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G6 m4-d-lesson 11-t G6 m4-d-lesson 11-t Document Transcript

  • Lesson 11: Factoring Expressions Date: 3/26/14 113 ©2013CommonCore,Inc. Some rights reserved.commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6•4Lesson 11 Lesson 11:Factoring Expressions Student Outcomes  Students model and write equivalent expressions using the distributive property. They move from an expanded form to a factored form of an expression. Fluency Exercise(5 minutes) GCF Sprint Classwork Example 1 (8 minutes) Example 1 a. Use the model to answer the following questions. How many fives are in the model? How many threes are in the model? What does the expression represent in words? The sum of two groups of five and two groups of three. What expression could we write to represent the model? b. Use the new model and the previous model to answer the next set of questions. Scaffolding: For students struggling with variables, you can further solidify the concept by having students replace the variables with whole numbers to prove that the expressions are equivalent. MP.7
  • Lesson 11: Factoring Expressions Date: 3/26/14 114 ©2013CommonCore,Inc. Some rights reserved.commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6•4Lesson 11 How many fives are in the model? How many threes are in the model? What does the expression represent in words? Two groups of the sum of five and three. What expression could we write to represent the model? or c. Is the model in part (a) equivalent to the model in part (b)? Yes, because both expressions have two s and two s. Therefore, . d. What relationship do we see happening on either side of the equal sign? On the left hand side, is being multiplied times and then times before adding the products together. On the right hand side, the and are added first and then multiplied by . e. In 5th grade and in Module 2 of this year, you have used similar reasoning to solve problems. What is the name of the property that is used to say that is the same as ? The distributive property. Example 2 (5minutes) Example 2 Now, we will take a look at an example with variables. Discuss the questions with your partner. What does the model represent in words? plus plus plus , two ’s plus two ’s, two times plus two times What does mean? means that there are s or . How many s are in the model? How many are in the model? MP.7
  • Lesson 11: Factoring Expressions Date: 3/26/14 115 ©2013CommonCore,Inc. Some rights reserved.commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6•4Lesson 11 What expression could we write to represent the model? How many s are in the expression? How many s are in the expression? What expression could we write torepresent the model? Are the two expressions equivalent? Yes, both models include and Therefore, Example 3 (8 minutes) Example 3 Use GCF and the distributive property to write equivalent expressions. 1. What is the question asking us to do? We need to rewrite the expression as an equivalent expression in factored form which means the expression is written as the product of factors.The number outside of the parentheses will be the GCF. How would Problem 1 look if we expanded each term? What is the GCF in Problem 1? How can we use the GCF to rewrite this? goes on the outside and will go inside the parentheses.  Let’s use the same ideas for Problem 2. Start by expanding the expression and naming the GCF. MP.7
  • Lesson 11: Factoring Expressions Date: 3/26/14 116 ©2013CommonCore,Inc. Some rights reserved.commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6•4Lesson 11 2. What is the question asking us to do? We need to rewrite the expression as an equivalent expression in factored form which means the expression is written as the product of factors. The number outside of the parentheses will be the GCF. How would Problem 2 look if we expanded each term? What is the GCF in Problem 2? The GCF is . How can we use the GCF to rewrite this? I will factor out the from both terms and place it in front of the parentheses.I will place what is left in the terms inside the parentheses: . 3. Is there a greatest common factor in Problem 3? Yes, when I expand I can see that each term has a common factor . Rewrite the expression using the distributive property. 4. Explain how you used GCF and the distributive property to rewrite the expression inProblem4. I first expanded each term. I know that goes into , so used it in the expansion. I determined that , or , is the common factor.So, on the outside of the parentheses I wrote , and on the inside I wrote theleftover factor, . Why is there a in the parentheses? When I factor out a number, I am leaving behind the other factor that multiplies to make the original number. In this case, when I factor out an from , I am left with a because . How is this related to the first two examples? In the first two examples, we saw that we could rewrite the expressions by thinking about groups. We can either think of as groups of and groups of or as groups of the sum of .This shows that is the same as . MP.7
  • Lesson 11: Factoring Expressions Date: 3/26/14 117 ©2013CommonCore,Inc. Some rights reserved.commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6•4Lesson 11 Exercises (12 minutes) If times allows, you could have students practice these questions on white boards or small personal boards. Exercises 1. Apply the distributive property to write equivalent expressions. a. b. c. d. e. f. g. h. 2. Evaluate each of the expressions below. a. and and b. and
  • Lesson 11: Factoring Expressions Date: 3/26/14 118 ©2013CommonCore,Inc. Some rights reserved.commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6•4Lesson 11 Lesson Summary An Expression in Factored Form: An expressionthat is a product of two or more expressions is said to be in factored form. c. and d. Explain any patterns that you notice in the results to parts (a)–(c). Both expressions in parts (a)–(c) evaluated to the same number when the indicated value was substituted for the variable. This shows that the two expressions are equivalent for the given values. e. What would happen if other values were given for the variables? Because the two expressions in each part are equivalent, they evaluate to the same number, no matter what value is chosen for the variable. Closing (3 minutes) How can use you use your knowledge of GCF and the distributive property to write equivalent expressions? Find the missing value that makes the two expressions equivalent. Explain how you determine the missing number. I would expand each term and determine the greatest common factor. The greatest common factor is the number that is placed on the blank line. Exit Ticket (4 minutes)
  • Lesson 11: Factoring Expressions Date: 3/26/14 119 ©2013CommonCore,Inc. Some rights reserved.commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6•4Lesson 11 Name Date Lesson 11: Factoring Expressions Exit Ticket Use greatest common factorand the distributive property to write equivalent expressions. 1. 2. 3.
