Division of Fractions: Balancing Conceptual and Procedural Knowledge Part 2

Division of Fractions: Balancing Conceptual and Procedural

Knowledge Part 2

January 15, 2013Common 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

Learning Intentions and Success Criteria

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

Components of Complete Understanding of Division

Estimate the answer

Think about

related operations

Draw a diagram

Write an equation

Use an strategy / algorithm Division

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

ESTIMATE

Estimate

• Greater than 5? • Equal to 5? • Less than 5?

435

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

Revisiting Division of Fractions

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

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

Juice Party

Quantity: 1/2 gallon of juiceHow can I divide that equally among:

2 friends 5 friends

• Individually solve each problem using reasoning and models

• As a group, take turns and share your reasoning

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

Looking at the Standards

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

Standard 5NF 7cApply 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?

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

Interpretations of Division

Group Size UnknownI 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.

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.

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

Standard 5NF 7a and 5NF 7bApply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.1 a. 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.

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

A Tricky Popcorn PartyServing Size: 3/4 cup of popcornHow many servings can be made from:

2 ¼ cups of popcorn

5 cups of popcorn

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

Now It’s Your turn

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?

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

Learning Intentions and Success Criteria

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

Computational Procedures

What procedure do you use to divide fractions?

Write an example of it on your slate.

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

Two Procedures for Division of Fractions

The common denominator method

Invert and Multiply

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

The Common Denominator Method

Have you ever used this?

Does it always work? Make up division problems to decide when you can use this algorithm.

123

124

41

31

121234

31134

134

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

Two Procedures for Division of Fractions

The common denominator method

Invert and Multiply

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

Invert and Multiply Method

• Have you ever used this?

WHY does it work?

871

815

25

43

52

43

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

Why can we “invert and multiply”?

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

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

Share your answer with the whole table.

Sample student work

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

Examine 6.NS.1

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?

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

Learning Intentions and Success Criteria

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.

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