1. Teaching Multiplication (and Division) Conceptually

2. Professional Learning Targets…
I can describe what it means and what it looks like to teach multiplication (and division) conceptually.
I can describe how standards progress across grade levels, giving details for the grade span in which I teach.

3. Agenda Warming up with Multiplication and Division
Number Strings
Quick Images
Number Talks with Multiplication and Division
Big Ideas of Multiplication and Division
Types of Multiplication and Division Problems
Multiplication/Division Games

4. A warm–up mental number string
100 x 13
2 x 13
102 x 13
99 x 13
14 x 99
199 x 34
•Present a seriesof related computation problems
•Use questioning and student discussion to explore mathematical ideas and increase computational fluency

5. A warm–up mental number string
100 x 13
2 x 13
102 x 13
99 x 13
14 x 99
199 x 34
What strategy does this string support?
What big ideas underlie this strategy?

6. Pictures for early multiplication

7. Small Group Discussions
What strategies would you expect to see?
How would you represent them?
You would expect kids to just count them.
You would expect kids to multiply them. What would you see? What property of multiplication is that?

8. How many apples? How many lemons?
Count by ones
Skip count
2 x 6 = 4 x 3 = 6 x 2
2 x 9 = 2 x 3 x 3 = 6 x 3
Associative property

9. How many tiles in each patio?
The furniture obscures some of the tiles possibly providing a constraint to counting by ones and supporting the development of the distributive property

11. Here’s one that I found.

12. Prior Understandings— Grades K-2
Counting numbers in a set (K)
Counting by tens (K)
Understanding the numbers 10, 20, 30, 40, …, 90 refer to one, two, three, four, …, nine tens (1)
Counting by fives (2)

13. Prior Understandings
2.G.2. Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

14. So What Else About Multiplication?

15. 3 5 3’ 5’

16. Array Cards

17. Scaling
This is probably the hardest multiplication structure since it cannot be understood by
counting though it is frequently used in everyday life in the context of comparing quantities
or measurements and in calculations of the cost of multiple purchases, for example.

20. So What About Division?
How many of our students understand dividing a number by 3 is the same as multiplying the number by 1/3?

21. 169 ÷ 14 =
To begin thinking about division, solve this problem using a strategy other than the conventional division algorithm. You may use symbols, diagrams, words, etc. Be prepared to show your strategy
Provide participants with 5-10 minutes to complete this work. You may need to provide scrap paper, colored pencils etc. Once completed, ask for volunteers to show their strategy via the document camera. Note how many participants use a multiplication strategy to solve.
Hedges, Huinker and Steinmeyer. Unpacking Division to Build Teachers’ Mathematical Knowledge, Teaching Children Mathematics, November 2004, p. 4-8.

22. This statement came from research conducted for the New Zealand Mathematical Society in the late 90’s.
Read slide and discuss

23. Forgiveness Method 21
132 12
0 21
Sometimes called the partial quotient method, this method allows students to use the distributive property to break down a number into it’s various parts. This method is formally called the distributive law of division over addition. For example, in the standard algorithm children must get the first digit correct in the quotient or no matter what additional work they do, the answer will be incorrect. This method allows a student to work through the problem based on “what they know”. In this example the child knows 12 times 10 is 120, therefore they are able to subtract this value without concern about getting the first digit in the answer correct. The child can than reason once again to subtract 120 by multiplying 12 times 10. They are then left with 12 x 1 and can quickly add to get the quotient of 21.

24. produces a product for the intermediate step.
Here is another example of decomposing the dividend into a sum of smaller numbers. This time, instead of calculating the answer to the side of the problem, the student stacks his answer above the dividend. For example, sharing 210 objects
among 8 people, each of the 8 people can get 20 objects first in the first round of sharing
(grouping), and the remaining objects are shared again, so that each of them get two more and finally five more.
There is no need to estimate the maximum but only a number (partition of quotient) that
produces a product for the intermediate step.
Issic Leung, Departing from the Traditional Long Division Algorithm: An Experimental Study. Hong Kong Institute of Education, 2006.

25. Here are two examples of students work who are using partial quotient
Here are two examples of students work who are using partial quotient. Do you notice anything different between the two examples?

