1. Ma & Pa Kettle http://www.youtube.com/watch?v=Bfq5kju627c
Gathering Activity - Play Ma & Pa Kettle Clip
Give out Learning to Think article – let them read now in preparation for later discussion

2. Fluency Hunt We’re going on A Multiplication and Division
Of Whole Numbers

3. MULTIPLICATION & DIVISION OF WHOLE NUMBERS
FLUENCY STANDARDS FOR
MULTIPLICATION & DIVISION OF WHOLE NUMBERS
For your assigned grade level, search for the words “Fluently” or “Fluency” as it relates to multiplication and/or division of whole numbers.
Assign a group for each of grades 2 through 6. 2nd grade group will discover that there are no fluency standards for Mult/Division in 2nd grade. 4th grade is a little tricky; there isn’t a specific standard that uses the word “fluent” but it shows up in the critical areas.
…highlight it!

4. MULTIPLICATION & DIVISION OF WHOLE NUMBERS
FLUENCY STANDARDS FOR
MULTIPLICATION & DIVISION OF WHOLE NUMBERS
3rd grade

5. MULTIPLICATION & DIVISION OF WHOLE NUMBERS
FLUENCY STANDARDS FOR
MULTIPLICATION & DIVISION OF WHOLE NUMBERS
3rd grade – 3.OA.7 Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.

6. MULTIPLICATION & DIVISION OF WHOLE NUMBERS
FLUENCY STANDARDS FOR
MULTIPLICATION & DIVISION OF WHOLE NUMBERS
3rd grade
4th grade

7. MULTIPLICATION & DIVISION OF WHOLE NUMBERS
FLUENCY STANDARDS FOR
MULTIPLICATION & DIVISION OF WHOLE NUMBERS
3rd grade
4th grade – Critical Area (1) Developing understanding and fluency with multi-digit multiplication, and developing understanding of dividing to find quotients involving multi-digit dividends.

8. MULTIPLICATION & DIVISION OF WHOLE NUMBERS
FLUENCY STANDARDS FOR
MULTIPLICATION & DIVISION OF WHOLE NUMBERS
3rd grade
4th grade
5th grade

9. MULTIPLICATION & DIVISION OF WHOLE NUMBERS
FLUENCY STANDARDS FOR
MULTIPLICATION & DIVISION OF WHOLE NUMBERS
3rd grade
4th grade
5th grade – 5.NBT.5 Fluently multiply multi-digit whole numbers using the standard algorithm.

10. MULTIPLICATION & DIVISION OF WHOLE NUMBERS
FLUENCY STANDARDS FOR
MULTIPLICATION & DIVISION OF WHOLE NUMBERS
3rd grade
4th grade
5th grade
6th grade

11. MULTIPLICATION & DIVISION OF WHOLE NUMBERS
FLUENCY STANDARDS FOR
MULTIPLICATION & DIVISION OF WHOLE NUMBERS
3rd grade
4th grade
5th grade
6th grade – 6.NS.2 - Fluently divide multi-digit numbers using the standard algorithm.

12. Multiplication and Division of whole numbers
Fluency Standards for
Multiplication and Division of whole numbers
3.OA.7
Multiply & Divide within 100
Strategies based on meaning and properties of operations (i.e. associative and commutative)
Inverse relationship of multiplication and division
By end of year, know one digit by one digit multiplication facts
4 – Critical Domain 1
Multi-digit Multiplication
Strategies based on place value and properties of operations
Supported by suitable representations
Also developing understanding of multi-digit division
5.NBT.5
Fluently multiply multi-digit numbers
Standard algorithm for multiplication
6.NS.2
Fluently divide multi-digit numbers
Standard algorithm for division

13. Delay! Delay! Delay!
“Students are unlikely to invent the standard algorithms. You will need to introduce and explain them, and help students to understand how and why algorithms work. No matter how carefully you introduce these algorithms into your classroom as simply another alternative, students may sense that ‘this is the one right way.’”
You may wish to refer them to their own books to read this quote. It’s on page 176 under the heading “DELAY! DELAY! DELAY!”
Teaching Student Centered Mathematics: Developmentally Appropriate instruction for grade 3-5; Van de Walle, et. al. 2014, page 176

