# More: In-Depth Discussion of the Reasoning Activities in "Teaching Fractions and Ratios for Understanding"

By Susan J. Lamon | Go to book overview

Chapter 4
Units and Unitizing
1. a. If 1 2/3 = 5/3 is 10 strips, then 1/3 is 2 strips, so 3/3 = 1 is 6 strips.
 a. Because 5 orks = 15 squares, 1 ork = 3 squares and 1/3 ork = 1 square. 1 2/3 = 5/3 orks must be 5 squares. b. If 1/5 is 4 stars, then 5/5 = 1 is 20 stars. 1/4 is 5 stars and 3/4 is 15 stars. c. If 9 triangles = 3/7, then 3 triangles = 1/7. So 5/7 must be 15 triangles. d. If 1/2 is 2 circles, then 2/2 = 1 must be 4 circles. 8/8 is 4 circles and 1/8 is 1/2 circle, so 3/8 is 1 1/2 circles.
2. For the child who is just beginning to study fractions with part-- whole comparisons, this picture is confusing in more than one respect. First of all, it uses two different units. The first two fractions are shaded on a unit consisting of 2 squares, but the third fraction is shaded on a unit twice the size of the first unit. For the child who knows that 1/2 and 2/4 are equivalent fractions, the picture is terribly confusing because equivalent fractions should cover the same area (and hence look the same) on the same area model. However, the addition is not correct and so it makes little sense to try to picture it anyway.

We would not want to use this picture with children whose only interpretation of fractions was the part--whole comparison. However, as we have already pointed out, the same fraction notation may be used to convey many different meanings. The fractions in this picture may be interpreted as ratios; ratio addition is different from addition with part--whole comparisons. If we interpret the first two pictures as "1 out of 2" then 1 out of 2 and another 1 out of 2 gives 2 out of 4. You can think about this in a baseball context. If you hit 1 ball out of 2 and later in the practice hit 1 ball out of 2, then for the day, you hit 2 out of 4 balls. But even if we limit the discussion to ratio situations, we still must take great care to make sure that pictures correspond with interpretations. For example, we could write 1:2 or 1/2 meaning a ratio of 1 shaded square to 2 nonshaded squares. For this interpretation, the pictures do not work.

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