A model lesson teachers can implement in their classroom. Clarifying Lessons combine multiple TEKS statements and may use several Clarifying Activities in one lesson. Clarifying Lessons help to answer the question "What does a complete lesson look like that addresses a set of related TEKS statements, and how can these TEKS statements be connected to other parts of the TEKS?"

Number, operation, and quantitative reasoning: 7.2A, B

Patterns, relationships, and algebraic thinking: 7.5A

Probability and statistics: 7.14A

- Overhead projector
- Blank transparencies
- Transparency pens
- 1 pad of easel paper or butcher paper
- Markers
- Masking tape
- Activity sheet 1 with teacher notes (pdf 384kb)
- Activity sheet 2 (pdf 364kb)
- Activity sheet 3 with teacher notes (pdf 396kb)
- Activity sheet 4 with teacher notes (pdf 392kb)
- Activity sheet 5 with teacher notes (pdf 400kb)

4 days

Glenda Lappan, James T. Fey, William M. Fitzgerald, Susan N. Friel, and Elizabeth Difanis Phillips, *Bits and Pieces II: Using Rational Numbers* (Dale Seymour Publications, 1998).

**EDITED Resources.** The resources on this page have been aligned with the
revised K-12 mathematics TEKS. Necessary updates to the resources are in
progress and will be completed Fall 2006. These revised TEKS were adopted by
the Texas State Board of Education in 2005–06, with full implementation
scheduled for 2006–07.

Given problems involving whole numbers, students analyze responses to two different types of division problems and discuss how the inclusion of the language of sharing and grouping will help them recognize situations that require division. Students then solve a sequence of problems organized to provide scaffolding for understanding of an algorithm for division of fractions. Emphasis will be placed on using pictures or diagrams and written descriptions of the thought processes involved to support solutions and the development of an algorithm.

Students build a conceptual understanding of the division of fractions through division of whole numbers and modeling of division of fractional parts.

- The teacher introduces students to types of division (sharing and
grouping) by posing the questions 1 and 2 in Activity 1, allowing
time for students to illustrate these problems on easel paper and
share them with the class.
It is important to use what students know about whole number division situations while dividing with fractions.

There are many kinds of situations that call for division whether with whole numbers or fractions. The first two problems in Activity 1 are designed to allow the teacher to have a conversation with the students about two major types of division problems—sharing problems and grouping problems.

In the sharing situation, some known quantity (amount) is shared equally among a known number of entities (people, boxes, packages, etc.). What is not known in a sharing situation is the amount of the given quantity per share. The quotient in this situation represents the amount per share, the size of each share or the unit rate.

In a grouping situation, the unknown is the number of groups of a given size that can be made from a given quantity (amount). The quotient in this situation tells how many groups of the specified size can be made from the given quantity.

- When the students have
an understanding of sharing and grouping, they can complete Activity
2, which provides a summary and a quick check for understanding.
Once the language of sharing and grouping is introduced in the whole number problems similar to those in Activities 1 and 2, teachers can continue to use this language and ask for such analyses in the fraction division problems as well. This language can be very helpful in understanding when division is an appropriate operation.

- Students model division using fractions in Activities 3, 4, and 5. They
begin with division of a whole number by a fraction, and move
to division of a fraction by a whole number and finally division
of a fraction by a fraction.
For Activity 3, students should work alone for a few minutes to formulate their thinking about how they might solve problem 1. Then they can work in small groups to discuss their thoughts and work through all parts of problems 1 and 2. Using transparencies on the overhead projector, students can share their answers with the class, with an emphasis on how they thought about each problem, pictured it, and processed it. After a discussion about these problems, students complete questions 3, 4, and 5 as a summary.

- Students follow the same procedure for Activities 4 and 5.

- How can you illustrate the problem? What operation are you using? Does the problem represent a sharing or grouping problem? Why do you think so?
- What is the difference between a sharing and grouping problem?
- What does the answer represent in a sharing problem? Grouping problem?
- Can you give another example of a sharing problem? Grouping problem?

