Unit 2: Linear Equations and Inequalities (5 Weeks)

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1 Page 1 of 5 Unit 2: Linear Equations and Inequalities (5 Weeks) UNIT OVERVIEW The material in this unit is the heart of algebraic thinking. Students write, simplify, evaluate, and model situations with linear expressions. Students then examine the concept of equality and use linear equations and linear inequalities to model and solve real-world problems. The properties of real numbers play a prominent role in this unit. The commutative, associative, and distributive properties are used when students simplify and evaluate expressions and solve multi-step equations. Opposites, reciprocals, and order of operations are used when students evaluate expressions and solve equations. Students revisit rational numbers when they solve equations and inequalities with rational number coefficients and rational number solutions. The activities in Unit 2 are designed to strengthen student understanding of prerequisite mathematical concepts. As you navigate through Unit 2, you may find that students need additional support. If this occurs, consider designing and implementing mini-lessons to review prerequisite skills that support the investigations. You are encouraged to pause, review prerequisite concepts, and then continue along. As the Common Core Standards are implemented over the next few years, Algebra I teachers may find that a good deal of the material in Unit 2 has already been learned. Each teacher will need to determine what material has been mastered and what material is new. Communication with middle school teachers will be essential. Teachers should omit material if it is unneeded, review as needed, and move through Unit 2 as quickly as the backgrounds of their students permit. The first investigation, Investigation 1, begins with number puzzles to encourage an understanding of algebraic expressions and the idea of working backwards to find a solution. Activities in this investigation ask students to use flowcharts and verbal descriptions to represent algebraic expressions. These techniques build students understanding of how the order of operations is used in evaluating an expression and solving an equation. In Investigation 2, students will apply their knowledge of solving one- and two-step equations and will write equations that model and solve real world problems such as bank fees, weight loss, and taxi cab charges. There is an emphasis on distinguishing between evaluating an expression and solving an equation. Activities involving algebra tiles and flow charts are available to support students with different learning styles. Investigation 3 focuses on modeling real world scenarios with equations that contain variables on both sides. Students are asked to justify their steps in the equation solving process, recognize the role of the commutative and associative properties, and check the reasonableness of their answers. You may continue to use algebra tiles to support student learning. Students are also introduced to identities and contradiction and learn how a table and graph can be used to solve an Unit 2 Plan CT Algebra I Model Curriculum Version 3.0

2 Page 2 of 5 equation. At the end of Investigation 3, a mid-unit test is provided to assess student mastery of solving two-step equations. In Investigation 4, students solve multi-step equations that require the distributive property and combining like terms. Throughout this investigation, students model situations in different ways, from hands-on to symbolic. Activities involving algebra tiles and pan balances are included to support different learning styles. There are many opportunities for students to write and solve equations to solve problems in contexts. Applications include walk-a-thons, pizza parties, geometry problems, and sports problems. Investigation 5 expands students equation solving skills to include the transformation of literal equations. Students learn to change the subject of a formula (literal equation) by algebraically solving for a variable. Flowcharts are reintroduced as a method of attack. Students solve for variables in common geometry formulas which show up on the CAPT test. The final investigation, Investigation 6, introduces the concept of linear inequalities. Students write and solve inequalities to solve a variety of contextual problems and are asked to represent solutions of inequalities on number lines. Activities focus student attention on the difference between inequalities and equations and on the justification for reversing an inequality symbol when a negative number is multiplied to both sides or divided by both sides. The Unit 2 Performance Task involves making consumer decisions about the purchase of ipods and downloads. In this task, and throughout this unit, students are encouraged to work cooperatively and share and compare their problem solving strategies. Essential Questions What is an equation? What is an expression? What does equality mean? What is an inequality? How can we use linear equations and linear inequalities to solve real world problems? What is a solution set for a linear equation or linear inequality? How can models and technology aid in the solving of linear equations and linear inequalities? Enduring Understandings To obtain a solution to an equation, no matter how complex, always involves the process of undoing the operations. Unit 2 Plan CT Algebra I Model Curriculum Version 3.0

3 Page 3 of 5 Unit Contents Investigation 1: Understanding Algebraic Expressions (2 days) Investigation 2: One-Step and Two-Step Linear Equations (4 days) Investigation 3: Combining Like Terms to Solve Equations (4 days) Mid-Unit Test (1 day) Investigation 4: Solving Equations Using the Distributive Property (4 days) Investigation 5: Formulas and Literal Equations (2 days) Investigation 6: Linear Inequalities (4 days) Performance Task: ipods (2 days) End of Unit Test (2 days including review) Common Core Standards Mathematical Practices #1 and #3 describe a classroom environment that encourages thinking mathematically and are critical for quality teaching and learning. Practices in bold are to be emphasized in the unit. 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Standards Overview Interpret the structure of Expressions Write expressions in equivalent form to solve problems Create equations that describe numbers or relationships Solve equations and inequalities in one variable Understand solving equations as a process of reasoning and explain the reasoning Reason quantitatively and use units to solve problems Standards with Priority Standards in Bold 8EE 7. Solve linear equations in one variable. a. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Unit 2 Plan CT Algebra I Model Curriculum Version 3.0

4 Page 4 of 5 A-SSE 1. Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients. b. Interpret complicated expressions by viewing one or more of their parts as a single entity... A-SSE 3. (part) Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. A-CED 1. (part) Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear... functions A-CED 4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm s law V = IR to highlight resistance R. A-REI 1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. A-REI 3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. N-Q 1 Use units as a way to understand problems and to guise the solution of multi-step problems; choose and interpret units consistently in formulas. N-Q 2 Define appropriate quantities for the purpose of descriptive modeling. N-Q 3 Choose a level of accuracy appropriate to limitations on measurements when reporting quantities. Vocabulary algebraic expression associative property coefficient constant commutative property distributive property Assessment Strategies Performance Task: ipods evaluate inequality symbol integers inverse operations linear inequalities literal equations order of operations properties of equality real numbers simplify variable Students will work on a two-day task that has them investigating file storage size and cost for various models of ipods. Students will share their findings with the class. Unit 2 Plan CT Algebra I Model Curriculum Version 3.0

