Finding Fractions Between Other Fractions

Math 131

Spring 04

Finding Fractions Between Other Fractions: Many Methods

Below are several students’ attempts to find a fraction that is between 1/3 and 1/2. Evaluate each student’s reasoning. If the reasoning is incorrect, try to fix it. If possible, use the student’s method (possibly modified) to find two fractions that are between 2/3 and 5/6. When you are done, answer the summary questions at the end.

Katia: My answer is 3/8. I used my fraction strips to solve this problem. I lined up the thirds strip and the halves strip, and then I tried some of the other strips. The one that worked was the 1/8's strip; I found that 3/8 is a little more than 1/3.

Demetria: My answer is 1/6. I also used fraction strips, and I lined them up like Katia did. I noticed that the space between 1/3 and 1/2 was 1/6, so that's my answer.

Beatriz: My answer is 41/100. I tried to think of my whole as a dollar. Then 1/3 of a dollar is between 33 and 34 cents, and 1/2 a dollar is 50 cents. So I picked 41 cents as a number in-between 1/3 and 1/2 a dollar. Since each cent is 1/100 of a dollar, I am talking about forty-one 1/100's or 41/100 as being between 34/100 and 50/100.

Tanicia: My answer is 1/4. I used the Cuisinaire Rods. I made the orange the whole. Then the yellow is 1/2, and 1/3 is a little bigger than the light green rod. So I figured the purple would be in-between. Then I lined up whites next to the purple, and there are 4 whites that make the purple, so my answer is 1/4.

Cecilia: My answer is 10/24. I used a common denominator. I remember this from grade school, although I don’t really know why it works. I picked 24 because both 3 and 2 are multiples of it. Then, to get 24 from 3, you have to multiply by 8, so I made 1/3 equal to 8/24 and then to get 24 from 2 you have to multiply by 12, so I made 1/2 equal to 12/24. Then since 10 is between 8 and 12, my answer is 10/24.

Carmel: My answer is 2/5. I first picked 1/2.5 because 2.5 is in-between the denominators 2 and 3. Then I wasn't sure if I could do this, and I didn’t want 2.5 in the denominator. So, I decided to double both the numerator and the denominator to get 2/5, but I wasn’t really sure if that was right.

Then I checked 2/5 as an answer another way. I decided to make 30 tiles my whole. I chose 30 because it is a multiple of 2, 3, and 5. Then 1/2 of my whole is 15 tiles, and 1/3 of my whole is 10 tiles. To find 2/5 of my whole, I first found that 1/5 of my whole is 6 tiles, and then I doubled that to find that 2/5 of my whole is 12 tiles. Since 12 tiles is between 10 and 15 tiles, 2/5 is in between 1/3 and 1/2.

Manuel: My answer is 3/6. I used fraction strips, and I saw that 3/6 is between 1/3 and 1/2.

Pauline: My answer is 5/12. I used pattern blocks, with two yellow hexagons as my whole. I first tried it with 1 yellow hexagon as the whole, but then the blue rhombus was 1/3 and the red trapezoid was 1/2, but there wasn't anything in-between them.

With two yellow hexagons as the whole, 1/2 is one yellow hexagon and 1/3 is a red trapezoid and a green triangle (or two blue rhombi or four green triangles). Then five green triangles would be in-between the fractions representing 1/3 and 1/2. Since the whole is 12 green triangles, the fraction is 5/12.

Terry: I figured that 1/2 = 4/8 and 1/3 = 4/12. Since 4/8 and 4/12 have the same numerators then 4/9, 4/10, and 4/11 would all be in between. Then I tried 1/2 = 9/18 and 1/3 = 6/18. So two more fractions that would be in-between 1/3 and 1/2 are 7/18 and 8/18. Then I decided to look for some crazier answers and I said 1/2 = 99/198 and 1/3 = 66/198, so I have a bunch more answers, 67/198, 68/198, 69/198, etc.

Summary Questions and Extensions:

1. Which methods did you like the best? Which didn’t you like?

2. Which methods are the easiest to explain conceptually (i.e. to explain why they work).

3. Which methods are the most general (i.e. they will work for many similar problems)? Which methods only work for some examples?

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