Anchor Chart · Fraction Division
Roots of Reason
Roots of Reason

The Meaning of Fraction Division

3 ÷ 1/2 asks: how many halves fit in 3?
1/2
1/2
1/2
1/2
1/2
1/2
1 whole1 whole1 whole
3 ÷ 12 = 6
six halves fit. count them.
1Same question as whole numbers
12 ÷ 3 asks how many 3s fit in 12. Nothing changed. Only the size of the piece did.
2Smaller piece, bigger answer
More small pieces fit than big ones. Dividing by 1/2 gives an answer larger than what you started with.
Division never stopped meaning how many fit. Hold onto that question.
Roots of Reason CA CCSS 6.NS.A.1 rootsofreason.org
Anchor Chart · Fraction Division
Roots of Reason
Roots of Reason

Area Models for Fraction Division

How many 1/8s fit in 3/4?
1/8
1/8
1/8
1/8
1/8
1/8
3/4 shaded = 6 eighths1/4 empty
34 ÷ 18 = 6
cut the whole into eighths. count the eighths inside the shading.
1Recut into the divisor's pieces
Dividing by 1/8? Cut the model into eighths. The common cut makes both fractions speak the same language.
2Then count
3/4 = 6 eighths. So six pieces of size 1/8 fit. The answer is a count, not a fraction of anything.
Recut, then count. The model does the dividing for you.
Roots of Reason CA CCSS 6.NS.A.1 rootsofreason.org
Anchor Chart · Fraction Division
Roots of Reason
Roots of Reason

Sharing a Fraction

1/2 a pan of brownies shared by 3 people. How much each?
1/2
→
1/6
12 ÷ 3 = 16
splitting the half into 3 equal shares makes sixths.
1This division shares
÷ 3 means split into 3 equal shares. Each new piece is a third of the half, and a third of a half is a sixth of the whole pan.
2Name pieces from the whole
The answer's name always comes from the whole pan, not from the half you started with.
Two questions live inside ÷ : how many fit, and how big is each share. Read the problem to know which one it is asking.
Roots of Reason CA CCSS 6.NS.A.1 rootsofreason.org
Anchor Chart · Fraction Division
Roots of Reason
Roots of Reason

Why the Reciprocal Strategy Works

Flip the divisor and multiply. Here is why that works.
start with one whole
1 ÷ 23 = 32
How many 2/3s fit in 1 whole? One full 2/3 fits, plus half of another. That is 11/2, or 3/2: the flip of 2/3.
then scale up
4 ÷ 23 = 4 × 32 = 6
If 3/2 of them fit in each whole, then 4 wholes hold 4 times as many. Dividing by a fraction is multiplying by its reciprocal.
✓Check with a model
If the strategy and your drawing disagree, trust the drawing and find the slip.
✓Estimate first
Dividing by a fraction under 1 should make the answer grow. If it shrank, something flipped that should not have.
Use the reciprocal strategy once you can explain it. Not before.