Solutions for Session 8, Part B

See solutions for Problems: B1 | B2 | B3

Problem B1

 a. You should use the yellow rod as "1," since you can make a five-car, one-color train out of white rods that is the same length as a yellow. Each white represents 1/5: b. The orange rod could be used as the unit, since it can be divided into a five-car train and a two-car train. Here, yellow would represent 1/2, and red would represent 1/5: c. The "main" unit (the "1") must have the numbers you're working with (the trains) as its factors, since each train must divide evenly into the total length.

Problem B2

Throughout this problem, we will use the orange rod as "1" (see Problem B1 for an explanation):

 a. One-half is represented as a yellow rod, and 2/5 is two red rods. Their sum is the same as the length of a blue rod. The blue rod is 9/10 of the length of "1" (the orange rod), so 1/2 + 2/5 = 9/10: b. Three-fifths is three red rods, and 1/2 is a yellow rod. Their difference is the same as the length of a white rod. The white rod is 1/10 of the length of "1," so 3/5 - 1/2 = 1/10: c. Three-fifths multiplied by 1/2 is modeled by counting 3/5 of the yellow rod (the rod representing 1/2). This is a light-green rod, and it represents 3/10: d. This is the equivalent of asking, "How many yellows (1/2) are there in a brown rod (4/5)?" The answer is 1 3/5 or 8/5:

Problem B3

To model thirds and fourths, you would need a rod of length 12 to represent "1." One way to do this is to combine an orange rod with a red rod and consider this a "rod" of length 12. Then a light-green rod represents 1/4, because four of these rods would equal the length of the orange-red rod. Similarly, the purple rod represents 1/3:

 a. Combining a purple rod and a light green rod gives a black rod of length 7/12. This is 7/12 of the overall length of "1," so 1/3 + 1/4 = 7/12: b. A blue rod has a length of 3/4 of "1," and a purple rod has a length of 1/3. Subtracting them gives us a yellow rod, which is 5/12 of the overall length of "1." Therefore 3/4 - 1/3 = 5/12: c. This is modeled by counting 3/4 of the purple rod (the rod representing 1/3), which is three white rods (or a light green rod), so the answer is 1/4: d. This is the equivalent of asking, "How many blues (3/4) are there in a brown rod (2/3)?" Expressing each of these in terms of white rods makes the question "How many nines are there in eight?," so the answer is 8/9: