Problems, Links and Tips
WEBCAST SEGMENTS
The Curriculum

A valuable learning experience is to hunt through the curriculum, at the various grade
levels and in all strands, to find expectations related to fractions.

Mindmaps can be effective for having students explore how fractions are used in our
daily lives. For more ideas about using mindmaps, refer to Dr. Cathy Marks Krpan’s
book Math Expressions: Developing Student Thinking and Problem Solving through
Communication, pages 120 to 124.
The Big Ideas

In addition to the big ideas highlighted in the curriculum, guide, and Marian Small’s
Making Math Meaningful, check out Van de Walle’s Big Ideas in Teaching StudentCentered Mathematics: Grades 3-5 on page 134. Look for similarities among all of the
big ideas presented in the various resources.
How Concepts Develop

“The development of mathematical knowledge is a gradual process. A continuous,
cohesive program throughout the grades is necessary to help students develop an
understanding of the “big ideas” of mathematics” (Ontario Ministry of Education - The
Curriculum, Grades 1 to 8, p. 4).

The PRIME developmental map shown in various segments is supported by documents in
the PRIME kit, which provide in-depth information of the various phases, concepts, and
skills, the content knowledge, and assessment strategies. They are designed to help
teachers assess and plan for students in the various developmental phases.

In addition to PRIME, there are other tools, which can help to gain an understanding of
how concepts develop. For example, Cathy Fosnot created her “Landscape of Learning”
for fractions. This can be accessed in Young Mathematicians at Work: Fractions,
Decimals, and Percents, on page 136. It can be helpful to check out a couple of the
developmental tools and find the commonalities.

In the Print and Video Resources section, there is a clip in which Doug Clements talks
more about Learning Trajectories. There are trajectories that relate to several strands and
mathematical concepts.
1
The Textbook

In the textbook’s teacher’s resource, “many of the problems that teachers will face have
been considered and planned for” (Small, 2009, p. 632).

Take advantage of the informative components that are in your textbook’s Teacher’s
Guide. In addition to ones mentioned in this segment, there are often additional questions,
assessment strategies, as well as suggestions for differentiating for struggling students
and for those who can benefit from more enriched challenges.
The Written Diagnostic

“The beginning [of planning a unit] is normally some sort of activity to set the stage for
the unit and to provide the teachers with important diagnostic information about their
students. (Small, 2009, p. 630).

To look at a diagnostic, which probes the understanding of one half, check out Gap
Closing Module 1, Number Sense, The Diagnostic at
www.edugains.ca/newsite/math2/gapclosing.html
The Oral Diagnostic

Use pattern blocks to reinforce the importance of knowing what the whole is. Have
students figure out what fractions other pieces are if the yellow hexagon is the whole.
Next, have them figure out the fractional values of pieces when the whole is two
hexagons. This ensures that they do not label one shape as always being the same
fraction.
o What other manipulatives could you use to consolidate the importance of
knowing what the whole is when investigating fractions?

Sample Problems to reinforce the importance of the whole
Fraction Introductions
Have students describe something about themselves using a fraction. (I make up 1/5 of
my family, I am ¾ through the month of February, I have read ½ of my book). Discuss
which scenarios let you know how much the part and the whole represent, and which are
ambiguous.
The Cake Dilemma
For Danielle’s birthday, her dad ate one fifth of the cake, her brother ate 2/10 of the cake,
Danielle ate 3/10 of the cake and her mom ate one piece. What fraction of the cake was
left? What happens if the students divide the cake into different amounts? (tenths,
twentieths etc.)
2
Building a Collaborative Team

“Providing all students opportunity to learn with understanding is fundamentally an issue
of equity.” (Empson, 2011, xiii).

To learn more about the role of the principal and professional learning conditions, view
the following clips that are in the Print Materials and Video Resources section.
- Cathy Bruce: Professional Learning - The Role of the Principal and Scaling Up
- Lucy West: Insights into Effective Practice - The Role of the Principal
The Language of Fractions

Grade 4 is the first time that students are expected to deal with standard fractional
notation. Before that, fractions are described in words.

To discover more ways to develop mathematic vocabulary, see Teaching Reading in
Mathematics, 2nd Edition by Mary Lee Barton and Clare Heidema.
Fractions as Division

“When fractions are introduced right from the start within fair-sharing contexts, as
division, a misconception … is usually avoided … that the greater the denominator the
greater the amount” (Fosnot, 2002, p. 56).

“Division into equal groups is a process that young children … understand intuitively …
children’s early thinking about multiplication and division provide the foundation for
learning fractions” (Empson, 2011, xix).

