# Conduct a Lesson Study Cycle on Fractions

## 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

• 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 1213 78.  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 Who Choose This Response 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 12 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 14     15 because 4 is bigger than 1 because 5 is bigger than 1 Student 2 34     78 are between 12 and 1.  Both are missing one part. Student 3 23     45 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 23    34 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 58     78 If you have 8 parts, then 4 parts are 12 so 5 parts and 7 parts are more then (sic) 12. But there (sic) not 1 because you need all the parts. Student 3 59     610 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?

##### ___
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:

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.