The Distributive Property


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1 The Distributive Property Objectives To recognize the general patterns used to write the distributive property; and to mentally compute products using distributive strategies. epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family Letters Assessment Management Common Core State Standards Curriculum Focal Points Interactive Teacher s Lesson Guide Teaching the Lesson Ongoing Learning & Practice Differentiation Options Key Concepts and Skills Apply equivalent names for sums and differences. [Number and Numeration Goal 4] Recognize patterns in number sentences of partial products. [Patterns, Functions, and Algebra Goal 1] Write special cases for basic arithmetic operations. [Patterns, Functions, and Algebra Goal 1] Recognize order of operations in using and applying distributive strategies. [Patterns, Functions, and Algebra Goal 3] Use distributive strategies to mentally compute products. [Patterns, Functions, and Algebra Goal 4] Key Activities Students apply the distributive property to simplify algebraic expressions and mentally calculate products. They also use the distributive property to factor expressions Playing Getting to One Student Reference Book, p. 31 Math Masters, p. 448 per partnership: calculator; overhead calculator (optional) Students practice comparing decimal numbers and apply proportional reasoning skills. Math Boxes 9 Math Journal, p. 39 straightedge Students practice and maintain skills through Math Box problems. Ongoing Assessment: Recognizing Student Achievement Use Math Boxes, Problem 3. [Number and Numeration Goal ] Study Link 9 Math Masters, p. 86 Students practice and maintain skills through Study Link activities. ENRICHMENT Writing Number Stories Math Journal, p. 38 Students write number stories that can be solved using the distributive property. EXTRA PRACTICE Applying the Distributive Property Math Masters, p. 8 Students use the distributive property to solve problems. ELL SUPPORT Building a Math Word Bank Differentiation Handbook, p. 130 Students add the term distributive property to their Math Word Banks. Ongoing Assessment: Informing Instruction See page 9. Key Vocabulary distributive property Materials Math Journal, pp B Student Reference Book, pp. 48 and 49 Study Link 9 1 slate Advance Preparation Teacher s Reference Manual, Grades 4 6 pp Unit 9 More about Variables, Formulas, and Graphs
2 Getting Started Mathematical Practices SMP1, SMP, SMP3, SMP, SMP6, SMP, SMP8 Content Standards 6.NS.4, 6.EE., 6.EE.b, 6.EE.3 Mental Math and Reflexes Students find the total number of objects in a set when a fractional part of the set is given. Suggestions: 1_ of the people in the room is _ of the books on a shelf is _ of the marbles in a bag is _ of the crayons in a box is _ of the pages in a book is _ of the questions on a test is Discuss students strategies. Some students may prefer solving the problems by first translating them to equations. For example, 4_ 1 of the pages in a book is 40 can be translated as 4_ 1 x = 40. Math Message Be ready to explain how to mentally find the following products: 4 36 =? 99 8 =? $11.0 =? Study Link 9 1 FollowUp Briefly review the answers. 1 Teaching the Lesson Math Message FollowUp WHOLECLASS DISCUSSION Students share solution strategies. Help them record their strategies as number sentences. Be sure to include distributive strategies as you record solutions. Examples: 4 36 = 4 (30 + 6) = (4 30) + (4 6) = = 144, or 4 36 = 4 (404) = (4 40)  (4 4) = = = (1001) 8 = (100 8)  (1 8) = = 9, or 99 8 = (90 + 9) 8 = (90 8) + (9 8) = 0 + = 9 $11.0 = ($ $0.0) = ($1.00 )  ($0.0 ) = $ $.0 = $.0, or $11.0 = ($ $0.0) = ($11.00 ) + ($0.0 ) = $.00 + $.0 = $.0 ELL Lesson 9 93
3 Algebra The Distributive Property You have been using the distributive property for years, probably without knowing it. For example, when you solve 40 * with partial products, you think of as 0 and multiply each part by 40. The distributive property says: 40 * (0 ) (40 * 0) (40 * ). The distributive property can be illustrated by finding the area of a rectangle. * * * * 80 Adjusting the Activity To extend the activity and review order of operations, write the expressions without parentheses. A U D I T O R Y K I N E S T H E T I C T A C T I L E V I S U A L Show how the distributive property works by finding the area of the rectangle in two different ways. Method 1 Find the total width of the rectangle and multiply that by the height. A 3 cm * (4 cm cm) 3 cm * 6 cm 18 cm Method Find the area of each smaller