Download Visual Fraction Models

Amy LeHew
Elementary Math Facilitator Meeting
February2013
Cut a sentence strip so that it is 5/6 units long.
Bring the ends of the sentence strip together to fold the
strip of paper in half.
Label this point.
Last time we looked closely at types of visual
fraction models and considered the
connection between the task and visual
model used to solve it.
Take a look at the 3rd grade fraction standards
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Highlight the phrase “visual fraction model”
everywhere it appears.
Underline the phrase “number line”
everywhere it appears.
Develop understanding of fractions as numbers.
1. Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b
equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
2. Understand a fraction as a number on the number line; represent fractions on a number line
diagram.
Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the
whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the
endpoint of the part based at 0 locates the number 1/b on the number line.
Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0.
Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b
on the number line.
3. Explain equivalence of fractions in special cases, and compare fractions by reasoning about
their size.
a. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a
number line.
b. Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why
the fractions are equivalent, e.g., by using a visual fraction model.
c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole
numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1
at the same point of a number line diagram.
d. Compare two fractions with the same numerator or the same denominator by reasoning about
their size. Recognize that comparisons are valid only when the two fractions refer to the same
whole. Record the results of comparisons with the symbols >, =, or <, and justify the
conclusions, e.g., by using a visual fraction model.
Turn and make a statement to your partner
about something you learned while
highlighting/underlining.
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Most widely used fraction model? Area
◦ Partition pizzas, brownies, etc.
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Limitations of area models
◦ Count pieces without attending to the whole (don’t
distinguish between fractional part of a set from
continuous quantity (area)).
What visual fraction model is used by most
students in our schools?
What visual fraction model is used the least?
Of the students who were successful, they
counted
“two-sixths” for the point depicted in
the figure below: it is “six spaces and
that’s two.”
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One-third is three pieces and one thing
There isn’t another fraction name
Don’t know
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Lay out 2-3 examples of number line work
from your school. Label them 3rd, 4th, or 5th
Gallery Walk
How can number lines help students
understand fraction concepts that are often
obscured by area models?
Support understanding of important properties
of fractions
◦ Numerical unit
◦ Relationship between whole numbers and fractions
◦ Density of rational numbers (infinite # between any
two)
◦ A number can be named infinitely many ways
http://lmr.berkeley.edu/docs/Pt3-Ch13NCTM%20Yearbook07-4.pdf
Read the classroom scenario on pages 20-22
◦ When you finish, reflect silently on the following:
 Do student in your building consider fractions as
numbers? Do teachers?
 What other generalizations do students and teachers
make about fractions? Are some of these
generalizations helpful?
Consider the following two statements. Declare
Always True, Sometimes True
If sometimes true, show an example and a counter
example.
1. The larger the denominator, the
smaller the fraction.
2. Fractions are always less than one.
3. Finding a common denominator is the
only way to compare fractions with
different denominators.
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How do we ensure students make meaning of
the numerator and denominator?
How can we make sure students think of
fractions as numbers?
Using your blank number line sheet and
Cuisenaire rods, label the lines as indicated
below
0
1st
2nd
3rd
5th
line in Halves
line in Fourths
line in Sixths
line in Twelfths
1
What number is halfway between zero and
one-half?
What number is one-fourth more than onehalf?
What number is one-sixth less than one?
What number is one-third more than one?
What number is halfway between one-twelfth
and three-twelfths?
What would you call a number that is halfway
between zero and one-twelfth?
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How might this impact student’s
understanding of fractions as numbers on a
number line? (using the Cuisenaire rods to
identify fractions on a number line)
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How does a number line diagram help students
make meaning of the numerator and denominator?
This activity found on page 23-26
Look at page 27+ for using 2 wholes
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http://www.telegraph.co.uk/news/newsvideo/weirdnewsvideo/9246505/Am
ericas-tallest-man-gets-measured-for-size-21-trainers.html
Amy plans to build an enclosure for her great
danes that is 1/8 of a mile wide and 1/4 of a
mile long. Her daughter argues, “that won’t
be enough room! And proves her point by
showing that…
Carefully cut out a strip of paper that has a
length of 5/6
- Bring the ends of the strip together to fold
the strip of paper in half. How long is half of
the strip? Use your strip to mark this point on
the number line.
- What two numbers can you multiply to find
the length of half the strip? Write an equation
to show this.
Unfold your paper strip so that you start with 5/6
again.
Now fold the strip of paper in half and then in half
again.
i.
How long is half of half of the strip? Use your
strip to mark this point on the number line.
ii. What numbers can you multiply to find the
length of half the strip? Write an equation to
show this
http://s3.amazonaws.com/illustrativemathematics/illustration_pdfs/0
00/000/965/original/illustrative_mathematics_965.pdf?134385689
5

How many 1" x 1" square tiles can you fit into
a 12" by 8" grid
How many 1" x 1" square tiles can you fit into
3/4 of a 12" by 8" grid
How many 1/2" x 1/2" square tiles can you fit
into a 7 1/2" x 9 1/2" grid
How many 1/4" x 1/2" rectangular tiles can
you fit into a 10 1/2" x 8 1/2" grid
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5.NF.4. Apply and extend previous understandings of
multiplication to multiply a fraction or whole number by a
fraction.
Interpret the product (a/b) × q as a parts of a partition of
q into b equal parts; equivalently, as the result of a
sequence of operations a × q ÷ b. For example, use a
visual fraction model to show (2/3) × 4 = 8/3, and create
a story context for this equation. Do the same with (2/3) ×
(4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)
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Find the area of a rectangle with fractional side lengths by
tiling it with unit squares of the appropriate unit fraction
side lengths, and show that the area is the same as would
be found by multiplying the side lengths. Multiply
fractional side lengths to find areas of rectangles, and
represent fraction products as rectangular areas.