Using the State Curriculum: Mathematics, Grade 3

The following multiplication properties are valuable when used in connection with learning the multiplication facts. This is NOT a list that should simply be handed out to students for their use as they learn their basic facts. Instead students should explore the individual properties within mathematics lessons in order to see the patterns that develop and make sense of each strategy.

Zero Property
Any number multiplied by zero equals zero.
6 × 0 = 0
0 × 8 = 0

Identity Property
Any number multiplied by one equals the number.
1 × 9 = 9
3 × 1 = 3

Commutative Property
The order of factors (numbers being multiplied) does not affect the product (answer in a multiplication problem).
If they know 9 × 7 = 63, then they also know 7 × 9 = 63

As they learn the properties and strategies, they can keep track of the facts they know on a multiplication grid.

Skip Counting by Twos
When students count by twos (two, four, six, eight, ten…), they are actually saying the multiples of two. All multiples of two are even numbers or numbers that end in 0, 2, 4, 6, or 8.

Skip Counting by Fives
Counting by fives (five, ten, fifteen, twenty…) reveals that all multiples of five end in a zero or a five.

Skip Counting by Tens
Counting by tens (ten, twenty, thirty,…) shows that the multiples of ten end in 0.

Squares
Multiplication squares are easy for some students to memorize.
1 × 1 = 1; 2 × 2 = 4; 3 × 3 = 9; 4 × 4 = 16; 5 × 5 = 25; 6 × 6 = 36; 7 × 7 = 49; 8 × 8 = 64; 9 × 9 = 81

Patterns with Nines
Multiples of nine create several patterns students can use to learn the products.

1. The tens digit of the product is always one less than the factor opposite the nine in the fact.
   9 × 3 = 27
   The 2 in the tens place is one less than 3, the factor being multiplied by 9.
2. The sum of the digits in the nines products is always nine.
   9 × 2 = 18; 9 × 7 = 63
   1 + 8 = 9 and 6 + 3 = 9

Sometimes students become overwhelmed when learning their basic facts. Every child does not need every strategy. This is an opportunity to differentiate meeting the needs of individual students as is appropriate.

As they learn the properties and strategies, they can update the facts they know on a multiplication grid.

Now there are only a few facts to learn and more sophisticated strategies can be used. Many of these are student devised.

Double and Double Again
If I know my two facts, the four facts can be learned by doubling a two fact. For example, to learn 4 × 3, think of 2 × 3, which is 6. Double that to get 4 × 3 which is 12.

Double and One More
If I know my two facts, the three facts can be learned by adding one more. For example, to learn 3 × 7, think of double 7 which is 14. Add another 7 to get 21.

Close to Facts are also helpful in mastering new facts. If I want to find 6 × 7 and know 5 × 7, find 5 × 7 which is 35 and add another 7 to get 42.

The important thing to remember is that students should not be left helpless when they don't know a given fact. There should be enough strategies available for them to find products that they have not committed to memory.

Another useful strategy for finding difficult facts is called splitting the array.

Splitting the Array
Any multiplication fact can be displayed as an array and then broken down into simpler facts a student already knows.
7 × 8 =56 can be broken into 7 × 4 and 7 × 4, so a student who already knows 7 × 4 = 28 could then double 28 to get 56 which equals 7 × 8.

Students should understand that there is not one way to split arrays. They will split the array dependent on the facts that they know.

The following division strategies may be modeled and then used by students to find division facts. Division strategies depend on a large part on the student's knowledge of multiplication facts and strategies.

Once again, this is NOT a list that should simply be handed out to students for their use as they learn their basic facts. Instead students should explore the individual strategies within mathematics lessons in order to see the patterns that develop and make sense of each strategy.

Inverse Operation
Division is the inverse, or opposite, of multiplication. A student who knows that 4 × 5 = 20 will quickly realize that:

20 ÷ 4 = 5
20 ÷ 5 = 4

Counting Multiples When asked to divide, students may count using fingers or tally marks by the multiples of the divisor to discover the quotient. For 56 ÷ 7, a student will say or think while keeping track of how many multiples he/she is listing, "seven, fourteen, twenty-one, twenty-eight, thirty-five, forty-two, forty-nine, fifty-six." The student named eight multiples of seven, so 56 ÷ 7 = 8.

Think-Multiplication Students may think of the corresponding multiplication fact to answer a division problem. If a student is asked, "Forty-eight divided by eight equals what?" the student must think, "What number times eight equals forty-eight?" He/she will think of six and be able to conclude that 48 ÷ 8 = 6.

Basic multiplication and division facts can be represented using number sentences, pictures, and drawings.

Number Sentence
A number sentence is an equation or inequality expressed using numbers and symbols. Examples of basic facts represented as number sentences:

4 × 3 = 12
81 ÷ 9 = 9

Picture
Students may visualize basic facts and create pictures to express the facts (especially useful for solving word problems).

Word Problem:
Miss Davis has 24 rulers for her students. If she'd like to keep the rulers in 4 baskets, how many rulers will be in each basket?

Drawing
Students may also use quick sketches such as tally marks, arrays, or other drawings to map out a basic fact.
3 × 2 may be drawn as