Mathematics Success Level F T105 [OBJECTIVE] The student will investigate greatest common factors and least common multiples. [MATERIALS] Student pages S39 S48 Transparencies T116, T118, T120, T122, T124, T126, T128 Centimeter cubes 30 to 40 per pair Colored pencils 3 different colors per pair [ESSENTIAL QUESTIONS] 1. What is the difference between greatest common factor (GCF) and least common multiple (LCM)? 2. What is the most important word in greatest common factor? Explain. 3. What is the most important word in least common multiple? Explain. [WORDS FOR WORD WALL] greatest common factor, least common multiple, factor, multiple [GROUPING] Cooperative Pairs (CP), Whole Group (WG), Individual (I) [LEVELS OF TEACHER SUPPORT] Modeling (M), Guided Practice (GP), Independent Practice (IP) [MULTIPLE REPRESENTATIONS] SOLVE, Verbal Description, Concrete Representation, Pictorial Representation, Graphic organizer [WARM-UP] (5 minutes IP, I, WG) S39 (Answers on T115.) Have students turn to S39 in their books to begin the Warm-Up. Students will solve simple multiplication problems and answer questions to review factors, products, and skip counting (multiples). Monitor students to see if any of them need help during the Warm-Up. Give students 3 minutes to complete the problems and then spend 2 minutes reviewing the answers as a class. {Verbal Description} [HOMEWORK] (5 minutes) Take time to go over the homework from the previous night.
T106 Mathematics Success Level F [LESSON] (50 60 minutes M, GP, IP, WG, CP, I) SOLVE Problem (1 minute GP, WG) T116, S40 (Answers on T117.) Have students turn to S40 in their books, and place T116 on the overhead. The first problem is a SOLVE problem. You are only going to complete the S step with students at this point. Tell students that during the lesson they will learn how to find the greatest common factor (GCF) and the least common multiple (LCM) of numbers. They will use this knowledge to complete this SOLVE problem at the end of the lesson. {SOLVE} Greatest Common Factor (10 minutes M, GP, IP, WG, CP) T116, S40 (Answers on T117.) Ask students what a factor is. (A factor is a number that divides evenly into another number.) Reference the problems in the Warm-Up on S39 (T115) and ask students to identify the factors in Problems 1 4. Talk with students about what the term greatest common factor means. Explain that the word common means something that is shared and that the greatest common factor is the largest factor shared by two or more numbers. 5 minutes M, GP, WG: Pass out the centimeter cubes to students. Use the hundred board on T116 (S40) and the following activity to model for students a concrete approach to finding the GCF of two numbers. {Concrete Representation, Graphic Organizer, Verbal Description}
Mathematics Success Level F T107 MODELING Greatest Common Factor Hands-On Step 1: Tell students that they are going to find the greatest common factor (GCF) of 12 and 18. Explain to students that first they need to find the factors of 12. Have students put a centimeter cube on each of the factors of 12: 1, 2, 3, 4, 6, and 12. Step 2: Next, explain to students that they need to find the factors of 18. Have students leave the cubes on the board and place centimeter cubes on all the factors of 18: 1, 2, 3, 6, 9, and 18. Step 3: Have students look at the factors and determine which ones are common to both 12 and 18, as you point them out on T116. Explain that these factors will have 2 cubes on them. (They are 1, 2, 3, and 6.) Ask, Which is the largest factor 12 and 18 have in common (or share)? (6) Explain that six is the greatest common factor of 12 and 18. Step 4: Record the answer on T116 as students record on S40. 3 minutes IP, CP: Have students work in partners with the centimeter cubes to find the greatest common factor of 14 and 42 and answer the questions on S40. {Concrete Representation, Verbal Description, Graphic Organizer} 2 minutes M, WG: Use centimeter cubes to model the answer for the GCF problem on T116 (S40) and go over the answers to Questions 1 3 as a group. GCF Pictorial (6 minutes M, GP, WG, CP, I) T118, S41 (Answers on T119.) 4 minutes M, GP, WG: Pass out the colored pencils to students. Have students turn to S41 in their books, and place T118 on the overhead. Use the following activity to model for students a pictorial approach to finding the GCF of two numbers. {Verbal Description, Graphic Organizer, Pictorial Representation}
