Solving Simple Equations Objective To use trial and error and a coverup method to


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1 Solving Simple Equations Objective To use trial and error and a coverup method to solve equations. 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 Use estimation strategies to solve problems. [Operations and Computation Goal ] Determine whether equalities are true or false. [Patterns, Functions, and Algebra Goal ] Apply the order of operations to solve equations. [Patterns, Functions, and Algebra Goal 3] Key Activities Students solve equations using the trialanderror method. They also learn how to solve equations using the coverup method. Ongoing Assessment: Recognizing Student Achievement Use journal page 8. [Patterns, Functions, and Algebra Goal ] Key Vocabulary variable open sentence solution trialanderror method coverup method Materials Math Journal, p. 8 Student Reference Book, pp. and 3 Study Link 6 7 Solving Map Problems Using Absolute Value Math Journal, pp. 9A and 9B Students use absolute value to solve problems involving distances on a map. Math Boxes 6 8 Math Journal, p. 9 Students practice and maintain skills through Math Box problems. Study Link 6 8 Math Masters, p. 0 Students practice and maintain skills through Study Link activities. READINESS Using the Broken Calculator Routine calculator Students use estimation strategies to find solutions to open sentences. ENRICHMENT Solving Challenging Equations Math Masters, p. 03 Students use trial and error to reason through relationships and solve equations. Advance Preparation If you plan to use the optional Readiness activity in Part 3, consider discussing Broken Calculator with a fourthgrade or fifthgrade teacher. Teacher s Reference Manual, Grades 6 pp. 8 9 Lesson 6 8 7
2 Getting Started Mathematical Practices SMP, SMP3, SMP, SMP6, SMP7, SMP8 Content Standards 6.NS.7, 6.NS.7c, 6.NS.7d, 6.NS.8, 6.EE.b, 6.EE., 6.EE.7 Mental Math and Reflexes Students solve number sentences mentally. Suggestions: 6 + x = x = 9 z = z = 6 n  () = 30 n =.b = 0 b = 00 t_  = 8 t = 96 p = p =.6 Math Message Given 3x  = 0, explain how you know the solution is 8. Study Link 6 7 FollowUp Go over the answers with the class. Teaching the Lesson Math Message FollowUp (Student Reference Book, pp. and 3) ELL WHOLECLASS DISCUSSION Algebraic Thinking Use the Math Message to discuss the following ideas. To support English language learners, write these ideas on the board along with examples. Number sentences may contain one or more variables letters or symbols that represent a missing or an unknown number. Such number sentences are called open sentences. An open sentence is a special kind of number sentence. Because 3x  = 0 contains the variable x, it is an open sentence. An open sentence is neither true nor false. However, when the variables in an open sentence are replaced by numbers, the resulting number sentence is either true or false. If the sentence is true, then the number used to replace the variable is called a solution of the sentence. Because 3(8)  = 0 is a true number sentence, 8 is the solution to the open sentence 3x  = 0. Do the Check Your Understanding problems on page 3 of the Student Reference Book as a class. Draw a chart on the board and record the results as follows: Equation 6 + c = 0 = 6 z Solution 7 Replace Variable with Solution 6 + = 0 true = 6 7 true 7 Unit 6 Number Systems and Algebra Concepts
3 Replacing the variable with the possible solution serves to check the solution. Throughout this unit, remind students to check solutions by substituting them for variables in open sentences. Briefly review what a number model is a number sentence that fits some real or hypothetical situation and how number models can be useful in solving problems and in recording solutions. Algebra Parentheses The meaning of a number sentence or expression is not always clear. You may not know which operation to do first. But you can use parentheses to make the meaning clear. When there are parentheses in a number sentence or expression, the operations inside the parentheses are always done first. Evaluate. ( 6) *? The parentheses tell you to ( 6) * subtract 6 first. 