  • Lesson 11: Factoring Expressions Date: 3/26/14 120 ©2013CommonCore,Inc. Some rights reserved.commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6•4Lesson 11 Exit Ticket Sample Solutions Use greatest common factor and the distributive property to write equivalent expressions. 1. 2. 3. Problem Set Sample Solutions 1. Use models to prove that is equivalent to . 2. Use greatest common factorand the distributive property to write equivalent expressions for the following expressions. a. or b. c. d.
  • Lesson 11: Factoring Expressions Date: 3/26/14 121 ©2013CommonCore,Inc. Some rights reserved.commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6•4Lesson 11 Greatest Common Factor – Round 1 Directions: Determine the greatest common factor of each pair of numbers. 1. GCF of and 16. GCF of and 2. GCF of and 17. GCF of and 3. GCF of and 18. GCF of and 4. GCF of and 19. GCF of and 5. GCF of and 20. GCF of and 6. GCF of and 21. GCF of and 7. GCF of and 22. GCF of and 8. GCF of and 23. GCF of and 9. GCF of and 24. GCF of and 10. GCF of and 25. GCF of and 11. GCF of and 26. GCF of and 12. GCF of and 27. GCF of and 13. GCF of and 28. GCF of and 14. GCF of and 29. GCF of and 15. GCF of and 30. GCF of and Number Correct: ______
  • Lesson 11: Factoring Expressions Date: 3/26/14 122 ©2013CommonCore,Inc. Some rights reserved.commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6•4Lesson 11 Greatest Common Factor – Round 1[KEY] Directions: Determine the greatest common factor of each pair of numbers. 1. GCF of and 16. GCF of and 2. GCF of and 17. GCF of and 3. GCF of and 18. GCF of and 4. GCF of and 19. GCF of and 5. GCF of and 20. GCF of and 6. GCF of and 21. GCF of and 7. GCF of and 22. GCF of and 8. GCF of and 23. GCF of and 9. GCF of and 24. GCF of and 10. GCF of and 25. GCF of and 11. GCF of and 26. GCF of and 12. GCF of and 27. GCF of and 13. GCF of and 28. GCF of and 14. GCF of and 29. GCF of and 15. GCF of and 30. GCF of and
  • Lesson 11: Factoring Expressions Date: 3/26/14 123 ©2013CommonCore,Inc. Some rights reserved.commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6•4Lesson 11 Greatest Common Factor– Round 2 Directions: Determine the greatest common factor of each pair of numbers. 1. GCF of and 16. GCF of and 2. GCF of and 17. GCF of and 3. GCF of and 18. GCF of and 4. GCF of and 19. GCF of and 5. GCF of and 20. GCF of and 6. GCF of and 21. GCF of and 7. GCF of and 22. GCF of and 8. GCF of and 23. GCF of and 9. GCF of and 24. GCF of and 10. GCF of and 25. GCF of and 11. GCF of and 26. GCF of and 12. GCF of and 27. GCF of and 13. GCF of and 28. GCF of and 14. GCF of and 29. GCF of and 15. GCF of and 30. GCF of and Number Correct: ______ Improvement: ______
  • Lesson 11: Factoring Expressions Date: 3/26/14 124 ©2013CommonCore,Inc. Some rights reserved.commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6•4Lesson 11 Greatest Common Factor– Round 2 [KEY] Directions: Determine the greatest common factor of each pair of numbers. 1. GCF of and 16. GCF of and 2. GCF of and 17. GCF of and 3. GCF of and 18. GCF of and 4. GCF of and 19. GCF of and 5. GCF of and 20. GCF of and 6. GCF of and 21. GCF of and 7. GCF of and 22. GCF of and 8. GCF of and 23. GCF of and 9. GCF of and 24. GCF of and 10. GCF of and 25. GCF of and 11. GCF of and 26. GCF of and 12. GCF of and 27. GCF of and 13. GCF of and 28. GCF of and 14. GCF of and 29. GCF of and 15. GCF of and 30. GCF of and