Division of Fractions: Balancing Conceptual and Procedural Knowledge Part 2 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, 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, School Year

Extending Meaning of Division to Fractions Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 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, School Year

ESTIMATE Estimate Greater than 5? Equal to 5? Less than 5? Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 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, School Year

Juice Party Quantity: 1/2 gallon of juice How 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, School Year

Looking at the Standards Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, School Year

Standard 5NF 7c 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? Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, School Year

Interpretations of Division Group Size Unknown 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 ou t. 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, School Year

Standard 5NF 7a and 5NF 7b Apply 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, School Year

A Tricky Popcorn Party Serving Size: 3/4 cup of popcorn How 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, 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, 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, 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, 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, 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. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 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, School Year

Invert and Multiply Method Have you ever used this? WHY does it work? Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, School Year

Why can we “invert and multiply”? Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 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, 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, 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, School Year