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Using the State Curriculum: Mathematics, Grade 4

Algebra | Geometry | Measurement | Statistics | Probability | Number | Processes
Clarifications: Each clarification provides an explanation of the indicator/objective to help teachers better understand the concept. Classroom examples are often included to further illustrate the concept. While classroom examples could be shared with the students, the intended audience for the explanation/clarification is the classroom teacher-not the student. In addition, classroom examples may or may not reflect the assessment limits.

Standard 6.0 Knowledge of Number Relationships and Computation/Arithmetic

Topic B. Number Theory

Indicator 1. Apply number relationships

Objective b. Identify factors

  • Use whole numbers (0 – 24)

Clarification

Number theory is the study of the properties of numbers and the relationship between the numbers. If a number (the dividend) is divided by a second number (the divisor) and there is no remainder, the second number is said to be a factor of the first number. Also the first number is a multiple of the second number.

To find the factors of a number, follow these steps.

  • Every number has at least two factors, 1 and the number.
  • Create a function table, starting with the first factors of every number, 1 and the number.
  • Use a calculator to decide if the number has a 2, 5, or 10 as a factor. If 2, 5, or 10 is a factor, divide the number by that factor. The quotient will be the other factor.
  • Divide the number by 3, 4 and 6 to see if they are factors of the number. The quotient will be the other factor in the function table.

Ask students to think back to their multiplication facts:

Think of 3 groups of 4 and 4 groups of 3.
How are they alike?
What are the factors that these groups show you?

If students are struggling with the concept of factors, have them create rectangular arrays of the number for which they are trying to find the factors. For example, if they are trying to find the factors of 15, they would make arrays with 15 in total. If the array can make a rectangle, then the numbers of columns and rows are the factors. The only rectangular arrays that can be made with 15 are 3 × 5 and 15 × 1. The factors of 15 are: 1, 3, 5, and 15.

To find the factors of 24, use the fact that factors generally are in pairs.

24
1 24
2 12
3 8
4 6
  • Start with 1 and the number as a pair of factors.
  • 2 is a factor of 24 because 2 divides 24 evenly. 2 × 12 is 24.
  • Check to see if 3 is a factor. Divide 24 by 3. There is no remainder. 3 is a factor of 24. Using multiplication facts, 3 × 8 = 24.  8, so 3 and 8 are another pair of factors.
  • Check to see if 4 is s factor. Four is also a factor because 4 × 6 = 24.  4 and 6 are another pair of factors.
  • Five is NOT a factor because 5 does not divide 24 evenly.
  • Next, check for 6. But since it is already on the chart, it is already listed as a factor.
  • Stop. Once, you have repeated a factor, you have found all the factors.

Also 24 is a multiple of all of the factors listed.

Classroom Example 1

What are the factors of 20?

Answer: Create a T table. 1 and 20 are automatically factors.  20 is divisible by 2.  2 × 10 = 20.  2 and 10 is another pair of factors.  20 can be divided by 5 evenly.  4 × 5 = 20.  4 and 5 is another pair of factors.  3 does not divide 20 evenly.

20
1 20
2 10
5 4

List the factors in order from least to greatest.

1, 2, 4, 5, 10, 20

Classroom Example 2

Mr. Ramirez can divide his class into groups of 5 students or groups of 4 students. Which of the following is the possible size of his class?

  1. 20
  2. 24
  3. 25
  4. 30

Answer: A is correct.  20 is the only number of which 4 and 5 are factors.  20 is a multiple of 4 and 5.

Classroom Example 3

Samantha has 18 dolls in her collection. What size groups can she divide her dolls into?

  1. 2
  2. 3
  3. 4
  4. 9

Answer: A, B and D are correct.  2, 3, and 9 are factors of 18.  4 is not a factor of 18.

Classroom Example 4

Use the following clues and logical deduction to determine the Mystery Number

Mystery Number

Clue 1. I am a number between 1 and 20.
Clue 2. 2 is one of my factors.
Clue 3. 4 is one of my factors.
Clue 4. I have exactly five factors.
Who am I?

Answer: 16.
First list all numbers that have 2 as a factor between 1 and 20.  [2, 4, 6, 8, 10, 12, 14, 16, 18]
Next eliminate all the numbers that do not have 4 as a factor.  [2, 4, 6, 8, 10, 12, 14, 16, 18]
That leaves 4, 8, 12 and 16.  16 is the only number left that has exactly five factors.  4 has three factors.  8 has four factors.  12 has six factors.
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