Teaching
Multiplication
(and Division)
Conceptually
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.
2
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
3
A warm–up mental number string
• 100 x 13
• 2 x 13
• 102 x 13
• 99 x 13
• 14 x 99
• 199 x 34
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
• What strategy does
this string support?
• What big ideas
underlie this
strategy?
5
Pictures for early
multiplication
6
Small Group Discussions
• What strategies
would you expect to
see?
• How would you
represent them?
7
How many apples? How many
lemons?
8
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
9
10
Here’s one that I found.
11
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)
12
Prior Understandings
• 2.G.2. Partition a rectangle into rows and columns of samesize squares and count to find the total number of them.
13
So What Else About
Multiplication?
X
Thinking that multiplication should always be
interpreted as repeated addition.
14
Let’s Review
Multiplication can be interpreted in a
variety of familiar ways.
3 x 5 = 15
array
repeated addition
3+3+3+3+3
area model
3
3’
5+5+5
5
5’
Array Cards
16
Scaling
17
Core Lesson
“Gary’s flashlight shines three times
farther than mine!”
My Flashlight
3 times farther than 5 feet
3 x 5 = 15
5 feet
Gary’s Flashlight
15 feet
Core Lesson
When can multiplication be interpreted
as scaling?
When it represents the relationship between
the size of the product and the factors.
3 x 5 = 15
15 is 3 times > 5
15 is 5 times > 3
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?
20
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
21
Hedges, Huinker and Steinmeyer. Unpacking Division to Build
Teachers’ Mathematical Knowledge, Teaching Children
Mathematics, November 2004, p. 4-8.
Forgiveness Method
21
12 252
- 120 10
132
- 120 10
12
- 12 1
0
21
Issic Leung, Departing from the Traditional Long
Division Algorithm: An Experimental Study. Hong Kong
Institute of Education, 2006.
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.
Lies my teacher told me…
Division is about “fair sharing”.
35 ÷ 8 =
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.
1.
2.
3.
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?
Near Facts…
Find the largest factor without
going over the target number
Partial Quotients
18 R 25
26 493
- 260 10
233
- 130
5
103
- 78
3
25 18
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.
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?
Division Vocabulary
Quotient
Divisor
34
35
THE PARTITIVE PROBLEM
36
Partitive
37
Example 1: Write a word
problem to represent this
model of division?
38
THE MEASUREMENT PROBLEM
39
Measurement
40
Example 2: Write a word
problem to represent this
model of division?
41
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?
42
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?
43
Measurement Model
44
Ratio
45
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.
46
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.
47
Distributive Property
& Area Models
3 x 7 =__
5
3
15
+
+
2
6
3x7=
3 x (5 + 2) =
(3 x 5) + (3 x 2)= 15 + 6 = 21
48
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.
49
Grade 3
Represent and solve problems involving multiplication and division (Glossary-Table 2)
Not until 4th Grade
50
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). 90 + 90 + 90 + 90 + 90 + 90 + 90 + 90 = (800-80) = 720
51
Sample Activity: Finding Factors
(Elementary and Middle School Mathematics: Teaching Developmentally by
Van de Walle, Karp, Bay-Williams)
1. With a partner, choose one of the following
numbers:
12, 18, 24, 30, 36, or 48
2. Use equal sets, arrays, or number lines to find
as many multiplication expressions as possible
to represent your number.
3. For each multiplication expression, write its
equivalent addition expression showing your
groupings.
52
Sample Activity: Finding Factors from
Elementary and Middle School Mathematics: Teaching
Developmentally by Van de Walle, Karp, Bay-Williams
53
Grade 4
A focus on teaching multiplication (and division)
conceptually
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.
55
Selected Standards…
• 4.NBT.5.
• Multiply a whole number of up to four digits by a onedigit 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)
56
Area Models
20 + 5
25 x 38= 950
30 + 8
150
40
600
160
750 + 200 = 950
57
Partitioning Strategies for
Multiplication
•
•
•
•
•
•
27 x 4
4 x 20 = 80
4 x 7 = 28
•
•
•
•
267 x 7
7 x 200 = 1400
7 x 60 = 420
7 x 8 = 56
108
27 x 4
27 + 27 + 27 + 27
54
108
----------------------------------1820
54
1876
58
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.
59
Selected Standards…
• 4.OA.3.
• Solve multistep word problems posed
with whole numbers and having wholenumber 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.
60
61
62
Grade 5
A focus on teaching multiplication (and division)
conceptually
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.
64
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.
65
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.
66
Scaling
• Recognize that 3 x (25,421 + 376)
is 3 times larger than
67
Gr. 5 Selected Standards
• 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
68
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
69
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.”
70
14 x 25: An Area Model
+
5
80
20
200
50
10
+
4
20
*Sketch is not drawn to scale.
71
Algebra 1: Multiplying Binomials
5
4
+
4x
20
x +
x
x2
5x
*Sketch is not drawn to scale.
72