5 Page 5 of 5 Other Evidence (Formative and Summative Assessments) Exit slips Class work Quizzes Homework assignments Math journals Mid-unit assessment End-of-Unit Tests (Versions 1 & 2) Unit 2 Plan CT Algebra I Model Curriculum Version 3.0

6 Page 1 of 3 Unit 2 Investigation 1 (2 Days) Understanding Algebraic Expressions CCSS: AA-SSE 1 Overview Students represent expressions using verbal descriptions and flowcharts. Students recognize that an algebraic expression involving a single variable term can be thought of as a sequence of operations on the variable term. Assessment Activities Evidence of Success: What Will Students Be Able to Do? Represent algebraic expressions by verbal descriptions and flowcharts. Convert verbal descriptions to algebraic expressions. Evaluate algebraic expressions. Assessment Strategies: How Will They Show What They Know? Exit Slip 2.1 requires students to represent algebraic expressions by verbal descriptions and flowcharts, and convert verbal descriptions to algebraic expressions. Journal Entry prompts students to describe what it means to evaluate an algebraic expression. Launch Notes Begin this investigation by presenting a magic trick to your students. Two options are described below. Option 1: Tell students you have a magic trick. Have students take out a calendar and draw a square around any four numbers in any month. Tell them that if they give you the sum of the four numbers, you can tell them what the lowest number is in their square. When a student gives you their sum, subtract 16, and then divide the difference by 4. The result will be the lowest number in their square (the number in the upper left corner of the square). After performing the trick several times, discuss how the trick works. Explanation of calendar trick: If x is the first number, x + 1 is the second, x + 7 is the third, and x + 8 is the fourth. The sum of the four numbers is x + (x +1) + (x + 7) + (x + 8) = 4x So, given the sum, find x by subtracting 16 from the sum, then dividing the difference by 4. The lowest number, or number in the upper left corner of the square, is x. Option 2: Present the magic trick on the first page of Activity The Magic of Algebra. Ask students to work in pairs or triples to complete the first two pages of the activity. Unit 2 Investigation 1 Overview CT Algebra I Model Curriculum Version 3.0

7 Page 2 of 3 Closure Notes Define an algebraic expression, define what it means to evaluate an algebraic expression, and reiterate the importance of applying order of operations when evaluating expressions. Teaching Strategies I. In Activity The Magic of Algebra, students explore the mathematics behind number tricks. Students see how performing operations on a number and then undoing the operations in a particular order results in the original number. Students represent number tricks symbolically and using diagrams, and create their own number trick. Group Activity Have students work in pairs or triples to complete Activity The Magic of Algebra. Encourage students to communicate with each other to identify the mathematics behind the magic trick and ask students to work together to create new magic tricks. II. III. In Activity Representing Expressions with Stories & Flowcharts, students represent algebraic expressions by verbal descriptions and flowcharts, and convert verbal descriptions to algebraic expressions. Verbal descriptions are called stories on x to emphasize that an expression indicates that something is happening to the variable term. The operations on x follow a certain order. Flowcharts allow students to capture the operations and the order in which they occur. This activity lays the foundation for using flowcharts to solve one- and two-step linear equations. In Activity Representing Expressions with Algebra Arrows, students represent algebraic expressions by verbal descriptions and algebra arrows, and convert expressions represented by algebra arrows to verbal descriptions. Students also practice evaluating expressions. Students can use an online applet to construct and evaluate algebra arrows by going to and clicking on Applets and then Algebra Arrows. Differentiated Instruction (For Learners Needing More Help) The flowcharts (Activity 2.1.2) and algebra arrows (Activity 2.1.3) provide students a visual representation of algebraic expressions. Differentiated Instruction (For Learners Needing More Help) To complement Activity and/or Activity 2.1.3, provide students additional practice using order of operations to simplify expressions. Worksheets containing order of operations problems are included in Unit 1 Investigation 2. Unit 2 Investigation 1 Overview CT Algebra I Model Curriculum Version 3.0

8 Page 3 of 3 IV. In Activity Evaluating Algebraic Expressions, students construct expressions for the perimeter and area of geometric figures which contain variable dimensions. Without simplifying the expressions, students evaluate the expressions using multiple inputs. Students recognize that once a variable is replaced by a number, simplifying the resulting expression requires applying the order of operations. Journal Entry What does it mean to evaluate an algebraic expression? Resources and Materials Activity The Magic of Algebra Activity Representing Expressions with Stories and Flowcharts Activity Representing Expressions with Algebra Arrows Activity Evaluating Algebraic Expressions Exit Slip 2.1 What s the Story WisWeb Algebra Arrows Student Journals Graphing calculators Unit 2 Investigation 1 Overview CT Algebra I Model Curriculum Version 3.0

9 Name: Date: Page 1 of 3 The Magic of Algebra Let s play a game. Complete the table below using the number on your card. The mathemagician predicts that your final result will be 3 no matter what number you start with. Step 1: Write down your number. This is the initial value. Step 2: Double your number. Step 3: Add six to the answer. Step 4: Divide the answer by 2. Step 5: Subtract the initial value from the answer in Step Did the mathemagician correctly guess the answer for Step 5? 2. Compare your answer for Step 5 to the other students. What do you notice? 3. Repeat the game again. This time start with the opposite of your initial value. Hint: If your initial value was 10, start with -10. If your initial value was -8, start with 8. Step 1: Write down your number. This is the initial value. Step 2: Double your number. Step 3: Add six to the answer. Step 4: Divide the answer by 2. Step 5: Subtract the initial value from the answer in Step Will the final result of this game always be 3? Explain why or why not. Activity CT Algebra I Model Curriculum Version 3.0

