Teaching Fractions Beyond Pizza and Pie Charts

By Sharri Zachary, Mathematics Research Coordinator

Research shows that “algebra is the [career] gateway to success for many students” (Williams, 2011, pg. 1). For students to understand the concepts and symbolic representations for everyday situations in algebra, students must master certain foundational skills and concepts at the elementary and middle school levels. According to the U.S. Department of Education (2006), a solid foundation in math in early grades will assist in the development of critical thinking skills necessary to pass algebra. The National Mathematics Advisory Panel (2008) identifies the skills that demonstrate algebra readiness as: (1) whole number computation, (2) fraction and decimal proficiency, (3) number concepts (including percents), (4) general concepts, proportions, and geometry, (5) problem-solving, and (6) basic understanding of integers, variables, and simple equations.

Fraction and decimal proficiency is often developed with the use of pizza and pie charts as the introductory piece. Visually, this allows students to see one “whole” partitioned into equal parts, which in turn, helps students understand the part-whole idea of fractions. However, this leaves out the idea that fractions are numbers with magnitude that can be compared (Siegler et al, 2010).

If students are to be truly algebra-ready, visual representations must extend to model fraction concepts that teach the part-whole relationship, as well as, fractions as a distance and magnitude. The IES practice guide recommends the use of number lines as the “central representational tool” to help students recognize fractions as numbers and expand student thinking beyond whole numbers (Siegler et al, 2010, p. 19).

The use of number lines can help students visualize and understand the magnitude of fractions, the relationship between fractions and whole numbers, and the relationship between fractions, decimals, and percents (Siegler et al, 2010). This conceptual understanding is foundational to understanding algebra (Siegler et al, 2010). Based on the recommendations put forth in the IES practice guide, teachers should center fraction instruction around the number line model and support that instruction with other models that include (but are not limited to) fraction circles and strip diagrams. If number lines are recommended as the “central representational tool” for fraction and decimal proficiency, then teachers must consider moving beyond pizza and pie charts in their instruction and assessments to prepare students for mastery of this component of algebra readiness.

National Mathematics Advisory Panel (Spring 2008). Final report, Washington, D.C. 

Siegler, R., Carpenter, T., Fennell, F., Geary, D., Lewis, J., Okamoto, Y., Thompson, L., & Wray, J. (2010). Developing effective fractions instruction for kindergarten through 8th grade (NCEE 2010-4039). Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education. Retrieved from http://ies.ed.gov/ncee/wwc/practiceguide.aspx?sid=15 

U.S. Department of Education. (2006). Math now: Advancing math education in elementary and middle school. February 2006. Retrieved from: http://www.ed.gov/about/inits/ed/competitiveness/math-now.html 

Williams, T. G. (2011). Reaching algebra readiness (RAR): Preparing middle school students to succeed in algebra – the gateway to career success. Rotterdam, The Netherlands: Sense Publishers.

Help Your Students Experience Fractions Conceptually

By Dawn Woods, RME Elementary Mathematics Coordinator

Many students find fraction concepts difficult to understand yet the understanding of fractions is essential for learning algebra and advanced mathematics (National Mathematics Advisory Panel, 2008). As an elementary mathematics educator, I noticed that many of my students struggled with fraction concepts across the curriculum. I wondered how I could help my students experience and understand fraction concepts conceptually so they could succeed not only in my classroom but also in advanced mathematics.

My search for answers began with research. I discovered that to understand fractions means to recognize the multiple meanings and interpretations of fractions. Furthermore, I needed to explicitly present these different constructs in a contextual way to build understanding. Mathematics educators generally agree that there are five main fraction constructs and that they are developmental in nature.

The first construct presents fractions as parts of wholes or parts of sets. Research suggests that this construct is an effective starting point for building fractions (Cramer & Whitney, 2010). However, it is important to realize that the part-whole relationship goes way beyond the shading of a region. For example, it could be part of a group of animals such as (¼ of the animals are dogs), or be part of a length, (we ran 1 ½ miles) (Van De Walle, Karp, Bay-Williams, 2013).

Researchers such as Cramer, Wyberg, and Leavitt suggest that the fraction circle manipulative is a powerful concrete representation since it helps to build understanding of the part-whole relationship as wells as the meaning of the relative size of fractions (2008). Here, they use fraction circle models to help build mental images that aid in the ability to judge relative sizes of fractions. It is also important to remember that the fractional parts do not need to be identical in shape and size, but must be equivalent in some other attribute such as area, volume, or number (Chapin & Johnson, 2006). However, it is important to teach beyond this first construct to include other fraction representations and models.