Pose problems that involve 4 people sharing 3 objects and then 3 people sharing 4
objects. Discuss how the situations are alike and different and how strategies may change
depending on the problem.

Read the book “Gator Pie” to your students, which deals with alligators trying to share a
pie. More and more gators keep showing up, making the sharing rather interesting.

To learn more about how students solve equal sharing problems, check out Cathy
Fosnot’s “The Sandwich Problem” described in Young Mathematicians at Work:
Constructing Fractions, Decimals, and Percents pages 2 to 16.

For more information about equal sharing problems, related examples, and the various
strategies that students use, see Chapters 1 and 2 in Extending Children’s Mathematics:
Fractions and Decimals, by Empson and Levi.
3
Representing Fractions

“Research indicated that students should be exposed to number lines and rectangular area
models in early grades. These representations supported students in understanding the
notion that a fraction is a number (for example 1/3 is a number) as well as enabled
students to create equal partitions using a variety of strategies” (edugains.ca).

“Sometimes it is useful to do the same activity with two quite different models; from the
viewpoint of the students, the activity is quite different” (Van de Walle, 2006, p. 134).
Area Models

Why would this be a good assessment question for students learning about fractions?
What possible responses might you get? What might this tell you about student
understanding of area models? What might be some next steps for both the
misconceptions and the correct responses?
What fraction of the shape is shaded? How do you know?
-
)
Linear Models

For more information on how to use a Mathematical Clothesline as a linear model, check
out pages 39 to 43 in Dr. Cathy Marks Krpan’s book Math Expressions: Developing
Student Thinking and Problem Solving through Communication.

To learn more about ways to use Fraction Towers, check out About Teaching
Mathematics: A K – 8 Resource 3rd Edition Pages 271 to 272, by Marilyn Burns.

Sample Problem for the Linear Model – Human Number Line
Have one student stand at one end of the room with a “0” sign. Have another student
stand at the other end of the room with a “1” sign. Give students various fractions and
have them place themselves on the number line. Through discussions with people around
them, they can determine whether they are in the correct place.
Another twist: Have a mystery person stand somewhere on the line and students figure
out what fraction is being represented. Do this again, but change the length of the number
line.
4
Visualization

“Number sense with fractions demands more – it requires that students have some
intuitive feel for fractions. They should know ‘about’ how big a particular fraction is and
be able to tell easily which of two fractions is larger” (Van de Walle, 2005, p. 74).

“Another useful skill for students is visualizing the whole when given the fraction… [it]
forces students to ‘think the other way around” (Small, 2009, p. 203).

Sample Problem
Tell students that the small triangle in a tangram set costs 5 cents to make. Have them
calculate how much each other piece would cost to make (Small, 2009, p. 203).
“I baked some cookies and let them cool on the counter. When I returned to the kitchen I
noticed that one fourth of the cookies (3 cookies) were gone! How many cookies did I
bake?”
Equivalent Fractions

“Teachers often confuse tools with strategies. Unifix cubes or fractions bars or paper and
pencil are not different strategies. They are different tools” (Fosnot, 2002, p. 27).
Comparing and Ordering Fractions

Ask students to estimate what fraction of a large tangram square is taken up by one of the
small triangles. They may visualize the entire square being divided into these triangles to
make their estimate.

“Fractions can be compared by renaming them with common denominators, or by
renaming them with common numerators” (Small, 2009, p. 203). “The strategy of using
common numerators to compare fractions is often overlooked” (p. 206).

Van de Walle recommends that students need to periodically draw representations of the
fractions they are comparing, which can uncover times when students get the correct
answer but in an erroneous way. For example, a student was asked to compare two thirds
and four fifths. The drawing reveals the misconception that the fractional pieces in the
two fractions had to be the same size rather than the wholes. (Van de Walle, 2006, p.
149).
Student: “Since it has more, four-fifths is bigger than two-thirds.”
5
Improper and Mixed Fractions

“In the Fourth National Assessment of Educational Progress about 80% of seventh
graders could change a mixed number to an improper fraction, but fewer than half knew
that 5 and 1/4 was the same as 5 + ¼.” (Van de Walle, 2005, p. 69). Many students revert
to using mindless rules if they lack conceptual understanding.
Debrief and Reflection

“Merely spotting when students are incorrect is relatively easy compared with
understanding the reasons behind their errors. The latter demands careful attention and a
deep knowledge of the mathematics concepts that students are learning” (Burns, 2007, p.
50).
6