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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

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 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.

Task 3

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 1-3 that you examined.
  • 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.”
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Common Challenges with Fractions