Simplifying Expressions: Combining Like Terms Objective To simplify algebraic expressions by combining like terms.


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1 Simplifying Expressions: Combining Like Terms Objective To simplify algebraic expressions by combining like terms. epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family Letters Assessment Management Coon Core State Standards Curriculum Focal Points Interactive Teacher s Lesson Guide Teaching the Lesson Ongoing Learning & Practice Differentiation Options Key Concepts and Skills Add and subtract signed numbers. [Operations and Computation Goal 1] Add and subtract decimals. [Operations and Computation Goal 1] Add and subtract fractions. [Operations and Computation Goal 3] Substitute values for a variable to check simplified expressions. [Patterns, Functions, and Algebra Goal 2] Use distributive strategies to simplify algebraic expressions. [Patterns, Functions, and Algebra Goal 4] Key Activities Students use the distributive property to combine like terms. They simplify expressions containing like terms. Key Vocabulary like terms combine like terms simplify an expression Materials Math Journal 2, pp. 330 and 331 Study Link 9 2 slate Estimating and Measuring in Millimeters Math Journal 2, p. 332 metric ruler Students measure line segments to the nearest millimeter. Math Boxes 9 3 Math Journal 2, p. 333 Geometry Template Students practice and maintain skills through Math Box problems. Ongoing Assessment: Recognizing Student Achievement Use Math Boxes, Problems 2a and 2b. [Patterns, Functions, and Algebra Goal 2] Study Link 9 3 Math Masters, p. 288 Students practice and maintain skills through Study Link activities. READINESS Playing Algebra Election Student Reference Book, pp. 304 and 305 Math Journal 1, Activity Sheets 3 and 4 Math Masters, pp. 434 and 435 per group: 4 counters or pennies, 1 sixsided die, calculator, index cards (optional) Students practice substituting values for variables and solving equations. ENRICHMENT Simplifying and Evaluating Expressions Math Masters, p. 289 Students find missing terms in expressions. They also simplify and evaluate expressions. EXTRA PRACTICE Generating Equivalent Expressions Differentiation Handbook, p. 135 Students record equivalent expressions in namecollection boxes. They substitute values for variables to check that expressions are equivalent. ELL SUPPORT Building a Math Word Bank Differentiation Handbook, p. 130 Students add the phrases combine like terms and simplify an expression to their Math Word Banks. Advance Preparation For the optional Readiness activity in Part 3, students will need Algebra Election Cards and the Electoral Vote Map from Lesson Teacher s Reference Manual, Grades 4 6 pp Unit 9 More about Variables, Formulas, and Graphs
2 Getting Started Mental Math and Reflexes Students write the value of each expression when x = 2. 3x x + 8x 24 x + 2x + 3x + 4x 20 x 2x x  ( 1_ 2 )x 7 (2x)  (9) 5 Mathematical Practices SMP1, SMP2, SMP3, SMP5, SMP6, SMP8 Content Standards 6.EE.2, 6.EE.2b, 6.EE.3, 6.EE.4 Math Message These expressions have two terms. Rewrite each expression as a single term. 4y + 7y 4y  7y Study Link 9 2 FollowUp Briefly review answers. 1 Teaching the Lesson Math Message FollowUp WHOLECLASS DISCUSSION 4y + 7y Algebraic Thinking Draw two namecollection boxes on the board, one for each expression. Record students answers in the namecollection boxes. (See margin for example.) Discuss their answers and explanations. Some students might use objects or amounts to justify their answers. For example: Think of 4y + 7y as 4 oranges + 7 oranges. That s 11 oranges, so 4y + 7y = 11y. 4 dollars minus 7 dollars is 3 dollars owed, so 4y  7y is 3y. Remind students that the multiplication symbol ( or ) is frequently omitted. For example, 7y means 7 y, and 5(2y + 3) means 5 (2y + 3). Ask students to name other expressions that are equivalent to the expressions in the Math Message. Record these in the namecollection boxes. Sample answers for 4y + 7y: 12y 1y; 88y 8; 3y 3 + 2y Combining Like Terms (Math Journal 2, pp. 330 and 331) WHOLECLASS Algebraic Thinking Read the first two paragraphs on journal page 330 as a class. Terms are the items that are