By Francis (Skip) Fennell, Posted February 16, 2015 – 

As noted in the two previous blog posts, Part 1 and Part 2 , perhaps the “signature expectation” of any pre-K–grade 6 mathematics experience is the development of a sense of number. Firmly establishing and maintaining flexibility with number is simultaneously ongoing and foundational to working with operations involving whole numbers and fractions.

Well, now it’s time think about fractions, in particular, fractions as numbers. The focus of the the next two installments of Math Tasks to Talk About is fraction equivalence and the number-sense connections involving representing, comparing, and ordering fractions. These related concepts and skills are a critically important aspect of the Number and Operations—Fractions content domain for grades 3 and 4 within the Common Core State Standards for Mathematics (CCSSM) (CCSSI 2010). I must emphasize that without flexible paths to determining fraction equivalence, and this includes related fraction-decimal-and-percent equivalence, next steps involving operations with fractions will be largely procedural and bereft of the levels of understanding associated with a sense of number. 

Representations

A critically important “step” to developing an understanding of fractions as numbers and fraction equivalence is regularly providing opportunities, via classroom activities or problem-based tasks, for students to represent fractions using a variety of representations (manipulatives, drawings) or representation tools (e.g., www.conceptuamath.com). Representation opportunities will include the use of circular and rectangular regions and the number line, as well as fractional parts of a set. Consider having students represent the following:

•     7/8 using a number line

•     3/4 using a circular region, a rectangular region, and a number line

•     4/12 using a circular region

•     3/2 using any representation 

•     0.4 using a rectangular region and a number line

•     0.34 using a hundred chart

•     1/6 and 5/6 using pattern blocks, color tiles, or counters

•     5/9 of a collection of objects and a region

•     11/12, 1/2, and 1/4 using a clock face

Students should expect to discuss each of their representations. 

Equivalence

It’s all about equivalence. If students develop flexibility in creating equivalent fractions and using “benchmark” fractions, extensions involving comparing and ordering fractions become just that—extensions based on student understanding of fractions as numbers. Flexibility with equivalence should extend to considering equivalent fractions, decimals and common percentages at the appropriate grade levels (4–6). Note that the suggestions here are about “seeing” and creating equivalent fractions before working with fraction operations. Have students do the following:

•     Create a number line equivalent fraction tool similar to the one below, which can be used as desktop resource as students build their understanding of common fraction equivalences. Note that this desktop resource serves as a linear mental image for equivalence as students begin their work with fractions as numbers within CCSSM at the third-grade level. At upper grade levels, offer expanded desktop number lines to directly relate to the role of fraction equivalence (e.g., denominators to include fifths, tenths, decimals from 0.1 to 1.0, etc.)

•     Show at least three equivalent fractions for 1/4 using a number line

•     Create fractions equivalent to 1/3 mentally

•     Use a clock face to show that 4/12 = 1/3 and that 6/12 = 3/6 = 1/2

•     Create a fraction and percent equivalent to 0.4

•     Use bar models to create three fractions equivalent to 8/10

•     Write a decimal and percent equivalent to 4/5

•     Determine the number of minutes equivalent to 1/4, 1/3, and 5/6 of an hour

Students should expect to discuss each of their responses and related representations to the equivalence activities. 

Your Turn

As with the place-value blogs, now it’s your turn. I have offered a rationale for the foundational importance of fraction equivalence to understanding fractions as numbers and activities to help consider the importance of representations (and their use) and equivalence. Take a look. Try them out. Send some of your favorites! How do you develop place fraction equivalence? How does what you do connect to developing the level of flexibility with number, particularly a/b fractions and decimals, so necessary for developing and establishing a sense of number? Consider this part 1 of a two-part blog on the importance of fraction equivalence as the foundational stepping stone for understanding fractions.

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

We 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.