Student Probe Least Common Multiple What is the least common multiple of 4 and 12? Answer: 12 Students will sometimes answer 4, especially if they have recently worked with finding the greatest common factor of numbers. Some will answer 48 with the misconception that the word multiple is telling them the action to perform on the two numbers. Lesson Description: The lesson is intended to help students in finding the least common multiple of two whole numbers less than or equal to 12. Rationale: In order to be able to find the least common multiple of two or more numbers, the student must be able to identify the factors and multiples of each number. Frequently a difficulty with multiplication facts will prevent the student from being successful. The factorization of whole numbers is a prerequisite skill for addition and subtraction of fractions. In addition, it is a prerequisite for algebra where students will be required to factor algebraic expressions. Preparation Create a display to show the prime factorization of 8, 10, 4, and 12. At a Glance What: Find the least common multiple of two whole numbers less than or equal to 100. Common Core Standards: CC.6.NS.4 Compute fluently with multi-­‐digit numbers and find common factors and multiples. Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2). Matched Arkansas Standard: AR.6.NO.3.5 (NO.3.6.5) Application of Computation: Find and use factorization (tree diagram) including prime factorization of composite numbers (expanded and exponential notation) to determine the greatest common factor (GCF) and least common multiple (LCM) Mathematical Practices: Make sense of problems and persevere in solving them. Who: Students who cannot find the least common multiple of two whole numbers less than or equal to 12. Grade Level: 6 Prerequisite Vocabulary: factor, prime, least, common, multiple, product Prerequisite Skills: finding prime factors of a whole number Delivery Format: individual, pairs, small group, whole group Lesson Length: 30 minutes Materials, Resources, Technology: None Student Worksheets: Factor tree sheet for each of the following numbers: 8, 10, 4, 12 Lesson The teacher says or does… 1. Let’s write 8 as a product of prime factors, using a factor tree. 2. What are all of the prime factors of 8? 3. So, is it important to name all of the prime factors of 8? Why or why not? 4.
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Expect students to say or do… If students do not, then the teacher says or does… 8 = 2 x 2 x 2 Monitor students, prompting when necessary. 2 and 2 and 2 Yes, because those are all the numbers that you need to multiply to get 8. Ask students to explain why they said no and base subsequent modeling on the students’ rationale for their answer. (e.g. 2 x 2 yields 4; 4 x the third 2 yields 8) Let’s write 10 as a product Write 10 = 2 x 5 (or equivalent Monitor students, prompting of prime factors. form) when necessary. (Write the following two Verify that the students write Work individually with the equations on the board.) the two equations in the student until he/she has set Write this along with me: correct manner on their up the two equations as the 8 = 2 x 2 x 2 paper. teacher has modeled it. 10 = 2 x 5 Do you see factors that Yes Let’s look at the two are in both equations? equations. Help me find the factors that they share with each other. Model by pointing to the 2 in 8 and then in 10. Ask students if they see another factor that is also shared (no). Draw a circle around each Circle the first 2 of the factors Model pair of common factors. for 8 and the (only) 2 of the factors for 10. Do you see any factors Yes, 10 has a 5 and 8 has a 2 Model and say, “This 2 is the that are NOT shared and a 2. only factor that is in both lists (common) to both of factors—that’s what numbers? (In other common means. It is what 8 words, factors that are not and 10 have in ‘common’ in circled.) their factorization.” What are they? The teacher says or does… Expect students to say or do… If students do not, then the teacher says or does… 9. How many pairs of factors One Point to the circled pair and have you found that are ask students to acknowledge common to 8 and to 10? that exactly one pair of factors is circled as teacher points. 10. Do you see any prime Yes, 2, 2, and 5. Point to all of the uncircled factors that are not shared prime factors of both 8 and 10 or common to 8 and 10? and discuss how those are all What are they? separate factors even though Circle EACH one of those they are the same number. unshared factors. (i.e. 2 x 2 x 2 is NOT the same thing as 2; the first is 8 and the last is 2) 11. What is the mathematical Multiplication If students’ response indicates operation that is taking a lack of understanding, place with the factors? conduct a short conversation about why multiplication is the operation between the factors. 12. When we are finding the Listen and watch for students’ If students indicate that they least common multiple, reactions, remarks and do not understand, determine we want the smallest questions that indicate their what portion of the discussion number that is a multiple understanding of the is in need of clarification and of both the numbers. So, teacher’s “talk” about the use a physical list of multiples in this problem we want multiples of 8 and 10. of 8 and 10 (going beyond 40 the smallest (or least) in the list for 10). Explain that number that both 8 and this is one way to find the 10 share (have in least common multiple but common) in their that the factor tree is often multiplication facts—that the easiest and quickest when is, their least common the numbers are large (e.g. 18 multiple. and 25). 13. To find the least common Students will write the If students do not write the multiple of 8 and 10, we following multiplication multiplication statement will write a multiplication statement: 2 x 2 x 2 x 5 (or correctly, refer back to the problem using the equivalent variation) prime factorization equations. common factors (one Have the student say and factor representing the write each shared (circled) circled pair) and all of the and unshared prime factor as unshared factors. the teacher points to it. The teacher says or does… Expect students to say or do… If students do not, then the teacher says or does… 14. Now that you have 2 x 2 x 2 x 5 = 40 Verbally guide the student written your through the multiplication, multiplication sentence, stopping after each let’s find the answer multiplication (e.g. 2 x 2 yields (product). 4; 4 x 2 yields 8; 8 x 5 yields 40; thus, the least common multiple of 8 and 10 is 40.) 15. Repeat with addition pairs of numbers 12. Teacher Notes: 1. When using the factor tree to perform prime factorization of a number, emphasize that the student must check the factor for its status as a prime or composite number. If students do not understand the concept of factorization or how to factor using a factor tree, the teacher will need to refer to Factor Pairs. 2. It is important that students have an understanding of the separate meaning of the words factor, prime number, least, common, and multiple within the context of this lesson. Spending a little bit of time on this vocabulary before beginning the lesson may clarify student misconceptions and misunderstanding of the term “least common multiple.” Variations None Formative Assessment Find the least common multiple of 6 and 9 using the factor tree method. Answer: 18 References Marjorie Montague, Ph.D. (2004, 12 7). Math Problem Solving for Middle School Students With Disabilities. Retrieved 4 25, 2011, from The Iris Center: http://iris.peabody.vanderbilt.edu/resource_infoBrief/k8accesscenter_org_training_resources_
documents_Math_Problem_Solving_pdf.html Russell Gersten, P. (n.d.). RTI and Mathematics IES Practice Guide -­‐ Response to Intervention in Mathematics. Retrieved 2 25, 2011, from rti4sucess: http://www.rti4success.org/images/stories/webinar/rti_and_mathematics_webinar_presentati
on.pdf