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Learning Math Home
Number and Operation Session 8: Rational Numbers and Proportional Reasoning
Session 8 Part A Part B Part C Homework
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Session 8 Materials:

A B C 


Solutions for Session 8, Part B

See solutions for Problems: B1 | B2 | B3

Problem B1


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:


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:


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.

<< back to Problem B1


Problem B2

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


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:


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:


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:


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:

<< back to Problem B2


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:


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:


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:


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:


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:

<< back to Problem B3


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