Exemplar 6 th Grade Math Unit: Prime Factorization, Greatest Common Factor, and Least Common Multiple


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1 Exemplar 6 th Grade Math Unit: Prime Factorization, Greatest Common Factor, and Least Common Multiple Unit Plan Components Big Goal Standards Big Ideas Unpacked Standards Scaffolded Learning Resources Appropriate Length of Time Aligned Assessment Comments This unit plan identifies a big goal for the overall unit assessment, as well as goals for each daily formative assessment. The big goal is written for all students, not a set percentage. Math content standards have been identified from the Common Core State Standards. Additionally, corresponding Mathematical have been selected for this unit. The Mathematical logically connect to the unit s content standards. Remediation standards have been identified from the fifth grade Common Core State Standards and mastery of them will support mastery of the first grade standards. Enrichment standards have not been identified because there are no corresponding standards in seventh grade. However, with the CCSS enrichment should be deepening student understanding of the grade level standards. The big ideas for this unit have been identified and written in with the unit s Big Goal. The CCSS have been unpacked into daily objectives. Each daily objective is written with a measurable verb and can be completed in a single lesson. The daily objectives build on one another in a logical sequence. Resources have been identified for the overall unit, as well as for individual objectives. The unit is designed for 13 days. There are 11 daily objectives, plus one buffer day and one day for the assessment. The buffer day is included to circumvent potential schedule changes. Each daily objective has one corresponding assessment item. Because all daily objectives build to a single standard, there is not a need for multiple assessment items per objective.
2 Prime Factorization, Greatest Common Factor and Least Common Multiple Common Core Standard: 6.NS.4 Mathematical 6. Notes and Resources 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 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express as 4(9 + 2) Construct viable arguments and critique the reasoning of others: In grade 6, students construct arguments using verbal or written explanations accompanied by expressions, equations, inequalities, models, and graphs, tables, and other data displays (i.e. box plots, dot plots, histograms, etc.). They further refine their mathematical communication skills through mathematical discussions in which they critically evaluate their own thinking and the thinking of other students. They pose questions like How did you get that?, Why is that true? Does that always work? They explain their thinking to others and respond to others thinking. There are resources located for individual objectives throughout the unit plan. The resources included in this section apply to multiple objectives. They should be used to help develop the teacher s content knowledge and to consider how topics could be introduced to students. Khan Academy  Greatest Common Factor and Least Common Multiple (including word problems) Inside Mathematics provides resources and videos that illustrate what a classroom focused on discussion and explanation could look like. Unit Goal Remediation Standards Enrichment Standards Unit Timing All students will score at least an 80% on the End of Unit Assessment. All students will score at least an 80% on daily formative assessments (e.g. observational notes, exit tickets, etc.) By the end of this unit students will be able to work with factors through 12 and multiples through 100, both in straight math and word problems. Additionally, students will develop the skill of explaining their mathematical reasoning (mathematical practices) as they apply the content. 5.NBT.5. Fluently multiply multidigit whole numbers using the standard algorithm. 5.NBT.6. Find wholenumber quotients of whole numbers with up to fourdigit dividends and twodigit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. There are no additional standards to push toward enrichment, however enrichment should be created through these standards and could include, more challenging word problems, larger numbers, or working with more than 2 numbers. This unit has been designed for 13 days. There are 11 daily objectives, plus one buffer day and a day for the assessment. Each objective is intended to be taught in a single day. Objectives are listed in the order they should be taught.