The second construct presents fractions as measures. Measurement (Van De Walle, Karp, Bay- Williams, 2013) involves identifying a length and then uses that length to determine a length of an object. The number line plays an important role in this construct by partitioning units into as many subunits that one is willing to create (Chapin & Johnson, 2006). For example, in the fraction ¾, you can use the unit fraction ¼ as the selected length and then measure to show that it takes three of those to reach ¾ (Van De Walle, Karp, Bay-Williams, 2013). Research suggests that students who develop an initial understanding of rational numbers as measures, develop ideas of unit, partitioning, order, addition and subtraction (Cramer & Whitney, 2010) while using the number line as a model. Essentially, this powerful construct illustrates that there are an infinite number of rational numbers on the number line as it focuses on how much rather than parts of a whole.

Fractions can also result from dividing two numbers. This construct is often called the quotient meaning, since the quotient is the answer to a division problem (Chapin & Johnson, 2006). Think about the number of cookies each person receives when 15 cookies are shared between 3 people. This problem is not a part-whole scenario (Van De Walle, Karp, Bay-Williams, 2013) but it still means that each person will receive one-third of the cookies expressed as ¹⁵⁄₃, ⁵⁄₁, or 5. Connecting division to fractions enables students to feel comfortable with seeing division expressed in multiple ways such as 16 ÷ 3, ¹⁶⁄₃, and 5¹⁄₃ and is important for continued success in advanced mathematics.

The fourth construct presents fractions as operators. In this construct, a fraction is a number that acts (or operates) on another number to stretch or shrink the magnitude of the number (Chapin & Johnson, 2006). For an example, a model of a car may be 1/16 the size of the original or a cell maybe magnified under a microscope to 400 times the actual size demonstrating a multiplicative relationship between the quantities. This construct takes fractions beyond representation to a place where students know how to use fractions to solve problems across the curriculum.

The fifth and final construct characterizes fractions as the ratio or comparison of two quantities. A ratio such as 1/3 can mean that the probably of an event is one in three (Van De Walle, Karp, Bay-Williams, 2013). Or a ratio can also represent part-whole relationships such as 11 children at the park compared to the total number of 18 people. We could write this part-to-whole relationship as the fraction ¹¹⁄₁₈. However, it is important that realize that all fractions are ratios but not all ratios are fractions (Chapin & Johnson, 2006). Part-to-part comparisons such as the number of children to the number of people at the park, 11:18, is not a fraction because this comparison does not name a rational number but presents a comparison of two numbers.

Fraction understanding, although a challenge to students, is a critical mathematics concept. For students to really understand fractions, they need to experience fractions across all five constructs in meaningful ways that build conceptual understanding. This conceptual understanding, in turn, provides students with mental representations that enable students to connect meaning to fractions across a variety of contexts.

Chapin, S.H., & Johnson, A. (2006). Math matters, 2nd edition. Sausalito: Math Solutions Publications.

Cramer, K., & Whitney, S. (2010). Learning rational number concepts and skills in elementary school classrooms. In D.V. Lambdin & F.K. Lester, Jr. (Eds.), Teaching and learning mathematics: Translating research for elementary school teachers (pp. 11-22). Reston, VA: NCTM

Cramer, K., Wyberg, T., & Leavitt, S. (2008). The role of representations in fraction addition and subtraction. Mathematics Teaching in the Middle School 13(8), 490-496.

National Mathematics Advisory Panel (2008). The final report of the national mathematics advisory panel. Jessup, MD: Education Publications Center. Retrieved from http://www2.ed.gov/about/bdscomm/list/mathpanel/report/final-report.pdf.

Van De Walle, J.A., Karp, K.S., & Bay-Williams, J.M. (2013). Elementary and middle school mathematics: Teaching developmentally, 8th edition. Boston: Pearson.

Comparing Fractions

By Cassandra Hatfield, RME Assessment Coordinator

4 × 5 is 20. 8 × 1 is 8.
So, ⁵⁄₈ is the greater fraction.

As a middle school math teacher, I found it interesting that when I would ask students to compare two fractions, many were quick to give me an answer, but when I asked them to order fractions it took those same quick students quite a bit more time and they often got the wrong answer. Intrigued by this I began to ask my students “how do you know?” I quickly found out that many students were comparing fractions using the “butterfly method” or cross multiplication. In this example, because the larger cross product is on the left, the larger fraction is on the left.

