Transcript CCLM - University of Wisconsin–Milwaukee

Division of Fractions: Balancing
Conceptual and Procedural
Knowledge Part 2
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.
January 15, 2013
Common Core Leadership in Mathematics2 (CCLM)
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
Use an
strategy /
algorithm
Think
about
related
operations
Division
Write an
equation
Draw a
diagram
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012-2013
School Year
ESTIMATE
Estimate
5
3
4
•
•
•
Greater than 5?
Equal to 5?
Less than 5?
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 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, 2012-2013
School Year
Looking at the Standards
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012-2013
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, 2012-2013
School Year
Interpretations of Division
Group Size Unknown
Number of Groups Unknown
I know the total number of objects.
I know the number of groups/shares.
How many objects are in each
group/share?
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 much chocolate will each
person get if 3 people share 1/2 lb of
chocolate equally
Example: How many 1/3-cup servings
are in 2 cups of raisins?
1
¸3=?
2
*Partitive division, sharing model, dealing
out.
1
2¸ =?
3
* 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 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, 2012-2013
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, 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?
1
3

1

4
4

12


3
12
43
12  12
43
1
 431
1
3
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, 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?
3
4

2
5

3

4
5
2

15
8
1
7
8
WHY does it work?
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012-2013 School
Year
Why can we “invert and multiply”?
Discuss this question with your
shoulder partner. Record your
answer on your slate
Share your answer with the whole table.
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012-2013 School Year
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