10 Name: Date: Page 2 of 3 We will now investigate how this magic trick works. We will first represent the trick using a diagram. Copy the numbers from one of your tables on the first page into the column labeled Numeric below. The column labeled Pictoral contains pictures representing what happened at each step. The square represents the initial value of the card and the circle represents the value of 1. Fill in the column labeled Symbolic based on what happens at each step. The initial value is represented by the variable x. KEY: = initial value = 1 SPECIFIC GENERAL Numeric Pictoral Symbolic Step 1: Initial Value x Step 2: Double the number. Step 3: Add six to the number. Step 4: Divide the number by 2. Step 5: Subtract the initial value. 5. Explain how the mathemagician s number trick works. 6. Would the trick work if the initial value was a decimal, fraction, or very large number? Explain why or why not. Activity CT Algebra I Model Curriculum Version 3.0

11 Name: Date: Page 3 of 3 The magic trick used a sequence of math operations in a particular order. We see that the order in which the operations are performed is very important. 7. Create your own magic trick. Your trick should include four mathematical operations. Write the magic trick below and make sure to include the final result of the trick. Use the table to check that your trick works. Step 1: Select any number. (Initial Value) Step 2: Step 3: Step 4: Step 5: FINAL RESULT OF TRICK = Represent your magic trick using the table below. Use a square to represent the initial value and a circle to represent the value of 1. KEY: = initial value = 1 SPECIFIC GENERAL Numeric Pictoral Symbolic Step 1: Initial Value x Step 2: Step 3: Step 4: Step 5: Activity CT Algebra I Model Curriculum Version 3.0

12 Name: Date: Page 1 of 4 Representing Expressions with Stories and Flowcharts When we think of stories we usually don t think of algebra, but mathematical expressions tell stories too! When we see an expression involving a variable, something is happening to that variable. In other words, something is being done to the variable. Let s first use a story to represent what is being done to the variable. Use the order of operations to decide what steps are taken to evaluate the expression. Expression Story of x! + 6!/3 8!! 5 2! + 6 6! + 3! 2 4! ! 7! ! ! 5 Activity CT Algebra I Model Curriculum Version 3.0

13 Name: Date: Page 2 of 4 We can also represent the story of x by a flowchart. The flowchart below displays the story of! for the expression 6! + 5. Multiply x Add 5 by 6 1. Use a flowchart to represent the following expressions involving two operations: A. 7! x B.!/ x C. 5! 11 x D.! + 5 x E. 5 3! x Activity CT Algebra I Model Curriculum Version 3.0

14 Name: Date: Page 3 of 4 2. Use a flowchart to represent the following expressions involving three operations: A. 2! x B. 3! x C. 4! x D. 6 2! 5 x Activity CT Algebra I Model Curriculum Version 3.0

15 Name: Date: Page 4 of 4 3. Let s now convert stories to mathematical expressions. Given the story on x, write a mathematical expression that describes the story. Use x to represent the unknown number. A. Multiply a number by 7. B. Add 14 to a number. C. Subtract a number from 12. D. Subtract 6 from a number. E. Divide a number by -3 F. Divide -8 by a number. G. Multiple a number by 4, then add 3. H. Multiply a number by 8, then add 11. I. Subtract 4 from a number, then multiply by 6. J. Add -1 to a number, then divide by 3. K. Divide a number by 2, then subtract 13. Activity CT Algebra I Model Curriculum Version 3.0

16 Name: Date: Page 1 of 2 Representing Expressions with Algebra Arrows Write the story of n for each of the following algebraic expressions. Carefully identify the order in which the operations occur on the variable term. Then build each story using the Algebra Arrows applet on the Freudenthal Institute website*. Sketch a chain of arrows for each expression and evaluate each expression when n = 3. Expression Story of n Algebra Arrows Chain Evaluate when n = 3 3n + 7 n 3 10 ( n ) 2 5 n ( n ) n + 2! + 3! 6 2 3n * Activity CT Algebra I Model Curriculum Version 3.0

17 Name: Date: Page 2 of 2 Fill in the expression created by each algebra arrow chain, and then tell the story of x. Then evaluate the expression when x = 5. Algebra Arrows Chain Story of x Evaluate when x = 5 Activity CT Algebra I Model Curriculum Version 3.0

18 Name: Date: Page 1 of 2 Evaluating Algebraic Expressions When we evaluate expressions, we follow the order of operations. We will find the perimeter and area of geometric figures that have an unknown dimension. 1. Given the following group of tiles: A. Write an expression for the perimeter of the outer square. B. Find the value of the perimeter when x = 3. C. Find the value of the perimeter when x = 5. D. Write an expression for the area of the outer square. E. Find the value of the area when x = 3. F. Find the value of the area when x = 5. Activity CT Algebra I Model Curriculum Version 3.0

19 Name: Date: Page 2 of 2 2. The figure on the right shows a pink square inside a larger triangle. A. Write an expression for the area of the square. B. Write an expression for the area of the large triangle. C. Write an expression for the area that is shaded in blue. D. Find the area of the blue region when x = 6. Activity CT Algebra I Model Curriculum Version 3.0

20 Page 1 of 4 Unit 2: Investigation 2 (4 Days) One-Step and Two-Step Linear Equations CCSS: 8EE 7, A-CED 1, A-REI 1, A-REI 3 Overview Students solve one-step and two-step linear equations and construct and solve linear equations to explore real world problems. Assessment Activities Evidence of Success: What Will Students Be Able to Do? Write linear equations that model real world scenarios, solve one- and two-step linear equations, and justify their steps using algebraic properties. Assessment Strategies: How Will They Show What They Know? Exit Slip 2.2 requires students to write a two-step linear equation that models a real world situation and to solve the equation. Journal Entry 1 asks students to explain how we can solve two-step linear equations and why the approach works. Journal Entry 2 asks students to explain which solution strategy (flowchart, algebra tiles, solving by undoing) they prefer to use to solve equations. Launch Notes Inform students that equations are extremely important mathematical tools which are used constantly to solve problems. Remind students of the hydrocarbon formula which they developed in Unit 1. In a simply hydrocarbon, the number of hydrogen atoms is two more than twice the number of carbon atoms. Write the formula h = 2! + 2 on the board, and ask students to find the number of carbon atoms if there are 24 hydrogen atoms. After substituting 24 for h, explain that 24 = 2! + 2 is an example of a two-step linear equation. Ask students to solve the equation and check their solution. Discuss the importance of using multiple approaches to solve linear equations. Four ways to solve a linear equation are: Make a table Guess and check Flowcharts Solving by undoing (Using the properties of equality) Unit 2 Investigation 2 Overview CT Algebra I Model Curriculum Version 3.0

