Download Developing Proficiency with Multiplication and Division

Developing Computational Proficiency with
Multiplication and Division
Students learn to add before they learn to multiply. Therefore, early attempts to multiply may be done
using repeated addition connected to a groups of equal quantities scenario. Since many students will
not be very accurate or speedy when it comes to repeated addition of numbers other than 2s, 5s, or
10s, it is necessary to provide explicit instruction in a variety of other computational strategies.
Sequence for Teaching Multiplication Facts and Strategies
Though there is no one correct way to sequence teaching of the multiplication facts, most researchers
and educators agree that 2s, 5s, and 10s come first, then 1s and 0s. Researcher Susan O’Connell
suggests 2, 10, 5, 1, 0, 3, 4, 6, 9, 8, 7 as the sequence for learning multiplication facts.
In its May 2015 Vol. 21, No. 9 publication, Teaching Children Mathematics, the National Council
of Teachers of Mathematics (NCTM) suggests a similar sequence, but adds square products as
foundational facts to be learned early. See Table 1.
Table 1: Sequence and Strategies for teaching multiplication facts.
Foundational Facts
2s, 5s, 10s
(begin these late in grade 2)
0s, 1s, multiplication squares (2x2, 3x3..)
Use story problems, arrays, skip counting,
and patterns on a hundreds chart, and a
multiplication table to learn these facts.
Derived Fact Strategies
Adding or subtracting a group
Halving and doubling
Using a square product
Decomposing a factor
Start with a nearby 2s, 5s, or 10s fact, then
subtract (or add) the group.
Example, I don’t know 9 x 6, so I think “10 x 6 =
60” and subtract one group of 6 to get 54.
Look for an even factor. Find the fact for half of
that factor, then double it.
Example, I don’t know 6 x 8, so I think “3 x 8 =
24” and double that to get 48.
Look for a nearby square. Find that fact then add
on or subtract off the extra group.
Example, I don’t know 7 x 6. I use 6 x 6 = 36 and
add one more 6 to get 42.
Partition one of the factors into a convenient
sum of known facts, find the known facts and
combine the products.
Example, I don’t know 7 x 6. I break the 7 into 2
and 5, because I know 2 x 6 and 5 x 6. Then I
add 12 and 30 to get 42.
Note: There are many interpretations of 3 x 2. In this document the supporting examples will
reference it as 3 groups of 2.
EduGAINS Mathematics K-12
Page 1 of 10
Knowing the foundational facts, along with properties and strategies, allows students to reason
the facts for 3, 4, 6, 7, 8, and 9. What follows is designed to support the teaching and learning of
foundational facts and computational strategies.
Multiplication: Establish the Foundational Facts and Strategies
Establishing 2s, 5s, and 10s
Effective ways to establish mastery of the 2s, 5s, and 10s foundational facts include frequent
opportunities involving:
• connecting products of 2 to skip counting by 2, products of 5 to skip counting by 5, products of 10
to skip counting by 10 and make use of the Commutative Property of Multiplication
Example: 5 x 3 may not be automatic when thought of as 5 groups of 3, however, thinking of
it as 3 groups of 5 may be determined by skip counting 5, 10, 15. See page 5 for more
detail on the Commutative Property of Multiplication.
• repeated addition on a number line
• observing patterns in a hundreds chart
• recitation during games, jumping rope, counting objects, etc.
Establishing 1s and 0s
a)Multiplying by 1
Once students are comfortable with interpreting 2 x 4 as 2 groups of 4, 5 x 3 as 5 groups of 3, etc.,
they can be led to interpret 1 x 2 as 1 group of 2, and 1 x 5 as 1 group of 5. After students have
worked with enough examples, support students in explicitly understanding multiplying any number
by 1 yields a product equal to the original number (i.e., a number keeps its original identity when
multiplied by 1). Hence, 1 is called the Multiplicative Identity.
b) Multiplying by Zero
Students can think of 0 x 3 = 0 or 3 x 0 as “no groups of 3” as well as “addition of 3 zeros” to generate
products of 0. They may even extend the product pattern.
