Getting Started |
| Misconception/Error The student is unable to find a common factor of two numbers. |
| Examples of Student Work at this Level The student is not able to identify a common factor of 36 and 42. The student:
- Begins or completes factor trees but does not identify common factors.
- Writes an equivalent expression without applying the Distributive Property (e.g., 40 + 38).
- List some pairs of factors for each number but is unable to identify a common factor.
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| Questions Eliciting Thinking What does the tem factor mean? Did you find any factors of these numbers?
What is a common factor? Can you find a number that divides evenly into both 36 and 42?
Do you know what a greatest common factor of two numbers means? |
| Instructional Implications Review the concepts of factor and common factor. Assist the student in developing a strategy for identifying all factors of each of two numbers and listing factors from least to greatest. Guide the student to identify all factors common to the two numbers.
Consider using The Factor Game, an instructional activity from NCTM, to reinforce factor concepts (http://illuminations.nctm.org/lesson.aspx?id=1010). NCTM lists the following objectives for this activity:
- Classify numbers as prime or composite.
- Recognize that some numbers are rich in factors while others have few factors.
- Recognize that factors come in pairs and when one factor is found, another can also be found.
- Discover the connection between dividing and finding factors of a number.
Review the meaning of greatest common factor and provide the student with examples of two or more numbers along with all of their factors. Guide the student to identify the greatest common factor. Give the student pairs of whole numbers less than 100 and challenge the student to find and list all of the factors of each number and to identify all common factors as well as the greatest common factor. Be sure to include pairs of numbers in which one number is a multiple of the other and in which the numbers are relatively prime (i.e., have no common factor other than one).
Model the process of rewriting the sum of two whole numbers in the form a(b + c). Choose a pair of numbers that has more than one common factor such as 12 and 18, so there is more than one way to rewrite the sum in the form a(b + c). Challenge the student to find every way to rewrite the sum in this form and to identify the expression in which a is the greatest common factor. Assist the student in understanding that if a is the greatest common factor, then b and c are relatively prime (i.e., have no factors in common other than one). Guide the student to evaluate each expression by multiplying a by the sum of b and c [e.g., 12 + 18 = 3(4 + 6) = 3(10) = 30] and to compare this product to the original sum (e.g., 12 + 18 = 30). |
Moving Forward |
| Misconception/Error The student is unable to rewrite an equivalent expression of the form a(b + c). |
| Examples of Student Work at this Level The student identifies a common factor of 2, 3, or 6 and writes an expression of the form a(b + c):
- That is not equivalent to 36 + 42.
- Whose value is 78 but does not result from factoring 36 + 42.
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| Questions Eliciting Thinking Can you explain how you found your answer?
Is the expression you wrote equivalent to 36 + 42? How can you tell? |
| Instructional Implications Model the process of identifying a common factor of a pair of addends and factoring the common factor from each addend. For example, rewrite 9 + 30 as 3(3) + 3(10) and then as 3(3 + 10). Show the student that all three of the expressions are equivalent.
Review the Distributive Property and apply it to the process of factoring. Model the process of rewriting the sum of two whole numbers in the form a(b + c). Choose a pair of numbers that has more than one common factor such as 12 and 18, so there is more than one way to rewrite the sum in the form a(b + c). Challenge the student to find every way to rewrite the sum in this form and to identify the expression in which a is the greatest common factor of the given addends. Assist the student in understanding that if a is the greatest common factor, then b and c are relatively prime (i.e., have no factors in common other than one). Guide the student to evaluate each expression by multiplying a by the sum of b and c [e.g., 12 + 18 = 3(4 + 6) = 3(10) = 30] and to compare this product to the original sum (e.g., 12 + 18 = 30). |
Making Progress |
| Misconception/Error The student rewrites the expression using a common factor other than the greatest common factor. |
| Examples of Student Work at this Level The student rewrites 36 + 42 in the form a(b + c) but a is not the greatest common factor.
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| Questions Eliciting Thinking Are there any other factors common to both 36 and 42? What is the greatest common factor of 36 and 42?
How would you write the expression if the value of a is 6? |
| Instructional Implications Ask the student to find the greatest common factor of 36 and 42 and to rewrite the sum using the greatest common factor. If needed, provide instruction on identifying the greatest common factor from among the factors common to a pair of whole numbers.
Provide the student additional opportunities to identify the greatest common factors of pairs of numbers and to rewrite the sum of the numbers in the form a(b + c). |
Almost There |
| Misconception/Error The student does not adequately show that the new expression has the same value as the original expression. |
| Examples of Student Work at this Level The student rewrites 36 + 42 as 6(6 + 7). However, the student:
- Does not show that the two values are equivalent.
- Calculates either 36 + 42 or 6(6 + 7) but not both.
 
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| Questions Eliciting Thinking How can you show that the expression you wrote is equivalent to the original expression?
What is 36 + 42? How can you evaluate your expression to be certain that it equals 78? According to the order of operations conventions, what should you do first? |
| Instructional Implications Model evaluating each expression using the order of operations conventions, that is, 36 + 42 = 78 and 6(6 + 7) = 6(13) = 78. Provide other pairs of numerical expressions in a variety of forms. Ask the student to evaluate each expression using the order of operations conventions in order to determine if the expressions are equivalent. |
Got It |
| Misconception/Error The student provides complete and correct responses to all components of the task. |
| Examples of Student Work at this Level The student rewrites 36 + 42 as 6(6 + 7) and shows that the two expressions are equivalent.
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| Questions Eliciting Thinking Can you explain how you found the greatest common factor and the values in parentheses (i.e., 6 and 7)?
What is the least common factor of 36 and 42? What is the least common factor of any two numbers? |
| Instructional Implications Ask the student to rewrite 36 + 42 in the form a(b + c) where a is a common factor in as many ways as possible. Then ask the student to compare the number of factors common to b and c when a is the greatest common factor to the number of factors common to b and c when a is not the greatest common factor. Introduce the student to the term relatively prime and use it to describe the relationship between b and c when a is the greatest common factor.
Challenge the student to find the greatest common factor of variable expressions (e.g., 9x and 6x). |
