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N OYCE, 7/07/11 Fractions: Equivalence and Representation

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W ARM - UP : W HAT APPROACHES TO FRACTIONS HAVE Y OU 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

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R EPRESENTATION OF F RACTIONS 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?

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M ORE 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?

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W HAT 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?

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S OME 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 L 1, L 2, and L 3. What relationships do you see between the lines? What is similar and what is different?

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W E NOW HAVE …

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C ONNECTIONS : F RACTIONS 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?

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C ONNECTIONS : F RACTIONS 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.

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

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

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C OMPARING F RACTIONS 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)

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C ONNECTION TO RIGHT TRIANGLES AND THE T ANGENT FUNCTION y 8 10 x 4 1 5

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C ONNECTION TO RIGHT TRIANGLES AND THE T ANGENT FUNCTION x y -5

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C ONNECTION TO RIGHT TRIANGLES AND THE T ANGENT 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 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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T ANGENT TABLE

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D EBRIEF 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?

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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.ttt press.com/pdf/CAMT-2004-Fraction-Finder- Handout.pdf

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