Problems for Section 0.1 on Dividing Fractions
1. A bread problem: If one loaf of bread requires
cups of our, then
how many loaves of bread can you make with 10 cups of our? (Assume
that you have enough of all other ingredients on hand.)
(a) Solve the bread problem by drawing a diagram . Explain your
reasoning.
(b) Write a division problem that corresponds to the bread
problem.
Solve the division problem by "inverting and multiplying." Verify
that your solution agrees with your solution in part (a).
2. A measuring problem: You are making a recipe that calls for
cup
of water, but you can't find your
cup measure. You can,
however,
find your 
cup measure.
How many times should you fill your
cup
measure in order to measure
of a cup of water?
(a) Solve the measuring problem by drawing a diagram. Explain your
reasoning.
(b) Write a division problem that corresponds to the measuring problem.
Solve the division problem by "inverting and multiplying."
Verify that your solution agrees with your solution in part (a).
3. Write a "how many groups ?" story problem for
and solve your
problem in a simple and concrete way without using the "invert and
multiply" procedure. Explain your reasoning. Verify that your solution
agrees with the solution you obtain by using the "invert and multiply"
procedure.
4. Write a "how many groups?" story problem for
and solve your
problem in a simple and concrete way without using the "invert and
multiply" procedure. Explain your reasoning. Verify that your solution
agrees with the solution you obtain by using the "invert and multiply"
procedure.
5. Jose and Mark are making cookies for a bake sale. Their recipe calls
for 
cups of our for each batch. They have 5
cups of our. Jose and
Mark realize that they can make two batches of cookies and that there
will be some our left. Since the recipe doesn't call for eggs, and since
they have plenty of the other ingredients on hand, they decide they can
make a fraction of a batch in addition to the two whole batches. But
Jose and Mark have a difference of opinion. Jose says that
and so he says that they can make
batches of cookies. Mark says
that two batches of cookies will use up 
cups of our, leaving 
left,
so they should be able to make
batches. Mark draws the picture in
Figure 6 to explain his thinking to Jose. Discuss the boys' mathematics:
Figure 6: Representing 
by Considering How Many 
Cups of Flour
are in 5 Cups of Flour
what's right, what's not right, and why? If anything is incorrect, how
could you modify it to make it correct?
6. Marvin has 11 yards of cloth to makes costumes for a play. Each
costume requires 
yards of cloth.
(a) Solve the fol lowing two problems :
i. How many costumes can Marvin make and how much cloth
will be left over?
ii. What is 
(b) Compare and contrast your answers in part (a).
7. A laundry problem: You need
of a cup of laundry
detergent to wash
one full load of laundry. How many loads of laundry can you wash with
5 cups of laundry detergent? (Assume that you can wash fractional
loads of laundry.)
(a) Solve the laundry problem by drawing a diagram. Explain your
reasoning.
(b) Write a division problem that corresponds to the laundry problem.
Solve the division problem by "inverting and multiplying." Verify
that your solution agrees with your solution in part (a).
8. Write a "how many groups?" story problem for
and solve your
problem in a simple and concrete way without using the "invert and
multiply" procedure. Explain your reasoning. Verify that your solution
agrees with the solution you obtain by using the "invert and multiply"
procedure.
9. Write a "how many groups?" story problem for
and solve your
problem in a simple and concrete way without using the "invert and
multiply" procedure. Explain your reasoning. Verify that your solution
agrees with the solution you obtain by using the "invert and multiply"
procedure.
10. Write a "how many groups?" story problem for
and solve your
problem in a simple and concrete way without using the "invert and
multiply" procedure. Explain your reasoning. Verify that your solution
agrees with the solution you obtain by using the "invert and multiply"
procedure.
11. Fraction division story problems involve the simultaneous use of different
wholes. Solve the following paint problem in a simple and concrete
way without using the "invert and multiply" procedure. Describe how
you must work simultaneously with different wholes in solving the problem.
A paint problem: You need
of a bottle of paint
to paint
a poster board. You have 
bottles of paint.
How many
poster boards can you paint?
12. An article by Dina Tirosh, [?], discusses some common errors in division.
The following problems are based on some of the findings of this
article.
(a) Tyrone says that 
doesn't make sense
because 5 is bigger than
and you can't divide a
smaller number by a bigger number. Give
Tyrone an example of a sensible story problem for
. Solve
your problem and explain your solution.
