How many groups of that size fit?
Compare an amount and group size in common units, including fractional numbers of groups.
Begin with the activity ↓How many eighth-metres fit into three quarters of a metre?
Rewrite 3/4 m as 6/8 m: six eighth-metres fit. Division asks how many of the specified group size the amount contains.
Cut 3/4 m of ribbon into pieces of 1/8 m. Express the ribbon as 6/8 m: there are six pieces. Thus 3/4÷1/8=6. The quotient is a number of pieces, not a length in metres.
A group of 1/2 m contains four eighths. Six eighths hold one full group with two eighths remaining. The remainder is half of a four-eighth group, so 3/4÷1/2=1 and 1/2.
Compare 1/2 m with a group size of 3/4 m. In quarters, the amount contains two pieces and a group contains three. The amount is 2/3 of such a group, so 1/2÷3/4=2/3. A quotient need not be a whole number.
Scale both quantities in 1/2÷3/4 by four to obtain 2÷3. Scaling amount and group size equally leaves the number of groups unchanged. This equals 1/2×4/3, connecting reciprocal multiplication with the common-unit meaning.
A divisor of zero does not define a usable group size. In situations with indivisible objects you may instead need full groups and a remainder. Here we compare divisible quantities such as lengths.
Listen one paragraph at a time
If you attached the leftover length directly to the quotient
If the quotient counts bottles, compare leftover liquid with one bottle’s capacity. A remaining 3/10 L is not automatically 3/10 of a bottle.
Now change it yourself
Change the amount and group size, then compare common-size pieces. Try quotients below and above one.
Amount 3/4 = 6/8 · One outlined strip = one equal-size whole
One group 1/8 = 1/8 · One outlined strip = one equal-size whole
One group contains 1 parts of size 1/8. Each outline below now represents this group size.
The divisor is positive and nonzero. A fractional quotient refers to a fraction of one divisor-sized group, not necessarily a fraction of the original metre.
Try with paper and objects
Draw two strips with the same unit spacing, labelling one as total amount and the other as one group size.
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
2 1/2 · A third is two sixths; five sixths contain two and a half such groups.
Does your explanation include these ideas?
Speak, point or draw on paper. Explain the reason, not just the numerical answer.
Continue, or pause here
Read decimal places as tenths and hundredths and explain why 0.3 equals 0.30.