14. Delay! Delay! Delay!
“So, first spend a significant time with invented strategies – months, not weeks. Note that the Common Core State Standards (CCSSO, 2010) suggests that students learn a variety of strategies based on place value and properties of operations one or two years before the standard algorithms are expected.”
Teaching Student Centered Mathematics: Developmentally Appropriate instruction for grade 3-5; Van de Walle, et. al. 2014, page 176

15. Let’s explore the path for Fluent Multiplication and Division

16. Path to Fluency Mult/Div of Whole numbers
Create a “visual” (words and/or pictures) to show the big ideas for your grade level(s)
Consider
Expected skills and understandings
Range of numbers
Role of supporting materials
Appropriate contexts
Expected level of mastery (i.e. developing vs. fluent)
Examples of appropriate instructional and/or assessment tasks
So that participants won’t have to spend a lot of time finding the relevant standards, we’ll give out the “Standards for Multiplication and Division” document created for this purpose. Have participants make a poster (using either chart paper or 11x17)
One poster for each of the following:
Primary (K-2)
3rd
4th
5th
Be sure to remind teachers to limit to whole number ideas for now. The standards interweave the fractional work. We’ll get there – just not yet.

17. Multiplication and Division of whole numbers
Fluency Standards for
Multiplication and Division of whole numbers
3.OA.7
Multiply & Divide within 100
Strategies based on meaning and properties of operations (i.e. associative and commutative)
Inverse relationship of multiplication and division
By end of year, know one digit by one digit multiplication facts
4 – Critical Domain 1
Multi-digit Multiplication
Strategies based on place value and properties of operations
Supported by suitable representations
Also developing understanding of multi-digit division
5.NBT.5
Fluently multiply multi-digit numbers
Standard algorithm for multiplication
6.NS.2
Fluently divide multi-digit numbers
Standard algorithm for division

18. Stem Completion Pair up with someone who worked on a different poster
Take turns responding to the following prompts
New thinking for me is….
Something I’m wondering about is….
Something we’ve discussed that I want the teachers in my building to know is…

19. Vocabulary And Conventions
Let’s discuss:
Key labels and standard vocabulary
Conventions – generally agreed meanings of symbols and positions

20. Vocabulary and Conventions
Dividend
Divisor
Quotient
Remainder
These labels are for division. Likely this will generate less discussion than the multiplication labels, so we start here. Someone might ask if the “remainder” is part of the quotient or you may wish to bring up the question. (In other words is 31/4 the quotient or is only the “7” part of the answer the quotient.) I looked it up and found that both are legitimate ways to define quotient. In some cases, only the integer part of the result is considered the “quotient” and in others the entire result is considered the quotient. Don’t let teachers get to hung up on this. If you prefer, refer to the “7” part as the “whole number part of the quotient”.

21. Vocabulary and Conventions
Red Book adopts the convention:
7 X 4 = 28
Multiplicand
Multiplier
Product
This is the more common convention outside of the united states. This red book has authors living in Australia, hence the choice here.
Might be read as “7 multiplied by 4” or
“7 added 4 times” (think )

22. Vocabulary and Conventions
In an “equal groups” context
Multiplicand: The size of each group
Multiplier: The number of groups
For the example “5 groups of 3 items”
The multiplicand is 3
The multiplier is 5
These definitions extend to other contexts. In particular, in a “compare” situation (i.e. I picked 3 zucchini yesterday. My neighbors garden is doing a lot better than mine. He picked 5 times as many zucchini as me. How many did he pick?)…. The multiplicand is 3 (the reference set) and the multiplier is 5. .