rectangle, and then add these areas. A (3 cm * 4 cm) (3 cm * cm) 1 cm 6 cm 18 cm Both methods show that the area of the rectangle is 18 cm. 3 * (4 ) (3 * 4) (3 * ). This is an example of the distributive property of multiplication over addition. The distributive property of multiplication over addition can be stated in two ways: a * (x y) (a * x) (a * y) (x y) * a (x * a) (y * a) Student Reference Book, p. 48 Ask students to look for patterns in the number sentences. Include the following points in your discussion: One of the factors is rewritten as a sum (or difference) of two numbers, each of which can be easily multiplied by the other factor. Parentheses are useful for keeping track of this sum (or difference) when it is rewritten. The original product becomes the sum (or difference) of two simple products. This strategy for making mental calculations is based on the distributive property. This property gets its name because the factor outside the parentheses is distributed to each of the terms within the parentheses. To support English language learners, model the meaning of the word distribute. (See margin.) The papers are distributed to each student. Summarizing the Distributive Property (Student Reference Book, pp. 48 and 49) WHOLECLASS DISCUSSION The factor is distributed to each term. The distributive property of multiplication over subtraction can also be stated in two ways: a * (x y) (a * x) (a * y) (x y) * a (x * a) (y * a) Show how the distributive property of multiplication over subtraction works by finding the area of the shaded part of the rectangle in two different ways. Method 1 Multiply the width of the shaded rectangle by its height. A 3 cm * (6 cm cm) 3 cm * 4 cm 1 cm Algebra Method Subtract the area of the unshaded rectangle from the entire area of the whole rectangle. A (3 cm * 6 cm) (3 cm * cm) 18 cm 6 cm 1 cm Both methods show that the area of the shaded part of the rectangle is 1 cm. 3 * (6 ) (3 * 6) (3 * ) This is an example of the distributive property of multiplication over subtraction. Algebraic Thinking Use the two examples on pages 48 and 49 of the Student Reference Book to discuss how the distributive property summarizes students work in Lesson 91. Then review the four different general patterns for the distributive property. Remind students that, as with other equations, they can interchange the left and right sides. Distributive Property of Multiplication over Addition a (x + y) = (a x) + (a y) (x + y) a = (x a) + (y a) Distributive Property of Multiplication over Subtraction a (x  y) = (a x)  (a y) (x  y) a = (x a)  (y a) Use the distributive property to solve the problems * (100 40). (3 1) * * (80 ) 4. Use a calculator to verify that 1.3 * (46 89) (1.3 * 46) (1.3 * 89). Check your answers on page 43. Student Reference Book, p Unit 9 More about Variables, Formulas, and Graphs
4 Make sure students understand that these general statements do not show four different properties but are different ways of stating the same general property. Demonstrate this by writing special cases. Examples: a (x + y) = (a x) + (a y) (30 + ) = ( 30) + ( ) = 4 (x + y) a = (x a) + (y a) (30 + ) = (30 ) + ( ) = 4 a (x  y) = (a x)  (a y) (40  ) = ( 40)  ( ) = 4 (x  y) a = (x a)  (y a) (40  ) = (40 )  ( ) = 4 Ongoing Assessment: Informing Instruction Students may recognize that they can use the Commutative Property of Multiplication to write the second general pattern and special case in each example. Although they can also change the order of numbers or expressions being added, watch for students who try to change the order in which numbers or expressions are subtracted. Date 9 Time The Distributive Property The distributive property is a number property that combines multiplication with addition or multiplication with subtraction. The distributive property can be stated in 4 different ways. Multiplication over Addition For any numbers a, x, and y: a º (x y) (a º x) (a º y) (x y) º a (x º a) (y º a) Use the distributive property to fill in the blanks r 4 6 n 13 n f x (6 º d) (6 º ) ( º 1) ( º h) 1. 4 º (0 8) (4 º ) (4 º ). 6 º 34 ( º 30) ( º 4) 3. (6 º 0) (6 º 4) º (0 ) 4. ( ) º 8 (40 º 8) (6 º ). 8 º (90 3) ( º 90) (8 º 3) 6. (0 º ) (8 º ) ( ) º. 9 º (0 ) (9 º ) ( º ) 8. 4 º ( 6) ( º ) ( º ) 9. (41 19) º ( º ) ( º ) 10. (18 4) º r (18 º ) ( º r ) 11. º (w ) ( º w) ( º 6) 1. n º (13 ) ( º ) ( º ) 13. (f 8) º 1 ( º ) ( º ) 14. (9 º x) (1 º x) ( ) º 1. 