- Are these sharing or grouping problems? Use a written explanation or diagrams to describe how you made sense of these problems.
- How might you write a generalization for the pattern you have found in these problems?
- Using your generalization, can you find the answer to problems such as 6 — 1/2? 5 — 1/9? 7 — 1/3?
- What about problems that involve division by a non-unit fraction, such as number 1d?
- What did you notice about these problems that might help you make sense of whole numbers divided by fractions?
- Why do your answers to these problems make sense?
- When we divide 9 by 1/4, we would say that there
are four 1/4s in each whole. So we would multiply 9 x 4 to get 36. However
if the problem is 9 — 3/4,
how can we reason the quotient?
The student might answer: "Using the same logic we can solve this problem, but now it takes only three 1/4s to make a whole. If we divide the 36 by 3 we get 12, which is the quotient."

- Will this reasoning work with all non-unit fraction divisors? Why?
- When
the divisor is a non-unit fraction what happens when you have a remainder
like you do in 2d? For example, 12 — 5/8?
The student might answer: It takes eight 1/8s to make a whole, so 12 x 8 is 72 in this situation in takes only five of the 1/8s to make the whole so 72 — 5 is 14 with a remainder of 2. The remainder 2 is 2 out of 5 it took to make the whole so the remainder is 2/5 not 2/8.

- How might you make a generalization for dividing a whole number by a unit fraction or dividing a whole number by a non-unit fraction?
- How might you describe a procedure for dividing any whole number by a fraction?

- Are these sharing or grouping problems? Use a written explanation or diagrams to describe how you made sense of these problems.
- How can you explain your reasoning and justify your answer?
- How did you label your diagram?
- What is the question asking you to find? Are you looking for unit rates or number of groups?
- What is another example of a problem like those you found in problems 1 and 2?
- Are the problems in number 2 different from the problems in number 1? How are they different? How are they the same?
- How did you reason through the problems in number 2?
- In number 2a will your answer be greater than or less than 1/4? How do you know?
- Is number 2c different than the others? Why or why not?
- What happens if you use the strategy of finding common denominators before you divide a fraction by a whole number?
- How might you briefly describe the process you used to solve the problems in Activity 4?
- How might you write a generalization for the pattern you have found in these problems?

- Are these sharing or grouping problems? Use a written explanation or diagrams to describe how you made sense of these problems.
- If you had 1/2 bag of flour and you wanted to bake a batch of cookies that called for 1/8 bag of flour, how many batches could you make?
- What are you trying to find out in this problem?
- What quantities do you know?
- Is this a sharing or grouping problem?
- How can you illustrate this problem?
- How can you label the parts of your illustration?
- Is there another way to find the same solution?
- How can common denominators help you find a solution?
- Is there a difference among the problems in numbers 1 and 2?
- Which are easier to solve? Why?
- Did the strategies you used for number 1 have to change when you answered number 2?
- What quantities are given in problems 1 and 2?
- What are the questions in problems 1 and 2 asking you to find? Are you looking for unit rates or number of groups?
- How will you deal with remainders?
- How might you briefly describe the process you used to solve the problems in Activity 5?
- How might you write a generalization for the pattern you have found in these problems?

- What is the difference between a sharing and grouping problem?
- How might you write a situation that would involve sharing?
- How might you write a situation that would involve grouping?
- Can you write an algorithm for dividing fractions? Demonstrate the algorithm on a problem for each situation given: a whole number divided by a fraction; a fraction divided by a whole number; a fraction divided by a fraction; a mixed number divided by a fraction.
- How is the answer to 20 — 1/5 related to the answer to 20 — 3/5?
- When will the quotient be greater than the dividend and divisor? What example can you give of this type of situation?
- When will the quotient be less than the dividend and divisor? What example can you give of this type of situation?
- When will the quotient be between the dividend and divisor? What example can you give of this type of situation?

The teacher asks students to write a story problem for each kind of situation (sharing and grouping), solve the problem showing justification, and explain why it is a sharing or grouping problem.

The teacher gives students the situation __ — 3/4 = 5/6, and asks them to complete the problem and write a story problem that can be solved by the division. Students should be able to explain why the calculation matches the story.