This “trick” will not take a student very far in their journey of mathematics. Although a student can arrive at the correct answer, the “trick” does not require any thought about the relative size of the fractions. If a student does not understand anything about the relative size of the fractions, how would the student order three or more fractions or think conceptually about this word problem: Max paid $12 for his portion of dinner. This was one-third of the total bill. How much was the total bill?

Instead of using the “trick” above, students should be given opportunities to compare fractions that have been intentionally selected by the teacher for investigation. This is not to say, that this should be a lesson titled “Many Ways to Compare Fractions.” Students should be given the opportunity to compare the fractions and share strategies with their classmates in a discussion. The teacher should listen for students who use specific strategies and poster those strategies in the classroom.

Below are some fractions I have intentionally selected for comparison. I’ve shared one strategy that could be used for comparison.

(Van De Walle, Karp, Bay-Williams, 2013, p. 310-311)

Summing it All Up
For students that are struggling with the abstractness of the above strategies, the use of number lines could be helpful. The IES Practice Guide, which is supported by research evidence, states, “conceptually, number lines and number paths show magnitude and allow for explicit instruction on magnitude comparisons" (IES Practice Guide, 2010).

First, see if the student understands how to partition a number line into equal parts and identify distances on the number line. If the student does, then models like the one shown below might be helpful for students to visually see the comparison strategy. 

Model of comparing ⁴⁄₅ and ⁹⁄₁₀ to a whole.

Now it’s your turn. Share with us a different strategy you would have used to compare the fractions above.

Van De Walle, J.A., Karp, K.S., & Bay-Williams, J.M. (2013) Elementary and middle school mathematics: Teaching developmentally. Upper Saddle River, New Jersey: Pearson.

Siegler, R., Carpenter, T., Fennel, F., Geary, D., Lewis, J., Okamoto, Y., Thompson, L., Wray, J. (2010) Developing effective fractions instruction for kindergarten through 8th grade: A practice guide. Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education. Retrieved from http://ies.ed.gov/ncee/wwc/publications/practiceguides/.

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Converting Fractions to Percentages

By Beth Richardson, RME High School Mathematics Coordinator

As a high school math teacher, I taught a wide range of students from ESL Algebra 1 and regular Geometry to Pre-AP Algebra 2. The resounding similarity I saw between all of my students was that, for some reason, students cringe when they see rational numbers. They feel like rational numbers automatically make the problem “hard”. I was amazed that by high school, students were still struggling with something as simple as converting from a fraction to a percent. Perhaps this is because, as teachers, we sometimes teach our students shortcuts that leave out the logic behind the scenes of the procedures they learn.

The IES Practice Guide, which is supported by research evidence, recommends that teachers “'help students understand why procedures for computations with fractions make sense’ and ‘develop students’ conceptual understanding of strategies for solving ratio, rate, and proportion problems before exposing them to cross-multiplication…’ (Siegler et al., 2010).”

Some common shortcuts teachers use are changing the fraction to a decimal then multiplying by 100 or changing the fraction to a decimal then moving the decimal to the right twice.

Examples:

Neither method above leads students to a percentage as the final answer, unless the student “remembers” to tag it on at the end. Units are crucial when converting in any context. In order for students to understand why they must multiply by 100% rather than 100 when converting from fraction to percent, units must be used properly.

Instead, students should be taught to set up proportional relationships, including units, between the fraction and unknown out of 100%. It is important that students understand that when the units of the numerator and denominator are the same, they cancel and the fraction is unit-less.

Example:
25 students went on a field trip and 5 wore a hat. What percentage of the students wore a hat?

20% of the students wore a hat on the field trip.

Through the process above, students see why they are multiplying by 100% and why the units in their answer must be a percentage. Also, students can use number sense to reason that x must be a percentage between 5 and 100.

Summing It All Up
Fellow teachers: it’s not safe to assume that our students understand why they are doing a particular procedure, even if it is one they “should” have mastered several grade levels ago. If we take a little more time to illustrate examples with labeled units and explanation, we will hopefully catch any previous misconceptions our students have and steer them on the right path towards math success.

Now it’s your turn. Share with us common misconceptions, similar to what we described above, that you’ve found in your classroom!

Resources:
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.

Siegler, R., Carpenter, T., Fennel, F., Geary, D., Lewis, J., Okamoto, Y., Thompson, L., & Wray, J. (2010) Developing effective fractions instruction for kindergarten through 8th grade: A practice guide. Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education. Retrieved from http://ies.ed.gov/ncee/wwc/publications/practiceguides/