6 º (d ) 16. º (1 h) Math Journal, p. 38 Multiplication over Subtraction For any numbers a, x, and y: a º(x y) (a º x) (a º y) (x y) º a (x º a) (y º a) Pose problems that students can solve mentally by applying the distributive property. Suggestions: , _ , ,98 Using the Distributive Property (Math Journal, p. 38) PARTNER Algebraic Thinking The problems on journal page 38 provide practice with four different ways of stating the distributive property. For most of the problems, there are many ways to fill in the blanks to obtain true sentences. Each problem, however, has a unique solution in the form of the distributive property. For example, in Problem 1, 4 (0 + 8) = (4 0) + (4 8) is a true sentence, but 4 (0 + 8) = (4 0) + (4 8) is the only solution in the form of the distributive property. When students have finished, go over their answers. Game Master Name Date Time 1 Getting to One Record Sheets Player s Name Player s Name Draw a line to separate each round. Draw a line to separate each round. Guess Display Result Guess Display Result on calculator Write: on calculator Write: (to nearest 0.01) L if too large (to nearest 0.01) L if too large S if too small if exact S if too small if exact Adjusting the Activity Divide journal page 38 into two sections problems without variables (Problems 1 9) and problems with variables (Problems 10 16). Have students use a blank sheet of paper to cover the second section while they work on the first section. A U D I T O R Y K I N E S T H E T I C T A C T I L E V I S U A L Math Masters, p. 448 Lesson 9 9
5 Factoring with the Distributive Property (Math Journal, pp B) WHOLECLASS Have students look at Problem on journal page 38. Remind them that this problem illustrates how the distributive property can be used to mentally solve the multiplication problem Tell students that when they apply the distributive property to solve multiplication problems mentally, they are distributing a number across a sum. In Problem, the 8 is distributed across the sum This process is called expanding the expression 8 (90 + 3). Now have students look at Problem 6. Ask them how Problem 6 is different from Problem. Sample answer: The left side of the equation in Problem 6 shows the expression expanded. Tell students that in this problem, the is undistributed from the expression (0 ) + (8 ). Applying the distributive property in this way is called factoring the expression (0 ) + (8 ). Because is a factor of both 0 and 8, it is a factor of the whole expression. In all the examples of factored expressions students have seen so far, the two addends have been written as products. Tell students that they can use their knowledge of greatest common factors to factor an expression even if the two addends are written as whole numbers. Write on the board. Walk students through the following steps to help them use the distributive property to factor this expression: Step 1: Find the greatest common factor of 33 and 1. List the factors of 33: 1, 3, 11, and 33. List the factors of 1: 1,, 3, 4, 6, and 1. From the lists, you can see that the greatest common factor of 33 and 1 is 3. Step : Write 33 and 1 as products, with their GCF as one of the factors. 33 = = 3 4 Step 3: Rewrite the original sum, substituting the products from Step for the addends = Step 4: Use the distributive property to factor the expression on the right side of the equal sign, or undistribute the = 3 (11 + 4) 9A Unit 9 More about Variables, Formulas, and Graphs
6 Point out that because you factored out the greatest common factor, the addends 11 and 4 have no common factor other than 1. Help students use mathematical language to express the meaning of the final number sentence. You might use language such as the following: 3 is a factor of the expression The sum is a multiple of the sum Factoring 3 out of the expression produces the expression 3 (11 + 4). Write other sums on the board. Ask students to identify the greatest common factor of the addends and use the GCF to factor the expression. Encourage students to check that the final two addends have no common factor other than 1. Suggestions: ; (9 + 13) ; 1 ( + 3) + 0 ; ( + 4) + 1 3; 3 (9 + ) ; (3 + ) ; ( ) 4 When students seem comfortable with the procedure, read journal page 38A as a class. Then have students work with a partner to complete the problems on journal page 38B. Date 9 Factoring Sums Time The Distributive Property of Multiplication over Addition a (x + y ) = a x + a y (a x ) + (a y ) = a (x + y ) (x + y ) a = x a + y a (x a) + (y a) = (x + y ) a The equation = 3 ( + ) is a special case of the distributive property in which the addends on the left side are written as whole numbers instead of products. This equation tells you a lot about these numbers. Here are some true statements about the relationships among the numbers and expressions in this equation: 3 is a factor of both 1 and 1. 