26. Change it UP!!!!
1. Deal each player five cards. The remaining cards are placed face down on the center of the table.
2. Player one places a card face up on the table reads the division problem and provides the quotient. The next player must place a card with the same quotient on the first card. If the player cannot match, he/she may place a “Math Wizard” card on top and then a card with a different quotient.
3. If the player in unable to make either move, he/she must draw from the deck until a match is made.
4. The first player to use all of his/her cards is the winner.

27. Lies my teacher told me…
Division is about “fair sharing”.
35 ÷ 8 =
How many of us have taught division in this context? A great deal of confusion comes when we teach division as fair sharing then we give them this type of problem.
Click
Remainders…which is NOT about fair sharing. What are we going to do with the 3 left over? According to Van De Walle, 99 percent of all life problems end with a remainder. So what do can we do to this “leftover”?

28. The Remainder Can be discarded.
The remainder can “force the answer to the next highest whole number.
The answer is rounded to the nearest whole number for an approximate result.
The remainder…
Click and read

29. Landon bought 80 piece bag of bubble gum to share with his five person soccer team. How many pieces did each player receive?
Brittany is making 7 foot jump ropes for the school team. She has a 25 foot piece of rope. How many can she make?
The ferry can hold 8 cars. How many trips will it need to make to carry 25 cars across the river?
In question number one, some pieces will be left over…so what do you do with them?
Click
In the next example, some of the rope will not be used or just discarded.
In the third example, the number must be forced to the next whole number. Can you think of a problem where the answer would need to be rounded?

30. Find the largest factor without going over the target number
Near Facts…
One example of helping students better understand the concept of the remainder, Van De Walle recommends an activity called, “Near Facts”. A student can use the facts in which they know to arrive at near facts…in other words it is getting them in the ballpark. Strategy development and number sense are the best contributors to fact mastery. By teaching students skills in which they can utilize their current tool box, we provide them with tools necessary to succeed. Think about it, how many of us have ever had someone at Walmart come up and ask what is 30 divided by 8…however we may encounter a problem where 30 cans of pop are 8 dollars and you can buy in smaller quantities at the same discount.
Find the largest factor without going over the target number

31. Partial Quotients 18 R 25 26 493 - 260 10 233 - 130 5 103 - 78 3 25 18
233
103
Here is the partial quotient method using a remainder.
Work through the problem with participants.

32. The Remainder Game
1. To begin the game, both players place their token on START.
2. Player one spins the spinner and divides the number beneath his/her marker by the number on the spinner. If there is a remainder, he/she is allowed to move his/her token as many spaces as the remainder indicates. If the division does not result in a remainder, he/she must leave his/her marker where it is.
3. The play alternates between the two players (a new spin must occur each time) until some reaches HOME.

33. Lies my teacher told me…
Any number divided by zero is zero!
6 ÷ 0 =
How many times can 0 be subtracted from 6?
How many 0 equal groups are there in six?
What does six divided into equal groups of 0 look like?
What number times 0 gives you 6?
Dividing by zero, unlike multiplication does not yield zero. In fact, some believe it is truly the end. Look at the following questions..
Click

34. Division Vocabulary Quotient Dividend Divisor
Does this example tell you what you need to know about division and it’s parts? Perhaps not, yet it came from a text commonly used in many classrooms. It treats the vocabulary independent of the operation and provides a multiple choice quiz on the terms. It is also here where many misconceptions begin. For example, we ask students how many 3’s go into 1, rather than 10-we ignore place value and encourage students to merely work through a procedure many do not understand.

36. THE PARTITIVE PROBLEM

37. Partitive
Both of the stories you made up for the animations can be recorded as 15 ÷ 5 = 3, but the numbers refer to quite different things. The first animation showed 15 ÷ 3 as “fifteen shared among three.” So the answer five, tells how many are in each group.

38. Example 1: Write a word problem to represent this model of division?
Have participants write a word problem describing this representation. Ask for volunteers and share out. You may also want to write examples on large chart paper.

39. THE MEASUREMENT PROBLEM
The most difficult issue for teachers is to distinguish between the two different concepts of division-those generally referred to as partition (fair shares) and measurement (repeated subtraction). It is not terribly important that teachers can name division situations as either partition or measurement but that they be able to understand and distinguish the two situations.

