Finding the Greatest Common Divisor and Least Common Multiple Using Manipulatives
Grade Level: 6th Grade
? Students will be able to determine the Greatest Common Divisor of a number using manipulatives evidenced by completing the common factors group task with 80% accuracy.
? Students will be able to determine the Least Common Multiple of a number using manipulatives by completing the school supplies task with 80% accuracy.
? Students will be able to orally express with a partner how they used manipulatives to determine the GCD and LCM.
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).
Engage effectively in a range of collaborative discussions (one-on-one, in groups, and teacher-led) with diverse partners on grade 6 topics, texts, and issues, building on others’ ideas and expressing their own clearly.
Materials: Integer Chips and Place Value Mats (Green=ones, Blue=Tens, Red=Hundreds, Yellow= Thousands), Promethean Board, Computer, Factor or Multiples Powerpoint, Paper, Pencil, Guided Practice Problems, Chart Paper, Markers, Finding Common Factors Task Sheet, School Supply Task Sheet.
Anticipatory Set/Warm-Up: The teacher will explain to students that today we will be learning how to find the least common multiple and greatest common divisor (factor) with manipulatives. The teacher will ask students to think about what a multiple is and then share their definition with their partner. The teacher will then ask students to share out. The teacher will ask students to think about what a factor (divisor) is and then share their definition with their partner. The teacher will then ask students to share out. As students are talking, the teacher will be cruising the room listening to conversations. Many students get factors and multiples mixed up, so the next natural teaching progression would be to show the Factors or Multiples Powerpoint.
The teacher will introduce the vocabulary of Least Common Multiple and Greatest Common Divisor (Factor) and their definitions.
The teacher will have students share the definitions of this vocabulary with their shoulder partners and then share out.
The teacher will have the students line up at the front of the classroom. The teacher will explain to the students that a chosen student is a concert promoter, and they (the students) will be receiving either a backstage pass, floor seats or balcony seats. The teacher will explain to the students that are in line as a multiple of 2 will receive a balcony seat, a multiple of 4 will receive a backstage pass, and every 5th person will receive a floor seat. The teacher will have the promoter pass out the tickets. The teacher will lead students in strategic discussion to help them determine that the lucky concert goer that received all three tickets is the Least Common Multiple. The teacher will then write out the problem on the board mathematically to model how finding the least common multiple is found. Students will sit down at their desks, which have integer chips and the place value mats out.
Next, the teacher will explain to students that they will become florists. Using their integer chips, they will be expected to divide up a set number of flowers equally in vases. The teacher will explain that each vase should hold the same number of flowers, as well as the same combination of flowers. The problem reads as such: If you have 18 lilies, 12 hydrangeas, and 9 peonies, how many vases can you make? How many of each type of flower will be in a vase? How many total flowers will be in each vase? The students can use the integer chips on their place value mats (the mats are to prevent the chips from sliding off the table, as they are felt and have some grab.) to work out the problem. The teacher will cruise the room while students are working to determine progress and to help when needed. After students have shown that they have completed the task, the teacher will lead a directed discussion to help the students understand that the number of vases is the GCF. The teacher will write the problem on the board mathematically to model how the GCF is found.
? The teacher will display the following problems on the Promethean board:
1. A gardener has 27 pansies and 36 daisies. He plants an equal number of each type of flower in each row. What is the greatest possible number of pansies in each row?
2. Fourteen boys and 21 girls will be equally divided into groups. Find the greatest number of groups that can be created if no one is left out.
3. A grocery store clerk has 16 oranges, 20 apples, and 24 pears. The clerk needs to put an equal number of apples, oranges, and pears into each basket. What is the greatest number of baskets that can be made so that no fruit is left?
4. The science department buys the equipment shown in the table. The bought all three items this year. In how many years will they have to buy all three items again?
Every 5 years
Every 4 years
Every 2 years
? The teacher will walk students through solving the first two problems, with students taking notes on their paper. Once both problems are solved, the teacher will have the students solve the other two problems. Students will have full use of the integer chips to aid in solving the problems.
