1. Multiplication with Fractions
Core Mathematics Partnership
Building Mathematical Knowledge and
High-Leverage Instruction for Student Success
Monday July 27, 2015

2. Isn’t this everything I need to know?

3. Unfortunately students also extend their whole number misunderstandings to fractions.

4. “The rush to tell students how to perform procedures prevents them from establishing a solid foundation of operation sense for fractions. It is time to shift the emphasis and redefine the goal of fraction instruction in elementary school from learning computational rules to developing fraction operation sense.” – Huinker (2014, 2002)

5. Operation Sense for Fractions
Understand the meanings of operations.
Recognize and describe real-world situations for specific operations.
Understand the meaning of symbols and formal mathematical language.
Translate easily among representations.
Understand relationships among operations.
Compose and decompose numbers and use properties of operations.
Understand the effects of an operation on a pair of numbers.
(Huinker, 2014, 2002)

6. Making Sense of Multiplication with Fractions
Moving from 4th grade to 5th grade

7. Learning Targets We are learning to…
Apply and extend our understanding of multiplication with whole numbers to multiplication with fraction through use of context (word problem situations) and use of the underlying mathematical structure.

8. The Fudge Shop
Moving from 4th grade to 5th grade

9. Sam loves fudge!! While at the Dells,
he bought 3 packages of fudge at the Fudge Shop.
Each package contained 4 pounds of fudge.
How many pounds of fudge did Sam buy?
3 people, 3 paper plates
12 parts, size of each part? one pound of fudge, so 12 parts of size 1 pound or 12 pounds.
Got 12 things, what is the size of each thing?

10. Each package contained 4/5 of a pound of fudge.
Morgan also loves fudge!! However, she didn’t have as much money as Sam. She bought 3 packages of fudge at the Fudge Shop.
Each package contained 4/5 of a pound of fudge.
How many pounds of fudge did Morgan buy?
3 people, 3 paper plates
12 parts, size of each part? 1/8 pound of fudge, so 12 parts of size 1/8 pound or 12/8 pounds.
Got 12 things, what is the size of each thing?

11. 3 x 4 = 12 Sam loves fudge!! While at the Dells,
he bought 3 packages of fudge at the Fudge Shop.
Each package contained 4 pounds of fudge.
How many pounds of fudge did Sam buy?
x = 12
Equation:
Contextual
Meaning:
Packages of fudge Sam bought.
Pounds of fudge in each package.
Pounds of fudge in all the packages combined.

12. Morgan also loves fudge
Morgan also loves fudge!! She bought 3 packages of fudge at the Fudge Shop. Each package contained 4/5 of a pound of fudge. How many pounds of fudge did Morgan buy?
x =
Equation:
Contextual
Meaning:
Packages of fudge Morgan bought.
Pounds of fudge in each package.
Pounds of fudge in all the packages combined.

13. The Underlying Mathematical Structure of Multiplication
Cluster: Represent and solve problems involving multiplication and division.
Standard 3.OA.1. Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each.

14. Standard for Mathematical Practice #7 “Look for and make use of structure”
Mathematically proficient students look closely to discern a pattern or structure.... They also can step back for an overview and shift perspective.
Mathematics has far more consistent structure than our language, but too often it is taught in ways that don’t make that structure easily apparent for students.

15. 3 x 4 = 12 Sam loves fudge!! While at the Dells,
he bought 3 packages of fudge at the Fudge Shop.
Each package contained 4 pounds of fudge.
How many pounds of fudge did Sam buy?
x = 12
Equation:
Packages of fudge Sam bought.
Pounds of fudge in each package.
Pounds of fudge in all the packages combined.
Contextual
Meaning:
Structural
Meaning:
Number of Groups
Size of
Each Group
Total
Amount

16. Morgan also loves fudge
Morgan also loves fudge!! She bought 3 packages of fudge at the Fudge Shop. Each package contained 4/5 of a pound of fudge. How many pounds of fudge did Morgan buy?
x =
Equation:
Contextual
Meaning:
Packages of fudge Morgan bought.
Pounds of fudge in each package.
Pounds of fudge in all the packages combined.
Structural
Meaning:
Number of Groups
Size of
Each Group
Total
Amount

17. However.... we are living in the world of fractions...
Structural
Meaning:
Number of Groups
Size of
Each Group
Total
Amount
Complete Groups
Partial Group

18. Professional Reading and Reflection (PRR)
Moving from 4th grade to 5th grade

19. PRR: A Unified Approach to Multiplying Fractions (Jack Ott, 1990)
Read the article (only 3 pages) and study the figures. Ott confirms the “relative difficulty of explaining to students the meaning of addition and multiplication of fractions” (p. 47) Choose one sentence from the article that impacted your thinking on language difficulty when understanding or explaining the multiplication of fractions. Re-write the sentence in your notebook and explain why you selected it.