T108 Mathematics Success Level F MODELING Greatest Common Factor Pictorial Step 1: Tell students that, for Problem 1, they are going to find the greatest common factor (GCF) of 12 and 30. Have students draw vertical lines with one colored pencil as shown below to show the factors of 12: 1, 2, 3, 4, 6, and 12. Step 2: Have students write the factors of 12 in the graphic organizer. Factors of 12 Factors of 30 Step 3: Next, have students draw diagonal lines as shown below with a different colored pencil to show the factors of 30: 1, 2, 3, 5, 6, 10, 15, and 30. Step 4: Have students write the factors of 30 in the graphic organizer. Factors of 12 Factors of 30
Mathematics Success Level F T109 Step 5: Have students look at the common factors of both numbers, 1, 2, 3, and 6, as indicated by the two different colored lines, and circle them on the hundred board. Then have students circle the common factors in the graphic organizer. Step 6: Finally, have students list the common factors on S41 as you do the same on T118. Ask students to identify the greatest common factor, 6, and record it. 2 minutes IP, CP, WG: Have students complete Problem 2 on S41 with a partner and then come back together as a class to discuss the answer. {Verbal Description, Pictorial Representation, Graphic Organizer} GCF Abstract (6 minutes M, GP, IP, WG, CP) T120, S42 (Answers on T121.) 3 minutes M, GP, WG: Have students turn to S42 in their books, and place T120 on the overhead. Use the following activity to model for students an abstract approach to finding the GCF of two numbers. {Verbal Description, Graphic Organizer} Greatest Common Factor Abstract Step 1: Tell students that, for Problem 1, they are going to find the greatest common factor (GCF) of 24 and 32. Explain that students will not be using a hundred board, but will use the graphic organizer they have used in previous problems to list the factors of 24 and 32. If students are struggling, have them think about the factor pairs: 1 and 24, 2 and 12, 3 and 8, and so on. Step 2: Model how to write the factors of 24 in the graphic organizer as students record on S42. Factors of 24
T110 Mathematics Success Level F Step 3: Model how to write the factors of 32 in the same graphic organizer as students record. Factors of 24 Factors of 32 Step 4: Have students circle and then list the common factors. Ask students what they are and record as students record. Factors of 24 Factors of 32 Step 5: Ask students what the greatest common factor of 24 and 32 is (8) and record as students record. 3 minutes IP, CP, WG: Have students complete Problems 2 4 on S42 with a partner and then come back together as a class to discuss the answers. {Verbal Description, Graphic Organizer} Least Common Multiple (8 minutes M, GP, IP, CP, WG) T122, S43 Ask students what a multiple is. (A multiple is a product of two numbers. The product is a multiple of each number.) Have students count by 5s to 50 to demonstrate multiples of 5. Talk with students about what the term least common multiple means. Explain that the word common means something that is shared and that the least common multiple is the smallest nonzero number that is a multiple of two (or more) whole numbers. {Concrete Representation, Graphic Organizer, Verbal Description}
Mathematics Success Level F T111 3 minutes M, GP, WG: Pass out the centimeter cubes to students. Use the hundred board on T122 (S43) and the following activity to model for students a concrete approach to finding the LCM of two numbers. {Concrete Representation, Graphic Organizer, Verbal Description} MODELING Least Common Multiple Hands-On Step 1: Tell students that they are going to find the least common multiple (LCM) of 4 and 5. Explain to students that first they need to find the multiples of 4. Have students put a centimeter cube on each of the multiples of 4, through 24: 4, 8, 12, 16, 20, 24. Step 2: Next, explain to students that they need to find the multiples of 5. Have students leave the cubes on the board and place centimeter cubes on all the multiples of 5, through 25: 5, 10, 15, 20, 25. Step 3: Have students look at the multiples and determine which ones are common to both 4 and 5, as you point them out on T122. Explain that these multiples will have 2 cubes on them. (There is only one. It is 20.) Ask, Which is the smallest multiple 4 and 5 have in common (or share)? (20) Explain that 20 is the least common multiple of 4 and 5. Step 4: Record the answer on T122 as students record on S43. 3 minutes IP, CP: Have students work in partners with the centimeter cubes to find the least common multiples for 7 and 3, 2 and 8, and 4 and 12. Have students answer the questions at the bottom of the page. {Concrete Representation, Verbal Description, Graphic Organizer} 2 minutes M, WG: Use centimeter cubes to model the answers for the LCM problems on T122 (S43) and go over the answers to Questions 1 3 as a group. LCM Pictorial (7 minutes M, GP, IP, WG, CP) T124, S44 (Answers on T125.) 5 minutes M, GP, WG: Pass out the colored pencils to students. Have students turn to S44 in their books, and place T124 on the overhead. Use the following activity to model for students a pictorial approach to finding the LCM of two or more numbers. {Verbal Description, Graphic Organizer, Pictorial Representation}
