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Private: Learning Math: Number and Operations
Rational Numbers and Proportional Reasoning Session 8: Homework
Problem H1
Show how to use Cuisenaire Rods to model 3/4 
2/3.
Problem H2
a. The shaded part is 5 1/4. Specify the unit:
b. Using that same unit, what would three small rectangles represent?
c. List other values that the shaded part could represent, and name the unit for each value.
Problem H3
Under what conditions does 2/3 + 3/4 = 5/7?
Problem H4
Which shows a greater change: a 6-by-6 square becoming a 4-by-8 rectangle, or a 4-by-8 rectangle becoming a 6-by-6 square? Explain how you know.
Problem H5
If you have a $1,000 investment that decreases by 50% in value and then increases by 50%, how much would it then be worth? Did you use relative or absolute reasoning to solve this problem?
Solutions
Problem H1
Using the same model as in Problem B3 (d) to divide 3/4 by 2/3, we need to ask, “How many browns (2/3) are there in a blue rod (3/4)?” The answer is 1 1/8, or 9/8:
Problem H2
a. The unit is four small rectangles.
b. Three small rectangles represent 3/4.
c. Some other values the shaded part could represent are 3 1/2 (the unit is six small rectangles), 21 (the unit is one small rectangle), and 10 1/2 (the unit is two small rectangles). Other values are also possible.
Problem H3
The equation will be satisfied when these fractions represent part-part ratios, and we are finding the proportion of the combined group by adding the numerators and denominators.
Problem H4
Changing a 4-by-8 rectangle into a 6-by-6 square shows a greater relative change.
Here is how we know: When a 6-by-6 square becomes a 4-by-8 rectangle, its area changes from 36 square units to 32 square units — a decrease of four square units. This decrease represents 4/36, or 1/9, of the square’s original area.
When a 4-by-8 rectangle becomes a 6-by-6 square, its area changes from 32 square units to 36 square units — an increase of four square units. This increase represents 4/32, or 1/8, of the rectangle’s original area.
Since 1/8 is greater than 1/9, the rectangle showed a greater change.
However, you could also say that in an absolute sense, the change for both was equal, as the area of both the rectangle and the square changed by four square units.
Problem H5
After it decreases 50% in value, the investment would be worth $500. Then, after the 50% increase, it would be worth $750. Solving this involves multiplicative, or relative type of reasoning.
