This course offers support for a lesson study cycle on fractions by providing additional details and materials to facilitate Study  Step 7: Study Research, Standards and Curricula.
Overview: Developing Students’ Number Sense for Fractions
The steps below provide support for a lesson study cycle on fractions that also builds students’ mathematical practices. When you reach Step 5 you will have a draft unit plan that captures your thinking about Teaching Through ProblemSolving.
Step 1: Fraction Unit Plan Template
Download and open the unit plan for fractions below, and use it to collect your ideas as you go along. Item 1 of the unit plan template provides a spot for you to record your goals for student mathematical practices and your “theory of action” about how to build student mathematical practices.
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?
___
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. 

Step 3: Curriculum Exploration
This step allows you to examine curriculum materials and fraction models. It provides fuel for thinking about items 25 of the unit plan for fractions that you have been working on.
Researcher: Which fraction is smaller? ^{6}_{4} or ^{6}_{5}
Chris: ^{6}_{4}. Because 4 takes longer to get to 6 than 5 does.* 
Like Chris, many students reason about fractions in ways that work in certain situations but not in others. Chris may have an image of a fraction as a quantity less than one and as a number of pieces in a whole with some missing. His reasoning would be helpful if he were comparing the size of ^{4}_{6} and ^{5}_{6}, but leads him to the wrong conclusion in the example above.
Students’ images and understandings of fractions are shaped by their encounters with mathematics textbooks and materials and by real life experiences. What images of fractions do your students typically draw when asked to make sense of a fraction problem? The illustration below shows the fraction 3/4 shaded in eight different models.
Models of the fraction ^{3}_{4}
As Tad Watanabe’s article “Initial Treatment of Fractions in Japanese Textbooks” notes, each of these models has different strengths and shortcomings for building student understanding, and there is not one single “best” model. Ultimately, students will need to understand that ^{3}_{4} represents several very different situations, such as ^{3}_{4} of a meter, 6 out of 8 squares colored on a checkerboard, and 3 girls in a group of 4 children. The resources here focus on the linear measurement context for fractions because it is often neglected in U.S. textbooks, although it is common in other countries and in researchbased programs.
Explore the linear measurement model with a challenging handson activity for adults!
To find out how to prepare for the mystery strip activity, click hereCreate one strip of paper exactly 1 meter long and do not mark the strip in any way, see example below:
Then create the blue mystery strip by downloading this pdf file: mystery strip.
You will need to save the file. Before printing the strip, check your pdf print settings are as described in the mystery strip pdf.
 Without using standard measuring devices, express the length of the mystery piece (a fractional part of a meter) in meters. How might students solve the problem and what they might find challenging? What understandings of fractions would help students solve the problem?
Read over “Fractions” (pp. 5764 of the Grade 3B Tokyo Shoseki textbook, 2006). It may be useful to know that 8 periods of 45 minutes are allocated for this unit, roughly one period per textbook page.
 Consider how this unit might help students:
 See nonunit fractions (fractions with a numerator other than 1) as accumulations of unit fractions (fractions with numerator of 1)?
 See the connection between whole numbers and fractions?
 Connect area, linear measurement, and number line models?
 Consider: How do the images and understandings of fractions that students might develop from the 3B textbook compare with what students might develop from your own curriculum’s introduction of fractions?
If you like this activity and want to do more of it, you can analyze the 4B Fractions Unit (pp.3851) in the same fashion.
In sections 25 of your fraction unit plan (unit flow and rationale), capture any key learnings from examining the curriculum.
*Adapted from Cramer, K. & Wyberg, T. (2007). When getting the right answer is not always enough: Connecting how stuents order frations and estimate sums and differences. pp. 205220. W. G. Martin & M. E. Strutchens (Eds.) The learning of mathematics. 69th Yearbook. Reston, VA:NCTM, p.215.
Step 4: Examine Video
Step 4 provides video to spark thinking about the design of your fractions unit.
 To start, revisit the beginning of the 3B fractions unit (pages 5764) and imagine how you might teach it.
 Read over Akihiko Takahashi’s lesson plan for a series of three lessons based on this textbook segment.
 Watch the video from Dr. T’s lessons (taught to grade 35 students in California). It will useful to print out the lesson plan and the table below, to have them as reference while watching.
 Watch Lesson 1 (stopping at 8:11 and 9:42 to consider questions in the chart below)
 Watch Lesson 2 (stopping at 4:40 and 8:40 to consider questions in the chart below)
 Watch Lesson 3 (stopping at 13:35 to consider the questions in the chart below)
If watching the video online, you can see the timecode by moving the cursor over the lower part of the video.
Analysis of TTP Lessons: Introduction to Fractions, Akihiko Takahashi
We suggest points to stop [STOP] and consider preceding questions.
Step 5: Plan Lessons
Researcher: Is 3 < 8 a true statement?
First Grade Student: If you have three really big units and 8 really small ones, 3 could be greater than 8. But if you’re working on a number line, then you know that 3 is less than 8 because all the units are the same.” [1]
Since the focus of this work is to learn about Teaching Through Problemsolving, we recommend that you base your lesson on the Japanese textbook excerpts and video provided, rather than devote time to writing a lesson from scratch. For most students at the elementary or middle school level, (re)introducing fractions using the linear measurement model found in the Japanese textbook (and in Dr. T.’s lesson) is likely to help them build important new insights into fractions. If you worry that the initial activities will be too simple for your students, you can move more quickly to the activities included in lessons two and three: finding nonunit fraction mystery strips, constructing fractions, and connecting to the number line. However, the initial experience of seeing, for example, that ^{1}_{3} goes into a meter 3 times will be important for the subsequent activities.
The unit plan provides a place to plan your research lesson. To strengthen the elements of TTP as you work on your research lesson plan, consider (re)visiting Steps 47 from the area of polygons: step by step guide, or use the quick links below:
You can see how Dr. T. built students’ journal writing skills by reviewing this 2minute video clip from the beginning of the first fractions lesson. Dr. T. asks students to look at examples of student journals from the prior day (a lesson about calendar numbers) and to add to their own journalwriting. Would a strategy like this be useful to your students?
Three additional linked resources may be useful as questions arise about fractions content:
 Translations of the Japanese teacher’s manual related to the 3B and 4B fractions units.
 List of Fractions Units in Japanese Elementary Curriculum
 Common Core State Standards, which can be searched for the word “fraction” to read about the trajectory of fraction learning in the U.S.
[1] Dougherty, B. J. & Zilliox, J. (2003). Voyaging from theory to practice in teaching and learning: A view from Hawai‘i. In N. Pateman, B. J. Dougherty, & J. Zilliox (Eds.), Proceedings of the 2003 joint meeting of PME and PMENA (pp. 17–31). Honolulu, HI: Curriculum Research & Development Group, University of Hawai‘i. pp. 1920. For additional information on the “Measure Up” curriculum, see Slovin, H. & Dougherty, B. J. (2004). Children’s conceptual understanding of counting. In M. JohnsenHoines & A. B. Fugelstad, Proceedings of the 2004 psychology of mathematics education, volume 4 (pp. 4209–4216). Bergen, Norway: Bergen University College.