3 is a factor of the expression The expression is a multiple of the expression +. When you use the distributive property to write as 3 ( + ), we say you factor 3 out of the sum You can use the distributive property to factor a sum of any two whole numbers. In this activity, you will factor out the greatest common factor of two addends. You will be rewriting the original sum as a multiple of another sum whose addends have no common factors other than 1. Complete the following steps for each sum. a. Find the greatest common factor of the two addends. b. Use the distributive property to factor the GCF out of the sum. Complete the number sentence to show the result. c. Fill in the blanks to give an example of what your number sentence shows. Example: a. Greatest common factor: b. Number sentence: = ( ) c. Fill in the blanks: The expression is a multiple of the expression. 38A_38B_EMCS_S_G6_MJ_U09_644.indd 38A Math Journal, p. 38A 3/9/11 11:13 AM Date 9 Time Factoring Sums continued Follow the directions on journal page 38A a. Greatest common factor: b. Number sentence: = ( + ) c. Fill in the blanks: is a factor of the sum a. Greatest common factor: b. Number sentence: 30 + = ( + ) c. Fill in the blank: The expression 30 + is a of the expression a. Greatest common factor: 4 b. Number sentence: ) = ( + c. Fill in the blanks: 4 is the greatest common factor of the numbers 48 and a. Greatest common factor: 9 b. Number sentence: c. Fill in the blanks: When the number 9 is factored out of the sum + 63, the result is the expression 9 (3 + ) a. Greatest common factor: b. Number sentence: c. Write your own sentence multiple Sample answer: + 63 = 9 (3 + ) 1 Sample answer: = 1 (3 + ) Sample answer: The sum is a multiple of the sum 3 +. Math Journal, p. 38B 38A_38B_EMCS_S_G6_MJ_U09_644.indd 38B 3/9/11 11:13 AM Lesson 9 9B
7 Getting to One Materials 1 calculator Players Skill Estimation Object of the game To correctly guess a mystery number in as few tries as possible. Directions 1. Player 1 chooses a mystery number that is between 1 and Player guesses the mystery number. 3. Player 1 uses a calculator to divide Player s guess by the mystery number. Player 1 then reads the answer in the calculator display. If the answer has more than decimal places, only the first decimal places are read. 4. Player continues to guess until the calculator result is 1. Player keeps track of the number of guesses.. When Player has guessed the mystery number, players trade roles and follow Steps 1 4 again. The player who guesses their mystery number in the fewest number of guesses wins the round. The first player to win 3 rounds wins the game. Player 1 chooses the mystery number 6. Player guesses: 4. Player 1 keys in: 4 6. Answer: 0.69 Too small. Player guesses: 3. Player 1 keys in: 3 6. Answer: 1.1 Too big. Player guesses: 6. Player A keys in: 6 6. Answer: 1. Just right! Advanced Version Allow mystery numbers up to 1,000. Student Reference Book, p. 31 Games For a decimal number, the places to the right of the decimal point with digits in them are called decimal places. For example, 4.06 has decimal places, 13.4 has 1 decimal place, and 0.80 has 3 decimal places. Links to the Future Students will apply their knowledge of the distributive property when they factor and multiply polynomials in future algebra courses. Ongoing Learning & Practice Playing Getting to One (Student Reference Book, p. 31; Math Masters, p. 448) PARTNER Divide the class into pairs. Distribute a calculator and a game record sheet (Math Masters, p. 448) to each partnership. Students read the directions on page 31 in their Student Reference Book. If an overhead calculator is available, ask a volunteer to demonstrate how to play the game. Encourage students to play a practice game. Math Boxes 9 (Math Journal, p. 39) INDEPENDENT Date 9 1 a. 1 Math Boxes 1. Use the distributive property to fill in the blanks. a. 8 º (30 4) ( 8 º 30 ) ( 8 º 4 ) 9 b. ( º) ( º6) 9º( 6) c. (0 6) º 10 (0 º 10) (6 º ) 9 d. (9 1) ()(9) ()(1) 3. Find the number. of what number is 1? 0 b. 3 of what number is? c. of what number is 14? 3 d. of what number is 9? 0 e of what number is 84? Time. Circle the expressions that represent the area of the rectangle. 