3 Mathematical Objective Notes Assessment Item Answer reasoning when determining if a number is prime or composite. An explanation of prime and composite numbers, and how to identify them are included at Khan Academy  Prime and Composite Numbers This is a remedial objective, found in the 5 th grade Common Core State Standards. However, because it is a foundational skill for later objectives, it is included in this unit plan. Teachers should use their judgment when analyzing diagnostic assessment data to determine the best setting for teaching this objective (whole class, small group, etc.). Determine if the numbers below are prime or composite. Then, justify your decision I know that 32 is a composite number because it has more than 2 factors. Factors of 32 include: 1, 32, 2, 16, 4, and 8. I know that 2 is prime because it only has two factors, 1 and itself. NOTE: Student must answer both parts of the question correctly to receive credit. Common Misconception: Students often think that all odd numbers are prime and all even numbers are composite. To circumvent this misconception, teachers should provide clear definitions and examples early in the lesson. finding the greatest common factor of two whole numbers less than 100 using factors lists. NOTE: 1 is neither prime nor composite. Explain how you would find the greatest common factor of 36 and 72 using factor lists. I know that factors are the numbers I multiply together to get a product. 36: 1, 2, 3, 4, 6, 9, 12, 18, and 36 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 Now I have to find the largest factor that they have in common. I see that it s 36. GCF = 36
4 Mathematical Objective Notes Assessment Item Answer finding the prime factorization of a number less than 100. finding the greatest common factor of two whole numbers less than 100 using prime factorization. rewriting addition sentences using the distributive property and greatest common factors. finding the least common multiple of two whole numbers less than or equal to 12 using multiple lists. There is not an explicit lesson on using exponents when writing the prime factorization of a number. Teachers should analyze the diagnostic assessment data to determine student proficiency and understanding of using exponents. A mini lesson on exponents may need to be included prior to this lesson. Explain how you would find the prime factorization of 36. Explain how you would find the greatest common factor of 50 and 48 using prime factorization. Explain how you would rewrite by using the distributive property. Explain how you would use multiple lists to find the least common multiple of 8 and 9. Possible explanation is included on the assessment attachment. Answer 36 = 3 x 3 x 2 x 2 or 3 2 x 2 2 Possible explanation included on the assessment attachment Answer The greatest common factor is 2. To rewrite an addition expression I need to find the GCF. Using factor lists I know that 7 is the GCF of 84 and 28. I put the GCF on the outside of the parenthesis and its other factor inside: 7 (12+4) I know that multiples are the To use multiple lists I have to list the multiples of 8 and 9. 8: 8, 16, 24, 32,40, 48, 56, 64, 72, 80, 88, 96 9: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108 Next I need to find the smallest multiple that is in both lists. I see that number is 72. LCM = 72
5 Mathematical Objective Notes Assessment Item Answer using prime factorization to find the least common multiple of two numbers that are less than or equal to 12.. solving word problems involving greatest common factor. solving word problems involving least common multiple. Students should explain how they knew it was a great common factor problem in their explanation. Although not explicit in the objective, in their explanation students should explain how they knew it was a least common multiple problem. Explain how you would use prime factorization to find the least common multiple of 12 and 10. Thayer bought 48 cupcakes and bought 32 pieces of pizza for his sister s birthday party. Each guest will receive the same number of cupcakes and pizza. What is the maximum number of guests that can attend the party? How many cupcakes will each guest receive? How many pieces of pizza will each guest receive? Explain your thinking as you solve this problem. Train A passes the Haddonfield Station every 10 minutes. Train B passes the station every 8 minutes. In how many minutes will they pass the station at the same time? Explain your thinking as you solve this problem. Possible explanation included on the assessment attachment Answer The least common multiple is 60. NOTE: Students could choose to solve using factor lists or prime factorization. Teachers should use their judgment and the explanations provided in previous GCF problems when determining student mastery. Sixteen guests can attend the party (GCF) Each guest will receive 2 pieces of pizza and 3 cupcakes. NOTE: Students could choose to solve using multiple lists or prime factorization. Teachers should use their judgment and the explanations in previous LCM problems when determining student mastery. The trains will pass the station at the same time in 40 minutes.
6 Score Level Mathematical Knowledge and Explanation The answer is correct and all steps necessary to solve are shown. If necessary, the answer is labeled (typically for word problems) The written or oral explanation includes details about what they did and why they did it with specific correct vocabulary If specified by the question, includes a picture or model with a clear explanation of all components The answer is correct and all steps necessary to solve are shown If necessary, the answer is labeled (typically for word problems) The written or oral explanation includes details about what they did, but may be less specific with why they did it If specified by the question, includes a picture or provides a model with a mostly clear explanation of all components The answer is correct, but the explanation provided is incorrect or minimal The answer is incorrect, but the explanation provided demonstrates understanding of most of the concept If the assessment item requires a picture or model, but it is not included, students score a 2 (even with a correct answer and explanation) 1 Both the answer and the explanation are incorrect 0 The answer is incorrect and an explanation is not provided
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