Math & History: Whole Number: Multiplication

I. Intro

The operation of multiplication is generally thought of as either repeated addition or scaling. Generally, multiplication is taught as repeated addition when learning about whole numbers and integers. This model starts to fall apart when multiplying fractions, which is when scaling is sometimes taught. As such, multiplication as repeated addition is shown here. Multiplication as scaling is introduced in the section multiplication of fractions.

The symbols for multiplication are ×, ∙, *, ()().

The numbers to be multiplied are called the factors (multiplicand and multiplier) and the answer the product.

The multiplicand is the number which is multiplied by the multiplier. Generally the multiplier is written first, then the multiplicand (Some people have learned the opposite). See here

Horizontal vs Vertical Form
When multiplying whole numbers we can write the problem horizontally or vertically.

Horizontal Form
4 x 3 = 12

Vertical Form
4
x 3
12

Verbally Expressing Multiplication Problems

There are many ways to say multiplication problems. Here are some common ways of saying 4 x 3:

Four times three
Four multiplied by three
Four times as much as three
Four groups of three
The product of four and three

Multiplication Tables
The basic multiplication table for digits one through ten.

II. Algorithms

1. Area Model (Box) Method Multiplication

Often the first method of multiplication currently taught to children around 4th grade. It is believed that it better helps children conceptually understand multiplication.

See Here

1. Partial Products

Often the second method of multiplication currently taught to children around 4th grade.

See Here

2. Standard Multiplication Algorithm
The standard method of multiplication involves multiplying every digit in the first number by every digit in the second and adding the results. It has five steps:

1) line up the numbers being multiplied by their place value putting the larger number on top.

2) Multiply the ones digit of the bottom number by each digit of the top number, starting from the right.

If a single-digit multiplication results in a two-digit number, write down the ones digit and carry over the tens digit to add to the next multiplication’s product in the next column. This carried value is added to the partial product of the next digit’s multiplication before writing it down.

3) Multiply the tens digit of the bottom number by each digit of the top number, starting from the right and add a zero placeholder since you are now multiplying by tens place

4) Repeat similarly for hundreds, thousands, etc., adding more zeros accordingly.

5) Add all the products calculated from the steps above for the final answer.

Example: Multiply 28 x 7

28
x 7

Multiply 8 x 7 = 56. Regroup the product placing the 6 ones below and the 5 tens over the 2.

5
28
x 7
6

Multiply 2 x 7 = 14. Add the 5 tens resulting in 19 tens.

5
28
x 7
196

Example: Multiply 846 x 12

846
x 12

Multiply the ones digits. 6 x 2 = 12. Regroup the product carrying the 1 ten.

1
846
x 12
2

Multiply 4 x 2 = 8, then add the 1 ten resulting in 9 tens.

1
846
x 12
92

Multiply 8 x 2 =16. Write below to complete the partial product aligned with the ones digit of the bottom number.

846
x 12
1692

Now we will multiply the tens digit of the bottom number with each digit of the top number. Multiply 6 x 1 = 6. Make sure the answer is aligned under the tens column below. You can first put a zero in the ones column to avoid mistakes.

846
x 12
1692
60

Then multiply 4 x 1 = 4

846
x 12
1692
460

Then multiply 8 x 1 = 8. Write below to complete the partial product aligned with the tens digit of the bottom number.

846
x 12
1692
8460

Finally, add the two partial products for the total product.

846
x 12
1692
8460
10152

IV. Conceptual Understanding
Repeated addition
When dealing with whole numbers, multiplication is often taught as repeated addition. Repeated addition is the process of repeatedly adding the same number. There are various ways this can be illustrated.

Set Model Approach

O O O O O O O O O
O O O O O O

Above are 3 groups of 5 circles which can be written as 5 + 5 + 5. The number 5 is repeated 3 times which would be written 3 x 5. This is read as "three times five".

So 3 x 5 = 3 groups of 5 = 5 + 5 + 5 = 15.

Number Line Approach

Multiplication can also be illustrated with the use of a number line. On the above example 3 x 5 is shown as three movements of five units ending on the number fifteen.

Area Model Approach

Multiplication can also be thought of in terms of an area model. The above area model shows three rows of five units or 3 x 5. This also demonstrates the commutative property of addition in that you could alternatively interpret it as five columns of three units or 5 x 3. Both give the same answer of 15.

Array Model Approach

Here's a nice video from Khan Academy Multiplication with arrays

V. Properties of Multiplication

Commutative Property
Multiplication (as well as addition) is commutative, meaning the order of the numbers being multiplied doesn't matter. For instance,

2 x 7 = 14 is the same as 7 x 2 = 14

Associative Property
Multiplication (and addition) is associative, meaning that when multiplying more than two numbers, the order in which multiplication is performed does not matter. For example:

(5 x 2) x 4 = 10 x 4 = 40 is the same as 5 x (2 x 4) = 5 x 8 = 40

Multiplicative Identity Property

The multiplicative identity property states that any number multiplied by 1 results in the number itself.

1 x 18 = 18

Distributive Property

The distributive property states that the sum of two numbers times a third number is equal to the sum of each addend times the third number.

Stated with variables, the distributive property states that x(y+z) = x(y) + x(z)

In arithmetic, this property can make it easier to solve a large multiplication problem by splitting it into two smaller ones and adding the products. For example:

17 x 101

Can be turned into

17(100 + 1)

Then multiply (distribute) the number outside the parentheses by each number inside the parentheses, and then add the products.

1700 + 17 = 1717

So when coming across a problem in the form of

a(b + c)

There are two ways you can go about it. You can use the normal order of operations and first add (b + c) and then multiply the sum by a. The other method is to use the distributive property and multiply (a x b) then (a x c) and then add the two products together. Why not just use the normal order of operations? Much of this is preparation for dealing with variables in Algebra.

Khan video

VI. Inverse Operations
Inverse operations are opposite operations that undo each other. Multiplication and division are inverse operations. For example:

4 x 5 = 20 then divide to get back to where we started 20 ÷ 5 = 4

VII. Miscellaneous

Multiplication & Division Problem Structures (Word Problems)

VII. Practice
Multiplication Practice at Lizardpoint

Math Games: Multiplication

Multi Digit Multiplication & Division Practice: Khan Academy

Multiplication Table Practice