1. Fractions and CCSSM joe georgeson february 26, 2014 UW-Milwaukee WSMI
integratED PDX

3. Learning Intentions......
Deepen conceptual understanding of division of fractions as presented in the Common Core
Unpack one standard relating to dividing fractions

4. We will know we are successful when we can
Justify our thinking when dividing fractions using reasoning and models.
Explain using models why “invert and multiply” works.

5. Not here to sell the Common Core-
rather to promote
Meaningful Mathematics!

7. Focus
do less but do it better
Coherent
progression from grade to grade
Rigor
develop skills
built on understanding
in the context of applications

8. CCSSM: New Structure and New Terminology
Standards For Mathematical Practice
(how you teach)
Standards for Mathematical Content
(what you teach)
Make sense of problems
Reason Quantitatively
Viable arguments
Model with Mathematics
Use tools strategically
Attend to Precision
Look for and use structure
Look for and use patterns
K-8 Standards organized by Grade Level
Domains
Clusters
Standards
High School Standards by Conceptual Category

9. K-8 Domains and HS Conceptual Categories
William McCallum, The University of Arizona

10. For as long as most of us can remember, the K-12 mathematics program in the U.S. has been aptly characterized in many rather uncomplimentary ways: underperforming, incoherent, fragmented, poorly aligned, narrow in focus, skill-based, and, of course, “a mile wide and an inch deep.” steve leinwand

11. “But hope and change have arrived
“But hope and change have arrived! Like the long awaited cavalry, the new Common Core State Standards for Mathematics (CCSS) presents us – at least those of us in the 44 states+ that have now adopted them (representing over 80% of the nation’s students) – a once in a lifetime opportunity to rescue ourselves and our students from the myriad curriculum problems we’ve faced for years.”

12. “Difficulty with learning fractions is pervasive and is an obstacle to further progress in mathematics and other domains dependent on mathematics, including algebra. It has also been linked to difficulties in adulthood, such as failure to understand medication regimens.”
(“Foundations for Success”
The National Mathematics Panel Final Report, 2008-page 28)

13. “These Standards are not intended to be new names for old ways of doing business.”
“They are a call to take the next step. It is time for states to work together to build on lessons learned from two decades of standards based reforms.”
“It is time to recognize that these standards are not just promises to our children, but promises we intend to keep.”

14. “You have just purchased an expensive Grecian urn and asked the dealer to ship it to your house.
He picks up a hammer, shatters it into pieces, and explains that he will send one piece a day in an envelope for the next year.
You object; he says “don’t worry, I’ll make sure that you get every single piece, and the markings are clear, so you’ll be able to glue them all back together. I’ve got it covered.”
Absurd, no? But this is the way many school systems require teachers to deliver mathematics to their students; one piece (i.e. one standard) at a time.
They promise their customers (the taxpayers) that by the end of the year they will have “covered” the standards.”

15. some “common” words from CCSSM

16. 6.NS.1 Apply and extend previous understandings of multiplication and division to divide fractions by fractions.
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?

17. Divide a whole number by a unit fraction
5th grade:
Divide a whole number by a unit fraction
Divide a unit fraction by a whole number
5th grade standard: 5.NF.7a and b
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.
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.

18. 4th grade:
Use a visual fraction model to show that 4/5 is 4 groups of 1/5 and that 10x3/4 is 10 groups of 3/4 which is 10 groups of 3 groups of 1/4
4th grade standard: 4.NF.4
Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4).
Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)

19. 3rd Grade
Know and understand that the unit fraction, 1/b, is one part of a whole that has been partitioned into b equal parts.
3rd Grade Standard: 3.NF.1
Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
Understand a fraction as a number on the number line; represent fractions on a number line diagram.
Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.
Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.

20. Unpack one of the NF standards: 5.NF.7
In your table group discuss:
what the standard is saying:
what are teacher behaviors
what student behavior would the teacher observe and know they knew and understood that standard.