Example:
4x3
3x3
2x3
1x3
0x3
= 12
=9
=6
=3
=0
Take away 3
Take away 3
Take away 3 Take away 3
c) Dividing by 1
Students who have had experience with the unknowns being written in all three of the possible
locations in a x b = c will appreciate that 2 ÷ 1 = ? means the same as 2 = 1 x ?. Their understanding
of the multiplicative identity, tells them that ? = 2. Like multiplication by 1, division by 1 also leaves a
number unchanged (e.g., 2 ÷ 1 = 2, 99 ÷ 1 = 99).
d) Divisions Involving 0
Students can reason about 0 ÷ 2 = ? by asking themselves, “How many groups of 2 yield a total of
0?” This quotative division question is answered by 0 since it takes zero groups of 2 to yield a total
of 0. If students ask the partitive division question, “2 groups of how many yield a total of 0?”, they
again, answer 0. Similarly, 0 divided by any number other than 0 yields a quotient of 0.
EduGAINS Mathematics K-12
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What is Partitive and Quotative Divison?
Example: 4 x 7 = 28, if 4 represents the number of groups and
7 represents the number of items in each group,
then the result of 28 ÷ 4 is the number of items in each group (partitive)
and the result of 28 ÷ 7 is the number of groups (quotative)
Depending on the context of a problem, you may be finding one or the other.
If not given a context, then one should be considering both types of divisions.
Some students may be curious about division by zero. Guide them to reason that 2 ÷ 0 = ? can be
a partitive division where the quantity per part is unknown. In that case, ask “Zero groups of what
quantity yield a total of 2?” Since this question does not define any number, we say that 2 ÷ 0 is
undefined. If 2 ÷ 0 is a quotative division, then 2 ÷ 0 = ? asks, “How many groups of 0 are in a total of
2?”. Again, this question does not define any number, so 2 ÷ 0 is undefined. Any number other than 0
can replace 2 in the argument above.
A few students may be curious about 0 ÷ 0 = ? By changing this division question to the equivalent
multiplication question, 0 = 0 x ?, students can reason that ? can be 1, or ? can be 2, or ? can be 99.
In fact, ? can be any number. Since the value for ? cannot be determined without further information
0 ÷ 0 is said to be indeterminate.
Establishing the Square Products
Effective ways to establish mastery of these foundational facts include frequent opportunities
involving:
•
observing patterns in a hundreds chart
•
drawing nested squares and observing the pattern in adding to one square to make the next.
EduGAINS Mathematics K-12
Page 3 of 10
The following visual summaries show the multiplication facts for 2, 5, 10, 0, and 1, and the facts
established using the commutative property.
Visual Summaries of Sequence for Teaching Multiplication Facts
2s, 5s, and 10s
Examples: 2 x 3, 5 x 4
Using Commutative Property
for 2s, 5s, and 10s
0s, 1s, and square products
Examples: 0 x 5, 4 x 4
Using Commutative Property
for 0s, and 1s
2s, 5s, 10s, 0s, 1s and using
the Commutative Property
altogether
Notice how few facts have
yet to be learned once the
foundational facts and the
Commutative Property have
been learned.
EduGAINS Mathematics K-12
Page 4 of 10
Multiplication: Naming and Using Properties
Commutative Property of Multiplication
It is helpful to demonstrate that an expression like 2 x 6 yields the same product as 6 x 2. The fact
that you get the same product when you switch the order of the numbers in a multiplication question
is called the Commutative Property of Multiplication.
Model of 2 x 6
Model of 6 x 2
Using the Commutative Property of Multiplication can reduce the cognitive load for students by
taking advantage of multiplication facts already mastered when learning a new multiplication fact.
For example, students typically learn certain facts before others (e.g., 2-times, and 3-times before
6-times). The models above convince students that 6 x 2 (a new fact) yields the same product as
2 x 6 (a known fact). Similarly, if the student knows 3 x 6 = 18, they can reason that 6 x 3 = 18. By
applying the Commutative Property to known facts, students can reduce the number of new facts they
need to learn.
It should be noted that the Commutative Property does not apply to division (e.g., 10 ÷ 2 does not
have the same value as 2 ÷ 10).
Associative Property of Multiplication
This property says that you can group numbers in whatever order you want when multiplying.
7x2x5
= (7 x 2) x 5
= 14 x 5
= 70
7 x 2 x 5 Most people find it easier to compute 7 x 10 than 14 x 5.
