The *Comparing Two Fractions* assessment is designed to elicit information about several common misconceptions that students have when comparing two fractions:

- ● Misconception 1 (M1): Viewing a Fraction as Two Separate Numbers / Applying Whole-Number Thinking
- ● Misconception 2 (M2): An Over-Reliance on Unit Fractions / A Focus on “Smaller Is Bigger”
- ● Misconception 3 (M3): Numerator and Denominator Have an Additive Relationship / A Focus on the Difference from One Whole

We strongly recommend that you reference the information below to learn more about these misconceptions, including how they appear in student work, and how to score the pre- and post-assessments.

**Topic:**Understanding Fractions**Date:**March 2014- Pre Assessment
- Post Assessment
- All Assessment Materials
- Research

Before they can successfully compare two fractions, students need to have a foundational set of conceptual understandings of fractions, for example:

The comparison of fractions is often initiated by using unit fractions in order to focus on the relationship between the size of the pieces when partitioning—for example, understanding that 1/3 is greater than ¼, because thirds are larger pieces than fourths. Learning often progresses to comparing other fractions with specific characteristics, for example:

*same denominator*(e.g., comparing 2/5 to 3/5) to understand that the number of parts matters when comparing fractions with same-size parts

*same numerator*(e.g., comparing 2/4 to 2/3) to understand that when you have the same number of pieces (same numerator), you must consider the size of the pieces (denominators)

*key benchmarks*, such as 1/4, 1/2 and 3/4 (e.g., 3/16, 3/5, and 7/8, respectively)

**Connections to Common Core Standards in Mathematics (CCSS)**

The CCSS outline specific understandings that students should be able to meet at each grade level.

__Grade 3__

At grade 3, students should be able to understand the size of a fraction in relation to the whole and to understand that fractions with different numbers can be equivalent to one another.

**Develop understanding of fractions as numbers.**

**3.NF.3.** Explain the equivalence of fractions in special cases, and **compare fractions by reasoning about their size.**

Compare two fractions **with the same numerator** or **the same denominator** by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols <, =, or >, and justify the conclusions (e.g., by using a visual fraction model).

__Grade 4__

At grade 4, students should be able to build on prior understandings of fraction size and comparison of fraction pairs with the same numerators or the same denominators.

**Extend understanding of fraction equivalence and ordering.**

**4.NF.2.** Compare two fractions **with different numerators** and **different denominators**, for example, by creating common denominators or numerators or by comparing to a benchmark fraction, such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols <, =, or >, and justify the conclusions (e.g., by using a visual fraction model).

When students are developing the understandings described above (see Topic Background), they can develop flawed understanding leading to misconceptions about how fractions are compared. Once students have been exposed to fraction pairs that have different numerators and different denominators and to a variety of strategies that can help them compare fractions, many overgeneralize, confuse, or misapply strategies.

Three particular misconceptions, noted in the research on students’ mathematical reasoning about fractions, are targeted in the Comparing Two Fractions assessment:

**Misconception 1: Viewing a Fraction as Two Separate Numbers / Applying Whole-Number Thinking**

Often, students do not perceive a fraction as a single quantity but rather see it as a pair of whole numbers, and they apply whole-number thinking by comparing the size of the numbers in the denominators, the numerators, or both.

Watch this brief video clip for a fuller description of this misconception.

**Misconception 2 (M2): An Over-Reliance on Unit Fractions / A Focus on “Smaller Is Bigger”**

Students with this misconception consistently compare only the denominators of the two given fractions. They apply what they know about unit fractions to reason that the larger the denominator, the smaller the value of the fraction (e.g., they see 1/3 as smaller than 3/5). These students have overgeneralized the concept that “smaller is bigger” to all cases, without consideration of the numerator.

Watch this brief video clip for a fuller description of this misconception.

**Misconception 3 (M3): Numerator and Denominator Have an Additive Relationship / A Focus on the Difference from One Whole**

Students with this misconception understand that it’s important to pay attention to the relationship between the numerator and denominator, but they believe that this relationship is expressed through addition or subtraction. As a result, they will compare fractions by focusing on the difference between the numerator and the denominator.

Many students apply this reasoning only when the numerator and denominator of each fraction have a difference of one. For example, when comparing 8/9 and 4/5, students reason that since 8/9 is only one piece away (1/9) from 9/9 or one whole, and 4/5 is also one piece (1/5) away from 5/5 or one whole, the two fractions must be equal, as they are each “one away” from a whole.

Watch this brief video clip for a fuller description of this misconception.

To see additional examples of student work illustrating this misconception, go to the “Sample Student Responses” tab on this page, and click on the button to download a PDF.

**Resources**

Hannula, M. S. (2003). Locating fractions on a number line. In N. A. Pateman, B. J. Dougherty, & J. Zilliox (Eds.), *Proceedings of the 2003 Joint Meeting of PME and PMENA, Vol. 3* (pp. 17–24). Honolulu, HI: CRDG, College of Education, University of Hawaii.

Harel, G., & Confrey, J. (1994). *The development of multiplicative reasoning in the learning of mathematics.* Albany, NY: State University of New York Press.

Hiebert, J., & Behr, M. (Eds.) (1988). *Number concepts and operations in the middle grades.* Reston, VA: National Council of Teachers of Mathematics.

Martinie, S., & Bay-Williams, J. (2003). Investigating Students’ Conceptual Understanding of Decimal Fractions Using Multiple Representations. *Mathematics Teaching in the Middle School*, 8(5), 244.

