NOYCE, 7/07/11•Fractions: Equivalence and Representation

WARM-UP: WHAT APPROACHES TO FRACTIONS HAVE YOU SEEN AND/OR USED? Conceptual approaches

What is the unit? Part to whole relationships Ratios and proportions Decimals, percents, and connections to fractions Comparison of fractions Connections with division

Representation Pattern blocks Fraction circles Fraction rectangles Fraction bars Number line

REPRESENTATION OF FRACTIONS AS POINTS IN THE COORDINATE PLANE 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

What do you notice?

12

,24

,36

,5

10, 2 4

, 4 8

MORE FRACTIONS REPRESENTED ON THIS LINE? 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?? 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 Now plot (on the same coordinate plane) the

representations of the following fractions:

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?

2 3

,8

12, 4 6

,1

1.5,

34.5

13

,1 3

,4

12, 39

, 515

WE NOW HAVE…

CONNECTIONS: FRACTIONS AND SLOPE OF LINES THROUGH THE ORIGIN 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 What can we say about the relationships

between the fractions represented by points on one line? 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.

What can we say about the relationships between fractions represented on different lines? These fractions are not equivalent.

SIMPLFIYING FRACTIONS TO LOWEST TERMS Use the fraction-line connection to simplify

the following fractions6

18

14 16

410

SIMPLIFYING FRACTIONS TO LOWEST TERMS Use the fraction-line connection to simplify

the following fractions6

18

13

14 16

78

410

25

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)

b)

c)

25

and 38

52

and 83

52

and 83

CONNECTION TO RIGHT TRIANGLES AND THE TANGENT FUNCTION

y

8 10

x 4 1 5

CONNECTION TO RIGHT TRIANGLES AND THE TANGENT FUNCTION

-1 -4 x

-10 -8

y

-5

CONNECTION TO RIGHT TRIANGLES AND THE TANGENT FUNCTION 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 ABCthat 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?

TANGENT TABLE

DEBRIEF 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.

Additional material using graph paper can be found at

https://docs.google.com/viewer?url=http://www.tttpress.com/pdf/CAMT-2004-Fraction-Finder-Handout.pdf