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Learning Math: Number and Operations

Rational Numbers and Proportional Reasoning Part A: Interpreting Fractions, Units, and Unitizing (45 minutes)

In This Part: Interpretation of Fractions
We know that a fraction, as a “rational” number, is a ratio of two numbers. See Note 2 below. In common usage, this ratio represents how many parts you have of a whole. But can a fraction have a different meaning?

There are actually two ways you might use fractional representation:
1. One or more parts of a unit that has been divided into some number of equal-sized parts, which is a “part-whole” interpretation. For example, for the fraction 3/4, you might represent three slices of a pie that’s been cut into four equal slices, or note that three out of every four balloons in a display are red. This is the interpretation most often used in the early and intermediate elementary grades.
2. 
One quantity in a whole compared to another quantity in a whole, which is a “part-part” interpretation. For example, to note that there are three red balloons for every four white balloons in a display, you would also use the fraction 3/4 (in this case, read as “three to four” rather than “three-fourths”). See Note 3 below.

For example, if Elizabeth has three red balloons and four green balloons, the ratio of red to green is 3/4. Suppose someone gives Elizabeth one more red balloon and two more green balloons (a ratio of 1/2). What is the ratio of red to green in Elizabeth’s new collection? Let’s explore this problem.Our standard rules for operations with fractions work perfectly with part-whole fractions, because the units are equivalent. These rules break down, however, when we look at part-part fractions.

Students will be amazed to hear that, to answer this problem, they can “add” fractions the way they’ve always wanted to — by simply adding the numerators and then adding the denominators!

3/4 + 1/2 = (3 + 1)/(4 + 2) = 4/6, or 2/3


Problem A1
Why does this work? How is this situation different from part-whole interpretation of fractions, which allows us to use our standard rules for operations with fractions?

Interpreting Fractions, Units, and Unitizing adapted from Lamon, Susan J. Teaching Fractions and Ratios for Understanding: Essential Content Knowledge and Instructional Strategies for Teachers (pp. 27, 41, 54-57). © 1999 by Lawrence Erlbaum Associates.

In This Part: Units and Unitizing
Math problems should either implicitly or explicitly define what unit you’re working with. Without this context, some problems are ambiguous. In such cases, you should identify the units before doing any computation. Problems A2-A4 are examples of problems with ambiguous units.


Problem A2

The shaded part could represent 5, or 2 1/2, or 5/8, or 1 1/4. Name the unit in each of these cases.


Problem A3
Given that the six points are evenly spaced on the number line, what common fraction corresponds to Point E?


Problem A4
a. The shaded part of the figure below is 3 2/3:

Specify the unit that is defined implicitly.
b. Using that same unit, how much would four small rectangles represent?
c. List three or more other values that the shaded part of the figure above could represent, and name the unit in each case.


Problem A5
Three turkey slices together weigh 1/3 pound. Jake is allowed only four ounces of turkey for his lunch. How many turkey slices can he eat?

Tip: There are 16 ounces in a pound.

Notes

Note 1
Materials Needed:
• 
Cuisenaire Rods (optional)
You can use the cut-outs of the rods from the Rod Template.

Note 2
We’re using the term “ratio” here to represent the comparison of one rational number to another in either a part-to-whole or a part-to-part situation, as you will see in Part A. In mathematical notation, a ratio can be written as a/b or a:b. Proportion is a fractional equation comprising two ratios (a/b = c/d). In common language, “proportion” is often used interchangeably with “ratio.”

Note 3
The distinction between part-part and part-whole relationships is seldom discussed. Usually we refer to part-part relationships as ratios rather than fractions. The ratio representation is a better way to write the relation, because you won’t be tempted to use the rules of fractions to work with them (for example, a ratio of 3 parts to 0 parts cannot be written as a fraction). However, the part-part relationship is sometimes written in fraction form (as it is in this section), even though it represents a ratio. So be sure to understand the difference.

Solutions

Problem A1
This works because the fractions 3/4 and 1/2 represent part-part relationships, and we are finding the ratio of the combined group (or “mixture”). In part-whole problems, adding fractions represents adding new parts to the same whole (more pieces of pie, for example); therefore, the denominator (the whole) stays the same. In part-part problems, adding these ratios (expressed as fractions) changes the whole — Elizabeth has more of each type of balloon now, both the numerator and the denominator, so you would add each part.

Here’s a similar example from baseball: In one game, a batter gets two hits in three at-bats. In a second game, he gets one hit in four at-bats. In total, he gets three hits in seven at-bats.

2/3 + 1/4 = 3/7

Problem A2
If the shaded part represents 5, the unit is one circle. If the shaded part represents 2 1/2, the unit is two circles. If the shaded part represents 5/8, the unit is all eight circles. If the shaded part represents 1 1/4, the unit is four circles. Other units are also possible.

Problem A3
Probably the easiest way to solve this problem is to find a common denominator so you can find the difference between the two named fractions. The lowest common denominator is 12: 3/4 – 1/3 = 9/12 – 4/12 = 5/12. There are five spaces between 1/3 and 3/4, so each space must represent 1/12. Then A is 4/12 (or 1/3), B is 5/12, C is 6/12 (or 1/2), D is 7/12, E is 8/12 (or 2/3), and F is 9/12 (or 3/4). Therefore, E is 2/3.

Problem A4
a. 
The unit would be three small rectangles, or 3/4 of one large rectangle.
b. 
Four small rectangles would represent 1 1/3 units in this situation.
c. 
The shaded part could represent 2 3/4 if one large rectangle is the unit. It could represent 11 if one small rectangle is the unit. It could represent 11/12 if all three large rectangles is the unit. Other units are also possible.

Problem A5
Jake can eat only 1/4 of a pound. Since three slices weigh 1/3 of a pound, the “unit” (one pound) is equivalent to nine turkey slices. Jake can eat 1/4 of that “unit,” so he can eat 9/4, or 2 1/4, turkey slices.

Notice there are many other ways to do this problem. The given solution, however, uses the concept of a “unit” and is probably the most concise.

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Learning Math: Number and Operations

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Produced by WGBH Educational Foundation. 2003.
  • ISBN: 1-57680-678-2

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