Roche, A., & Clarke, D. (2004). When does successful comparison of decimals reflect conceptual understanding? In I. Putt, R. Faragher, & M. McLean (Eds.), *Mathematics Education for the Third Millennium: Towards 2010*. Proceedings of the 27th Annual Conference of the Mathematics Education Research Group of Australasia, Townsville (pp. 486–493). Sydney, Australia: MERGA.

Stafylidou, S., & Vosniadou, S. (2004). The development of students’ understanding of the numerical value of fractions. *Learning & Instruction*, 14(5), 503–518. doi:10.1016/j.learninstruc.2004.06.015

Steinle, V., & Stacey, K. (2004). A longitudinal study of students’ understanding of decimal notation: An overview and refined results. In I. Putt, R. Faragher, & M. McLean (Eds.), *Mathematics Education for the Third Millennium: Towards 2010*. Proceedings of the 27th Annual Conference of the Mathematics Education Research Group of Australasia, Townsville (pp. 541–548). Sydney, Australia: MERGA.

**About This Assessment**

These EM2 diagnostic formative pre- and post-assessments are composed of items with specific attributes associated with student conceptions that are specific to comparing fractions. Each item within any EM2 assessment includes a selected response (multiple choice) and an explanation component.

While many different fraction pairs can be compared, this assessment targets proper, non-unit fractions. (Proper fractions are those where the numerator is less than the denominator; non-unit fractions are those with numerators not equal to 1.) This is due to the particular difficulties that these pairs elicit, as identified in the mathematics research cited under Research Foundations. The fractions being compared in this assessment are confined to the following:

The first numerator is 1, which is less than the second numerator, 3, **and** the first denominator is 2, which is less than the second denominator, 4.

The learning target for the Comparing Two Fractions assessment is as follows:

*The learner will accurately compare two fractions with different numerators and different denominators when the two fractions refer to the same whole.*

**Prior to Giving the Pre-Assessment**

**Administering the Pre-Assessment**

*Today you will complete a short individual activity, which is designed to help me understand how you think about comparing fractions.*

*The activity includes seven problems. For each problem, choose your answer by completely filling in the circle to show which answer you think is correct. Because the goal of the activity is to learn more about how you think about fractions, it’s important for you to include some kind of explanation in the space provided. This can be a picture, or words, or a combination of pictures and words that shows how you chose your answer. *

*You will have about 20 minutes to complete all the problems. When you are finished please place the paper on your desk and quietly [read, work on ____] until everyone is finished.*

*You have a few minutes left to finish the activity. Please use this time to make sure that all of your answers are as complete as possible. When you are done, please place the paper face down on your desk. Thank you for working on this activity today.*

Pre-Assessment Administration Process

**After Administering the Pre-Assessment**

Use the analysis process (found in the Scoring Guide PDF document under the “Scoring Process” tab) to analyze whether your students have one or more of three possible misconceptions:

*Misconception 1: Viewing a Fraction as Two Separate Numbers / Applying Whole-Number Thinking*

*Misconception 2: An Over-Reliance on Unit Fractions / A Focus on “Smaller Is Bigger”*

*Misconception 3: Numerator and Denominator Have an Additive Relationship / A Focus on the Difference from One Whole*

The *Comparing Two Fractions* assessment is composed of seven items with specific attributes associated with different misconceptions that are directly related to comparing two fractions. We encourage you to carefully read the Scoring Guide to understand these specific attributes and to find information about analyzing your students’ responses.

Click on the button to download a PDF of the Scoring Guide.

To determine the degree of understanding and misunderstanding in the student work, it’s important to consider both the answer to the selected response and the explanation text and representations. The example above is one of many student work samples that provide insight into student thinking about one of three different misconceptions targeted in these diagnostic assessments (see Research Foundations for more information and videos about these misconceptions).

We encourage you to look at the collection of student work examples provided here. Click on the button to download a PDF.

If the *Comparing Two Fractions* pre-assessment shows that any of your students have one or more of the misconceptions outlined in the Scoring Guide, plan and implement instructional activities designed to increase students’ understanding. The post-assessment provided here can then be used to determine if the misconception has been addressed.

**Prior to Giving the Post-Assessment**

**you should avoid using or reviewing items from the post-assessment before administering it.**

**Administering the Post-Assessment**

*Today you will complete a short individual activity, which is designed to help me understand how you think about comparing fractions, a topic we have been working on in class.*

*This activity includes seven problems. For each problem, choose your answer by completely filling in the circle to show which answer you think is correct. Because the goal of the activity is to learn more about how you think about fractions, it’s important for you to include some kind of explanation in the space provided. This can be a picture, or words, or a combination of pictures and words that shows how you chose your answer.*

*You will have about 15 minutes to complete all the problems. When you are finished, please place the paper on your desk and quietly [read, work on ____] until everyone is finished.*

*You have a few minutes to finish the activity. Please use this time to make sure that all of your answers are as complete as possible. When you are done, please place the paper face down on your desk. Thank you for working on this activity today.*

Click on the button to download a PDF of the Post-Assessment Administration Process.

Post-Assessment Administration Process

**After Administering the Post-Assessment**

Use the analysis process (found in the Scoring Guide PDF document under the “Scoring Process” tab) to analyze whether your students have one or more of three possible misconceptions:

*Misconception 1: Viewing a Fraction as Two Separate Numbers / Applying Whole-Number Thinking*

*Misconception 2: An Over-Reliance on Unit Fractions / A Focus on “Smaller Is Bigger”*

*Misconception 3: Numerator and Denominator Have an Additive Relationship / A Focus on the Difference from One Whole*

Some students who previously had the misconception will no longer have it—the ideal case. Consider your instructional next steps for those students who still show evidence of the misconception.