21 Page 2 of 4 Closure Notes Ask students the following questions to summarize the key points in this lesson: What does it mean to solve by undoing? How can you check that your solution is the correct solution of an equation? Can every real world situation be reduced to a two-step linear equation? Teaching Strategies I. In Activity Solving Equations using Flowcharts, students extend their understanding of flowcharts to solve one-step and two-step linear equations. This activity highlights the fact that solving by undoing involves undoing the operations that have been done to the variable. Help students to complete their flowcharts. Once the inverse operations are identified, check that students are correctly applying these inverse operations to the equation. Differentiated Instruction (For Learners Needing More Help) The flowchart method (Activity 2.2.1) is recommended for students who have difficulty identifying the appropriate steps for solving one-step and two-step linear equations. Students who understand the flowchart method tend to gradually transition into solely using the undoing method. This activity can be done immediately after Activity from Unit 2 Investigation 1. II. In Activity Solving Equations with Algebra Tiles, students use algebra tiles to model and solve one-step and two-step equations. Students model linear expressions and one-step and two-step linear equations with algebra tiles. Differentiated Instruction (For Learners Needing More Help) Algebra tiles (Activity 2.2.2) provide students a tactile experience. The activity can be performed with or without algebra tiles. III. IV. In Activity Solving One-Step Linear Equations, students convert verbal descriptions of one-step equations to symbolic equations, create equations to model situations, solve equations, and check solutions. In Activity Equations in Education, students explore formulas for elementary and secondary per pupil spending in the United States. Students evaluate linear expressions, and construct and solve a variety of two-step linear equations. This activity provides students an opportunity to explore data from the U.S. Census Bureau. Unit 2 Investigation 2 Overview CT Algebra I Model Curriculum Version 3.0

22 Page 3 of 4 V. In Activity New York City Cab Fares, students construct an equation to represent the cost of a cab ride using cab fare data. This is a short activity which can be assigned inside or outside of class. VI. In Activity Station Problems, students move around the class to different stations to solve a variety of problems involving one-step and two-step equations. Group Activity Activity Station Problems has eight pages, one for each station. Set up eight stations throughout the classroom and place a page at each station. If possible, print the pages on colored paper. You could also post the pages around the room. They do not need to be arranged in any order. At the beginning of the activity, have students count off by 8 s. The number each student says is the station they will start at. Each student should have a piece of paper and pencil so they can solve the problems at the stations. Students can work with other students at the same station. Once they solve the problem, they should look around the room to find the station with their answer; then they should do the problem above it. They continue in this fashion, until they have been to and solved the problems at all eight stations. You can tell that they completed the activity correctly, because every student should have done the stations in the same order, despite which problem they started at. Instruct students to label each problem on their paper with the corresponding station number. VII. Activity Solving Two-Step Linear Equations provides students additional practice solving two-step linear equations. For each equation, students identify the operations being done on the variable and the order in which the operations are done. Students then identify the inverse operations and use these operations to solve for the variable. Journal Prompt 1. Explain to a student who missed class how we can solve a two-step linear equation? Why does our approach work? 2. Which method do you prefer to use when solving equations: flowchart, undoing, or algebra tiles? Why? Differentiated Instruction (For Learners Needing More Help) Show students the Algebra Balance Scales found at the National Library of Virtual Manipulatives (NLVM) website. This activity helps students visualize linear equations and understand what happens to an equation when an operation is performed on both sides. Unit 2 Investigation 2 Overview CT Algebra I Model Curriculum Version 3.0

23 Page 4 of 4 Resources and Materials Activity Solving Equations using Flowcharts Activity Solving Equations with Algebra Tiles Activity Solving One-Step Linear Equations Activity Equations in Education Activity New York City Cab Fares Activity Station Problems Activity Solving Two-Step Linear Equations Exit Slip 2.2 Pizza Problems Student Journals Graphing calculators Algebra tiles Algebra Balance Scales (NLVM) applet: ategory_g_4_t_2.html Unit 2 Investigation 2 Overview CT Algebra I Model Curriculum Version 3.0

24 Name: Date: Page 1 of 4 Solving Linear Equations using Flowcharts You will now use flowcharts to solve one-step and two-step linear equations. Apply the corresponding steps to the equation on the right side. Check your solution. 1. Solve! 5.4 = 19.8! 5.4 = 19.8 x Right Hand Side Check: 2. Solve:!/4.6 = 3.5!/4.6 = 3.5 x Right Hand Side Check: 3. Solve:! + 9 = 5! + 9 = 5 x Right Hand Side Check: Activity CT Algebra I Model Curriculum Version 3.0

25 Name: Date: Page 2 of 4 4. Solve 3! + 4 = 11 3! + 4 = 11 x Right Hand Side Check: 5. Solve 2! 8 = 14 2! 8 = 14 x Right Hand Side Check: 6. Solve 2! + 12 = 20 2! + 12 = 20 x Right Hand Side Check: Activity CT Algebra I Model Curriculum Version 3.0

26 Name: Date: Page 3 of 4 Model the situations below with a linear equation. For each problem, identify the unknown, create an equation, solve the equation using a flowchart, and then check your solution. 7. Kevin bought seven tickets to the Haunted Graveyard at Lake Compounce for $ How much does one ticket cost? Identify the unknown: Equation: x Right Hand Side Check: 8. Verizon charges $18.75 per month for phone service and $0.08 per minute. Last month my bill was $ How many minutes did I use? Identify the unknown: Equation: x Right Hand Side Check: Activity CT Algebra I Model Curriculum Version 3.0

27 Name: Date: Page 4 of 4 9. Jose spent $ of his birthday money. He bought an ipod for $159 and 21 songs from itunes. How much did each song cost? Identify the unknown: Equation: x Right Hand Side Check: 10. Your school band needs to buy new recording equipment. The equipment will cost $3000. The band has collected $1200 from previous fundraisers. If the band sells sandwiches at $5 each, how many sandwiches must they to sell to raise the remaining funds? Identify the unknown: Equation: x Right Hand Side Check: Activity CT Algebra I Model Curriculum Version 3.0