? The teacher will give the students one minute and thirty seconds to solve the problems. While they are solving the problems, the teacher will cruise the classroom to note progress, and correct any misconceptions.
? Once the three minutes are up, the teacher will go over the problems, pointing out what students were doing well or performing incorrectly.
Students will be broken up into homogeneous groups of 4 for their independent practice group activity. Groups will be given one of two tasks: Finding Common Factors, or the School Supply Task. The teacher will provide students with chart paper and markers. The teacher will explain to students that they will complete their task using the chart paper, and then explain their process to the rest of the class.
Common Factors Task
1. Two students are having a party. They want to make treat bags for their guests. They want each bag to be identical with nothing leftover. They have 36 Silly Bandz and 72 pieces of bubble gum to put in the bags. What is the greatest number of treat bags they can make and how many of each item will be in each treat bag?
2. Mitzi is making trail mix out of 48 bags of nuts and 32 bags of dried cranberries. She wants each new portion of trail mix to be identical containing the same combination of nuts and cranberries with nothing left over. What is the greatest number of portions of trail mix Mitzi can make and how much of each ingredient will be in each portion?
3. The Junior Beta Club is making food baskets for the local homeless shelter. They asked for donations and they received 88 cans of food and 44 loaves of bread. If they want all the baskets to be the same with nothing left over, how many baskets can they make and how many of each item will be in each basket?
4. Keesha baked 4 dozen oatmeal cookies and 30 chocolate chip cookies. She wants to divide the cookies into plastic containers with the same amount of cookies in each container. If she wants the container to hold the greatest number of cookies possible how many containers does she need and how many of each cookie will be in each container?
School Supply Task
The Parents Teachers Association (PTA) at your school donated school supplies to help increase student creativity and student success in the classroom. Your teacher would like you to create kits that include one package of colored pencils, one glue stick, and one ruler. When you receive the supplies, you notice the colored pencils are packaged 12 boxes to a case, the rulers are packaged 30 to a box, and glue sticks are packaged 4 to a box.
1. What is the smallest number of each supply you will need in order to make the kits and not have supplies left over? Explain your thought process.
2. How many packaged rulers, colored pencils, and glue sticks will you need in order to make the kits? Explain the process you used to determine how many packages are needed for each supply.
Check for Understanding: There are multiple occasions to check for understanding built into the lesson. The teacher has students think-pair-share in the beginning of the lesson. The teacher also leads directed discussion to help students determine they have found the LCM and GCF in the input section of the lesson. The teacher frequently roams the room during portions where students are working in order to squash misconceptions and to help when needed.
During input, the teacher will have students think-pair-share the definitions of greatest common divisor (factor) and least common multiple. The teacher will have the students share out to determine understanding.
During guided practice, the teacher will roam the room to informally assess students and make sure they understand how to complete the practice problems.
The students will present how their presentations for the task they were assigned and be graded on the following rubric.
Displays errors in knowledge of mathematical concept.
Gives rudimentary explanation of mathematical concept.
Clearly articulates mathematical concepts.
Clearly articulates mathematical concepts and develops connections between concepts. .
Difficulty explaining procedures.
Explains procedures without difficulty.
Explains procedures without difficulty and provides partial explanations as to why procedures are appropriate.
Explains procedures without difficulty and provides full explanation why procedures are appropriate.
Inappropriate examples or representations.
Examples or representations are appropriate but no connection made.
Examples or representations are appropriate and explained. Connections are explained.
Examples or representations are appropriate and explained.Connections are deeply discussed and explained.
Inappropriate use of mathematical vocabulary.
Adequate use of mathematical vocabulary and symbols.
Appropriate use of mathematical vocabulary and symbols.
Excellent use of mathematical vocabulary and symbols.
No structure or chaotic.
Some structure including an introduction and conclusion.
Clearly defined structure with some transitions.
Well defined structure with nice transitions and effective introduction and conclusion.
Finding the greatest common divisor and least common multiple using manipulatives enhances students understanding of place value through the use of integer chips that represent the different place values. Students continue to use what they know about place value to multiply or divide using the manipulatives.