20. CCSSM Standards

21. Standard 3.NF.1 Standard 4.NF.3
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.
“6 parts of size one fifth”
Standard 4.NF.3
Understand a fraction a/b
with a > 1 as a sum of fractions 1/b.

22. = = 6 x 6 x 1 5 Standard 3.NF.1 “6 parts of size one fifth”
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.
“6 parts of size one fifth”
Standard 4.NF.3
Understand a fraction a/b
with a > 1 as a sum of fractions 1/b.
Note: Any number is a multiple of 1.
Standard 4.NF.4a
Understand a fraction a/b as a multiple of 1/b. =
6 x 1 5
= 6 x

23. 1. State what each quantity means in the story.
I am making 3 batches of bean soup. Each batch calls for 2/5 cup of dried beans. How many cups of beans will I need for the three batches?
1. State what each quantity means in the story.
2. Using your “1/5” measuring cup, demonstrate how to measure out the correct amount of beans for one batch.
3. Write at least two expressions to represent the amount of beans needed for one batch of soup.
4. Demonstrate the scoops for 2 batches? 3 batches?
5. Write at least three expressions to represent the amount of beans needed for 3 batches of soup.
Look for both 1/5 + 1/5 = 2/5 as well as 2 x 1/5. Get both up on chart paper and surface relationship between the two expressions.

24. Bean Soup: Keeping Track of Quantity
How many groups of 2/5 cup did you measure out?
How do you know?
How many total scoops of 1/5 cup did you measure out? How do you know?
4.NF.4b. 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.)
3x2 scoops of size 1/5 is the same as 6 scoops of size 1/5

25. 4.NF.4b. 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.) =
3 x
3 x 2 5 =
(3 x 2) x =
Three batches; 2-fifths of a cup in each group.
Three batches with two scoops in each batch; size of a scoop is one-fifth of a cup.
Six scoops of size one-fifth of a cup.

26. A Glimpse into a Classroom: Ribbons
Moving from 4th grade to 5th grade

27. Extend Understanding of Repeated Addition to Multiplication of a Fraction by a Whole Number
Video possibility #1 .
Works for 4.NF.4b We would need to stop the video at 9:26 and ask teachers to analyze the task cards with various representations.
Video timing:
0:00 -> 1:04 Teacher statement of lesson goal (transitioning from repeated addition to multiplication)
1:04 -> 1:30 Statement of problem context
2:00 (approx.) First student solution
2:50 “It’s OK. You just made a mistake.”
3:19 -> 5:00 Student misconception on rounding to nearest yard
4:18 -> 5:11 Teacher explanation of how “the lesson took a slightly different turn”.
5:11 -> 6:29 Discussion of array solution
6:29 -> 8:00 Discussion of circle solution (“cover up the threes”)
8:00 -> 9:26 Teacher summary and next steps

28. Possible Solutions to 4 × ⅔
15 is a multiple of 5, of 3, of 1.

29. How did Ms. Spies create opportunities for students to share and learn from each other?
What did students learn by critiquing the teacher’s solutions?
How did Ms. Spies help her students further their understanding of multiplication as repeated addition?

30. Moving towards Those “Partial” Groups
Moving from 4th grade to 5th grade

31. Another Classroom: Fraction Number Talk
0:00 -> 1:57 Number talk, ending with “Should we be scared if the numerator is not 1?”

32. Mental Math
½ of 20 1/4 of 20 1/5 of 20 3/5 of 20 1/8 of 20 1/7 of 21 3/7 of 21 6/7 of 21
Elicit conceptual understanding as teacher explain how they did each problem.
Prelude to Video 1

33. Making Lasagna
Moving from 4th grade to 5th grade

34. In order to encourage her family to eat more vegetables Melissa decides to include spinach in her lasagna. The recipe calls for ¾ pound of spinach for each batch of lasagna.
Referent Whole: What quantity is represented by one paper strip?
Size of Each Group: What quantity comprises one “complete” group?
Three different lasagna problems

35. Context: Restate the problem in your own words.
In order to encourage her family to eat more vegetables Melissa decides to include spinach in her lasagna. The recipe calls for ¾ pound of spinach for each batch of lasagna. How many pounds of spinach does Melissa need to make 2 batches of lasagna?
Context: Restate the problem in your own words.
Physical  Visual Model: Use new fraction strips! Partition as needed, then cut or tear them apart to just show the amount of spinach in each batch. Make a sketch to record your reasoning.
If time allows, also solve using a number line.
Discuss, share, demonstrate.

