A part of a part: multiplying fractions
Use an overlap diagram to find a fraction of a fraction of the original whole.
Begin with the activity ↓What fraction of the original whole is half of 2/3?
Choose 2/3 of a whole, then take half of that part. The result is 1/3 of the original whole.
Divide a rectangle into three equal-width columns and shade two: 2/3. To take half of that region, split it into two equal-height parts and keep one.
Extend the horizontal line across the whole rectangle. There are six equal cells and two overlap, giving 2/6=1/3 of the original whole. Thus 2/3×1/2=1/3.
The denominator product, three times two, counts all cells in the whole. The numerator product, two times one, counts selected overlap cells. Multiplying numerators and denominators counts those cells.
Multiplication does not always make a positive amount larger. Taking half makes it smaller. Multiplying by a positive fraction below 1 reduces the amount; multiplying by 1 preserves it. This extends, rather than contradicts, whole-number multiplication.
State the final reference whole. We took half of the selected 2/3, but the answer 1/3 is measured against the original full rectangle. Changing that reference can give a different fraction name for the same amount.
Listen one paragraph at a time
If the reference whole became unclear
Compare overlap cells with every cell in the original rectangle. Treating only the first shaded part as a new whole answers a different question.
Now change it yourself
Change horizontal and vertical fraction selections and count the overlap. A distinct hatch marks the overlap without relying on colour alone.
Select 2/3 of the original whole, then 1/2 of that part. Count cells where both hatch directions overlap.
The diagram uses two fractions between 0 and 1. Products beyond a whole extend the same arithmetic, but this activity begins with parts of parts.
Try with paper and objects
Divide a rectangle equally in both directions and mark selected parts using different hatch directions.
Check your idea with another example
Check your idea 1
Check your idea 2
Try a different setting
Write an answer yourself
Return to the explanationCompare with an explanation
3/10 · Six of twenty cells gives 6/20=3/10.
Does your explanation include these ideas?
Speak, point or draw on paper. Explain the reason, not just the numerical answer.
Continue, or pause here
Compare an amount and group size in common units, including fractional numbers of groups.