Step 2: Math Content
There’s something weird about fractions. If the numerator, which is the top, is the same, the bigger the denominator the smaller the fraction. – Student from Grade 5 research lesson
Like Jordan, many students feel that “there’s something weird about fractions.” The three tasks below help you explore students’ challenges with fractions.
 If working with a team, we recommend that team members individually solve the three tasks and anticipate student thinking before you share and discuss your solutions and ideas about how student solutions. Each task is downloadable.
Task 1: Estimate the answer to ^{12}_{13 }+ ^{7}_{8}. You will not have time to solve the problem using paper and pencil.
Student solutions to task 1 and discussion prompts
Student Response 
Percent of Age 13 Students 
1 
7 
2 
24 
19 
28 
21 
27 
I don’t know 
14 
Post, T. R. (1981, May). Fractions: Results and implications from National Assessment. The Arithmetic Teacher.
Discussion Questions: How did you solve the problem, and how might students solve the problem? What insights into fractions make quick work of solving this problem? Discuss why students chose each of the responses shown. Why do so many students find this problem difficult?
Task 2: Find two fractions between ^{1}_{2} and 1.
Student solutions to Task 2 and discussion prompts
Below are six examples of student work on this task. Three are from students who used a basal textbook and three are from students who learned fractions in a measurement context (the “Measure Up” curriculum):
Context 1: Basal Text [*]
Student 1 ^{1}_{4} ^{1}_{5} 
because 4 is bigger than 1 because 5 is bigger than 1 
Student 2 ^{3}_{4} ^{7}_{8} 
are between ^{1}_{2} and 1. Both are missing one part. 
Student 3 ^{2}_{3} ^{4}_{5} 
2 is more than 1 3 is more than 2 4 is more than 1 5 is more than 2 
Context 2: “Measure Up” Curriculum (Measurement Context) [**]
Student 1 ^{2}_{3} ^{3}_{4} 
If there is a lot of parts, there are smaller pieces. So you have to use a lot of them to get close to 1. So 2 out of 3 parts is close to 1 and so is 3 out of 4 parts 
Student 2 ^{5}_{8} ^{7}_{8} 
If you have 8 parts, then 4 parts are ^{1}_{2} so 5 parts and 7 parts are more then (sic) ^{1}_{2}. But there (sic) not 1 because you need all the parts. 
Student 3 ^{5}_{9} ^{6}_{10} 
What I did was think of the number of parts and then what is haf (sic). Then I added one to it. 
Discussion Questions: How did you solve the problem, and how might students solve the problem? What understandings and misunderstandings about fractions might this problem reveal? What do the student responses suggest that each student understands and does not understand about fractions? Do you notice any differences in the responses of students who used the basal textbook and the students who participated in the “Measure Up” curriculum?
[*]Student work from Work Session presented by Barbara Dougherty and Barbara Fillingim, NCTM Annual Meeting Research Presession, April 21, 2009, Washington D.C., reproduced by permission of first author. [**]The “Measure Up” curriculum emphasizes use of units of length, area, and volume to explore basic mathematical ideas such as equivalence. For example, students might compare two lengths by using a third length. Students using this curriculum become very attuned to asking, “What is the unit?” since different units (such as a hexagon and six triangles) might be used to create equivalence.
Jim has of 3/4 a yard of string which he wishes to divide into pieces, each 1/8 of a yard long. How many pieces will he have?
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Discussion Questions: Explain what 70% of responding 4th grade students might have been thinking when they answered the above question incorrectly on a national assessment. (27% of 4th students answered correctly; 3% did not respond).
The Nation’s Report Card (2003). Sample Questions from the National Assessment of Educational Progress. National Center for Education Statistics. Washington, D.C.: Institute of Education Sciences. Retrieved on January 12, 2009 from http://www.nces.ed.gov/nationsreportcard/itmrls/startsearch.asp.
Now that you have solved three different fraction tasks and considered student solutions, we suggest that you:
 Briefly summarize your insights by answering the question “What is difficult for students about fractions?”
The table below lists six different aspects of fraction number sense, with examples of student understanding of each.

 Read through the table and focus on the connection between the two columns (the type of knowledge and the examples of student difficulty or understanding). Identify any that:
 Are puzzling or particularly interesting to you.
 Help you think about any of the student solutions to tasks 13 that you examined.
 Read through the table and focus on the connection between the two columns (the type of knowledge and the examples of student difficulty or understanding). Identify any that:
 Discuss these with your colleagues and fill in items 26 of your fraction unit plan template.
Type of Understanding or Knowledge 
Example of Student Difficulty or Understanding 
A Fraction is a Number
A fraction represents an amount, not just pieces (such as 2 of 3 pieces of a pizza) or a situation (such as 2 of 3 shirts are red). 

Partitioning Fractions


The Meaning of the Denominator


Knowing What is the Whole


Fraction Size


Fractions Can Represent Quantities Greater Than One May be difficult for students who have a strong image of a fraction as a piece of something. 