28 Name: Date: Page 1 of 2 Solving Linear Equations with Algebra Tiles KEY: = positive number = positive variable = negative number = negative variable Write the mathematical expression represented by each group of tiles Model each expression below with your tiles. Then, draw a picture of what your tiles look like. 4.! ! ! 5 Write the equation represented by each group of tiles Activity CT Algebra I Model Curriculum Version 3.0

29 Name: Date: Page 2 of 2 Model each equation with your tiles. Then, draw a picture of what your tiles look like below j + 1 = = 4 5x = 2w 1 Model each equation with your algebra tiles. Then solve each equation by making zero pairs and rearranging the groups as necessary. Check your solutions. 13.! + 5 = h 1 = Yellow Cab Company charges $7 for picking you up and $2 per mile. How many miles can you ride if you have $20? Identify the unknown: Equation: 16. Mary has 6 CD s. If she has 7 less than John, how many CD s does John have? Identify the unknown: Equation: Activity CT Algebra I Model Curriculum Version 3.0

30 Name: Date: Page 1 of 2 Solving One-Step Equations Solve each problem by defining the unknown, writing an equation, solving it, and checking your answer. Show your work! Problem Identify the unknown Write and solve an equation Check it Four more than a number is negative six. Find the number. The product of a number and 2.5 is 375. Find the number. A number divided by negative six is five. Find the number. Three less than a number is negative ten. Find the number. A number divided by 3 2 is Find the number. Activity CT Algebra I Model Curriculum Version 3.0

31 Name: Date: Page 2 of 2 Problem Identify the unknown Write and solve an equation Check A corn stalk grew 3.5 inches in one week in July. If the stalk was 42.1 inches at the end of the week, how tall was it at the start of the week? A box of 24 candles is packed in rows, and the rows are stacked three deep. How many candles are in each row? Beth ran a 400 meter dash in 57.2 seconds. This was 2.4 sec faster than her previous time. What was her previous time? Four friends share the cost of a meal equally. If each pays $12.50, what was the cost of the meal? Dan bought 6 boxes of candy for his teachers at holiday time. If he spent $41.70, how much did each box cost? Activity CT Algebra I Model Curriculum Version 3.0

32 Name: Date: Page 1 of 4 Equations in Education The bar graph below shows the increase in elementary and secondary per pupil spending from 2005 to Elementary and secondary education refers to kindergarten through high school. Per pupil spending is the amount of money a school district spends on each student. The data is described by the mathematical model! = 406.9! where P is the per pupil spending for elementary and secondary students in the United States and t is the number of years since Use the formula to estimate United States per pupil spending in the year (Hint: 2008 corresponds to t = 3, since 2008 is 3 years after 2005.) 2. Let s use this model to make some predictions. We would like to know when per pupil spending in the United States will reach $15,000. To find this, set P = 15000, then solve the equation for t. Identify the actual year. Activity CT Algebra I Model Curriculum Version 3.0

33 Name: Date: Page 2 of 4 3. When will per pupil spending in the United States reach $20,000? Identify the actual year. In 2010, Connecticut was the 7 th highest state in the nation in per pupil spending. Connecticut s per pupil spending from 2005 to 2010 can be described by the mathematical model! = 690.4! where P is Connecticut s elementary-secondary per pupil spending and t is the number of years since Use the formula to estimate Connecticut s per pupil spending in Use the formula to predict Connecticut s per pupil spending in When will per pupil spending in Connecticut reach $18,000? Identify the actual year. Activity CT Algebra I Model Curriculum Version 3.0

34 Name: Date: Page 3 of 4 For each situation below, identify the unknown quantity and assign it a variable, create an equation, solve the equation, and then check your solution. 7. In a recent year, the number of elementary and secondary students in Connecticut decreased by 3800 students. If there are 543,000 elementary and secondary students in Connecticut, and the number of students decreases by 3800 per year, how many years will it take for the number of students to fall to 516,400? Identify the unknown: Write an equation: Solve the equation: Check your solution: 8. Wyoming has the lowest number of elementary and secondary students in the nation. If there are 87,900 students, and the number of students increases by 950 each year, how many years will it take for the number of students to reach 99,300? Identify the unknown: Write an equation: Solve the equation: Check your solution: Activity CT Algebra I Model Curriculum Version 3.0

35 Name: Date: Page 4 of 4 9. Maine has 188,700 elementary and secondary students. This population is 7300 more than two times Vermont s elementary and secondary student population. What is Vermont s elementary and secondary student population? Identify the unknown: Write an equation: Solve the equation: Check your solution: 10. Kansas has 470,000 elementary and secondary students. A member of the state department of education in Kansas would like to see the student population grow to 485,000 over the next twelve years. How much should the population increase by each year? Identify the unknown: Write an equation: Solve the equation: Check your solution: Activity CT Algebra I Model Curriculum Version 3.0

36 Name: Date: Page 1 of 1 New York City Cab Fares Use the following information about New York City cab fares to answer the questions below. Initial fare...$2.50 Each 1/5 mile (4 blocks) $0.40 Each 1 minute idle...$0.40 Additional riders... FREE 1. Zarela, who is an actuary, travels to work at 7 a.m. Her cab travels different routes depending on traffic, but it is usually idle for a total of 6 minutes. Write an expression to determine the cab fare, F, Zarela will pay if she travels m miles to work. 2. Use your expression to find the cab fare if Zarela has a 1.8 mile commute to work. 3. Zarela knows her trip is 1.8 miles long. On the way home from work at 5 p.m., she never knows how long the cab will be idle. If she has $15.00 for the cab ride home and wants to include a $2.00 tip, how long can the cab be idle on her way home from work? Use your expression from question (1) to answer this question. 4. Estimate how much it would cost you to take a cab ride from Grand Central Station to the Empire State Building. Use the Internet to find the distance. Assume that the cab would be idle for 3 minutes. Activity CT Algebra I Model Curriculum Version 3.0

