Getting Started 
Misconception/Error The student does not understand the concept of a multiple. 
Examples of Student Work at this Level The student:
 Confuses multiples with factors and identifies common factors.
 Finds the greatest common factor rather than the least common multiple.
 Finds the least common factor.

Questions Eliciting Thinking What does it mean to be a multiple of a number? Can you give me an example of a multiple of the number 10?
What does it mean to be a factor of a number? Can you give me an example of a factor of the number 10?
How are multiples different from factors?
What does common mean? What does least mean? 
Instructional Implications Review the definitions of factors and multiples emphasizing the difference between them. Provide the student with an effective strategy for finding multiples of a number. For example, give the student a number such as eight and guide the student to multiply eight by each of the first six whole numbers: 0, 1, 2, 3, 4, and 5 to generate the first six multiples: 0, 8, 16, 24, 32, and 40. Assist the student in understanding that the list of multiples is infinitely long, so it would never be practical to find all of the multiples of a number, but the list can be extended by continuing to multiply by the whole numbers.
Consider using The Product Game, an instructional activity from NCTM, to reinforce factor and multiple concepts (http://illuminations.nctm.org/lesson.aspx?id=4101). NCTM lists the following objectives for this activity:
 Review multiplication facts.
 Develop understanding of factors and multiples and of the relationships between them.
 Understand that some products are the result of more than one factor pair.
 Develop strategies for winning The Product Game.
Review the meaning of least common multiple. Provide the student with examples of two or more numbers along with their first several multiples to include a common multiple and guide the student to identify the least common multiple. Give the student pairs of whole numbers less than 100 and challenge the student to find and list enough of the multiples of each number, so the least common multiple can be identified. Be sure to include pairs of numbers in which one number is a multiple of the other.

Moving Forward 
Misconception/Error The student understands the concept of a multiple but errs in identifying the least common multiple. 
Examples of Student Work at this Level The student lists multiples of 8 and 12 but does not choose the least common multiple. Upon questioning the student is unsure which multiple to select as the least common multiple.

Questions Eliciting Thinking What does it mean to be the common multiple of two numbers?
What does it mean to be the least common multiple of two numbers?
Can there be more than one least common multiple? 
Instructional Implications Review strategies for finding multiples of numbers and encourage the student to be systematic. Using the numbers on this worksheet, 8 and 12, ask the student to list the first 10 multiples of each. Have the student identify all common multiples as well as the least common multiple. Ask the student whether it was necessary to find 10 multiples of each number in order to find the least common multiple.
Give the student pairs of whole numbers less than 100 and challenge the student to find and list enough of the multiples of each number, so the least common multiple can be identified. Be sure to include pairs of numbers in which one number is a multiple of the other. 
Almost There 
Misconception/Error The student’s explanation is unclear or reveals a misconception. 
Examples of Student Work at this Level The student correctly identifies the least common multiple as 24 but:
 Provides an explanation that is incomplete, vague, nonmathematical, or unclear.
 Conveys a misconception. For example, the student explains that he or she listed all of the multiples of the two numbers or indicates that there is more than one least common multiple.
 Explains why 24 is a common multiple but does not justify its being the least common multiple.

Questions Eliciting Thinking What do you mean by “they meet together?” Is there another way that you can express this idea more clearly?
How many common multiples do 8 and 12 have?
Can two numbers have more than one least common multiple?
You explained why 24 is a common multiple, but can you explain why it is the least common multiple? 
Instructional Implications Assist the student in correcting and clarifying his or her explanation. Be sure the student understands that two numbers have infinitely many common multiples but there is only one least common multiple. Challenge the student to find several more common multiples of 8 and 12 and to devise a way to describe these common multiples (e.g., as multiples of 24).
Encourage the student to use a strategy based on prime factorization for finding the least common multiple of a pair of numbers. For example, have the student rewrite 8 as 2 x 2 x 2 and 12 as 2 x 2 x 3. Then show the student that the least common multiple contains the greater number of factors of each unique prime factor that appears in either factorization (i.e., 2 x 2 x 2 x 3 = 24). For example, two appears as a prime factor three times in the factorization of eight but only two times in the factorization of 12, so the least common multiple contains three factors of two. Likewise, three appears as a prime factor once in the factorization of 12 but not at all in the factorization of eight, so the least common multiple also contains one factor of three. Give the student additional pairs of whole numbers less than 100 and challenge the student to use this strategy to find the least common multiple. Be sure to include pairs of numbers in which one number is a multiple of the other. Have the student consider under what conditions each approach (listing multiples versus using prime factorizations) might be preferable. 
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 uses a mathematically correct strategy for identifying the least common multiple. For example, the student systematically lists multiples of each number (from least to greatest) until a common multiple is found and identifies it as the least common multiple.

Questions Eliciting Thinking What is the difference between factors and multiples?
What strategy did you use to find multiples of these number?
Are there any other strategies you can use to find the least common multiple?
Can two numbers have more than one common multiple? Can they have more than one least common multiple?
Does it make sense to attempt to identify the greatest common multiple? Why or why not? 
Instructional Implications If the student is not already doing so, encourage the student to use a strategy based on prime factorization for finding the least common multiple. For example, have the student rewrite 8 as 2 x 2 x 2 and 12 as 2 x 2 x 3. Ask the student to use exponents to express the factorizations as and . Then show the student that the least common multiple is the product of each unique prime factor raised to the greater power found in the prime factorizations (i.e., ). Give the student additional pairs of whole numbers less than 100 and challenge the student to use this strategy to find the least common multiple. Be sure to include pairs of numbers in which one number is a multiple of the other. Have the student consider when each approach (listing multiples versus using prime factorizations) to finding the least common multiple might be most appropriate. 