Here are five strategies that students have been taught to use to solve multiplication problems. Note: students may call the "equal groups" method the "circles and stars" method.
Multiplication & Division Vocabulary:
Factor - The whole numbers that are multiplied to get a product. For example, in the problem 3 x 2 = 6, 3 and 2 are the factors and 6 is the product.
Product - The answer to a multiplication problem. In the problem 3 x 4 = 12, 12 is the product.
Divisor - The number you are dividing by. For example, in the problem 12 ÷ 3 = 4, 3 is the divisor.
Dividend - The big number that is being divided. For example, in the problem 12 ÷ 3 = 4, 12 is the dividend.
Quotient - The answer to a division problem. For example, in the problem 12 ÷ 3 = 4, 4 is the dividend.
Fact Family - A group of 4 facts that are all related. If you learn one of the facts you know all four.
Example: 3 x 4 = 12
4 x 3 = 12
12 ÷ 3 = 4
12 ÷ 4 = 3
Operation - The four operations are addition, subtraction, multiplication, and division.
Equation - (number sentence) An equation says that two things are equal and will include an equal sign (=). So an equation is like a statement "this equals that". Equations can be vertical (up and down) or horizontal (side to side). Here are some different examples of equations:
12 - 3 = 9 6 + 2 = 12 - 4 148 + 376 = 514 17 + b = 25 7 x 8 = 56
Skip counting - repeatedly adding the same number over and over. For example, skip counting by 5 sounds like this: 5, 10, 15, 20, 25, 30… We can solve multiplication problem by using skip counting. We can use number lines or hundred charts to keep track as we skip count.
Area Model - Pictorial model representing multiplication through the formula for area, usually represented as rectangles, often built with base ten blocks.
Array - A set of objects or pictures in equal rows and equal columns. We use them to represent a multiplication problem.
Base ten blocks (flats, rods, and units) - blocks used to represent ones, tens, hundreds and thousands. We call these units, rods, flats and blocks. We expect students may sometimes solve problems using pictures of these blocks.
Factor - The whole numbers that are multiplied to get a product. For example, in the problem 3 x 2 = 6, 3 and 2 are the factors and 6 is the product.
Product - The answer to a multiplication problem. In the problem 3 x 4 = 12, 12 is the product.
Divisor - The number you are dividing by. For example, in the problem 12 ÷ 3 = 4, 3 is the divisor.
Dividend - The big number that is being divided. For example, in the problem 12 ÷ 3 = 4, 12 is the dividend.
Quotient - The answer to a division problem. For example, in the problem 12 ÷ 3 = 4, 4 is the dividend.
Fact Family - A group of 4 facts that are all related. If you learn one of the facts you know all four.
Example: 3 x 4 = 12
4 x 3 = 12
12 ÷ 3 = 4
12 ÷ 4 = 3
Operation - The four operations are addition, subtraction, multiplication, and division.
Equation - (number sentence) An equation says that two things are equal and will include an equal sign (=). So an equation is like a statement "this equals that". Equations can be vertical (up and down) or horizontal (side to side). Here are some different examples of equations:
12 - 3 = 9 6 + 2 = 12 - 4 148 + 376 = 514 17 + b = 25 7 x 8 = 56
Skip counting - repeatedly adding the same number over and over. For example, skip counting by 5 sounds like this: 5, 10, 15, 20, 25, 30… We can solve multiplication problem by using skip counting. We can use number lines or hundred charts to keep track as we skip count.
Area Model - Pictorial model representing multiplication through the formula for area, usually represented as rectangles, often built with base ten blocks.
Array - A set of objects or pictures in equal rows and equal columns. We use them to represent a multiplication problem.
Base ten blocks (flats, rods, and units) - blocks used to represent ones, tens, hundreds and thousands. We call these units, rods, flats and blocks. We expect students may sometimes solve problems using pictures of these blocks.
Properties of Multiplication - Students learn these properties but are not expected to memorize the names. They should know how to apply them.
Distributive Property - breaking apart problems in two simpler or known problems Example: (3 x 7) = (3 x 5) + (3 x 2) note: students are not necessarily expected to correctly use parentheses and the order of operations. It is more appropriate to expect a student to verbalize that they can solve 3 x 7 by knowing 3 x 5 and 3 x 2 and adding together the products. Note: Students may refer to this as the "break-apart strategy". We also sometimes talk about breaking apart numbers as "decomposing" the numbers.
Zero Rule - Any number times zero is zero.
Identity Property - Any number times one is itself.
Commutative Property - The order of multiplication is not important. 3 x 5 is the same as 5 x 3. This is a great property because it cuts our work in half when learning facts!
Distributive Property - breaking apart problems in two simpler or known problems Example: (3 x 7) = (3 x 5) + (3 x 2) note: students are not necessarily expected to correctly use parentheses and the order of operations. It is more appropriate to expect a student to verbalize that they can solve 3 x 7 by knowing 3 x 5 and 3 x 2 and adding together the products. Note: Students may refer to this as the "break-apart strategy". We also sometimes talk about breaking apart numbers as "decomposing" the numbers.
Zero Rule - Any number times zero is zero.
Identity Property - Any number times one is itself.
Commutative Property - The order of multiplication is not important. 3 x 5 is the same as 5 x 3. This is a great property because it cuts our work in half when learning facts!
Division
Division is usually introduced after multiplication. Again, students start off modeling division in a variety of ways. It is very beneficial to use real objects to solve problems. There are some helpful explanations and examples here.
Helpful strategies include a circles and stars diagram (often called “fair share” or “partitioning”) which is a lot like dealing cards. To solve 24÷6 students would draw 6 circles and then put one star in each circle, then another, until a total of 24 stars have been drawn. They then count the number of stars in each circle.
_
Division is usually introduced after multiplication. Again, students start off modeling division in a variety of ways. It is very beneficial to use real objects to solve problems. There are some helpful explanations and examples here.
Helpful strategies include a circles and stars diagram (often called “fair share” or “partitioning”) which is a lot like dealing cards. To solve 24÷6 students would draw 6 circles and then put one star in each circle, then another, until a total of 24 stars have been drawn. They then count the number of stars in each circle.
_
Another strategy is repeated subtraction. To solve 24÷4 students would start with 24 and keep subtracting 4 until they reach zero. They keep track of the number of times they subtract the divisor, which is their answer.
A really helpful way to think about division is as the reverse of multiplication. This is where all the hard work with multiplication facts pays off. Once you know one multiplication fact it can lead to a total of four facts.
Example:
4x7=28 is the same as 7x4=28 and is also 28÷4=7 and 28÷7=4
We call this a fact family.
Example:
4x7=28 is the same as 7x4=28 and is also 28÷4=7 and 28÷7=4
We call this a fact family.
A final strategy for solving a division fact is skip counting. This is the reverse of repeated subtraction. To solve 35 ÷ 5 simply count by 5s until you reach 35 and keep track of how many times you count.