= 7 x (2 x 5) The brackets do not have to be included. They are used here to show
= 7 x 10
which numbers are associated first.
= 70
Distributive Property of Multiplication Over Addition or Subtraction
This property says you can multiply a sum by multiplying each of the addends and then add the
partial products. For example, 6 x (3 + 2) is equal to 6 x 3 + 6 x 2. The examples below show how
decomposing a number allows us to take advantage of the Distributive Property.
Example 1:
7x7
= (5 + 2) x 7
= (5 x 7) + (2 x 7)
= 35 + 14
= 49
When learning the fact
7 x 7 = 49, a student
may think of 5 groups
of 7 and 2 more
groups of 7.
Example 2:
7 x 23
= 7 x (20 + 3)
= (7 x 20) + (7 x 3)
= 140 + 21
= 161
This property is
especially useful for
multiplying larger
numbers. Here, a
student can think “7
groups of 20 and 7
groups of 3”.
For more detail about the Distributive Property and visual justification for using this strategy, see
Pictorial Strategy for Multiplying All Types of Numbers on page 9.
EduGAINS Mathematics K-12
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Multiplication: Connecting Computational Strategies and Properties
*Many steps are included for understanding but these steps do not have to be written. Students may
use less formal notation.
Strategy
Examples*
Additional Information
Skip Counting
6x2
2, 4, 6, 8, 10, 12
Students may use their
fingers to keep track of
the six 2s as they skip
count.
Repeated Addition
6x2
=2+2+2+2+2+2
= 12
7 x 2 may seem easier
as
Commutative Property = 2 x 7
Half then Double
Anchor of 2s, 5s, or
10s plus a group
Anchor of 2s, 5s, or
10s minus a group
Anchor of 2s, 5s, or
10s plus or minus
more than one group
6x7
=3x7x2
= 21 x 2
= 42
3x7
= (2 +1) x 7
=2x7+1x7
= 14 + 7
= 21
9x7
= (10 - 1) x 7
= 10 x 7 – 1 x 7
= 70 – 7
= 63
8x7
= (10 – 2) x 7
= 10 x 7 – 2 x 7
= 70 – 14
= 56
EduGAINS Mathematics K-12
As the number of addends increases, the
student is less likely to use repeated addition
or skip counting. Many students will use skip
counting only for 2s, 5s, and 10s, and prefer
one of the strategies outlined below for the
other facts.
It is helpful to support students with
understanding the connection between the
partial sums 2, 2 + 2, 2 + 2 + 2, ... to the skip
count “2, 4, 6, 8, 10, 12”.
Switching the order of a multiplication question
is sometimes done to determine unknown
facts.
If a quantity is halved, then doubled, the end
result is the same as the starting quantity (e.g.,
6 halved is 3, then 3 doubled is 6). This makes
strategic use of the Associative Property and
decomposes an unknown multiplier.
Start with a nearby anchor of a 2s, 5s, or 10s
fact, then add or subtract one or more groups.
This makes strategic use of the Distributive
Property and decomposes an unknown
multiplier.
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Multiplication: Choosing Which Computational Strategy to Use
Being flexible in using strategies is a great skill. Many computational strategies depend on breaking
larger numbers down to smaller or friendly numbers (decomposing the number), then applying the
Distributive or the Associative Property. Which particular strategy a student chooses will depend on
which particular facts the student has already mastered. The best strategy for decomposing factors
depends on which number facts the student knows. The following chart illustrates that multiple
strategies can be used for the same product.
Multiple Strategies for the Product 6 x 7
Strategy
Anchor of 5
Plus a Group
Perfect
Square
Plus a Group
Perfect
Square
Minus a
Group
Anchor of 5
Plus Two
Groups
Half then
Double
Example
6x7
6x7
6x7
6x7
6x7
Decompose
= (5 + 1) x 7
= 6 x (6 + 1)
= (7 – 1) x 7
= 6 x (5 + 2)
=3x7x2
Use the
=5x7+1x7
Distributive or = 35 + 7
Associative = 42
Property
6 is near the
Student
anchor of 5
Thinking
=6x6+6x1
= 36 + 6
= 42
=7x7–1x7
= 49 – 7
= 42
=6x5+6x2
= 30 + 12
= 42
= 21 x 2
= 42
6 x 7 is near
the square
product 6 x 6
6 x 7 is near
the square
product 7 x 7
7 is near the
anchor of 5
6 is even, so
take half, then
double
Visual summaries of the various decomposing strategies are shown below and illustrate the fact that
more than one strategy can be used for some of the facts. If the Commutative Property was used as
well, the first and third charts will have more facts filled in.