T112 Mathematics Success Level F MODELING Least Common Multiple Pictorial Step 1: Tell students that, for Problem 1, they are going to find the least common multiple of 2, 4, and 5. Have students draw vertical lines with one colored pencil to show the multiples of 2, through 26. Have students write the multiples of 2 in the graphic organizer. Multiples of 2 Multiples of 4 Multiples of 5 Step 2: Next, have students draw diagonal lines (from top right to bottom left) with a different colored pencil to show the multiples of 4, through 24. Have students write the multiples of 4 in the graphic organizer. Multiples of 2 Multiples of 4 Multiples of 5 Step 3: Have students draw different diagonal lines (this time from top left to bottom right) with a different colored pencil to show the multiples of 5, through 25. Have students write the multiples of 5 in the graphic organizer. Multiples of 2 Multiples of 4 Multiples of 5 Step 4: Have students circle and then list the multiple that is common to all three numbers (just 20) and identify the least common multiple of 2, 4, and 5 (20).
Mathematics Success Level F T113 2 minutes IP, CP, WG: Have students complete Problem 2 on S44 with a partner and then come back together as a class to discuss the answer. {Verbal Description, Pictorial Representation, Graphic Organizer} LCM Abstract (7 minutes M, GP, IP, WG, CP) T126, S45 (Answers on T127.) 3 minutes M, GP, WG: Have students turn to S45 in their books, and place T126 on the overhead. Use the following activity to model for students an abstract approach to finding the LCM of two numbers. {Verbal Description, Graphic Organizer} MODELING Least Common Multiple Abstract Step 1: Tell students that, for Problem 1, they are going to find the least common multiple of 4 and 6. Explain that students will not be using a hundred board, but will use the graphic organizer they have used in previous problems to list the multiples of 4 and 6. Ask students what they think the LCM of 4 and 6 will be and why. (Based on the work they have already completed, many students may think that the LCM will be 24, because they simply multiplied 4 and 6.) Step 2: Model how to write the multiples of 4 in the graphic organizer as students record on S45. Then model how to write the multiples of 6 in the graphic organizer as students record. Step 3: Have students circle the common multiples. Ask students what they are and record as students record. Ask students to determine what the least common multiple of 4 and 6 is, and record as students record. Step 4: Refer to the graphic organizer on T126. Ask students if they needed to continue to write the multiples once they found the least common multiple. (No, they can stop when they determine the LCM.). 4 minutes IP, CP, WG: Have students complete Problems 2 4 on S45 with a partner and then come back together as a class to discuss the answers. {Verbal Description, Graphic Organizer}
T114 Mathematics Success Level F SOLVE Problem (3 minutes GP, WG) T128, S46 (Answers on T129.) Remind students that the SOLVE problem is the same one from the beginning of the lesson. Complete the SOLVE problem with your students. Ask them for possible connections from the SOLVE problem to the lesson. (Students will be using least common multiples to solve the problem. {SOLVE, Verbal Description} If time permits (10 minutes IP, I) S47 (Answers on T130.) Have students solve Problems 1 5 on S47 independently. Review the answers as a class. [CLOSURE] (2 minutes) To wrap up the lesson, go back to the essential questions and discuss them with students. What is the difference between greatest common factor (GCF) and least common multiple (LCM)? (The greatest common factor is the largest number that is a factor of 2 (or more) whole numbers. The least common multiple is the smallest number that is a multiple of two (or more) whole numbers. It can not be 0.) What is the most important word in greatest common factor? Explain. ( Greatest is the most important word because we are looking for the largest number that is a factor of 2 (or more) whole numbers.) What is the most important word in least common multiple? Explain. ( Least is the most important word because we are looking for the smallest number that is a multiple of two (or more) whole numbers except for 0.) [HOMEWORK] Assign S48 for homework. (Answers on T131.) [QUIZ ANSWERS] T132 The quiz can be used at any time as extra homework or to see how students did on working with greatest common factors and least common multiples.
