division of fractions: balancing conceptual and procedural knowledge part 2
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Division of Fractions: Balancing Conceptual and Procedural Knowledge Part 2. January 15, 2013 Common Core Leadership in Mathematics2 (CCLM). - PowerPoint PPT PresentationTRANSCRIPT
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