23. Vocabulary and Conventions
However, 3.OA.1 states “Interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each.”
5 X 7 = 35
Multiplicand
Multiplier
Note: The standards don’t actually use the terms “multiplier” and “multiplicand”.
I’m not sure how much you want to go into this. But… for leader reference, here’s what I’ve figured out digging into this topic:
The standards do not use multiplier or multiplicand as far as I could tell. However, on the word problem chart, they are consistent with the “multiplier x multiplicand” format. The problem “A blue hat costs $6. A red hat costs 3 times as much as the blue hat. How much does the red hat cost?” is represented by the equation “3 x 6 = ?”. The progressions document use the word “multiplier” but I could not find a progressions document that used “multiplicand”
For example, in the OA progressions document (top of page 24) it states: “In Equal Groups, the roles of the factors differ. One factor is the number of objects in a group (like any quantity in addition and subtraction situations), and the other is a multiplier that indicates the number of groups.”
Product

24. Vocabulary and Conventions
In the context of arrays
3 X 5
# of rows # of columns
(Think: 3 “rows of” 5)
This slide uses the array to illustrate the commutative property. So although there is a convention of rows x columns, we know that result is equal to the result of column x row.
We can rotate the array to make 5 x 3

25. Vocabulary and Conventions
“The usual convention in the United States is that 4x8 refers to four sets of 8, not eight sets of four. There is no reason to be so rigid about this convention that you would mark a student as incorrect. The important thing is that the student can tell you what each factor in their equation represents.”
Internationally, the convention of the red book is more standard. As Dr. Wright is in Austraila, this explains why they are using this convention.
Interesting note - I found in my research that some places define the SMALLLER number to be the multiplier, regardless of order. As this is the way most us think to solve multiplication – this makes a lot of sense. For example, most of us solve 23 x 2 the same way we’d solve 2 x 23– i.e. thinking 2 groups of 23. (either = 46 or 2 groups of 20 is 40, 2 groups of 3 is 6 so 26) By thinking “2 groups of 23” we are effectively making 23 the multiplicand and 2 the multiplier.
Teaching Student Centered Mathematics: Developmentally Appropriate instruction for grade 3-5; Van de Walle, et. al. 2014, page 112

26. Vocabulary and Conventions
Going back to our example: “5 groups of 3 items” Is it 5 x 3 or 3 x 5? Allow students to write either! Just be sure they understand why both are correct. (Associative Property)

27. Vocabulary and Conventions
Ultimately, we want a student to know that, regardless of how the multiplication arises in a context, the student can choose how to think of it.
Consider the example:
“Mrs. Sampson’s class had a bake sale. They put cookies in packages of 2. If they sold 28 packages, how many cookies did they sell?
In this context, what is the multiplicand? What is the multiplier?
When you solved 28 x 2, how did you think about the numbers?

28. National Council of Teachers of Mathematics
Principles and Standards for School Mathematics states, “Computational fluency refers to having efficient and accurate methods for computing. Students exhibit computational fluency when they demonstrate flexibility in the computational methods they choose, understand and can explain these methods, and produce accurate answers efficiently.

29. Vocabulary and Conventions
7 x 4 = 28
28 is a multiple of both 4 and 7
4 and 7 are factors of 28

30. Vocabulary and Conventions
We can avoid most of the confusion of multiplier and multiplicand by using the label “factors”.
7 X 4 = 28
Factor
Factor
Product

31. Vocabulary and Conventions
3 + (7 x 4) = 31
Order of Operations for 3rd graders (3.OA.8 footnote)
Do the operations inside of the parentheses first
Do multiplication and division before doing addition & subtraction

32. Vocabulary and Conventions
Symbols for Multiplication
5 X   (5)(7)
Symbols for Division
35  / 5
Note – I moved the “obelus” vocab to here. I found and added the term “viniculum” to the division bar in the fraction. I could not find a name for the “house” symbol for long division but found many statements that there ISN’T a name for that symbol… other than “long division symbol”. Not sure if the backslash divider in “35/5” is also a viniculum. These are NOT terms students need – but I thought it would be fun to share.
“obelus”
“viniculum”