How to Use Factors & Multiples in Middle School Math

Knowledge of basic facts assists in middle school math.

The factorization of a number refers to viewing the number as a product of two or more factors. This requires breaking a number apart or dividing it to find factors. Finding the multiples of a numeral involves multiplying the number by factors to obtain a list of products. Finding and using factors and multiples entails basic fact mastery of multiplication and division. It also requires the understanding of fact families. Middle school math draws upon knowledge of factors and multiples to solve algebraic equations that use variables. This level of math also requires the ability to manipulate fraction problems to figure out the greatest common factor or least common multiple.

Things You'll Need

• Pencil
• Paper

Instructions

Finding Algebraic Variables

1. Read an algebraic equation that implements a product or quotient, a known factor--which is included in the problem, and a variable--an unknown factor represented by a letter such as "x" or "y". This equation must require you to find the numeric equivalent for the variable to solve the problem.

2. Factorize the product or quotient, by listing factors that multiply together to create the product or quotient. Factors that multiply or divide together to create a product or quotient are called a fact family. For example, the numbers four, five and 20 create a fact family because the factors four and five multiply together to make 20 and 20 divided by either factor equals the other factor.

3. Look at the product and the known factor written within the problem. Divide the product by the known factor to find the variable--unknown factor. You can also find the number that the variable represents by looking at the fact families created in Step 2 and finding the missing number. For example, if an equation has the numbers 20, five and a variable in it such as (5 x ?) = 20, the variable is the missing fact from the fact family. In this case the missing fact is four.

4. Follow the finding variables example:

   Equation: (20 divided by x) = 5

   Factorization: 20: (1 x 20), (2 x 10), (4 x 5)

   Solution: (20 divided by 5) = 4, so x = 4

Reducing Fractions

1. Look at the numerator and denominator of the fraction. List the factors of the numerator and denominator. Circle the factors they have in common.

2. Find the greatest common factor (GCF) --- the largest number that both the numerator and denominator can be divided by evenly.

3. Divide the numerator and denominator by the GCF to put the fraction in simplest form.

4. Follow the reducing fractions example:

   Problem: Reduce 12/16

   Factorization of 12: (1 x 12), (2 x 6), (3 x 4)

   Factorization of 16: (1 x 16), (2 x 8), (4 x 4)

   GCF: 4

   Solution:12/16 divided by 4 = 3/4

Least Common Multiple

1. Compare the denominators of two fractions that you are adding or subtracting to find a common denominator.

2. Find the least common multiple that both denominators can multiply up to by thinking about the fact families of both denominators.

3. Multiply the denominators by each other to find a common multiple, if you can't find the least common multiple. This may require more reducing after finding the sum or difference.

4. Multiply the numerators of each fraction by the same number by which you multiplied the denominators. Add or subtract the numerators. Place the common denominator under the fraction line.

5. Reduce the fraction by finding the greatest common factor that can divide into both the numerator and denominator.

6. Follow the Least Common Multiple (LCM) example:

   Problem: 2/3 + 1/4

   Finding Least Common Multiple: (2/3 x 4) = 4/12 and (1/4 x 3) = 3/12

   Solution: 4/12 + 3/12 equals 7/12

References

• Math: Factors and Multiples
• Online Math Learning: Factors and Multiples