T116 Mathematics Success Level F TRANSPARENCY MASTER Directions: Complete the following SOLVE problem with your teacher. You will only complete the S step. Maddie and Elli are selling tickets to the school play. The tickets cost $3.50 each. They sold tickets for several hours over the weekend. If Maddie sold 5 tickets per hour and Elli sold 8 tickets per hour, what is the least number of tickets they would need to sell to have sold exactly the same amount of tickets? S Underline the question. This problem is asking me to find. Directions: Complete the rest of this page with your teacher. GCF of 12 and 18 = GCF of 14 and 42 = 1. What is a factor? 2. What does common mean? 3. What does greatest common factor mean?
Mathematics Success Level F T117 Here is the key to S40. Directions: Complete the following SOLVE problem with your teacher. You will only complete the S step. Maddie and Elli are selling tickets to the school play. The tickets cost $3.50 each. They sold tickets for several hours over the weekend. If Maddie sold 5 tickets per hour and Elli sold 8 tickets per hour, what is the least number of tickets they would need to sell to have sold exactly the same amount of tickets? S Underline the question. This problem is asking me to find the number of tickets Maddie and Elli have to sell to have sold the same amount of tickets. Directions: Complete the rest of this page with your teacher. GCF of 12 and 18 = 6 GCF of 14 and 42 = 14 1. What is a factor? a number that divides evenly into another number; a number that can be multiplied by another number to equal a product 2. What does common mean? shared 3. What does greatest common factor mean? the largest factor shared by two or more numbers
T118 Mathematics Success Level F TRANSPARENCY MASTER Directions: Complete this page with your teacher. 1. Find the GCF of 12 and 30. Factors of 12 Factors of 30 Common Factors of 12 and 30: Greatest Common Factor: 2. Find the GCF of 21 and 42. Factors of 21 Factors of 42 Common Factors: Greatest Common Factor:
Mathematics Success Level F T119 Directions: Complete this page with your teacher. Here is the key to S41. 1. Find the GCF of 12 and 30. Factors of 12 Factors of 30 Common Factors of 12 and 30: 1, 2, 3, 6 Greatest Common Factor: 6 2. Find the GCF of 21 and 42. Factors of 21 Factors of 42 Common Factors: 1, 3, 7, 21 Greatest Common Factor: 21
T122 Mathematics Success Level F TRANSPARENCY MASTER Directions: Complete this page with your teacher. LCM: 4 and 5 = 7 and 3 = 2 and 8 = 4 and 12 = 1. What is a multiple? 2. What does common mean? 3. What does least common multiple mean?
Mathematics Success Level F T123 Directions: Complete this page with your teacher. Here is the key to S43. LCM: 4 and 5 = 20 7 and 3 = 21 2 and 8 = 8 4 and 12 = 12 1. What is a multiple? The product of two numbers is a multiple of each number. 2. What does common mean? something that is shared 3. What does least common multiple mean? the least number that is a multiple of two or more numbers
T124 Mathematics Success Level F TRANSPARENCY MASTER Directions: Complete this page with your teacher. 1. Find the LCM of 2, 4, and 5. Multiples of 2 Multiples of 4 Multiples of 5 Common Multiples: Least Common Multiple: 2. Find the LCM of 6 and 8. Multiples of 6 Multiples of 8 Common Multiples: Least Common Multiple:
Mathematics Success Level F T125 Directions: Complete this page with your teacher. Here is the key to S44. 1. Find the LCM of 2, 4, and 5 Multiples of 2 Multiples of 4 Multiples of 5 Common Multiples: 20 Least Common Multiple: 20 2. Find the LCM of 6 and 8. Multiples of 6 Multiples of 8 Common Multiples: 24 Least Common Multiple: 24