MULTIPLICATION

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```					       MULTIPLICATION
Definition – Multiplication of whole

3x5=5+5+5

In this example 3 and 5 are factors; 15
is the product.
Three Models of Multiplication

Set model: 3 x 5 is interpreted as 3 groups
of 5 objects.
Measurement model

3 x 5 is interpreted as a bar diagram
with 3 sections of length 5. Or using
a number line, start at 0 and take 3
hops of length 5.
Rectangular array model
3 x 5 is interpreted as 3 rows of 5 objects.
The objects could be dots, stars or boxes.

While this is often called the area model, you should not use this term with
students until you have introduced the concept of area in 3rd or 4th grade.
Properties of Multiplication
1. Multiplicative Identity: 1 x 8 = 8

1 group of 8

2. Commutative Property: 3 x 5 = 5 x 3

Which model would you use to explain this?
3. Associative Property:
(2 x 3) x 4 = 2 x (3 x 4)

See graphic with 2 shelves each with 3
boxes of 4 flowers on page 26.

• Together the associative and
commutative properties give us an “any
order” property for multiplication of a list
of factors
4. Distributive Property

3 x (12) = 3 x (10 +2) = 3 x 10 + 3 x 2

This can be explained nicely using
rectangular arrays.
3

10   2
Important
These properties should be
ways numbers behave, illustrated
with models and practiced with
mental math exercises and word
problems. They are not simply
rules to memorize.
Teaching Multiplication
Three Overlapping Stages

Before starting students should be able to:

1. Count to 100.
2. Know basic addition doubles 1+1= up to
10 + 10 =
3. Be able to skip count by 2, 3, 5, and 10.
Stage 1 - (End of grade 1) –Focuses on the
meaning of multiplication as repeated

Stage 2 – (Grades 2 and 3) – Focuses on
increased computational proficiency and
conceptual understanding. Use of word
problems, mental math and practice work
sheets.

Stage 3 – (End of third grade) Mastery of
single digit multiplication.
Example – Mr. Bisk buys 2 boxes of pies.
Each box has 4 pies. If the pies cost \$3
each, how much did he pay.

Note that this is a two step problem.
Examples – Mental Math

•   6 x 20 = 6 x (2 x 10) = (6 x 2) x 10

= 12 tens = 120

•   8 x 9 = 8 x (10 – 1) = (8 x 10) – (8 x 1)

= 80 – 8 = 72
DIVISION

Definition – Division is defined by missing
factors. The number 56 ÷ 8 is the missing
factor in ____ x 8 = 56.

Multiplication and Division are opposite
operations.
Interpretations of 20÷4

• Partitive Model: “20 is 4 groups
of what unit?”

• Measurement Model: “20 is how
many groups of 4?”
It is important to use both models.

Which model works for division by a
fraction?

Example: 12 pounds of flours is put
into bags weighing 1 ½ pounds each.
How many bags are there?
Quotient Remainder Theorem

Pretend you are in third grade. Find:
Quotient Remainder Theorem
For any two whole numbers A and
k with k≠0, there are unique
whole numbers q and r such that

A=k·q+r       where 0 ≤ r < k
How would you illustrate 20 ÷
3 with a “rectangular array
with remainder?”
Division by Zero
10 ÷ 0 = ?

0÷0=?

Why is it undefined?
Case 1:
Suppose 10 ÷ 0 = ?.

Then ? x 0 = 10.

Can’t be!
Case 2:
Suppose 0 ÷ 0 = ?.

Then     ? x 0 = 0.

Anything works here.
But our answer is supposed to be
unique.
Homework
Read sections 1.5, 1.6 and 1.7
HW Set 5 - Problems 1, 2cd, 4, 6, 7, 8, 9a
HW Set 6 - Problems 1, 2ab, 3ab, 6
1. Write as a base 10 number: 431five

2. Write short word problems illustrating:

a. The comparison model for 12-5
b. The part-whole model for 17-9

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