4 a. 4m 8 b. 4 º m c. 8m d. (m ) º 4 e. 4( m) f. 8 m 4. Write,, or. a. 8 (1) 36 () m b. 1 (3 4 ) 3 º 8 c. 400 º 3 0 d. 1 / 3 º 10 1 e Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 94. The skill in Problem previews Unit 10 content. Writing/Reasoning Have students write a response to the following: Explain how you found the coordinates of the midpoint of AB in Problem d. Sample answer: For the x value of the midpoint, I found the mean of the xcoordinates. For the y value of the midpoint, I found the mean of the ycoordinates. Ongoing Assessment: Recognizing Student Achievement Math Boxes Problem 3 Use Math Boxes, Problem 3 to assess students ability to find the total number of objects in a set when a fractional part of the set is given. Students are making adequate progress if they complete parts a e. Some students might be able to mentally solve these problems. [Number and Numeration Goal ] Study Link 9 (Math Masters, p. 86) INDEPENDENT Plot and label points on the coordinate grid as directed. a. Plot (4,). Label it A. b. Plot (4,). Label it B. c. Draw line segment AB. B y x Home Connection Students practice using the distributive property. d. Name the coordinates of the midpoint of AB. ( 0, 0 ) A 34 Math Journal, p Unit 9 More about Variables, Formulas, and Graphs
8 3 Differentiation Options Name Date Time STUDY LINK 9 Study Link Master Using the Distributive Property Reminder: a º (x y) (a º x) (a º y) a º (x y) (a º x) (a º y) ENRICHMENT INDEPENDENT Writing Number Stories (Math Journal, p. 38) 1 30 Min To further explore applications of the distributive property, students choose expressions on journal page 38 and make up number stories that fit those expressions. Example: 4 (0 + 8) Four friends shared a pile of coins. Each person received dimes and 8 pennies. How much money was originally in the pile? Have students share their stories with other students. 1. Use the distributive property to rewrite each expression. a. (3 4) ( º ) ( ) b. (3 π) ( º ) ( ) c. (3 y) ( º ) ( ) d. (3 ( 4)) ( ) ( ( 4)) e. (3 ( π)) ( ) ( ( )) f. (3 ( y)) ( ) ( ( )). Use the distributive property to solve each problem. Study the first one. a. (110 ) ( 110)+( ) b. 0 (4 19) c. (3 0) 40 d. (90 8) 11 e. 9 (1 ) 4 y (0 4) (0 19) (3 40) (0 40) 1,80,000 3,80 (90 11) (8 11) (9 1) (9 ) Circle the statements that are examples of the distributive property. a. (80 ) (10 ) (80 10) b. 6 (3 0.) (6 3) 0. c. 1(d t) 1d 1t d. (a c) n a n c n e. (16 4m) f. (9 º 1 ) (1 3 º 1 ) (9 1 3 ) (4m 9.) º 1 Practice y EXTRA PRACTICE INDEPENDENT Applying the Distributive Property (Math Masters, p. 8) 1 Min Write each quotient in lowest terms Math Masters, p To provide extra practice applying the distributive property, have students write number models to show how they solved number stories. ELL SUPPORT SMALLGROUP Building a Math Word Bank (Differentiation Handbook, p. 130) 1 Min To provide language support for vocabulary terms, have students use the Word Bank template found on Differentiation Handbook, page 130. Ask students to write distributive property and represent the term with a picture and other words that describe it. See the Differentiation Handbook for more information. Name Date Time 9 Teaching Master Applying the Distributive Property 1. Cheng and of his friends are buying lunch. Each person gets a hamburger and a soda. How much money will they spend in all? Write a number model to show how you solved the problem. Answer Sample answer: $ ( ) c $1.00 $.90 Explain how the distributive property can help you solve Problem 1. Sample answer: Using the distributive property, you can first add the two values 1.10 and 0.90 and then multiply the sum by 6.. Minowa signed her new book at a local bookstore. In the morning she signed 36 books, and in the afternoon she signed 1 books. It took her minutes to sign each. How much time did she spend signing books? Sample answer: Write a number model to show how you solved the problem. (36 1) t Answer hours and 1 minutes 3. Ms. Hays bought fabric for the school musical chorus. She bought 4 yards each of one kind for 30 group costumes and 4 yards each of another kind for 6 soloists. How many yards did she buy in all? Sample answer: Write a number model to show how you solved the problem. Answer (30 4) (6 4) y 144 yards 4. Mr. Katz gave a party because all the students got 100% on their math test. He had budgeted $1.1 per student. It turned out that he saved $0. per student. If there are 30 students, how much did he spend? Sample answer: Write a number model to show how you solved the problem. (30 1.1) (30 0.) n Answer $.00 Fill in the missing numbers according to the distributive property ( 0 8 ) 6 6. ( 0 6) ( 8 6) (0 8) 6 Math Masters, p. 8 Lesson 9 9
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