Using Visualization to Extend
Students’ Number Sense and
Problem Solving Skills
in Grades 4-6 Mathematics (Part 2)
LouAnn Lovin, Ph.D.
Mathematics Education
James Madison University
Comparing Fractions
How do you compare fractions?
2
5
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Comparing Fractions
Comparing Fractions
using “size” of parts
Which is larger? 3/5 or 2/5?
Which is larger? 1/8 or 1/9?
Which is larger? 12/17 or 12/19?
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Comparing Fractions
Comparing Fractions by thinking of benchmark
fractions (and size of parts) (0, ½, 1)
Which is larger? 2/5 or 5/8?
Which is larger? 9/10 or ¾?
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Comparing Fractions
1. Same-size parts
3
8
5
8
Common Denominator
2. Same number of parts but different kind of parts
5
5
Common Numerator
7
8
3. More or less than one-half or one whole
4. Distance from one-half
or one whole
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10
2
5
5
8
3
4
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A large candy bar is section off:
If Pat gets ⅓ of what is left, then Pat gets ____ of the
candy bar.
So, ⅓ of ¾ is _____.
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Draw a (proportional) diagram to solve the problem.
(Label your diagram.)
There was ¾ of a pie left in the refrigerator. John ate ⅔ of
what was left. How much pie did he eat? What is the
whole (the unit) for each given fraction and for the
answer?
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Draw a (proportional) diagram to solve the problem.
(Label your diagram.)
Mrs. Green bought 3/5 lb of sugar. She used 2/3 of it to
make a cake. How much sugar did she use?
Mrs. Green bought 3/5 lb of sugar. She used ¾ of it to
make a cake. How much sugar did she use?
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Draw a (proportional) diagram to solve the problem.
(Label your diagram.)
Mrs. Smith bought 5/6 lb of hamburger. She cooked 2/3 of
it. How much hamburger did she cook?
Can you see 10 = 5 in your diagram?
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Act these problems out.
Pay attention to your actions.
Mark has 12 apples. He wants to share them
equally among his 4 friends. How many apples
will each friend receive?
Mark has 12 apples. He put them into bags
containing 4 apples each. How many bags did
Mark use?
What is known in each situation?
What
are
you
trying
to
find
in
each
situation?
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What is known in each problem?
(What is known determines the action used by the student.)
Mark has 12 apples. He put them into bags
containing 4 apples each. How many bags did
Mark use?
Measurement (know # in each group)
Want to find how many groups
Types ofHe
word
problem
handout
Mark has 12 apples.
wants
to share
them
equally among his 4 friends. How many apples will
each friend receive?
Partitive or Sharing (know # of groups)
Want to find how many is in each group
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Fraction Division
Draw a (proportional) diagram to solve the problem.
(Label your diagram.)
 4 boys shared 2/3 of a pie equally. What fraction of the
pie did each boy receive?
 A string of length 4/5 m is cut into 2 equal pieces. What
is the length of each piece?
 Sara poured 2/5 pint of fruit juice equally into 4 cups.
How much fruit juice was there in each cup?
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I have 1 yd of ribbon and it takes ⅕ yd of
ribbon to make a bow. How many bows can I
make?
1 yd
⅕ yd
1÷⅕
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(How many ⅕s are in 1?)
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I have 1 yd of ribbon and it takes 3/5 yd of
ribbon to make a bow. How many bows can I
make?
1÷⅗
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(How many 3/5s are in 1?)
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A recipe calls for ½ of a cup of sugar. You
have ¾ of a cup of sugar. How many recipes
can you make (assuming you have the other
ingredients on hand)?
How many ½s are in ¾?
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Draw a (proportional) diagram to solve the problem.
(Label your diagram.)
Kathleen had 3/4 of a gallon of milk. She gave each of
her cats 1/12 of a gallon to drink. How many cats got
milk?
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Repeating Decimals
Why do fractions like 1/3, 1/6, 1/7, and 1/9 repeat?
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Base Ten uses Powers of Ten
1 .
1
10
100
10
102
101 100 . 10-1
1
1000
1
100
10-2
0.1
10-3
0.01
0.001
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Repeating Decimals
Why do fractions like 1/3, 1/6, 1/7, and 1/9 repeat?
Shade ½ of 1.
Shade ¼ of 1.
Shade 1/5 of 1.
Shade 1/8 of 1.
Shade 1/3 of 1.
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 Take a minute and write down two things you are
thinking about from this afternoon’s session.
 Share with a neighbor.
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Take Aways
 Emphasis is on helping students think about quantities
and their relationships to each other (developing
number sense).
 Visualization strategies can make significant ideas
explicit (color coding, highlighting significant ideas in
students’ work)
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Not everyone sees what you may see.
An old man’s
face or two lovers
kissing?
Cat or mouse?
So we must listen to our students and make sense of
the sense they are making of the mathematics.
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References

Diesmann, C., & English, L. (2001). Promoting the use of diagrams as tools for thinking. In A. Cuoco & F. Curcio (Eds.), The Roles
of Representation in School Mathematics, pp. 77-89. Reston, VA: NCTM.

Dolk, M. & Fosnot, C. (2005). Fostering Children’s Mathematical Development CD. Portsmouth, NH: Heinneman.

Hersch, S., Fosnot, C., & Cameron, A. (2005). Fostering Children’s Mathematical Development: Landscape of Learning Grades 35. Portsmouth, NH: Heinneman.

Fosnot, C. & Dolk, M. (2001). Young Mathematicians at Work: Constructing Multiplication and Division. Portsmouth, NH:
Heinneman.

Fosnot, C. & Dolk, M. (2002). Young Mathematicians at Work: Constructing Fractions, Decimals, and Percents. Portsmouth, NH:
Heinneman.

Gersten, R. & Clarke, B. (2007). Research Brief: Effective Strategies for Teaching Students with Difficulties in Mathematics.
NCTM: Reston, VA.

Ministry of Education Singapore. (2009). The Singapore Model Method. Panpac Education: Singapore.

NCTM (2000). Principles and Standards of School Mathematics. NCTM: Reston, VA.

Parrish, S. (2010). Number Talks: Helping Children Build Mental Math and Computation Strategies. Math Solutions: Sausalito, CA.

VandeWalle, J. & Lovin, L. (2005). Teaching Student-Centered Mathematics: Grades 3-5. Boston: Pearson.
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