40. Measurement
The second animation showed 15 ÷ 3 as “fifteen separated (measured) into sets of three.” In this case the answer five tells the number of sets that could be made.

41. Example 2: Write a word problem to represent this model of division?
Have participants write a word problem describing this representation. Ask for volunteers and share out. You may also want to write examples on large chart paper. Compare how each example of the division problem differs.

42. Two basic types of problems in division
Partitive (Sharing): You have a group of objects and you share them equally. How many will each get?
Example: You have 15 lightning bugs to share equally in three jars. How many will you put in each jar?
PD Provider Notes: The purpose of this slide is to show the “measurement” type of division problem. The “lightning bug” problem is solved by repeatedly subtracting 3 bugs at a time, and counting how many times you do this.
Here, one could ask the participants to explain why division by zero doesn’t make sense (is undefined). Indeed, subtracting away 0 from 15 can be done infinitely many times.
 ACTIVITY: Division Sort
Have participants work in table groups to sort the Division Sort Cards into partitive and measurement examples.

43. Two basic types of problems in division
Measurement: You have a group of objects and you remove subgroups of a certain size repeatedly. The basic question is—how many subgroups can you remove?
Example: You have 15 lightning bugs and you put three in each jar. How many jars will you need?
PD Provider Notes: The purpose of this slide is to show the “measurement” type of division problem. The “lightning bug” problem is solved by repeatedly subtracting 3 bugs at a time, and counting how many times you do this
Here, one could ask the participants to explain why division by zero doesn’t make sense (is undefined). Indeed, subtracting away 0 from 15 can be done infinitely many times.
Reference: Carpenter, T.P., Fennema, E., Franke, M.L., Levi, L., & Empson, S.B. (1999). Children’s mathematics: cognitively guided instruction. Portsmouth, NH: Heinemann.

44. Measurement Model

45. Ratio

46. Grade 3 Introduction
In Grade 3, instructional time should focus on four critical areas: (1) developing understanding of multiplication and division and strategies for multiplication and division within 100; …
Students develop an understanding of the meanings of multiplication and division of whole numbers through activities and problems involving equal-sized groups, arrays, and area models;
multiplication is finding an unknown product, and division is finding an unknown factor in these situations.
For equal-sized group situations, division can require finding the unknown number of groups or the unknown group size.
Students use properties of operations to calculate products of whole numbers, using increasingly sophisticated strategies based on these properties to solve multiplication and division problems involving single-digit factors.
By comparing a variety of solution strategies, students learn the relationship between multiplication and division.

47. Commutative Property
It is not intuitively obvious that 3 x 8 = 8 x 3. A picture of 3 sets of 8 objects cannot immediately be seen as 8 piles of 3 objects. Eight hops of 3 land at 24, but it is not clear that 3 hops of 8 will land at 24.
The array, however, can be quite powerful in illustrating the commutative property.

48. Distributive Property & Area Models
3 x 7 =__ 3 15 + 6
3 x 7 =
3 x (5 + 2) =
(3 x 5) + (3 x 2)= = 21

49. Grade 3
3.MD.7. Relate area to the operations of multiplication and addition.
Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths.
Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning.
Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning.
Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems.

50. Grade 3Represent and solve problems involving multiplication and division (Glossary-Table 2)
Not until 4th Grade

51. Connections to other 3rd Grade Standards
3.NBT.3: Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations.
9 x 80:
80 is ten 8’s. So, if I know that 8x9 is 72, then I have ten 72’s. That equals 720.
Or..80 is 8 tens. So, if 10 x 9 = 90, then I know I have 8 of those (90s) = (800-80) = 720

52. Sample Activity: Finding Factors (Elementary and Middle School Mathematics: Teaching Developmentally by Van de Walle, Karp, Bay-Williams)
With a partner, choose one of the following numbers:
12, 18, 24, 30, 36, or 48
Use equal sets, arrays, or number lines to find as many multiplication expressions as possible to represent your number.
For each multiplication expression, write its equivalent addition expression showing your groupings.

