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Division of Fractions: Balancing Conceptual and Procedural Knowledge Part 2

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January 15, 2013

Common Core Leadership in Mathematics2 (CCLM)

This material was developed for use by participants in the Common Core Leadership in Mathematics (CCLM^2) project through the University of Wisconsin-Milwaukee. Use by school district personnel to support learning of its teachers and staff is permitted provided appropriate acknowledgement of its source. Use by others is prohibited except by prior written permission.

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012-2013 School Year

We are learning to …

- apply and extend understandings of division to fractions that includes a focus on unit fractions in the context of real-world problems.
We will be successful when we can…

- explain and provide examples of standard 5.NF.7 using visual models, contexts, and concept-based language to divide unit fractions by whole numbers and whole numbers divided by unit fractions.

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012-2013 School Year

Extending Meaning of Division to Fractions

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012-2013 School Year

CCLM

Division

CCLM

Estimate

- Greater than 5?
- Equal to 5?
- Less than 5?

- Review to Popcorn Problems for last class
- What were the big ideas from these problems?
- What representations did we use?

Quantity: 1/2 gallon of juice

How can I divide that equally among:

- 2 friends
- 5 friends

Looking at the Standards

CCLM

Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.1

c. Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?

Group Size Unknown

Number of Groups Unknown

I know the total number of objects. I know the number of objects in each group/share. How many equal groups/shares can be made?

Example: How many 1/3-cup servings are in 2 cups of raisins?

* Quotative division, measurement division, grouping, subtractive model.

I know the total number of objects. I know the number of groups/shares. How many objects are in each group/share?

Example, How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally

*Partitive division, sharing model, dealing out.

CCLM

Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.1

- Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.
b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.

CCLM

Serving Size: 3/4 cup of popcorn

How many servings can be made from:

2 ¼ cups of popcorn

5 cups of popcorn

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In pairs, solve each problem using reasoning and models (don’t forget the tape diagram).

- How many ¾ cups servings of popcorn are in 4 ¼ cups of popcorn?
- A serving is ½ of a cookie. How many servings can I make from 3/8 of a cookie?

We are learning to …

- apply and extend understandings of division to fractions that includes a focus on unit fractions in the context of real-world problems.
We will be successful when we can…

- explain and provide examples of standard 5.NF.7 using visual models, contexts, and concept-based language to divide unit fractions by whole numbers and whole numbers divided by unit fractions.

CCLM

What procedure do you use to divide fractions?

Write an example of it on your slate.

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The common denominator method

Invert and Multiply

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Have you ever used this?

Does it always work?

Make up division problems to decide when you can use this algorithm.

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The common denominator method

Invert and Multiply

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- Have you ever used this?
WHY does it work?

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Discuss this question with your shoulder partner. Record your answer on your slate

- Share your answer with the whole table.

CCLM

CCLM

Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?

- Reread this standard. Do the examples and tasks make more sense to you now?

We are learning to …

- apply and extend understandings of division to fractions that includes a focus on unit fractions in the context of real-world problems.
We will be successful when we can…

- explain and provide examples of standard 6.NS.1 using visual models, contexts, and concept-based language to divide unit fractions by whole numbers and whole numbers divided by unit fractions.