 4 ninths left over.
   139 = 149

Check. Take away one of the fractions from the answer: 13 − 6 = 7. That is the top of the other fraction. ✓

Watch out. Do not add the bottom numbers. The pieces stay ninths, so the bottom number stays 9.

Key words. An improper fraction has a top number at least as big as the bottom, like 7/4. A mixed number is a whole number and a fraction, like 1 3/4.

Method. Change both numbers into improper fractions with the same bottom number. Then compare the top numbers.

1. Write any mixed numbers as improper fractions. Wholes become parts: whole × bottom number + top number.
   112 = (1 × 2 + 1) over 2 = 32
2. Rewrite them with the same bottom number, 4. Multiply the top and the bottom by the same number.
   32 = 64
3. Compare the top numbers. The bottom numbers are the same (4).
   7 > 6
4. Write the answer.
   74 > 112

Check. Draw both on a number line and see which is further from 0. ✓

Watch out. Compare the whole numbers first only when they are different. If the whole numbers are the same, compare the fractions.

Key words. An improper fraction has a top number at least as big as the bottom, like 7/4. A mixed number is a whole number and a fraction, like 1 3/4.

Method. Change both numbers into improper fractions with the same bottom number. Then compare the top numbers.

1. Write any mixed numbers as improper fractions. Wholes become parts: whole × bottom number + top number.
   135 = (1 × 5 + 3) over 5 = 85
2. Compare the top numbers. The bottom numbers are the same (5).
   9 > 8
3. Write the answer.
   95 > 135

Check. Draw both on a number line and see which is further from 0. ✓

Watch out. Compare the whole numbers first only when they are different. If the whole numbers are the same, compare the fractions.

Key words. An improper fraction has a top number at least as big as the bottom, like 7/4. A mixed number is a whole number and a fraction, like 1 3/4.

Method. Change both numbers into improper fractions with the same bottom number. Then compare the top numbers.

1. Write any mixed numbers as improper fractions. Wholes become parts: whole × bottom number + top number.
   123 = (1 × 3 + 2) over 3 = 53
2. Rewrite them with the same bottom number, 6. Multiply the top and the bottom by the same number.
   53 = 106
3. Compare the top numbers. The bottom numbers are the same (6).
   11 > 10
4. Write the answer.
   116 > 123

Check. Draw both on a number line and see which is further from 0. ✓

Watch out. Compare the whole numbers first only when they are different. If the whole numbers are the same, compare the fractions.

Key words. An improper fraction has a top number at least as big as the bottom, like 7/4. A mixed number is a whole number and a fraction, like 1 3/4.

Method. Change everything into improper fractions with the same bottom number. Order the top numbers.

1. Write the mixed numbers as improper fractions. Then every number is a fraction of the same kind.
   134 = 74
   112 = 32
2. Rewrite every fraction as quarters. Multiply the top and the bottom by the same number.
   94 = 94
   134 = 74
   54 = 54
   112 = 64
3. Put the top numbers in order. We want smallest first.
   5, 6, 7, 9
4. Write the ORIGINAL numbers in that order.
   54,  112,  134,  94

Check. Check on a number line: the numbers go right along the line. ✓

Watch out. Write the original numbers in your answer.

Key words. An improper fraction has a top number at least as big as the bottom, like 7/4. A mixed number is a whole number and a fraction, like 1 3/4.

Method. Change everything into improper fractions with the same bottom number. Order the top numbers.

1. Write the mixed numbers as improper fractions. Then every number is a fraction of the same kind.
   134 = 74
   118 = 98
2. Rewrite every fraction as eighths. Multiply the top and the bottom by the same number.
   118 = 118
   134 = 148
   32 = 128
   118 = 98
3. Put the top numbers in order. We want biggest first.
   14, 12, 11, 9
4. Write the ORIGINAL numbers in that order.
   134,  32,  118,  118

Check. Check on a number line: the numbers go left along the line. ✓

Watch out. Write the original numbers in your answer.