The *Representing Fractions 1* assessment is designed to elicit information about a common misconception that students have when representing fractions visually:

- ● Misconception 1 (M1): Seeing Fractions as Part-to-Whole with Unequal Parts

We strongly recommend you reference the information below to learn more about this misconception, how it appears in student work, and how to score the assessment.

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

The topic of “representing fractions” includes many different concepts for students to understand and many places for potential misconceptions to develop. Here, we focus on the aspect of representing fractions that is targeted in these diagnostic assessments.

The primary conceptual understanding on which this diagnostic assessment focuses is the idea that a pictorial representation of a fraction must show parts that are the same size, meaning that they have equal area. To work with pictorial representations of fractions, students must learn to distinguish between correct representations and incorrect representations.

As a prerequisite foundation for this understanding, students also need to understand the meaning of the numerator and the denominator in a fraction. This becomes a little more challenging if we think about the different concepts that a fraction can represent, including part-whole relationships, division, ratios, or measurements. Students need to understand the meaning of the numerator and denominator across all these uses of fractions.

Students benefit from working with a variety of pictorial representations of fractions. Students should have opportunities to work with representations of fractions in which the parts are all the same size and shape as well as those with parts that have equal area but are different shapes.

They should also work with representations in which the parts are grouped together as well as those with parts that are not grouped together.

Finally, they should work with representations in which the partitioning is clearly indicated, those where the partitioning needs to be inferred from the shape, and those where the partitioning may appear unequal.

Opportunities to work with this variety of pictorial representations will deepen students’ understanding of the meaning of the numerator and denominator as well as their understanding of the necessity of same-size (equal-area) parts.

This diagnostic assessment is restricted to pictorial representations that are called “simple representations”; here, the total number of parts in the shape matches the denominator of the fraction, and the number of shaded parts is associated with the numerator.

**(Note that the Representing Fractions 2 diagnostic assessment includes a wider variety of pictorial representations.)**

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 use fractions along with visual fraction models to represent parts of a whole.

**Develop understanding of fractions as numbers.**

**3.NF.1: **Understand a fraction *1/b* as the quantity formed by 1 part when a whole is partitioned into *b* equal parts; understand a fraction *a/b* as the quantity formed by *a* parts of size *1/b*.

At grade 3, students should also be able to:

**Reason with shapes and their attributes.**

**3.G.2: **Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. *For example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape.*

Many learners have difficulty moving from whole-number understandings to rational-number understandings. These conceptual understandings and misunderstandings become more visible as students begin to use pictorial and symbolic representations, such as area models, linear models, and set models.

When the representation is a part/whole area or measurement model in which the total number of equal-size pieces in the whole matches the denominator and the shaded part of the whole equals the numerator, this is referred to as a “simple representation” of the fraction. When the representation has a whole where the total number of equal parts is a multiplicative factor of the denominator of the fraction in symbolic form, this is considered an “equivalent representation” (Wong & Evans, 2007).

One of the critical attributes of a simple or equivalent representation when partitioning a whole is that the parts need to be equal in size (area), although not necessarily the same shape. During instruction, however, students often work with models that are already the same size and shape (Van de Walle, Karp, & Bay-Williams, 2010). This has led to some students thinking that the pieces must be the same shape, and others failing to even consider size as an attribute. Students also have numerous difficulties as they move from simple representations to equivalent representations of fractions (Gould, 2005; NRC, 2001; Pearn, Stephens, & Lewis, 2003). While some students can determine equivalent fractions procedurally when given the fraction in symbolic form, many cannot connect the symbolic and pictorial representations (Wong & Evans, 2007).

The following common misunderstandings and misconceptions when representing fractions based on mathematics research will be targeted in the Representing Fractions I assessment:

**Misconception 1 (M1): Seeing Fractions As Part-to-Whole with Unequal Parts**

Students with this misconception consistently associate the number of pieces shaded with the numerator and associate the total number of pieces for the entire figure with the denominator, but do not pay attention to the size of the pieces (or region).

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.

References

Gould, P. (2005). Drawing sense out of fractions. In M. Coupland, J. Anderson, & T. Spencer (Eds.), *Making mathematics vital.* Proceedings of the 20th Biennial Conference of the Australian Association of Mathematics Teachers (pp. 133–138). Adelaide, Australia: AAMT.

National Research Council. (2001). *Adding it up: Helping children learn mathematics.* Washington, DC: The National Academies Press.

Pearn, C., Stephens, M., & Lewis, G. (2003). Assessing rational number knowledge in the middle years of schooling. In M. Goos & T. Spencer (Eds.), *Mathematics: Making waves.* Proceedings of the 19th Biennial Conference of the Australian Association of Mathematics Teachers (pp. 170–178). Adelaide, Australia: AAMT.

Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2010). *Elementary and middle school mathematics: Teaching developmentally* (7th ed.). Boston, MA: Allyn & Bacon.

Wong, M., & Evans, D. (2007). Students’ conceptual understanding of equivalent fractions. In J. Watson & K. Beswick (Eds.), *Mathematics: Essential Research, Essential Practice.* Proceedings of the 30th Annual Conference of the Mathematics Education Research Group of Australasia (vol. 2, pp. 824–833). MERGA: Hobart, Australia.

About This Assessment

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

The fractions in these assessments are all fractions less than 1.

The learning target for the *Representing Fractions 1* assessment is:

When considering visual representations of fractions as a part out of a whole, the learner will understand that all the pieces must be the same size (area).

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 representing fractions as a picture.

The activity includes four 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 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.

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

Pre-Assessment Administration Process

After Administering the Pre-Assessment

Use the __analysis process__ (found in the Scoring section of the support guide) to analyze whether your students have this misconception:

__Misconception 1__: Seeing Fractions As Part-to-Whole with Unequal PartsThe *Representing Fractions 1* assessment is composed of four items with specific attributes associated with different misconceptions that are directly related to representing fractions visually. 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 assessment materials. The Scoring section starts on page 8.

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 the particular misconception targeted in these diagnostic assessments (see the “Research-Based Misconceptions” tab for more information and a video about this misconception).

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

If the Representing Fractions 1 pre-assessment shows that one or more of your students have the misconception 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 now think about representing fractions as a picture, a topic we have been working on in class.

Like before, the activity includes four 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.

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.

After Administering the Post-Assessment

Use the analysis process (found in the Scoring Guide) to analyze whether your students have this misconception:

__Misconception 1__: Seeing Fractions As Part-to-Whole with Unequal Parts

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.