Task 12
Improper Fractions to Mixed Numbers
Count in unit fractions, regroup complete wholes and see why division with a remainder gives the mixed number.
Mathematical focus
In a fraction a⁄b, the denominator tells us how many equal bths make one whole. The numerator tells us how many of those parts we have.
To rewrite an improper fraction as a mixed number, we group the unit fractions into complete wholes. Each complete group contains the number of parts named by the denominator.
The rods make the grouping visible before we connect it to the formal division method.
We do — Count in halves
Let the red rod represent one whole. Then white is one-half of red:
w = ½r and 2w = r
Build a train one white rod at a time and count in halves:
½, 2⁄2 = 1, 3⁄2 = 1½, 4⁄2 = 2, 5⁄2 = 2½
Now build five white rods. Group them into pairs. Replace each complete pair with one red whole.
5w = 2r + w
So:
5⁄2 = 2½
What did we actually do? We asked how many groups of two are in five. There are two complete groups with one half left over.
5 ÷ 2 = 2 remainder 1
We do — Now thirds
Let the light green rod represent one whole. White is now one-third of the whole:
w = ⅓g and 3w = g
Build eight white rods. How many complete groups of three can you make?
Replace each complete group of three whites with one light green whole.
8w = 2g + 2w
Therefore:
8⁄3 = 2⅔
Again, the formal calculation mirrors the grouping:
8 ÷ 3 = 2 remainder 2
The quotient tells us the number of complete wholes. The remainder tells us how many thirds are left.
Your turn — Regroup into wholes
Write each improper fraction as a mixed number or whole number.
Predict first. Then use the rods to model and check your answer.
Choose a suitable rod to represent one whole, build the numerator using unit-fraction parts, then replace each complete group with a whole.
- 5⁄2
- 7⁄3
- 7⁄4
- 8⁄2
- 9⁄4
- 11⁄5
For at least two examples, record what the rods show using colour letters as well as numbers.
Be ready to explain: what does the quotient represent? What does the remainder represent?
Show answers
- 5⁄2 = 2½
- 7⁄3 = 2⅓
- 7⁄4 = 1¾
- 8⁄2 = 4
- 9⁄4 = 2¼
- 11⁄5 = 2⅕
Useful whole rods: red for halves, light green for thirds, pink for quarters and yellow for fifths. With these choices, white represents the unit fraction.
Feedback / teacher prompts
- Why are we grouping in twos for halves, threes for thirds and fours for quarters?
- What does the denominator tell us about the size of each group?
- What does the quotient represent physically?
- What does the remainder represent?
- Why does the denominator stay the same in the fractional part of the mixed number?
- What happens when there is no remainder?
Connect to the formal method
The rods are not a different method. They show why division is the formal method.
For example:
11 ÷ 5 = 2 remainder 1 therefore 11⁄5 = 2⅕
Exit strategy: once pupils understand that the numerator is being grouped into denominator-sized wholes, the rods can be withdrawn and division can do the same work abstractly.
Rehearse the teaching
For a task worth practising, work in pairs. One person teaches while the other responds as a pupil using the rods. Model enough to get the pupil started, then stop explaining and question their thinking. Swap roles.
Reflect: What was harder to explain than you expected? Which wording or question worked well?