## Fractions Task 2

Task 2: Find two fractions between ** ^{1}/_{2}** and 1.

#### 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?
- Look at the student solutions to Task 2. 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?

##### Task 2 Student Solutions

Context 1: Basal Text

Student Response |
Rationale |

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 |

*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.*

Context 2: “Measure Up” Curriculum (Measurement Context)

Student Response |
Rationale |

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. |

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.