(b) Kim says that 
can't be equal to 12
because when you divide,
the answer should be smaller. Kim thinks the answer should be
because that is less
than 4. Give Kim an example of a story
problem for and explain why it makes sense
that the answer
really is 12, not .
13. Write a story problem for 
and another
story problem for 
(make clear which is which). In each case, use elementary reasoning
about the story situation to solve your problem. Explain your reasoning.
14. Sam picked
of a gallon of
blueberries. Sam poured the blueberries
into one of his plastic containers and noticed that the berries filled
the container full.
Solve the following problems in any way you like
without using a calculator . Explain your reasoning in detail.
(a) How many of Sam's containers will 1 gallon of blueberries fill?
(Assume Sam has a number of containers of the same size.)
(b) How many gallons of blueberries does it take to fill Sam's container
completely full?
15. A road crew is building a road. So far,
of the road has been
completed
and this portion of the road is
of a mile long. Solve
the following
problems in any way you like without using a calculator . Explain your
reasoning in detail.
(a) How long will the road be when it is completed?
(b) When the road is 1 mile long, what fraction of the road will be
completed?
16. Will has mowed of
his lawn and so far it's taken him 45 minutes.
For each of the following problems, solve the problem in two ways:
1) by using elementary reasoning about the story situation and 2) by
inter preting the problem as a division problem (say whether it is a
"how many groups?" or a "how many in one group?" type of problem)
and by solving the division problem using standard paper and pencil
methods. Do not use a calculator. Verify that you get the same answer
both ways.
(a) How long will it take Will to mow the entire lawn (all together)?
(b) What fraction of the lawn can Will mow in an hour?
17. Grandma's favorite muffin recipe uses 
cups of our for one batch of
12 muffins. For each of the following problems, solve the problem in
two ways: 1) by using elementary reasoning about the story situation
and 2) by interpreting the problem as a division problem (say whether
it is a "how many groups?" or a "how many in one group?" type
of problem) and by solving the division problem using standard paper
and pencil methods . Do not use a calculator. Verify that you get the
same answer both ways.
a) How many cups of our are in one muffin?
(b) How many muffins does 1 cup of our make?
(c) If you have 3 cups of our, then how many batches of muffins
can you make? (Assume that you can make fractional batches of
muffins and that you have enough of all the ingredients.)
18. Write a "how many in one group?" story problem for
and use
your story problem to explain why it makes sense to solve
by
"inverting and multiplying," in other words by multiplying 4 by
.
19. Write a "how many in one group?" story problem for
and use
your story problem to explain why it makes sense to solve
by
"inverting and multiplying," in other words by multiplying 4 by
.
20. Write a "how many in one group?" story problem for
and use
your story problem to explain why it makes sense to solve
by
"inverting and multiplying," in other words by multiplying 9 by
.
21. Write a "how many in one group?" story problem for
and use
your story problem to explain why it makes sense to solve
by
"inverting and multiplying," in other words by multiplying
by
.
22. Write a "how many in one group?" story problem for
and use
your story problem to explain why it makes sense to solve
by
"inverting and multiplying".
23. Give an example of either a hands-on activity or a story problem for
elementary school children that is related to a fraction division problem
(even if the children wouldn't think of the activity or problem as
fraction division). Write the fraction division problem that is related to
your activity or story problem. Describe how the children could solve
the problem by using logical thinking aided by physical actions or by
drawing pictures.
24. Buttercup the gerbil drank
of a bottle of water
in days. Assuming
Buttercup continues to drink water at the same rate, how many bottles
of water will Buttercup drink in 5 days? Use multiplication and
division to solve this problem, explaining in detail why you can use
multiplication when you do and why you can use division when you do.
25. If you used
truck loads of mulch
for a garden that covers
of an acre,
then how many truck loads of mulch should you order for a garden that
covers 
acres? (Assume that you will spread
the mulch at the same
rate as before.) Use multiplication and division to solve this problem,
explaining in detail why you can use multiplication when you do and
why you can use division when you do.
26. If pints of jelly
filled jars, then how many jars will you
need
for 12 pints of jelly? Will the last jar of jelly be completely full? If
not, how full will it be? (Assume that all jars are the same size.) Use
multiplication and division to solve this problem, explaining in detail
why you can use multiplication when you do and why you can use
division when you do.
