77 Enhancing the teaching of problem solving involving fractions
Saturday
9:45 AM-11:15 AM
Cheryl McAllister ([email protected])
Southeast Missouri State University
Truman
Middle-Workshop
This workshop will focus on how to improve student performance when dealing with fractions in
problem solving situations. Emphasis will be given to use of manipulatives, drawing diagrams, and
writing and solving word problems related to fractions.
I. Conceptualizations of fractions
There is no standard way to define a fraction
A. Some of the ways to conceptualize or define fractions include:
1. Part-whole:
2. Operator:
3
means 3 parts out of 4 equal parts of a unit whole
4
3
of something – multiply by 3 and divide by 4 OR divide by 4, then multiply
4
by 3.
3
means 3 parts compared to 4 parts
4
3
3
4. Quotient:
means 3 divided by 4.
is the amount each person receives when 4
4
4
3. Ratio and Rates:
people share a 3-unit of something (concrete model)
5. Measure:
3
means a distance of 3 one-fourth units from 0 on the number line or 3
4
one-fourth units of a given area
II. Focus on a solid understanding of the definition of a fraction – word problems that require only
definition and making drawings.
Exploring the Part-whole Concept of Fractions
The most common way fractions are presented to students in school mathematics is by the concept of equal parts of
a whole. The following activity is designed to help you see the importance in identifying the ‘whole’ or ‘unit’ when
dealing with fraction representations.
3
? What is the whole? What do the smaller subdivisions of the rectangle
5
3
represent? In the symbolic representation , what does the 5 mean? What does the 3 mean? Could this figure
5
2
represent ?
5
1. How does the figure above represent
5
of something? What is the whole in this case?
3
5
What do the smaller subdivisions of the rectangle represent? Interpret the symbolic representation in this
3
2. Can you now look at the same figure and see it representing
situation.
3. Can you see
5
3
of
in this situation? Explain.
3
5
4. Can you see
2
3
of
in this situation? Explain.
3
5
5. Can you see 1 
3
in this situation? Explain.
5
7. Jenny has a large piece of fabric that she has cut into two pieces: a larger piece and a smaller piece. By laying the
pieces of fabric next to each other, Jenny can tell that the smaller piece of fabric is the same size as
2
of the larger
3
piece. What fraction of the original whole piece of fabric (before cutting) is the small piece of fabric? Use the partwhole meaning of fractions and a picture to help you explain your answer.
8. Each of the eight pairs of rectangles shown below represents what is left of two cakes, one with pink frosting (the
lighter shading) and one with chocolate frosting (the darker shading). For each pair, look at what is left of the two
cakes and tell which portion is more or whether they are the same.
9. Benton used
3
cup of butter to make a batch of cookie dough. Benton rolled his cookie dough out into a
4
rectangle, as shown below. Now Benton wants to cut off a piece of the dough so that the portion he cuts off contains
1
cup of butter. How could Benton cut the dough? Explain.
4
3
of the recipe. Instead of figuring out
4
3
3
3
what number
of 5 is, Mira measures
of a cup of flour 5 times, and uses this amount of flour for
of the
4
4
4
10. Mira has a recipe that calls for 5 cups of flour. Mira wants to make
recipe. Use the part-whole meaning of fractions and the following picture to help you explain why Mira’s strategy is
valid:
This activity was adapted from activities in Class Activities to Accompany Mathematics for Elementary Teachers by
Sybilla Beckmann (2005). Boston: Pearson Education, Inc. and Making Sense of Fractions, Ratios, and Proportions:
2002 Yearbook, pp. 103 – 106.NCTM (2002). Reston, VA: NCTM.
III. Concrete models of operations with fractions – have students write word problems to go with the
models.
Problem Solving and Dr. Loyd’s Fraction Kit
I. Find the unit (whole) given the following information. Explain how you solved the problem by drawing
a picture of the fraction pieces.
a) The pink piece is ¼ of some amount. Find that the color and size of that piece. ___________________
b) The pink piece is 1/6 of some amount. Find that the color and size of that piece.
__________________
c) The smallest dark green piece is 1/3 of some amount. Find that piece. _________________
II. If the pink piece is ¼ what value do these pieces have. Explain your reasoning.
a) 1 orange piece
b) 1 small dark green piece
c) 1 smoky gray piece
Extra challenge: If the smoky gray piece is 2/3 what value does one pale green piece have?
IV. Writing good fraction word problems.
Contexts sheet and pretest assignment.
Exploring basic types of fraction word problems
MA318 – McAllister
Addition – Join Situations
1. Betty picked 4 2/3 lb of green beans. John picked 3 ¼ lb. of green beans. How many pounds
of green beans did they pick together?
Subtraction – Take away Situation
2. Gary has 7/8 yard of string. He cuts off ½ yard and gives it to Tom. How much string does he
have left?
Subtraction – Comparison Situation
3. Linda has a garden that is 9/10 of an acre. Bobby has a garden that is 5/6 of an acre. How
much bigger is Linda’s garden?
Subtraction – Missing Addend Situation
4. Jackie’s teacher assigned 7 ¾ pages of her science book to read for a quiz tomorrow. So far,
she has read 5 ½ pages. How many more pages does she need to read to complete the
assignment?
Subtraction – ‘what’s left of the whole’ situation
5. Ben has a Hershey’s bar. He eats 5/8 of it. How much of the candy bar is left?
Subtraction – missing subtrahend situation
6. Robert bought 4/5 lb of chocolate candies. He leaves them in his room and goes out to play.
When he comes back to his room, there is only ½ lb of chocolate candy left. How much candy
did his mom ‘borrow’?
Addition – missing minuend situation
7. Margot’s grandmother gave her some fancy ribbon to use for a craft project. She used 4 2/3
yard for her project. She has 2 ¼ yards of the ribbon left. How much did her grandmother give
her?
Add/Subtract Combination problems
8. Patty’s quilt has three colors of squares- red, green and yellow. If the quilt is 2/5 red squares
and 1/3 green squares, what fraction of the quilt is composed of yellow squares?
Multiplication – area context
9. Billy has a cow pasture that is 7/8 mile in length and 4/5 mile in width. What is the area of the cow
pasture in square miles?
Multiplication – scaling context – enlarging.
10. Betty is making cookies for a large gathering. Her recipe calls for 1 ¼ cup flour and 3/4 cup of sugar.
She wants to make 3 ½ times the recipe. How much flour and sugar will she need for this?
Multiplication – scaling context – shrinking.
11. Tommy grandmother has a 7/8 acre garden. She gives Tommy 1/5 of her garden for a 4-H project.
How much of a whole acre is Tommy’s part of the garden?
Division – measurement situation
12. Joe pours 4 ½ cups of Sugar Pops into his bowl. He happens to look at the side of the box. It
indicates that 1 serving of the cereal is ¾ cup. How many servings of Sugar Pops is Joe planning to eat?
Division – fair share (partitive) situation
13. Susan is picking berries for a pie. So far she has 4/5 pint of berries. This is 1/3 of the berries her
recipe calls for. How many pints of berries does she need altogether?
For each of the following fraction problems:
1. Solve the problem showing your steps. No calculators allowed. Final answers should be
simplified and improper fractions changed to mixed numerals
A.
3 2