53. Sample Activity: Finding Factors from Elementary and Middle School Mathematics: Teaching Developmentally by Van de Walle, Karp, Bay-Williams

54. A focus on teaching multiplication (and division) conceptually
Grade 4
A focus on teaching multiplication (and division) conceptually

55. Grade 4 Introduction
In Grade 4, instructional time should focus on three critical areas: (1) developing understanding and fluency with multi-digit multiplication, and developing understanding of dividing to find quotients involving multi-digit dividends
They apply their understanding of models for multiplication (equal-sized groups, arrays, area models), place value, and properties of operations, in particular the distributive property, as they develop, discuss, and use efficient, accurate, and generalizable methods to compute products of multi-digit whole numbers.
Depending on the numbers and the context, they select and accurately apply appropriate methods to estimate or mentally calculate products.
They develop fluency with efficient procedures for multiplying whole numbers; understand and explain why the procedures work based on place value and properties of operations; and use them to solve problems.
Students apply their understanding of models for division, place value, properties of operations, and the relationship of division to multiplication as they develop, discuss, and use efficient, accurate, and generalizable procedures to find quotients involving multi-digit dividends.
They select and accurately apply appropriate methods to estimate and mentally calculate quotients, and interpret remainders based upon the context.

56. Selected Standards… 4.NBT.5.
Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations.
Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
(Area models for this standard are directly linked to the understanding of partitioning a rectangle into equal parts and 3.MD.7c)

57. Area Models
25 x 38=
950
30 + 8
150 40
20 + 5
160
600
= 950

58. Partitioning Strategies for Multiplication
27 x x 4
4 x 20 =
4 x 7 =
108
267 x 7
7 x 200 =
7 x 60 =
7 x 8 = 56

59. Selected Standards… 4.NBT.6.
Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division.
Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

60. Selected Standards… 4.OA.3.
Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted.
Represent these problems using equations with a letter standing for the unknown quantity.
Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

63. A focus on teaching multiplication (and division) conceptually
Grade 5
A focus on teaching multiplication (and division) conceptually

64. Grade 5 Introduction
(2) extending division to 2-digit divisors, integrating decimal fractions into the place value system and developing understanding of operations with decimals to hundredths, and developing fluency with whole number and decimal operations;
Students develop understanding of why division procedures work based on the meaning of base-ten numerals and properties of operations.
They finalize fluency with multi-digit addition, subtraction, multiplication, and division.
They apply their understandings of models for decimals, decimal notation, and properties of operations to add and subtract decimals to hundredths. They develop fluency in these computations, and make reasonable estimates of their results. Students use the relationship between decimals and fractions, as well as the relationship between finite decimals and whole numbers (i.e., a finite decimal multiplied by an appropriate power of 10 is a whole number), to understand and explain why the procedures for multiplying and dividing finite decimals make sense. They compute products and quotients of decimals to hundredths efficiently and accurately.

65. Grade 5 Selected Standards
5.NBT.2 and 5.NBT.5
Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
Fluently multiply multi-digit whole numbers using the standard algorithm.

66. Gr. 5 Selected Standards 5.NF.5
Interpret multiplication as scaling (resizing), by:
a. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.
b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b =(n×a)/(n×b) to the effect of multiplying a/b by 1.

67. Scaling
Recognize that 3 x (25, ) is 3 times larger than

68. Gr. 5 Selected Standards 5.NBT.6.
There must be a clear connection between multiplication and division using manipulatives, before this understanding takes place.
5.NBT.6.
Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models

69. Zero and Identity Properties
Rules with no reasons?
No…ask students to reason.
Ex: How many grams of fat are there in 7 servings of celery? Celery has 0 grams of fat.
Ex: Note that on a number line, 5 hops of 0 land at 0. Also, 0 hops of 20 also stays at 0.
Arrays with factors of 1 are also worth investigation to determine the identity property

70. Distributive Property Activity
Slice it Up…
Each pair, please use the grid paper to make a rectangle that has a total area greater than 10 square units.
Make a slice through the rectangle and write an equation that matches using the lengths and widths of the smaller rectangles created.
Continue this process until you have found all the ways to “slice it up.”

71. 14 x 25: An Area Model
*Sketch is not drawn to scale. 80 20 50
200

72. Algebra 1: Multiplying Binomials
*Sketch is not drawn to scale. x x 4x 20 5x x2