By Francis (Skip) Fennell, Posted March 2, 2015 – 

Welcome to the fourth and final installment in our Critical Foundations series, with a continued look at fraction equivalence. See the full introduction to our look at fraction equivalence . This entry provides additional thinking, activities, and tasks and, as usual, we hope that you will respond with comments, questions, thoughts, activities, and ideas.

I noted in the three previous postings ( Place Value, Part 1 and Part 2) and Fraction Equivalence, Part 1 that to me, the “signature expectation” of any pre-K–grade 6 mathematics experience is the nurturing of a sense of number, and that instruction related to the understanding of, and flexible proficiency with, place value and equivalent fractions are foundational pillars to number sense and the prerequisites to work with operations involving whole numbers and fraction, including decimals.

The initial Fraction Equivalence entry discussed the importance of understanding fractions as numbers. In particular, that blog post examined and offered activities designed to underscore the importance of the use of varied representations and equivalence. We know that instruction that helps students move flexibly within and between varied representations (e.g., fraction regions, the number line) will help them in literally “seeing” equivalence and will eventually deepen their understanding to the point at which physical models can be replaced by a student’s mental images. The focus of the current post is extending representations and equivalence to comparing and ordering fractions. These activities and tasks present a natural progression from the activities and related tasks found in the previous blog entry, recognizing that success with comparing and ordering is rooted in prior knowledge and experiences involving fraction equivalence.

As referenced in the previous post, representation, equivalence, and connections involving comparing and ordering are all important components of the Number and Operations: Fractions content domain for grades 3 and 4 within the Common Core State Standards for Mathematics (CCSSI 2010). Without flexible paths to determining fraction equivalence, and this includes related fraction-decimal-and-percent equivalence, instructional “steps” involving operations with fractions will be largely procedural and bereft of the levels of understanding associated with a sense of number. OK, here we go. Let’s move forward with Fraction Equivalence, Part 2.

Comparing and Ordering

As students complete activities and solve problems that involve comparing and ordering of fractions, they use their understanding of equivalence, common fraction benchmarks, and flexible use of varied representations. Such activities and tasks should, as noted, also include decimals, and common percentages. Consider the following, always thinking about how you might adapt these activities and related tasks to meet the instructional needs of your own students. 

Have students—

• use varied representations and determine if the amount represented is closest to the benchmarks below.

Close to 0                    Close to 1/2                 Close to 1 

Examples: Use the representation noted, create the fraction, then select the appropriate benchmark.

          +  Circular region—shade 5/8 of the region

          +  Rectangular region—shade 1/4 of the region*

          +  Number line—locate 9/10 on a number line

          +  Counters—pick up 5 of 6 counters

          +  Circular region (clock-like)—shade 1/12 of the region; also 10/12

          +  Rectangular region—shade 4/10 of the region; also 7/10

          +  Counters—pick up 3 of 8 counters

          +  Number line—locate 3/5 on a number line

* Have students discuss why 1/4 is equally close to both 0 and 1/2.

    Have students create their own representations and then determine if their fraction was closest to 0, 1/2, or 1.

    Students should be prepared to discuss their benchmark decision making.

• Compare 1/2 to 1/3 (as fractions). Ask which is greater.
• Then show the following:

Discuss the importance of the following statement: Comparisons are valid only when the two fractions refer to the same whole.

• Complete the following, using representation, and then discuss them:

          +  Which is greater, 5/16 or 7/16?

          +  Which is less, 4/5 or 4/7?

          +  How do you know? What helps you when comparing fractions with the same denominator? With the same numerator?

Use double bar models to compare 3/4 and 7/8.

• Use a number line to compare 1/2; 0.7; 0.45; 4/5; and 25%.
• Determine and discuss how students know which of the following is closest to 1:

          +  24/25

          +  15/16

          +  5/6

          +  19/20

• Order from greatest to least: 44.4; 0.44; 0.4; 0.404; 4.0; 0.444

Using a number line show a fraction > 1/2. Then show a fraction between 1/2 and 3/4.

Discuss the following: If you increase the numerator of the fraction 4/5 what happens? What happens if you increase the denominator of the fraction 4/5? What can you say about the value of a fraction when the denominator is increased?

Try this: Starting with the fraction 1/2, what happens to the value of the fraction if the numerator and denominator increase by one (e.g., 1/2 becomes 2/3, then 3/4, 4/5, 5/6, etc.)?

          +  What would happen if the starting fraction was 2/1 and the numerator and denominator was increased by one (e.g. 2/1 becomes 3/2, 4/3, 5/4, 6/5, etc.)?

• Find the fraction or decimal:

          +  I am a decimal greater than 1/2 and less than 7/8; what am I?

          +  I am a fraction greater than 0.25 and less than 0.5; what am I?

          +  Create your own.

• Name the number:

          +  A fraction between 0.5 and 1

          +  A decimal between 2/3 and 4/5

          +  A fraction between ½ and 0.75

          +  A decimal between 0 and 1/5

          +  Create your own

          +  Consider the Cookie Monster problem in “ Fractions are Numbers, Too!” (from Fennell, Kobett, and Wray. 2014. Mathematics Teaching in the Middle School 19 [April]: 490.) 

As with all previous activities, students should discuss their thinking. Encourage them to use a variety of representations.

A final reminder

These critical foundations, place value and fraction equivalence, are both essential pillars of a sense of number so critical as students extend these understandings to work with operations involving whole numbers and fraction.

Your Turn      

I have provided a rationale for the foundational importance of fraction equivalence, and more generally, understanding fractions as numbers, and activities to help in considering the importance of comparing and ordering fractions and decimals. Take a look. Try them out. Send some of your favorites. How do you connect and extend fraction equivalence to comparing and ordering fractions and decimals? How does what you do connect to developing the level of flexibility with number, particularly a/bfractions and decimals, so necessary for developing and establishing a sense of number? It’s my hope that Fraction Equivalence, Parts 1 and 2 help in framing your own commitment to ensuring that this critical foundation is developed and also provides you with activities and tasks helpful in the classroom—your classroom!

Thanks for this opportunity to blog. And, we so want to hear from you. Post your comments below or share your thoughts on Twitter @TCM_at_NCTM using #TCMtalk. Contact me personally on Twitter @SkipFennell or at ffennell@mcdaniel.edu. Feel free to visit the following websites for information, resources, or just for fun. My personal site is www.ffennell.com. The project site for the Elementary Mathematics Specialists and Teacher Leaders Project is www.mathspecialists.org.

Francis (Skip) Fennell, ffennell@mcdaniel.edu ,is the L. Stanley Bowlsbey Professor of Education and Graduate and ProfessionalStudies at McDaniel College in Westminster, Maryland, where he directs theBrookhill Foundation-supported Elementary Mathematics Specialists andTeacher Leaders Project (http://www.mathspecialists.org). He is a pastpresident of NCTM and a recipient of NCTM’s Lifetime Achievement Award. He isinterested in the work of mathematics specialists, implementation of CCSSM,teacher education, number and fraction sense, and educational policy.