Learning Math: Number and Operations
Fractions and Decimals Part C: Ordering Fractions (25 minutes)
There are several ways to compare fractions, many of which use benchmarks or intuitive methods and do not require computation of common denominators or converting to decimal form. See
When ordering fractions, use 0, 1/2, and 1 as benchmarks for comparison. That is, first determine whether the fraction is more or less than 1. If it is less than 1, check to see if it is more or less than 1/2. Then further refine the comparisons to see if the fraction is closer to 0, 1/2, or 1.
a. What quick method can you use to determine if a fraction is greater than 1?
b. What quick method can you use to determine if a fraction is greater or less than 1/2?
Organize the following fractions according to these benchmarks: 0 to 1/2, 1/2 to 1, greater than 1:
After you organize fractions by benchmarks, you can use these intuitive methods:
• Same denominators: If the denominators of two fractions are the same, just compare the numerators. The fractions will be in the same order as the numerators. For example, 5/7 is less than 6/7.
• Same numerators: If the numerators of two fractions are the same, just compare the denominators. The fractions should be in the reverse order of the denominators. For example, 3/4 is larger than 3/5, because fourths are larger than fifths.
• Compare numerators and denominators: You can easily compare fractions whose numerators are both one less than their denominators. The fractions will be in the same order as the denominators. (Think of each as being a pie with one piece missing: The greater the denominator, the smaller the missing piece, thus, the greater the amount remaining.) For example, 6/7 is less than 10/11, because both are missing one piece, and 1/11 is a smaller missing piece than 1/7.
• Further compare numerators and denominators: You can compare fractions whose numerators are both the same amount less than their denominators. The fractions will again be in the same order as the denominators. (Think of each as being a pie with x pieces missing: The greater the denominator, the smaller the missing piece; thus, the greater the amount remaining.) For example, 3/7 is less than 7/11, because both are missing four pieces, and the 11ths are smaller than the sevenths.
• Equivalent fractions: Find an equivalent fraction that lets you compare numerators or denominators, and then use one of the above rules.
Arrange these fractions in ascending order:
Use benchmarks and intuitive methods to arrange the fractions below in ascending order. Explain how you decided. (The point of this exercise is to think more and compute less!):
Computing and then comparing common denominators can be extremely tedious, as is changing everything to decimals and then comparing the decimals. Try these common-sense methods — you’ll like them!
It is important to be able to picture fractions in your mind (that’s one reason using manipulatives is important) — at least the ones with single-digit denominators. For example, if you use a pie to visualize, let’s say, 2/3 and 4/5, it’s pretty easy to quickly tell which one is larger.
a. If the numerator is larger than the denominator, the fraction is greater than 1.
b. If twice the numerator is larger than the denominator, the fraction is greater than 1/2. If twice the numerator is smaller than the denominator, the fraction is less than 1/2.
Here are the fractions within each range:
• 0 to 1/2: 17/35, 2/9
• 1/2 to 1: 4/7, 14/15
• Greater than 1: 25/23
a. These fractions have the same denominator. The order is 4/17, 7/17, 12/17.
b. These fractions have the same numerator. The order is 3/8, 3/7, 3/4.
c. These fractions all have numerators that are one less than their denominators. The order is 3/4, 5/6, 7/8.
d. These fractions all have numerators that are five less than their denominators. The order is 1/6, 8/13, 12/17.
e. These fractions all have numerators that are one less than their denominators. The order is 2/3, 5/6, 10/11.
First divide the list into fractions larger and smaller than 1/2:
|•||Smaller than 1/2: 2/5, 1/3, 1/4|
|•||Larger than 1/2: 5/8, 3/4, 2/3, 4/7|
For those smaller than 1/2, 2/5 is larger than 1/4 (numerators three less than denominators), 1/3 is larger than 1/4 (same numerator), and 2/5 is larger than 1/3 (compare using equivalent fractions 6/15 and 5/15). The order is 1/4, 1/3, 2/5.
For those larger than 1/2, convert 3/4 and 2/3 to fractions where the numerator is three less than the denominator. (Or you may be able to visualize that 3/4 is greater than 2/3.) The list becomes 5/8, 9/12, 6/9, 4/7. This makes it easy to put the list in order: 4/7, 5/8, 6/9, 9/12.
The complete list is 1/4, 1/3, 2/5, 4/7, 5/8, 2/3, 3/4.
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.