36. (1) Contextual meaning of the numbers.
In order to encourage her family to eat more vegetables Melissa decides to include spinach in her lasagna. The recipe calls for ¾ pound of spinach for each batch of lasagna. How many pounds of spinach does Melissa need to make 2 batches of lasagna?
Equation:
Write the equal that models the problem situation. Label each number with phrases to describe:
(1) Contextual meaning of the numbers.
(2) Structural meaning of the numbers.
Discuss, share, demonstrate.

37. Total of 6 parts of size ¼ of a pound or 6/4 pounds of spinach.
A fraction strip represents 1 whole pound of spinach
1 batch of lasagna
uses ¾ of a pound of spinach
1 batch of lasagna
uses ¾ of a pound of spinach
Total of 6 parts of size ¼ of a pound
or 6/4 pounds of spinach.

38. Context: Restate in your own words. Define the Referent Whole.
Janine just called she will be joining them for dinner. Melissa decides to make 2 ⅓ batches of her lasagna. Given that the recipe calls for ¾ pound of spinach for each batch, now how many pounds of spinach does Melissa need to buy?
Context: Restate in your own words.
Define the Referent Whole.
Describe the Size of Each Group.
Use the fraction strips!! (Make a drawing to record; try a number line if time allows.)
Write a “descriptive” equation.
Three different lasagna problems

39. ⅓ batch of lasagna uses ¼ ¾ of a pound of spinach
A fraction strip represents 1 whole pound of spinach
1 batch of lasagna uses
¾ of a pound of spinach
1 batch of lasagna uses ¾ of a pound of spinach
⅓ batch of lasagna uses
____ of a pound of spinach
Total of 7 parts of size ¼ of a pound or 7/4 pounds of spinach.

40. Context: Restate in your own words. Define the Referent Whole.
Melissa rethinks her decision. She finally settles on making 2 ⅔ batches of lasagna. The recipe calls for ¾ pound of spinach for each batch, now how many pounds of spinach does she need?
Context: Restate in your own words.
Define the Referent Whole.
Describe the Size of Each Group.
Use the fraction strips!! (Make a drawing to record; try a number line if time allows.)
Write a “descriptive” equation.
Three different lasagna problems

41. Total of 8 parts of size ¼ of a pound,
A fraction strip represents 1 whole pound of spinach
1 batch of lasagna uses
¾ of a pound of spinach
1 batch of lasagna uses ¾ of a pound of spinach
2/3 batch of lasagna uses
____ of a pound of spinach
2/4
Total of 8 parts of size ¼ of a pound,
which is 8/4 or 2 pounds of spinach.

42. Summing Up
Moving from 4th grade to 5th grade

43. “Symbols can become tools for thinking when students use them as records of actions and things they already know. Without this understanding, students manipulate symbols without meaning rather than thinking of symbols as quantities and actions to be performed or records of actions already performed. Huinker (2002) p. 73

44. Learning Targets We are learning to…
Apply and extend our understanding of multiplication with whole numbers to multiplication with fraction through use of context (word problem situations) and use of the underlying mathematical structure.

45. Day 6 Reflections (Log) Revisiting our learning intentions:
A key connection in extending understanding of multiplication with whole numbers to multiplication with fractions.
The importance of connecting representations to support students’ understanding of multiplication with fractions.

46. How about your students?
Moving from 4th grade to 5th grade

47. Analyzing Student Reasoning
Felicia is running on the track at school on a Saturday. The distance around the track is ¾ of a mile. Felicia ran 2 1/3 laps around the track. How many miles did she run?
Solve using a visual model.
Solve using an equation.

51. “I’m introducing multiplication with fractions today
“I’m introducing multiplication with fractions today. I hope at least half the class will understand the lesson.”

52. Disclaimer Core Mathematics Partnership Project
University of Wisconsin-Milwaukee,
This material was developed for the Core Mathematics Partnership project through the University of Wisconsin-Milwaukee, Center for Mathematics and Science Education Research (CMSER). This material may be used by schools to support learning of teachers and staff provided appropriate attribution and acknowledgement of its source. Other use of this work without prior written permission is prohibited—including reproduction, modification, distribution, or re-publication and use by non-profit organizations and commercial vendors.
This project was supported through a grant from the Wisconsin ESEA Title II, Part B, Mathematics and Science Partnerships.