37 STATION 1 PROBLEM 3 2m = 21 Answer to Another Station 20 Activity Algebra 1 Model Curriculum Version 3.0

38 STATION 2 PROBLEM Better Buy sells laptops that had 4GB of memory for $495. You can buy additional memory for $97 per GB. If you have $980 to buy a laptop and additional memory, how much additional memory can you buy? Answer to Another Station 39 Activity Algebra 1 Model Curriculum Version 3.0

39 STATION 3 PROBLEM 48 = 8m Answer to Another Station 27 Activity Algebra 1 Model Curriculum Version 3.0

40 STATION 4 PROBLEM 2 m 7 = 19 3 Answer to Another Station 12 Activity Algebra 1 Model Curriculum Version 3.0

41 STATION 5 PROBLEM Mr. Smith was so pleased with the amount of homework his students completed that he bought sandwiches, fries, and drinks for the whole class. The total cost was $100. The sodas and fries cost $ If the sandwiches cost $0.99 each, how many sandwiches did he buy? Answer to Another Station 7 Activity Algebra 1 Model Curriculum Version 3.0

42 STATION 6 PROBLEM 1 12 = m Answer to Another Station 14 Activity Algebra 1 Model Curriculum Version 3.0

43 STATION 7 PROBLEM 9 = 16 m Answer to Another Station 5 Activity Algebra 1 Model Curriculum Version 3.0

44 STATION 8 PROBLEM Friendly s has a special offer on milkshakes right now. If you buy their limited edition glass for $5.80, then you can refill it (with a milkshake) for $1.25 anytime during the month. If you have $23.30, how many milkshakes can you buy this month? Answer to Another Station 6 Activity Algebra 1 Model Curriculum Version 3.0

45 Name: Date: Page 1 of 4 Solving Two-Step Equations In the equation 4x 5 = 23, the story of x contains the operations done to x to get 23 (following the order of operations). In this activity, you will work backwards to solve each equation. When working backwards, you will perform an inverse operation to undo each operation in the story of x. The operation done first in the story of x gets undone last. Remember that what you do to one side of the equation must be done to the other side! Equation Story of x Work it backwards Solution Check 4x 5 = 23 x + 8 = 2 3 Multiply by 4, then subtract 5. Divide by 3, then add 8. Add 5, then divide by 4. Subtract 8, then multiply by 3. 4x = x = 28 4 x 28 = 4 4 x = 7 x = x = 10 3 x ( 3) = (3)( 10) 3 x = 30 4(7) 5 = = = 23 ( 30) + 8 = = 2 2 = 2 3x + 6 = 24 Activity CT Algebra I Model Curriculum Version 3.0

46 Name: Date: Page 2 of 4 Equation Story of x Work it backwards Solution Check 2 x 1 = 19 3 x + 3 = x = 31 Rearrange: 25 x = 4 49 Rearrange: Activity CT Algebra I Model Curriculum Version 3.0

47 Name: Date: Page 3 of 4 Complete the following chart, filling-in the missing steps in the process of solving two-step equations. Equation Story of x Work it backwards Solution Check 2x + 5 = 37 Divide by 4, then add 7, to get x = Multiply by -5, then add 6, to get -9 Activity CT Algebra I Model Curriculum Version 3.0

48 Name: Date: Page 4 of 4 Equation Story of x Work it backwards Solution Check x 3 ( 9) = (9)( 2) 9 x 3 = -18 x = x = x = 32.3 Rearrange: x 17 + = 4 7 Rearrange: Activity CT Algebra I Model Curriculum Version 3.0

49 Page 1 of 6 UNIT 2 INVESTIGATION 3 (4 days) Combining Like Terms to Solve Equations CCSS: 8EE 7, A-SSE 3, A-CED 1, A-REI 1, A-REI 3 Overview Students solve multi-step equations in a variety of real-life contexts. To solve the equations students must combine like terms on one side of an equation, and collecting variable terms on one side and collect constants on the other side. Students also solve equations that have no solution or an infinite number of solutions. Assessment Activities Evidence of Success: What Will Students Be Able to Do? Students will be able write linear equations that model real world scenarios, solve equations with variables on both sides, and justify their steps using the properties of equality. Students will also recognize equations for which there is no solution or an infinite number of solutions. Assessment Strategies: How Will They Show What They Know? Exit Slip asks students to model a contextual problem with a linear equation and then solve the equation, and to solve a multi-step equation. To solve both equations students must combine like terms. Exit Slip asks students to identify the steps which occurred in the process of solving an equation. Journal Entry 1 asks students to identify how the commutative and associative properties allow us to solve a multi-step linear equation. Journal Entry 2 asks students to explain how they can tell if an equation has no solution or an infinite number of solutions. Launch Notes Inform students that they will encounter equations which require that they combine like terms. To describe the process of combining like terms, you may introduce this investigation by one of the following discussions. 1. Discuss a scenario involving students attending a concert. Tickets cost $30 but there is also a $5 for handling. Suppose n students attend the concert. Write an expression for how much money they will spend altogether. There are two possible approaches. One is to find the cost for one person, which is $35. The total spent is therefore 35n dollars. The other approach is to find how much is spent for the tickets (30n dollars) and add it to the amount spent on handling fees (5n dollars). Hopefully both methods will be suggested; if not, you can lead a discussion to discover the other one. Conclude that the two algebraic expressions, 35n and Unit 2 Investigation 3 Overview CT Algebra I Model Curriculum Version 3.0

