Divide It!
What is 6÷3/4?
6÷3/4=6×4/3=24/4=8
How to solve: Keep 6, change division to multiplication, and flip 3/4 to 4/3.
A student solves:
6÷3/4=6×3/4
What is the error?
The student did not use the reciprocal.
Correct:
6÷3/4=6×4/3=8
Error: When dividing fractions, flip the second fraction, not the first.
Convert 3 1/2 to an improper fraction.
3 1/2=3(2)+1 over 2=7/2
Without calculating, will
8×3/4
be greater than or less than 8?
Less than 8
Why: 3/4<1, so multiplying by it decreases the original quantity.
You have 3 pizzas. Each student gets 1/2 pizza. How many students can receive a serving?
3÷1/2=3×2=6
A ribbon is 4 feet long. Each piece is 1/2 foot. How many pieces can be cut?
4÷1/2=4×2=8 pieces
How to solve: Dividing by 1/2 asks how many halves fit into 4.
A student says:
12÷3/4=12×4/3=16
What did the student do correctly, and what does the answer mean?
The procedure is correct.
12×4/3=16
So the answer is:
16
There are 16 groups of 3/4 in 12.
Convert 5 3/4 to an improper fraction.
5 3/4=5(4)+3 /4=23/4
Without calculating, will
10×5/4
increase or decrease 10?
Increase
Why: 5/4>1, so the product is greater than 10.
A baker has 4 pounds of dough. Each loaf requires 2/3 pound. How many complete loaves can be made?
4÷2/3=4×3/2=6
At summer camp, each basketball game lasts 3/4 hour. There are 6 hours available. How many games can be played?
6÷3/4=6×4/3=24/4=8
How to solve: Use the reciprocal of 3/4. There can be 8 complete games
For the wedding-mint problem, a student writes:
12÷3/4=12×4/3=16
The student says, "There will be 16 pounds of mints in each box."
What is the error?
The operation and calculation are backward.
The problem has 3/4 pound of mints being divided into 12 boxes:
3/4÷12
not
12÷3/4
Correct:
3/4÷12=3/4×1/12=3/48=1/16 pound
Error: The student reversed the quantities in the division problem.
2 1/2×3/4
2 1/2=5/2
5/2×3/4=15/8=1 7/8
1 7/8
Which will be greater than the original number?
A. 15×2/3
B. 15×5/4
C. 15×1
B
Why: 5/4>1, so the product increases.
A teacher has 3/4 pound of candy and divides it equally among 12 students. How many pounds does each student receive?
3/4÷12=3/4×1/12=3/48=1/16
Each student receives:
1/16 pound
This is the same mathematical structure as the wedding-mint problem.
Alex has 8 feet of wood. Each piece must be 5/6 foot long. What is the greatest number of complete pieces he can use?
8÷5/6=8×6/5=48/5=9 3/5
Since Alex needs complete pieces, he can use:
9 pieces
How to solve: Divide using the reciprocal, then interpret the answer in the context. You cannot make a tenth complete piece.
A student solves the wood problem:
8÷5/6=8×5/6=40/6
Explain the error.
The student multiplied by 56 instead of its reciprocal.
Correct:
8÷5/6=8×6/5=48/5=9 3/5
So the greatest number of complete pieces is:
9
4 2/3÷7/8
4 2/3=14/3
14/3÷7/8=14/3×8/7
Simplify:
=16/3=5 1/3
5 1/3
Without calculating, compare:
24×7/8
to 24. Is the product greater than, less than, or equal to 24?
Less than 24
Why: 7/8<1.
If calculated:
24×7/8=21
So:
21<24
A runner normally runs 4 1/2 miles. Today she runs 2/3 of her normal distance. How far does she run?
4 1/2=9/2
9/2×2/3=18/6=3
3 miles
A roll of biscuit dough is 10 1/2 inches long. Each slice is 3/8 inch thick. How many slices can Nathan cut?
Convert the mixed number:
10 1/2=21/2
Then divide:
21/2÷3/8=21/2×8/3
Simplify:
21×8 over 2×3 =28
28 slices
How to solve: Convert the mixed number to an improper fraction, multiply by the reciprocal, and simplify.
For the biscuit problem, a student writes:
10 1/2÷3/8=12/2×8/3=16
Identify the error and solve correctly.
The mixed number was converted incorrectly.
The student treated 10 1/2 as 12/2.
Correct conversion:
10 1/2=10(2)+1 over 2=21/2
Then:
21/2×8/3=28
28 slices
Error: The mixed number was not converted correctly.
A recipe requires 2 1/4 cups of flour per batch. How much flour is needed for 3 1/2 batches?
2 1/4=9/4
3 1/2=7/2
Multiply:
9/4×7/2=63/8=7 7/8
7 7/8 cups
How to solve: Convert both mixed numbers to improper fractions, multiply, then convert the answer back to a mixed number.
A number is multiplied by 3/5, and the result is smaller than the original number. Explain why this must happen without calculating the product.
3/5<1
A positive number multiplied by a fraction less than 1 gives only a part of the original amount.
Therefore:
The quantity decreases.
A construction company has an 8-foot board. Each step stool piece must be 5/6 foot long. What is the greatest number of complete pieces that can be cut? Explain why the decimal/fraction remainder matters.
8÷5/6=8×6/5=48/5=9 3/5
The quotient says 9 complete pieces plus 3/5 of another piece.
Because the question asks for complete pieces
Reasoning: The 3/5 leftover is not enough to make another complete 5/6-foot piece.