8 * Then multiply by. 36 ( 6) * 36 Evaluate (6 * ). The parentheses tell you (6 * ) to multiply 6 * first. Then subtract. (6 * ) Finding Solutions by Trial and Error Links to the Future WHOLECLASS DISCUSSION Algebraic Thinking Write the following question on the board and ask students to find the correct answer. What is the solution to  x = 7 + x? A) B) C) 3 D) Have volunteers share how they found the solution. B Because most students do not know an algorithm to solve the problem, they probably tried each choice until they found the correct answer. The trialanderror method is a problemsolving strategy that all students have used at one time or another. For example, trial and error is used to complete a jigsaw puzzle. If one piece doesn t fit, try another. Trial and error is not guessing; it is a legitimate problemsolving skill. Trial and error is an appropriate strategy to use when taking a timed multiplechoice test, where it may be more efficient to rule out choices than to work the problem. Record the following open sentences on the board. Have students work in pairs to find each solution. 7 = 30 + y y = x  8 = 0 x = 0 = ( + 3m) m = 7 When most students have finished, have volunteers share their strategies for limiting the number of solutions that they needed to try. The activities in this lesson focus on two methods for solving equations the trialanderror method and the coverup method. A third method, a systematic approach for solving equations, is explored in Lessons 60 and 6 and then applied in Units 8 and 9. Evaluate (3 * C) H if C and H. Replace C with and replace H with. The result is (3 * ). The parentheses tell you to multiply 3 * first. (3 * ) Then add. 7 If C and H, then (3 * C) H is 7. Open Sentences In some number sentences, one or more numbers may be missing. In place of each missing number is a letter, a question mark, or some other symbol. These number sentences are called open sentences. A symbol used in place of a missing number is called a variable. For most open sentences, you can t tell whether the sentence is true or false until you know which number replaces the variable. For example, x is an open sentence in which x stands for some number. If you replace x with 3 in x, you get the number sentence 3, which is false. If you replace x with 7 in x, you get the number sentence 7, which is true. Student Reference Book, p. If a number used in place of a variable makes the number sentence true, this number is called a solution of the open sentence. For example, the number 7 is a solution of the open sentence x because the number sentence 7 is true. Finding the solution(s) of an open number sentence is called solving the number sentence. Many simple equations have just one solution, but inequalities may have many solutions. For example, 9, 3.,, and 8 are all solutions of the inequality x 0. In fact, x 0 has infinitely many solutions any number that is less than 0 is a solution. Number Models In Everyday Mathematics, a number sentence or an expression that describes some situation is called a number model. Often, two or more number models can fit a given situation. Suppose, for example, that you had $0, spent $8.0, and ended up with $.0. The number model $0 $8.0 $.0 fits this situation. The number model $0 $8.0 $.0 also fits. Number models can be useful in solving problems. For example, the problem Juan is saving for a bicycle that costs $9. He has $. How much more does he need? can be modeled by $9 $ x or by $9 $ x. The first of these number models suggests counting up to find how much more Juan needs; the second suggests subtracting to find the answer. Other kinds of mathematical models are discussed in the section on problem solving. To evaluate something is to find out what it is worth. To evaluate an algebraic expression (like the last example), first replace each variable with its value and then carry out the operations. Some open sentences are always true. 9 y y 9 is true if you replace y with any number. Some open sentences are always false. C C is false if you replace C with any number. Algebra Number models are not always number sentences. The expression $0 $8.0 is not a number sentence, but it is another model that fits the situation. Find the solution of each equation.. 6 c 0. 6 * z 3. ( * f ) 6 Write a number model that fits each problem.. Hunter used a $0 bill to pay for a CD that cost $.9. How much change did he get? Check your answers on page 3.. Eve earns $0 a week babysitting. How many weeks will it take her to earn $90? Student Reference Book, p. 3 Lesson