2s, 5s, or 10s Anchor
plus or minus
1 group
Square Product
Anchor plus or minus
1 group
EduGAINS Mathematics K-12
2s 5s, or 10s Anchor
plus or minus
2 groups
Half
then Double
Page 7 of 10
Multiplication and Division: Strategies for Larger Numbers
Rename Numbers to Multiply and Divide by 10, 100, 1000 Mentally
Students learn early that 10 ones can be traded for a ten, 10 tens can be traded for a hundred, and
10 hundreds can be traded for a thousand.
Gap Closing J/I Representing and Renaming Whole Numbers Student Book p.18
Using trading of units and flexibility in renaming numbers, students can make sense of multiplication
and division by 10, 100, and 1000.
52 tens
can be written as 520
52 x 10
can mean 52 tens
630 x 100
can mean
630 hundreds
630 hundreds
can be written as
63000
can mean how many
hundreds are in
63 000?
63 000 can be thought
of as 630 hundreds
so the answer is 630
can mean how many
tens are in 520?
520 can be thought of
as 52 tens
so the answer is 52
63 000 ÷ 100
520 ÷ 10
so the answer is 520
so the answer is 63 000
Multiplying Double-Digit Numbers
The following strategies are related to strategies shown earlier for single-digit multiplication. Again,
several strategies can be used for the same fact.
*Many steps are included for understanding but these steps do not have to be written. Students may
use less formal notation.
Strategy
Decompose to Use Known
Facts
Examples*
14 x 25
= 14 x (20 + 5)
= 14 x 20 + 14 x 5
= 280 + 70
= 350
14 x 25
= 14 x 100 ÷ 4
Scaling Up and Scaling Down = 1400 ÷ 4
= 350
EduGAINS Mathematics K-12
Additional Information
Use a known multiplier as an anchor then
use the distributive property.
“Quadruple then Quarter” and “Half then
Double” strategies are both based on
keeping equivalence while using numbers
that are easier to work with mentally.
Page 8 of 10
Pictorial Strategy for Multiplying All Types of Numbers
The following chart contains examples of how an open partitioned array can be used to find a product
by decomposing larger whole numbers, decimals, fractions, and algebraic expressions.
Many steps are included for understanding but these steps do not have to be written
Whole Numbers
Decompose
using
friendly
numbers
Decimal Numbers
14 x 25
7.1 x 8.5
=(10 + 4)x(20 + 5) = (7 + 0.1) x (8 + 0.5)
Fractions
Algebra
2 1/4 x 3 1/2
= (2 + 1/4) (3 + 1/2)
(c + 2)(c + 3)
6 + 1 + 3/4 +1/8
= 7 7/8
c² + 3c + 2c + 6
= c² + 5c + 6
Use an open
array
Determine
partial
products
Add partial
products
200 + 80 + 20 +
50
= 350
EduGAINS Mathematics K-12
56 + 3.5 + 0.8 + 0.05
= 60.35
Page 9 of 10
Standard Algorithm
Whole Numbers
Decimal Numbers
Fractions
Algebra
2 1/4 x 3 1/2
= 9/4 x 7/2
= 63/8
= 7 7/8
•
•
•
Multiply 4 x 25.
Multiply 1 x 25 and
move 25 over 1
place value position
since it is really 10 x
25 or 250.
Add lined up digits.
•
•
•
•
Multiply 5 x 71
Multiply 8 x 71 and
move over one
place value position
Add lined up digits
Put the decimal
in by adding
the number of
decimal places in
the question and
counting over that
many places from
the right.
EduGAINS Mathematics K-12
•
•
•
•
Convert to improper
fractions.
Multiply numerators
and multiply
•
denominators
Simplify fractions if
necessary
Convert the
improper fraction to
a mixed fraction if
desired or required. •
c² + 3c + 2c + 6
= c² + 5c + 6
Multiply using
“FOIL”
Multiply the
F irst terms,
O uter terms,
I nner terms,
L ast terms,
Collect like terms.
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