Worksheet · Fraction Division · 1 of 8
Roots of Reason
Name
Date
Fraction Bars
The bars are drawn and cut. Your only job: count the pieces.
1.2 ÷ 1/3. How many thirds fit in 2 wholes?
2 ÷ 13 =
2.4 ÷ 1/2
4 ÷ 12 =
3.3 ÷ 1/4. Cut the bars into fourths yourself, then count.
3 ÷ 14 =
4.Look at your three answers. When you divide by a fraction smaller than 1, is the answer bigger or smaller than the number you started with? Why?
Division still asks the old question: how many fit?
Worksheet · Fraction Division · 2 of 8
Roots of Reason
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Date
Number Line Jumps
Each jump is one divisor. Count the jumps to reach the dividend.
1.3 ÷ 1/2. The first two jumps are drawn. Finish the jumps, then count them.
3 ÷ 12 = jumps
2.2 ÷ 1/4. Mark the fourths yourself, draw the jumps, count.
2 ÷ 14 =
3.Write the division equation this story matches, then solve it with jumps in your head: "A 4-meter board is cut into 1/2-meter shelves. How many shelves?"
equation:
shelves =
A number line turns division into counting jumps.
Worksheet · Fraction Division · 3 of 8
Roots of Reason
Name
Date
The Common Denominator Strategy
Fraction ÷ fraction: cut the bar into the divisor's pieces, count the shaded ones.
1.1/2 ÷ 1/8. The bar is cut into eighths and 1/2 is shaded. Count the shaded eighths.
12 ÷ 18 =
2.2/3 ÷ 1/6. The bar is cut into sixths. Shade 2/3 of it first, then count.
2/3 = sixths
23 ÷ 16 =
3.5/6 ÷ 1/6. Draw your own bar this time.
56 ÷ 16 =
The answer is a count of pieces, so it can be bigger than 1 even when both fractions are small.
Worksheet · Fraction Division · 4 of 8
Roots of Reason
Name
Date
Equal Shares
Fraction ÷ whole number asks: how big is each share? Name the answer from the whole.
1.1/2 ÷ 3. The half is split into 3 equal shares. What is one share, named from the whole bar?
12 ÷ 3 =
2.1/3 ÷ 2. Draw the model.
13 ÷ 2 =
3.3/4 ÷ 3. Draw the model.
34 ÷ 3 =
4.1/2 ÷ 3 and 3 ÷ 1/2 use the same numbers. Why are the answers so different?
How many fit counts pieces. How big is each share names a piece. Different questions, different answers.
Worksheet · Fraction Division · 5 of 8
Roots of Reason
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Date
Choosing a Model
No bars given. Pick your model: bar, number line, or area. Label everything.
1.2 ÷ 2/3. Careful: the last piece may not be whole.
2 ÷ 23 =
2.3/4 ÷ 1/4
34 ÷ 14 =
3.1/2 ÷ 4
12 ÷ 4 =
A model you drew yourself is proof, not decoration.
Worksheet · Fraction Division · 6 of 8
Roots of Reason
Name
Date
Predicting the Quotient
Before touching a pencil: will the answer be bigger or smaller than the first number?
1.Circle your prediction, then compute. Use a model or the reciprocal strategy.
5 ÷ 12
bigger smaller
=
12 ÷ 5
bigger smaller
=
4 ÷ 23
bigger smaller
=
6 ÷ 34
bigger smaller
=
2.One of your predictions above surprised most students the first time. Which one, and what does it teach?
Dividing by less than 1 grows the answer. Dividing by more than 1 shrinks it.
Worksheet · Fraction Division · 7 of 8
Roots of Reason
Name
Date
Fraction Division Stories
For each story: write the equation, draw a quick model, answer with a unit.
1.A 3-yard ribbon is cut into 1/2-yard pieces for bows. How many bows?
equation: answer:
2.2/3 pound of trail mix is packed into 1/6-pound snack bags. How many bags?
equation: answer:
3.1/2 gallon of paint covers 4 equal walls. How much paint per wall?
equation: answer:
Problems 1 and 2 count pieces. Problem 3 shares. Read before you divide.
Worksheet · Fraction Division · 8 of 8
Roots of Reason
Name
Date
The Reciprocal Strategy
Flip the divisor and multiply. Every answer still needs a reason.
1.Use the reciprocal strategy: flip the divisor, then multiply.
34 ÷ 18 =
5 ÷ 23 =
25 ÷ 4 =
2.Find the flaw. Leo computes 1/2 ÷ 1/4 = 1/8 "because half of a fourth is an eighth." Draw the model that proves him wrong, then explain what he did instead of dividing.
12 ÷ 14 =
Leo actually found:
3.Explain to a 5th grader why 1 ÷ 2/3 = 3/2. You may use words, a drawing, or both.
A shortcut you can defend is yours. One you cannot defend belongs to someone else.