4 Date 6 8 Solving Equations Find the solution to each equation. Write a number sentence with the solution in place of the variable. Check that the number sentence is true. Equation Solution Number Sentence. x 3. y b 3. m p ( 9) 6. 7 (a ) 7. (9 w) / 6 (6 / 6) 8. (3n 6) (3 6) Find the solution to each equation t 0. 9 ( c) = 8. 7 k / 8. (m ) / / 9 p. 3 a. (3 p) 6 6. d 3 7. Make up four equations whose solutions are whole numbers. Ask your partner to solve each one. Equation a. b. c. d. x y b 7 m p 9 a 0 w n ( 9) 7 (0 ) (9 ) / 6 (6 / 6) ( 6) (3 6) t c k 8 m 0 p a 8 p 7 d 0 Answers vary. Math Journal, p. 8 Solution 0 3 Writing and Solving Equations (Math Journal, p. 8) PARTNER Algebraic Thinking Write the equation 8 (  c) = 7 on the board and cover the expression ( c) with your hand or a sheet of paper. Ask: What multiplied by 8 equals 7? 9 Then write  c = 9. Now cover up c and ask: Eleven minus what equals 9? Tell students that you have just used the coverup method to solve an equation. Assign Problems 8, telling students to solve as many equations as they can. When most students have completed the problems, bring the class together to go over solution strategies before assigning Problems 9 7. Ongoing Assessment: Recognizing Student Achievement PROBLEM SOLVING Journal Page 8 Problems 9 6 Use journal page 8, Problems 9 6 to assess students abilities to use the trialanderror or coverup method to solve equations. Students are making adequate progress if they are able to solve the equations in Problems 9 6. [Patterns, Functions, and Algebra Goal ] Adjusting the Activity Provide students with additional parameters for the equations they create for Problem 7. For example, require that one equation includes division of fractions or that one equation contains exponents. 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 Date 6 8 Absolute Value on a Map Two cousins, Sarah and Bev, live in the same town. A map with the locations of some important places in the town is shown below. Each square represents one block. y. Write the coordinate pair for each location. a. Sarah s house (,3) b. Bev s house (,) c. Park (,3) d. Grocery store (0,0) e. Post office (,0) f. Grandma s house (,3) Grandma s house 3 Grocery store Sarah s house Bev s house 3 Park Post office. The two major roads in Sarah s town are State Street and Bay Road. State Street is represented by the xaxis and Bay Road is represented by the yaxis. How could you use absolute value to find the distance of the locations shown on the map from each of the major roads? (Hint: Sarah s house is block from Bay Road, because  =.) Sample answer: The absolute value of the xcoordinate gives the distance from the yaxis, or Bay Road, and the absolute value of the ycoordinate gives the distance from the xaxis, or State Street. 3. Use absolute value to find the distance of each location from Bay Road and State Street. x Ongoing Learning & Practice Solving Map Problems Using Absolute Value (Math Journal, pp. 9A and 9B) INDEPENDENT Review with students how to find the absolute value of a number, and situations in which it makes sense to compare the absolute values of numbers. Then have students complete journal pages 9A and 9B. They find the coordinates of different locations on a grid map and use absolute value to find distances on the map. a. Sarah s house: Bay Road blocks State Street 3 blocks c. Park: Bay Road blocks State Street 3 blocks e. Post office: Bay Road blocks State Street 0 blocks Math Journal, p. 9A b. Bev s house: Bay Road blocks State Street blocks d. Grocery store: Bay Road 0 blocks State Street 0 blocks f. Grandma s house: Bay Road blocks State Street 3 blocks 7 Unit 6 Number Systems and Algebra Concepts
5 Math Boxes 6 8 (Math Journal, p. 9) INDEPENDENT Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 66. The skill in Problem previews Unit 7 content. Study Link 6 8 (Math Masters, p. 0) INDEPENDENT Home Connection Students solve equations and translate word sentences into equations. They also write expressions equivalent to target numbers. Date 6 8 Absolute Value on a Map continued. Bev told Sarah, My house is closer to State Street than Grandma s house, because  < 3. Do you agree with Bev? Use absolute value to help justify your answer. Sample answer: I do not agree with Bev. Distance is always 0 or positive. Bev needs to use the absolute value of the coordinates.  > 3, so Bev s house is actually farther from State Street than Grandma s house.. Use absolute value to find the distance between the following locations. Write a number sentence to show how you found your answer. a. Sarah s house and Bev s house: blocks 3  ( ) = b. Sarah s house and the park: blocks   = c. The park and the post office: 3 blocks 30 = 3 d. The grocery store and the post office: blocks 0  = Try This 6. The streets in Sarah and Bev s town are shown by the gridlines. To walk from one location to the next, they must walk along the streets. Use absolute value to find the distance they would travel if they walked the following routes. Write a number sentence to show how you got your answer. a. From Bev s house to the park: 6 blocks (3) = 6 Sample number sentences are given. b. From the park to Grandma s house: blocks  () = Sample number = sentences are given. 