added and subtracted in expressions. Like terms are either number terms (constant terms) or terms that contain the same variables raised to the same power (variable terms). To support English language learners, discuss the meaning of like in this context. Remind students that the number part of a variable term is called the coefficient. Example 1 shows how to use the distributive property to combine like terms, which means to rewrite the sum or difference of like terms as a single term. ELL Date 9 3 Combining Like Terms Algebraic expressions contain terms. For example, the expression 4y + 2x  7y contains the terms 4y, 2x, and 7y. The terms 4y and 7y are called like terms because they are multiples of the same variable, y. To combine like terms means to rewrite the sum or difference of like terms as a single term. In the case of 4y + 2x  7y, the like terms 4y and 7y can be combined and rewritten as 3y. To simplify an expression means to write the expression in a simpler form. Combining like terms is one way to do that. Reminder: The multiplication symbol ( ) is often not written. For example, 4 y is often written as 4y, and (x + 3) 5 as (x + 3)5. Example 1: Simplify the expression 5x  (8)x. Use the distributive property. 5x  (8)x = (5 x)  (8 x) = (5  (8)) x = (5 + 8) x = 13 x, or 13x Check your answer by substituting several values for the variable. Check: Substitute 5 for the variable. Check: Substitute 2 for the variable. 5x  (8)x = 13x 5x  (8)x = 13x (5 5)  (8 5) = 13 5 (5 2)  (8 2) = (40) = (16) = = = 26 If there are more than 2 like terms, you can add or subtract the terms in the order in which they occur and keep a running total. Example 2: Simplify the expression 2n  7n + 3n  4n. 2n  7n = 5n 5n + 3n = 2n 2n  4n = 6n Therefore, 2n  7n + 3n  4n = 6n. Simplify each expression by rewriting it as a single term. 1. 6y + 13y 19y 2. 7g  12g 4x _ 2 x  1 1_ x 4. 3c  (5)c 2 13y 5n 15x 11y 5. 5y  3y + 11y 6. 6g  8g + 5g  4g 7. n + n + n + n + n 8. n + 3n + 5n  7n 9. 2x + 4x  (9)x x + 2x + 3x Math Journal 2, p g 8c g 2n 2x _369_EMCS_S_G6_U09_ indd 330 2/26/11 1:19 PM Lesson
3 Date 9 3 Combining Like Terms continued An expression such as 2y 6 4y 8 9y (3) is difficult to work with because it is made up of 6 different terms that are added and subtracted. There are 2 sets of like terms in the expression. The terms 2y, 4y, and 9y are 1 set of like terms. The constant terms 6, 8, and (3) are a second set of like terms. Each set of like terms can be combined into a single term. To simplify an expression that has more than one set of like terms, combine each set into a single term. Example 3: Simplify 2y 6 4y 8 9y (3) by combining like terms. Step 1 Combine the y terms. 2y 4y 9y 6y 9y 3y Step 2 Combine the constant terms. 6 8 (3) 2 (3) 5 Final result: 2y 6 4y 8 9y (3) 3y (5) 3y 5 Check: Substitute 2 for y in the original expression and the simplified expression. 2y 6 4y 8 9y (3) 3y 5 (2 º 2) 6 (4 º 2) 8 (9 º 2) (3) (3 º 2) (3) Simplify each expression by combining like terms. Check each answer by substituting several values for the variable y x 3x n 6 8n 2 3n 14. n π 2n 1 2 π x 9 1.4x d 2a (6a) 3d 15d 7y 24 2x 8 6n 4 3n 1 2, or 3n 2 1.1x 9.6 3d 8a Math Journal 2, p Redo the Math Message problems using the distributive property to combine like terms. To simplify 4y + 7y, think: 4y + 7y = (4 y) + (7 y) = (4 + 7) y = 11 y = 11y So, 4y + 7y = (4 + 7)y = 11y. To simplify 4y  7y, think: 4y  7y = (4 y)  (7 y) = (47) y = 3 y = 3y So, 4y  7y = (47)y = 3y. To simplify an expression means to write the expression in a simpler form. To support English language learners, explain that even though one expression is in simpler form, the expressions are equivalent. Combining like terms is one way to simplify an expression. Removing parentheses is another way. This method will be the focus of Lesson 94. Example 2 shows how to extend the method in Example 1 to combine more than two like terms. The first two terms are combined. The result is then combined with the next term and so on. Adjusting the Activity At the board, demonstrate a factoring procedure for combining like terms. Write the expression 2n  7n + 3n  4n. Enclose the expression in parentheses and cross out (or erase) each occurrence of the