50 Page 2 of 6 30n + 5n represent the same situation. Inform students that in this investigation we will combine like terms to simplify expressions when we solve equations. 2. Ask students why 2x + 3x = 5x? Is this really true for all values of x. Use a table to show that 2x + 3x = 5x, for every value of x. x 2x 3x 2x + 3x 5x Students can discover that for whatever values are chosen for x in the first column, the values in the last two columns are the same. Discuss other examples of combining like terms such as 8! 5! = 3! or! 6! = 5!. Use these examples to introduce the procedure combining like terms. Stress the fact that when we combine like terms we have two equivalent expressions containing the same variable(s). This means that no matter what values are substituted for the variable, the values of the expressions are equal. Closure At the end of this investigation we may step back and look at the bigger picture. Much of what we do in solving an equation is to find a simpler equation. For example, combining like terms makes the expression on one side of the equation simpler. When a variable appears on both sides of an equation, we can eliminate the variable from one side. This also makes an equation simpler. The underlying problem solving strategy which we have used is sometimes called solve a simpler problem. Explain to students that even though some equations may look complicated, if we keep this principle in mind we can often transform a complex equation into a simpler equation. Teaching Strategies I. In Activity Combining Like Terms with Algebra Tiles, students use algebra tiles to simplify algebraic expressions by combining like terms. Students could use algebra tiles or draw pictures of algebra tiles to model and simplify expressions. Differentiated Instruction (For Learners Needing More Help) Algebra tiles (Activity 2.3.1) provide students a hands-on activity for modeling and simplifying algebraic expressions. Unit 2 Investigation 3 Overview CT Algebra I Model Curriculum Version 3.0

51 Page 3 of 6 Group Activity Students can work together in small groups using algebra tiles (Activity 2.3.1) to model and simplify algebraic expressions. II. In Activity Solving Equations that Contain Like Terms, students model contextual problems with linear equations of the form!" +!" =! or the form!" +!" +! =!. The first problem in this activity asks students to solve an equation using a table, by graphing the expressions on both sides and finding the intersection point, and by solving the equation. Note: If you still have students who use the flow-chart method for solving equations, that is fine. Make sure to advise them they can continue to use that method, but they must first simplify the equation (distribute, combine like terms, and get the variable on only one side of the equation) before they put the equation into the flowchart. III. In Activity Solving Equations with Variables on Both Sides, students model contextual problems with linear equations of the form!" +! =!" +!. The first problem in this activity asks students to solve an equation using a table, by graphing the expressions on both sides and finding the intersection point, and by solving the equation. The final problem allows students an opportunity to create their own an equation and solve it. Prior to distributing this activity, you may present students the following strategy for solving multi-step equations that contain variable terms on both sides. 1. Simplify each side of the equation by combining like terms. 2. Collect all the variable terms on one side, and collect all constants on the other side. 3. Multiply or divide by the same number on both sides to solve for x. Discuss how to solve an equation with variables on both sides. Show an equation like 3! + 10 = 7! 6. Ask the class, What can we do to collect the x terms on one side? You may want to show that the above equation may be solved in at least two ways. 3! + 10 = 7! 6 3x 3x 10 = 4! = 4! 16 4 = 4! 4 4 =! 3! + 10 = 7! 6 7x 7x 4! + 10 = ! = 16 4! 4 = 16 4! = 4 Unit 2 Investigation 3 Overview CT Algebra I Model Curriculum Version 3.0

52 Page 4 of 6 Students should be able to justify each step in terms of the addition, subtraction, multiplication, and division properties of equality. Most students will prefer the first method and will chose the first step so that the coefficient of the x term is positive. Also, point out that the associative and commutative properties of addition allow us to combine like terms on each side of an equation. For example, the equation 1 2w + 3 = 8 can be rewritten as (1 + 3) 2w = 8. This simplifies to 4 2w = 8. Group Activity Put students in small groups to write their own step-by-step procedures for solving multi-step equations. Then have them share their procedures with the class. IV. In Activity Practice Solving Equations, students solve non-contextual multistep equations. V. In Activity Solving Equations with Balance Scales, students are introduced to an online virtual manipulative, Algebra Balance Scales, at the National Library of Virtual Manipulative (NLVM) website. Students can use the applet to model and solve linear equations of the form!" +! =!" +!. The applet emphasizes keeping an equation balanced through performing the same operation to both sides of an equation. Differentiated Instruction (For Learners Needing More Help) Algebra Balance Scales (Activity 2.3.5) introduces students to a visual manipulative for solving multi-step equations. The applets are located on the National Library of Virtual manipulative (NLVM) website. One applet only models equations with positive coefficient, and the other models equations with negative coefficients. VI. In Activity How Many Solutions, students practice solving equations which contain no solutions or infinite number of solutions. Before distributing the activity, you may introduce this topic by asking students to solve the following equations: Solve: (1) 5x 6 = 5x 2 4 (2) x = x 30x (3) 2x x = 12x + 2 Have a class discussion about what happened with each equation. For equation (1), have students input!! = 5! 6 and!! = 5! 2 4 into their graphing calculators. Then, using the ASK feature in the table, have students evaluate both functions (Y 1, Y 2 ) try different for several whole numbers, integers, decimals and fractions. They should see that for all values of x the two expressions are equal. The solution of equation (1) is the set of all real numbers. We call this type of equation an identity. Unit 2 Investigation 3 Overview CT Algebra I Model Curriculum Version 3.0

53 Page 5 of 6 Repeat the process for equation (2). In most cases the two sides of the equation are not equal. However if you try x = 8 you will find that Y 1 and Y 2 are both equal to 270. This is the typical situation in which we find one unique solution to the equation. Repeat the process for equation (3). When we make a table for this equation, we find that the two expressions are never equal. In fact they always differ by 6. This type of equation is called a contradiction. Students will notice that in solving equations (1) and (3) the x-terms end up cancelling out to give 0x. For equation (1) in the last step we have 0x = 0. Since 0 times any number is equal to zero, every possible value of x will satisfy this equation. For equation (3) we end up with 0x = 6. There is no value of x that satisfies this equation. VII. In Activity Comparing Cab Fares, students model the cost of using several taxicab companies and identify when two companies charge the same amount. The equations are of the form!" +! =!" +!. Differentiated Instruction (For Enrichment) Some students may want to consider problem situations which lead to equations whose solutions are valid mathematically, but do not make sense in the context of the problem. For example, in Activity 2.3.7, suppose the fuel surcharge for Fast Cabs is $6.00 (and all other charges are the same). For how many miles will the cost of a ride with Fast Cabs be equal to the cost of a ride with Speedy Cabs? The equation would be 0.7x = 0.95x + 6. The solution is x = 2, which does not make sense. For all non-negative values of x, Speedy Cabs is the less expensive option. Challenge students to modify some of the other problems encountered in this investigation to produce similarly inappropriate solutions. Journal Entry 1. When you solve 5 + 2x + 7 = 5x + 4, how can the commutative property and the associative property for addition help you simplify the left side of the equation? 2. How can you tell if an equation has no solution or an infinite number of solutions? Resources and Materials Activity Combining Like Terms with Algebra Tiles Activity Solving Equations that Contain Like Terms Activity Solving Equations with Variables on Both Sides Activity Practice Solving Equations Activity Solving Equations with Balance Scales Activity How Many Solutions Activity Comparing Cab Fares Unit 2 Investigation 3 Overview CT Algebra I Model Curriculum Version 3.0

