# Fractions – Revised

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.

5 hours/week

##### MATERIALS:

Textbook Name ## 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 Problem-Solving.

## 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

• 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.
• Discuss these with your colleagues and fill in items 2-6 of your fraction unit plan template.

What’s Hard About Fraction Number Sense?

 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). When asked to put the fraction 23  on a number line, a student said “you can’t put it on a number line, because it’s two pieces out of three pieces, it’s not a number.” Or “ 23 is not a number, it’s two numbers.” [*] Partitioning Fractions A whole can be divided into smaller and smaller equal parts. The same fractional quantity can be represented by different fractions. Difficulty seeing how to divide a whole into equal parts. Difficulty seeing that 12 is equal to 24, 36, 48, 510 and so on. The Meaning of the Denominator Different units (such as 13  and 15) are different sizes. The more units a whole is partitioned into the smaller each one is. 1n fits exactly n times into the whole. Students add 13 + 15 and get 28, without realizing they are adding two different things (thirds and fifths) sort of like adding apples and hammers. Students may think “ 15 is bigger than 14 because 5 is bigger than 4.” Difficulty seeing that 13 fits in the whole 3 times, 14 fits in the whole 4 times.  Has trouble seeing that 33, 44 etc. equal 1. Knowing What is the Whole Constructing the whole when given a fractional part. Keeping track of the whole. Difficulty making the whole when you give them a fractional part, e.g.: “This paper is 23; show me the whole.” Sees that the magnitude of a fraction depends on the magnitude of the whole (e.g., half of a small cookie is not the same as half of a large cookie) Confusion about whether the two drawings below together represent 38 of a pie or 316 of a pie.  Fraction Size Understands that fraction size is determined by the (multiplicative) relationship between numerator and denominator – not just by the numerator, not just by the denominator, and not by the differencebetween numerator and denominator. Sees non-unit fraction as an accumulation of unit fractions. [A unit fraction has a numerator of 1; a non-unit fraction has a numerator other than 1.] May think 49 is bigger than 34 because 4 is bigger than 3 (comparing numerators), or 49 is bigger than 34 because 9 is bigger than 4 (comparing denominators), or 35 is the same size as 57 because the difference between the top and the bottom in both fractions is 2. Sees that equivalent fractions have the same multiplicative relationship between numerator and denominator. In 24, 48, 36, etc. denominator is two times numerator. Sees 58 is made up of 5 18 ‘s or 5 times 18, that 98 is made up of 9 eighths or 9 times 18, etc. 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. “You can’t have 65 because there’s only 55 in a whole.”

## Step 3: Curriculum Exploration

This step allows you to examine curriculum materials and fraction models. It provides fuel for thinking about items 2-5 of the unit plan for fractions that you have been working on.

 Researcher: Which fraction is smaller? 64 or 65 Chris: 64. 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  46 and 56, 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 34 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 34 represents several very different situations, such as 34 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 research-based programs.

Explore the linear measurement model with a challenging hands-on 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: 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. 57-64 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 non-unit 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.38-51) in the same fashion.

In sections 2-5 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. 205-220. 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 57-64) 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 3-5 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. ## 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.” 

Since the focus of this work is to learn about Teaching Through Problem-solving, 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 non-unit fraction mystery strips, constructing fractions, and connecting to the number line.  However, the initial experience of seeing, for example, that 13 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 4-7 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 2-minute 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 journal-writing.  Would a strategy like this be useful to your students?