variable n. Then write the variable n to the right of the parentheses, saying that you have undistributed the variable. Evaluate the numerical expression within parentheses. The simplified expression is 6n. 2n  7n + 3n  4n = (2n/  7n/ + 3n/  4n/ )n = ( )n = 6n 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 Provide additional practice by writing several more expressions containing only one kind of term on the board or overhead. Then have students simplify them on their slates. Suggestions: 2b  (6b) + 4b 12b 4p  7p  (4p) 1p or p 1_ 2 m + 3_ 4 m  ( 5_ m) 10_ m, 5_ m, or 21_ w + 9.6w  7.5w 0.6w Partners complete the problems on journal page 330. When most students have finished, briefly go over the answers. 2 m 800 Unit 9 More about Variables, Formulas, and Graphs
4 Read the paragraphs at the top of journal page 331 and go over Example 3, which shows how to combine like terms when there is more than one kind of term. Demonstrate Example 3 on the board as shown below. Remind students that they identified variable terms and constant terms in their work with equations (Lesson 611). In the expression 2y y  89y + (3), the variable terms 2y, 4y, and 9y are one set of like terms. the constant terms 6, 8, and 3 are a second set of like terms. the variable terms are combined into one single term. the constant terms are combined into a second single term. variable terms variable terms constant terms Date 9 3 Estimating and Measuring in Millimeters All measurements are approximations. The pencil shown below measures about 4 inches. A more precise measurement is about inches. An even more precise measurement 6 is about 93 millimeters. The smaller the unit you use to measure an object, the more precise the measurement will be. 1. Explain why 93 millimeters is a more precise measurement than inches for the 16 length of the pencil. Measure each line segment below to the nearest millimeter. 2. A B Length of AB 3. D E Length of DE 4. Measure PQ. Then, in the space provided, draw a line segment that is 5 the length 8 of PQ. Label the segment RS. Length of P œqœ 56 P Q R Length of RœSœ 5. Measure FG. Then, in the space provided, draw a line segment that is 125% the length of FG. Label the segment JK. Length of FG 60 F G J 0 INCHES 1 S 2 0 CM Sample answer: Millimeters are smaller units than sixteenths of an inch. K 3 4 Length of JK constant terms Assign the problems on journal page 331. Go over the answers. Note that an instruction to check answers is included with the problem set. Remind students to check their answers by substituting at least two different values for the variable in each expression. Math Journal 2, p Ongoing Learning & Practice Estimating and Measuring in Millimeters (Math Journal 2, p. 332) PROBLEM SOLVING SMALLGROUP Date Students measure line segments to the nearest millimeter. Math Boxes 9 3 (Math Journal 2, p. 333) INDEPENDENT Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lessons 91 and 95. The skills in Problems 4 and 5 preview Unit 10 content. Writing/Reasoning Have students write a response to the following: Explain how the ratio in Problem 3c is related to the ratio of the corresponding sides. Sample answer: The ratio of the perimeters is equivalent to the ratios of the corresponding sides The area of the rectangle is 238 units 2. Write a number sentence to find the value of x. Number sentence Solve for x. 4. I am a regular polygon with all obtuse angles. I have the smallest number of sides of any polygon with obtuse angles. How many sides do I have? 5 Math Boxes x 9 x 238 7(9 x) 25 units 3. Triangles ABC and DEF are similar. a. Length of AB units b. Length of DF units 8 Perimeter of triangle ABC c. 7 Perimeter of triangle DEF units 2. Solve. a w 4 Solution 7 w 8 b. 24 p 8 4 Solution c. 4j 24 6j 120 Solution E 28 D p 20 j Use the diagonals to find the sum of the interior angle measures of the octagon below. C F A B Use your Geometry Template to draw this polygon below. Math Journal 2, p. 333 Sum of the interior angle measures 1, Lesson