0  () (3) = c. From Grandma s house to the grocery store: blocks d. From the grocery store to Sarah s house: blocks Math Journal, p. 9B Study Link Master Name Date STUDY LINK 6 8 Solving Simple Equations Date 6 8 Math Boxes. Find the solution to each equation. b 9 y 3 a. b 7 b. 3 n 9 c. / y d. m º 3 3 n m. Translate the word sentences below into equations. Then solve each equation. 3. Solve. Simplify your answers.  7 _ 7 a. 3 =  _ 3_ a. (3) 3 =  _ 70. _ b. 6 ( _ 6 ) = c. 33 = 3 _ 8 ( _ 8 ). Multiply or divide. b. 0.(0.) = c. 3 = _3  Word sentence Equation Solution a. If you divide a number by 6, x the result is b. Which number is 7 less than 00? c. A number multiplied by 8 is equal to,98. d. 7 is equal to 3 increased by which number? 3. For each problem, use parentheses and as many numbers and operations as you can to write an expression equal to the target number. You may use each number only once in an expression. Write expressions with more than two numbers. Sample answers: a. Numbers: 3, 9,,, 9 Target number: 36 (3 º ) ( 9) b. Numbers:,, 6,, 8 Target number: 0 º 8 c. Numbers:,, 8,, 7 Target number: (7 ) d. Numbers: 6, 7,,, 0 Target number: (0 ) 7 Practice 00 7 n b º 8, n x 60 n 93 b 6 n 3. Triangles DAB and BCD are congruent. A B Which is a pair of corresponding angles? Circle the best answer. A. ABD and ABC B. BAD and DCB C. BCD and CDB D. ADC and ADB. You spin the spinner shown at the right. a. How many equally likely outcomes are there? 0 b. What is the probability that the spinner will land on a factor of 0? c. What is the probability that the spinner will land on a multiple of 3 or a multiple of? D C Label the axes of this mystery graph and describe a situation it might represent. _ 0, or _ Sample answers: xaxis (minutes) yaxis Temperature ( F) Situation Cake is taken out of an oven and set on a countertop _ Complete Math Masters, p. 0 Math Journal, p. 9 Lesson 6 8 7
6 3 Differentiation Options READINESS Using the Broken Calculator Routine SMALLGROUP 30 Min To provide experience estimating solutions to open sentences, use the Broken Calculator routine. Broken Calculator is used in Fourth and Fifth Grade Everyday Mathematics for applying estimation skills to solve open sentences. Students generate values for a variable and test them as solutions. They observe how close the test values are to the solution and then use this information to select new test numbers, attempting to get closer to the actual solution. Write equations on the board as shown below and ask students to solve them on their calculators without using the specified operation key. Suggestions: Equation Broken Key Solution 8 + x = z + 67 =, d  68 =,387 +,8 p = 9 Æ,66 y 33 = 76 Æ,08 7 t = Teaching Master Name Date Solving Challenging Equations. x x 0 n a 8 3 y y 0 g Which of the above sentences have a. no solution? b. more than one solution? c. a solution that is a negative number?. Find the solution to each equation below. a. x (x ) (x ) 90 x 9 (Hint: Think of this equation as a sum of three numbers.) b. a (a ) (a ) (a 3) (a ) Whole numbers are said to be consecutive if they follow one another in an uninterrupted pattern. For example,, 6, 7, 8, 9, and 0 are six consecutive whole numbers. a. Find three consecutive whole numbers whose sum is 90. (Hint: Replace each variable x in Problem a with the solution of the equation.) b. Find five consecutive whole numbers whose sum is y y ; 0 g x x 0; a 8 3 n ; a 8 3; x x 0 a 6 s 39 =,833 7 ENRICHMENT PARTNER Solving Challenging Equations (Math Masters, p. 03) 30 Min To further explore solving equations using a trialanderror method, students work in partnerships to solve the problems on Math Masters, page 03. Tell students that trial and error will be an effective strategy for these problems. They may need some encouragement when reasoning through the relationships in Problem. For example, M must be 0, L + U must equal P, A must be less than G, and P must be _ (R + 0). c. Find four consecutive whole numbers whose sum is Each letter in the subtraction problem below represents a different digit from 0 through 9. The digits 3 and do not appear. Replace each letter so the answer to the subtraction problem is correct. GRAPE G 9 R A 8 P 7 PLUM APPLE E L U 6 M 0 Math Masters, p Unit 6 Number Systems and Algebra Concepts
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