5 7
E.
5 7

7 9
B.
1
1
1 2
5
4
F.
3 3
1 1
8 5
C.
G.
2 4

3 7
1 5

6 9
D.
3
3
4 2
5
4
H.
5 2
2 
6 5
2. Solve each of the following word problems that involve fractions. You may use any method.
A. Jenny has a large piece of fabric that she has cut into two pieces: a larger piece and a smaller
piece. By laying the pieces of fabric next to each other, Jenny can tell that the smaller piece of fabric is
the same size as
3
of the larger piece. What fraction of the original whole piece of fabric (before cutting)
4
is the small piece of fabric?
B. Greg plants three-fifths of his garden in potatoes and one-sixth in carrots. What fraction of
the garden remains for his other crops? How much more of his garden is planted in potatoes than in
carrots.
C. Jake ate a third of his cookie and gave the rest of the cookie to his dog Barnie. Barnie ate
one-half of the piece of cookie that Jake gave him. How much of a cook is left?
D. A recipe for cookies will prepare enough for four-sevenths of Ms. Jordan’s class of 28
students. If she makes three batches of cookies, how many extra students can she feed?
E. Mrs. Martin is making 4 bridesmaid dresses and 1 dress for the flower girl. Each bridesmaid
dress takes 3 3/4 yards of fabric and the dress for the flower girl requires 1/2 as much fabric as the dress
for a bridesmaid. What is the total amount of fabric Mrs. Martin needs to make all of the dresses
(assuming she doesn’t need any extra fabric)?
F. You are eating Crispy Oat Crunchies for breakfast. You pour 2 1/3 cup of the cereal in your
bowl. You notice on the box that 3/4 cup is considered the serving size for this cereal. According to the
box, how many servings of Crispy Oat Crunchies are you having for breakfast?
G. Bob is making a rectangular table top. The table is 11/12 yard wide and 1 2/3 yard long.
What is the area of the table in square yards?
H. Sam buys 2 4/5 lb of jelly beans. He wants to share them evenly among his 3 sisters. How
many pounds of jelly beans will each sister get?
I. The county is paving the gravel road in front of my house. So far they have paved 5 3/4 miles.
This is 5/6 of the total length the final paved road section will be. What will be the length of the finished
paved road?