A fraction does not have to be smaller than one. The denominator tells us how many equal pieces make one whole. So if the denominator is 3, then three thirds make one complete whole: 33 = 1. Once we have more than three thirds, we have more than one whole.
Watch thirds build a whole
13
23
33
1 whole
1/3 → 2/3 → 3/3 → 1 whole
Start from the fraction
What does 113 actually mean?
It means eleven pieces, each one third of a whole. To make sense of that amount, group the thirds into complete wholes.
113
Each outlined circle is one whole. Each coloured piece is 1/3. Count the eleven pieces, then watch groups of three build whole circles.
11 coloured pieces, each 1/3 of a whole.
3/3 + 3/3 + 3/3 + 2/3 = 11/3
This is division.
11 ÷ 3 = 3 remainder 2
Three complete groups of 3 thirds make 3 wholes, with 2 thirds left.
113
=
323
Look first
One whole and three quarters
Start with one whole and three quarters. It is one quantity, shown in two parts.
1 whole + 3/4 = 7/4
Split the whole into four quarters, then count those four quarters and the three extra quarters.
One whole circle and three quarters of another circle.
4 quarters + 3 quarters = 7 quarters. 134 = 74. The amount stays the same.
Notice
The amount has not changed
1 whole + 34 = 74
Watch the notation
134
≠
134
The whole number and the fraction are separate parts of a mixed number. Do not join the digits together.
Mixed number
It tells us how many complete wholes we have, and then how much of another whole remains.
Improper fraction
It counts every equal-sized piece together, even when there are enough pieces to make one or more whole ones.
Why mixed numbers?
A different way to think about the same quantity
The improper fraction counts all the equal pieces. The mixed number tells us how many complete wholes those pieces make, and what is left over.
113=323
These are two equivalent ways to describe exactly the same amount.
Sometimes one form is easier to understand; sometimes the other is easier to calculate with. So we need to be able to move in both directions.
Reverse the process
Mixed number → improper fraction
Now go backwards. First count how many fraction pieces are hidden inside the wholes. Then add the extra pieces.
Keep this one still
257
2×7→14 sevenths
2 wholes × 7 sevenths in each whole = 14 sevenths
14+5→19 sevenths
Then add the 5 extra sevenths.
197
134
1 whole × 4→4 quarters
4 + 3→7 quarters
74
323
3 × 3 = 9→9 thirds
9 + 2 = 11→11 thirds
113
The pattern
1. Multiply the whole number × denominator.
2. Add the numerator.
3. Keep the same denominator.
Interactive lab
What does the picture show?
Count complete wholes and leftover pieces, then choose the matching number.
Choose an answer.
Method Trainer
Move between the two forms
Try fresh examples in either direction, or mix both directions. Results feed into your Practice Lab log on this device. Each question is counted once; after feedback, you can correct your working or move on. Changing direction starts a new set.
How would you like to answer?
1. To convert the whole number into parts, multiply.
×=
2. Add the numerator to the count of parts from step 1.
?+=
3. Keep the denominator
??
1. Divide the numerator by the denominator
÷=remainder
2. Read the result as wholes and leftover pieces
???
You have learned
A mixed number and an improper fraction can describe exactly the same amount.
To move between them, think in equal-sized pieces: build wholes from the pieces, or break wholes back into those pieces.
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