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
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
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.
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.
Given that the six points are evenly spaced on the number line, what common fraction corresponds to Point E?
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.
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.
• Cuisenaire Rods (optional)
You can use the cut-outs of the rods from the Rod Template.
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.”
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.
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
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.
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.
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.
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.
Session 1 What Is a Number System?
Understand the nature of the real number system, the elements and operations that make up the system, and some of the rules that govern the operations. Examine a finite number system that follows some (but not all) of the same rules, and then compare this system to the real number system. Use a number line to classify the numbers we use, and examine how the numbers and operations relate to one another.
Session 2 Number Sets, Infinity, and Zero
Continue examining the number line and the relationships among sets of numbers that make up the real number system. Explore which operations and properties hold true for each of the sets. Consider the magnitude of these infinite sets and discover that infinity comes in more than one size. Examine place value and the significance of zero in a place value system.
Session 3 Place Value
Look at place value systems based on numbers other than 10. Examine the base two numbers and learn uses for base two numbers in computers. Explore exponents and relate them to logarithms. Examine the use of scientific notation to represent numbers with very large or very small magnitude. Interpret whole numbers, common fractions, and decimals in base four.
Session 4 Meanings and Models for Operations
Examine the operations of addition, subtraction, multiplication, and division and their relationships to whole numbers. Work with area models for multiplication and division. Explore the use of two-color chips to model operations with positive and negative numbers.
Session 5 Divisibility Tests and Factors
Explore number theory topics. Analyze Alpha math problems and discuss how they help with the conceptual understanding of operations. Examine various divisibility tests to see how and why they work. Begin examining factors and multiples.
Session 6 Number Theory
Examine visual methods for finding least common multiples and greatest common factors, including Venn diagram models and area models. Explore prime numbers. Learn to locate prime numbers on a number grid and to determine whether very large numbers are prime.
Session 7 Fractions and Decimals
Extend your understanding of fractions and decimals. Examine terminating and non-terminating decimals. Explore ways to predict the number of decimal places in a terminating decimal and the period of a non-terminating decimal. Examine which fractions terminate and which repeat as decimals, and why all rational numbers must fall into one of these categories. Explore methods to convert decimals to fractions and vice versa. Use benchmarks and intuitive methods to order fractions.
Session 8 Rational Numbers and Proportional Reasoning
Begin examining rational numbers. Explore a model for computations with fractions. Analyze proportional reasoning and the difference between absolute and relative thinking. Explore ways to represent proportional relationships and the resulting operations with ratios. Examine how ratios can represent either part-part or part-whole comparisons, depending on how you define the unit, and explore how this affects their behavior in computations.
Session 9 Fractions, Percents, and Ratios
Continue exploring rational numbers, working with an area model for multiplication and division with fractions, and examining operations with decimals. Explore percents and the relationships among representations using fractions, decimals, and percents. Examine benchmarks for understanding percents, especially percents less than 10 and greater than 100. Consider ways to use an elastic model, an area model, and other models to discuss percents. Explore some ratios that occur in nature.
Session 10 Classroom Case Studies, K-2
Watch this program in the 10th session for K-2 teachers. Explore how the concepts developed in this course can be applied through case studies of K-2 teachers (former course participants) who have adapted their new knowledge to their classrooms.
Session 11 Classroom Case Studies, 3-5
Watch this program in the 10th session for grade 3-5 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 3-5 teachers (former course participants) who have adapted their new knowledge to their classrooms.