5 Name Date STUDY LINK 9 3 Combining Like Terms Simplify each expression by rewriting it as a single term. 1. 3x 12x 2. (1 3 5 )y (1 3 )y (5t) 6t 4. 4d (3d) Complete each equation. 6 3p 5. 15k (9 )k p p 0.4p 3 15x 11t 7. (8 ) m 5m 8. j 4.5j 3.8j Simplify each expression by combining like terms. Check your answers by substituting the given values for the variables. Show your work on the back of this sheet. Example: 18 6m 2m 26 Combine the m terms. 6m 2m 8m Combine the number, or constant, terms So, 18 6m 2m 26 8m 44. Check: Substitute 5 for m. 18 (6 5) (2 5) 26 (8 5) b 9 4b 3b (2b) (5) Check for: b a 3 4 t 2 3 a (1 2 t) Check for: a 2 and t 2 Practice Study Link Master a 1 4 t 7b (32) (11) Math Masters, p y 1 0 d Ongoing Assessment: Recognizing Student Achievement Math Boxes Problems 2a and 2b Use Math Boxes, Problems 2a and 2b to assess students ability to solve equations. Students are making adequate progress if they can use a method (trial and error or equivalent equations) to find a solution and then check the solution by substituting it for the variable. Some students might be able to combine like terms to solve Problem 2c. [Patterns, Functions, and Algebra Goal 2] Study Link 9 3 (Math Masters, p. 288) Home Connection Students simplify algebraic expressions by combining like terms. 3 Differentiation Options INDEPENDENT READINESS Playing Algebra Election (Student Reference Book, pp. 304 and 305; Math Journal 1, Activity Sheets 3 and 4; Math Masters, pp. 434 and 435) PARTNER Min NOTE A state s electoral votes equal the total number of its representatives and senators in the United States Congress. The number of representatives depends on the state s population as determined by the decennial Census. The vote distribution on this map is based on the 2000 Census. Following the 2010 Census, it is likely that some states will lose votes and some will gain votes because of changes in the population. Algebraic Thinking Students practice substituting for variables in equations by playing Algebra Election, which was introduced in Lesson In this game, students travel through the United States capturing electoral votes by solving a variety of problems. They may play a shorter version of the game by going through the 32 cards only one time. You can tailor the game to address any skills you wish to review by writing additional problems on index cards. The team with the most votes after going through the cards once wins the game. NOTE: Alaska and Hawaii are not drawn to scale. Math Masters, pages 434 and 435 show the electoral votes for each state based on the 2000 census. 802 Unit 9 More about Variables, Formulas, and Graphs
6 Teaching Master ENRICHMENT INDEPENDENT Simplifying and Evaluating Expressions (Math Masters, p. 289) 5 15 Min Students extend their knowledge of simplifying expressions by finding missing terms in expressions. They also evaluate simplified expressions. Name Date 9 3 Simplifying and Evaluating Expressions Each expression on the left of the equal sign can be simplified to the expression on the right. Fill in the missing variable or constant terms. 1. 9x = 7x m  8n = 4m + 32n t  (2v) + 2w = 4t  2v + 4w 4. 3c + 2c + 6d  4d = 2d f  + 4g = f + g + 10 Simplify each expression below. Then evaluate the expression for x = 2, y = 3, and z = y  6z + 4x  2z  8y  12x 7. 3x x  9y + 6x  4z Sample answers for Problems 3 5: 2x 38m 40n 122t 4v 2w 5c 8 9f 3g 38x  6y  8z; 30 5x  9y  4z 10; 31 EXTRA PRACTICE PARTNER Generating Equivalent Expressions (Differentiation Handbook, p. 135) 5 15 Min To provide additional practice with finding equivalent expressions, have students complete namecollection boxes for several different expressions. Students exchange papers with a partner and check that their partner s expressions are equivalent by substituting a value for the variable in each of the expressions. Suggestions: 24x 1_ y 34.2d d 3 py g g p _EMCS_B_G6_MM_U09_ indd x + 96z + 3y  6z z  5x + 9y + x  y + 5z 10. 1_ 2 x  2_ 3 y + 3_ 4 z  3_ 2 x Math Masters, p x + 3y  12z  6; 474x + 8y + 8z; 0 x + 2_ 3 y + 3_ 4 z; 3 2/26/11 3:11 PM ELL SUPPORT SMALLGROUP Building a Math Word Bank (Differentiation Handbook, p. 130) 5 15 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 the phrases combine like terms and simplify an expression and then to represent these phrases using pictures and examples. See the Differentiation Handbook for more information. Lesson
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