21. Example:
It depends on the context!
Make up a situation in which this division would apply.
John has 10 candy bars. He is going to give each of his friends of a candy bar. How many friends could he give this candy to?

22. Here is a picture and the related multiplication equation-which one?
John has 10 candy bars. He is going to give each of his friends of a candy bar. How many friends could he give this candy to?
Here is a picture and the related multiplication equation-which one?

23. Another Context for
My tank was empty. I put 10 gallons in and it is now 2/3 full. How much does the whole tank hold?
In this case I am asking for the whole when I know 2/3 of it.
Supply a picture and related multiplication for this example.

24. My tank was empty. I put 10 gallons in and it is now 2/3 full
My tank was empty. I put 10 gallons in and it is now 2/3 full. How much does the whole tank hold?

25. Are these the same or different? It depends on the context.
There are 24 students. Three-fourths of them play a sport. How many play a sport?
There are 24 people at a party. Everyone got three fourths of a small pizza. How many pizzas were needed?

26. Understanding fractions requires knowing what a fraction is.
UNIT FRACTIONS
Understanding fractions requires knowing what a fraction is.
Using the precise language of the Common Core:
“one part of a whole that was partitioned into 8 equal parts”

27. Two different interpretations of 5/8:
One: A whole was partitioned into 8 equal parts. You have 5 of those parts, or 5/8.
What would the picture look like?
What situations would lead to this interpretation?

28. What would this picture look like?
Two: There are 5 wholes. They are to be shared equally among 8 groups. How much in each group?
What would this picture look like?
What situation would lead to this interpretation?

29. Could you explain it in a meaningful context?
The algorithm, multiply by the reciprocal, is not done until 6th grade.
That is the capstone. It only comes after students understand multiplication and division in a meaningful way.
Could you explain, without using mathematical jargon, why this is what is done?
Could you explain it in a meaningful context?
to a student?

30. I have 5/6 of a cake left after a party
I have 5/6 of a cake left after a party. My freezer containers each hold 1/3 of the cake. How many containers can I fill? Can I fill part of another container? If so, how full will it be?
Solve this and represent it with a tape diagram or number line, or other way to show it.

31. How many groups of 1/3 can I fit into 5/6?

32. I have 3/4 oz. of gold. I want to make rings that have 1/8 of an ounce
I have 3/4 oz. of gold. I want to make rings that have 1/8 of an ounce. How many rings could I make?
Show this with a linear model (tape diagram or number line).
Write a series of equations that reflect your reasoning.

33. I have 10 feet of rope and I want to make smaller pieces, each 2/3 of a foot in length. How many of these smaller-sized pieces could I make?
How would you figure this out without using any algorithm? Pretend you are looking at this without learning how to divide fractions.
Explain your reasoning to the person next to you.

34. The algorithm says to multiply by the reciprocal.
What does the 10 times 3 represent in the context about the rope?
What does the divide by 2 represent?
How has the meaning and role of the “3” and the “2” changed as you solved the problem?

35. The “common denominator” algorithm:
Does this work all the time?
Does it make sense?

36. John has 2/3 of a gallon of lemonade
John has 2/3 of a gallon of lemonade. How many water bottles can he fill if each bottle holds 1/12 of a gallon?
Draw a visual model of the problem.
Solve using both the invert and multiply algorithm and the common denominator algorithm.
Explain the meaning of each step in the algorithm as represented in the diagram and context of the actions with the lemonade.

43. Learning Intentions......
Deepen conceptual understanding of division of fractions as presented in the Common Core
Unpack one standard relating to dividing fractions

44. We will know we are successful when we can
Justify our thinking when dividing fractions using reasoning and models.
Explain using models why “invert and multiply” works.

45. “It’s the story that is important in learning mathematics.”
My final thought:
“It’s the story that is important in learning mathematics.”

46. Common Core Standards for School Mathematics
Resources:
Common Core Standards for School Mathematics
Progressions Documents for the Common Core:
Fordham Institute Interview with Jason Zimba