NOYCE, 7/07/11
•Fractions:
Equivalence and Representation
WARM-UP: WHAT APPROACHES TO
FRACTIONS HAVE YOU SEEN AND/OR
USED?
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Conceptual approaches
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What is the unit?
Part to whole relationships
Ratios and proportions
Decimals, percents, and connections to fractions
Comparison of fractions
Connections with division
Representation
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Pattern blocks
Fraction circles
Fraction rectangles
Fraction bars
Number line
REPRESENTATION OF FRACTIONS AS
POINTS IN THE COORDINATE PLANE
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On graph paper, create a coordinate plane with
the origin in the middle of the page.
Plot the following points, so that the
Numerator is the y-coordinate, and
 Denominator is the x-coordinate

1 2 3 5 2 4
, , , , ,
2 4 6 10 4 8

What do you notice?
MORE FRACTIONS REPRESENTED ON THIS
LINE?
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Find another 2 fractions that are represented on
this line. Plot their points.
Find another 2 points on this line, which do not
have positive coordinates. What fractions do they
represent?
Find another 2 points on this line which do not
have integer coordinates. What fractions do they
represent?
WHAT ABOUT THE ORIGIN??
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Did you choose the origin as one of your points?
Why or why not?
Does the origin also represent a fraction as the
other points do? Why or why not?
SOME MORE LINES
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Now plot (on the same coordinate plane) the
representations of the following fractions:
2 8 4 1 3
, , ,
,
3 12 6 1.5 4.5
1 1 4 3 5
, ,
, ,
3 3 12 9 15
Connect the points in each set with a line. Let’s
call the lines L1, L2, and L3.
 What relationships do you see between the lines?
What is similar and what is different?
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WE NOW HAVE…
CONNECTIONS: FRACTIONS AND SLOPE OF
LINES THROUGH THE ORIGIN
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What can we say about the relationships between
the fractions represented by points on one line?
What can we say about the relationships between
fractions and slope?
What can we say about the relationships between
fractions represented on different lines?
CONNECTIONS: FRACTIONS AND SLOPE OF
LINES THROUGH THE ORIGIN
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What can we say about the relationships between
the fractions represented by points on one line?
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All fractions represented by points on one line
through the origin are equivalent.
What can we say about the relationships between
fractions and slope?
Any fraction represented by a point on a line is the
slope of the line.
 All the slopes thus noted are equivalent, since all
fractions represented on the same line are
equivalent.
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What can we say about the relationships between
fractions represented on different lines?
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These fractions are not equivalent.
SIMPLFIYING FRACTIONS TO LOWEST
TERMS
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Use the fraction-line connection to simplify the
following fractions
6

18
14

16
4

10
SIMPLIFYING FRACTIONS TO LOWEST
TERMS
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Use the fraction-line connection to simplify the
following fractions
6 1

18 3
14 7

16 8
4
2

10
5
COMPARING FRACTIONS
Take a new sheet of graph paper, and create a
coordinate plane.
 Use the ideas we’ve just discussed to compare the
values of the following pairs of fractions:

a)
2
3
and
5
8
a)
5
8
and
2
3
b)
5
8
 and 
2
3
CONNECTION TO RIGHT TRIANGLES AND
THE TANGENT FUNCTION
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y
8
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x
4
5
1
10
CONNECTION TO RIGHT TRIANGLES AND
THE TANGENT FUNCTION
-5
-1
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-4
x
-10 -8
y
CONNECTION TO RIGHT TRIANGLES AND
THE TANGENT FUNCTION
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Bruce drew triangle ABC using the following
coordinates: (0,0), (5,0), and
(5,-4). Give the coordinates of the vertices for
two other triangles similar to triangle ABC
that are in different quadrants created by
rotation about the origin.
What are the coordinates of the vertices for
each triangle?
 What is the scale factor of each triangle?
 What are the angle measurements for each
triangle?
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TANGENT TABLE
DEBRIEF
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In what ways does this approach to the
relationship between fractions and lines enhance
your students’ understanding of one or the other
or both?
What standards at your grade level can be
addressed either by this approach?
Where might you include this kind of activity in
your teaching?
RESOURCES
Lombard, B. and Fulton, B. Simply Great Math
Activities; Fractions Decimals, and Percents.
 Schuster, L. and Anderson, N. Good Questions for
Math Teaching.
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Additional material using graph paper can be
found at
https://docs.google.com/viewer?url=http://www.ttt
press.com/pdf/CAMT-2004-Fraction-FinderHandout.pdf