54 Page 6 of 6 Exit Slip Tickets for a Play Exit Slip What Did They Do? Student Journals Graphing calculators Algebra tiles Algebra Balance Scales (NLVM) applet (positive coefficients): ategory_g_4_t_2.html Algebra Balance Scales (NLVM) applet (negative coefficients): egory_g_4_t_2.html. Unit 2 Investigation 3 Overview CT Algebra I Model Curriculum Version 3.0

55 Name: Date: Page 1 of 4 Combining Like Terms with Algebra Tiles Key = 1 = x = x 2 Rule to Remember Like terms in an equation have the same variable and exponent. They do not need to have the same coefficient. = -1 = -x = -x 2 Algebra tiles can be used to model algebraic expressions and simplify expressions by combining like terms. Carefully look at the example below. Each term in Example 1 is positive. Example 1: Simplify 4x 2 + 2x x 2 + 3x + x using algebra tiles. First represent each term with algebra tiles. 4x 2 2x 2 2x 2 3x x 2 1 Then group the similar tiles together and state the result. 7x 2 5x 3 The resulting expression is 7x 2 + 5x + 3. Activity CT Algebra I Model Curriculum Version 3.0

56 Name: Date: Page 2 of 4 1. Use algebra tiles to model the expression and combine like terms. a) 4x x + 5 b) 2 + 3x + 5x + 4x + 1 c) 2 + x 2 + 3x + 2x d) 2x + 3x 2 + 3x + 2x x 2 + x e) x 2 + 2x + x x 2 + x x f) x 2 + 2x + x + 2x 2 + x x x 2 + 3x 2 Activity CT Algebra I Model Curriculum Version 3.0

57 Name: Date: Page 3 of 4 Algebra tiles can also be used when an expression contains negative terms. Black tiles represent the negative terms. Negative terms are terms that are being subtracted. Example 2: Simplify 2!! 2! + 3 +!!! using algebra tiles. First represent each term with algebra tiles. 2x 2 2x 3 x x 2 Group the similar sized tiles together. Remove pairs that equal zero. zero zero Show the result. The resulting expression is!!! + 3. Activity CT Algebra I Model Curriculum Version 3.0

58 Name: Date: Page 4 of 4 2. Use algebra tiles to model the expression and combine like terms. a) 3! 2 2! + 4 b)! + 3 2! 2! c) 1!! + 2! + 2!! 2 d)!! 2! 2!! +!! 3 + 3! + 1 Activity CT Algebra I Model Curriculum Version 3.0

59 Name: Date: Page 1 of 5 Solving Equations that Contain Like Terms 1. Two computer technicians are upgrading software on 51 computers in a school. On average, Marissa upgrades 5 computers in 1 hour and Ryan upgrades 7 computers in 1 hour. We want to know how long it will take for both of them to upgrade the 51 computers. We can use several strategies to solve this problem. a) What is unknown in this problem? b) Complete the table that shows the number of computers upgraded by each technician and the total number of computers upgraded after 0, 1, 2, 3, 4, and 5 hours. Number of Hours Number of Computers Marissa Upgraded Number of Computers Ryan Upgraded Total Computers Upgraded t c) Make a graph that represents the total number of computers upgraded after each hour d) Use the graph to estimate the number of hours it took to upgrade all the computers. Activity CT Algebra I Model Curriculum Version 3.0

60 Name: Date: Page 2 of 5 e) Let t equal the number of hours that Marissa and Ryan upgrade computers. Write an expression for the total number of computers upgraded in t hours. f) Write an equation to determine the amount of time it will take Marissa and Ryan to upgrade 51 computers. Then solve the equation. g) Write your answer in a sentence. h) In your own words, why do you think we were able to combine the technicians work? Activity CT Algebra I Model Curriculum Version 3.0

61 Name: Date: Page 3 of 5 2. A skateboard park charges $7 per session to skate and $4 per session to rent safety equipment. Jared skates safely and rents safety equipment every time he skates. He bought a new skateboard for $125 in the spring. During the year, he spent $224 for his skateboard, skating charges, and equipment rentals. How many skating sessions did he attend? a) Define your variable(s). b) Write an equation that can be used to find the number of sessions Jared attended. c) Solve the equation. List your steps and show your work. d) Write your answer in a sentence. e) When Jared s friend Rocco tried to solve this problem he took the following steps. 7! + 4! = ! = 224! = 1.65 Rocco concluded that since Jared can t skate 0.65 times, he skated once. Explain to Rocco what he did wrong and why it is mathematically illegal. Also explain what he should he have done to solve the problem. Write your explanation below. Dear Rocco, Activity CT Algebra I Model Curriculum Version 3.0

62 Name: Date: Page 4 of 5 3. The crazy game show called Mud-Tower begins with a pile of mud that weighs 20 pounds. The Blue-Team has to add mud to that pile, while the Red-Team has to remove mud from the same pile at the same time. The Red-Team is scooping-away an average of 0.75 lbs of mud every second. The Blue-Team is piling-on an average 0.55 lbs of mud every second. a) Write an expression that represents the final weight of the mud pile based on how many seconds the two teams have been competing. b) When the game ended there was a total of 10 pounds of mud left in the pile. Write and solve an equation to determine the number of seconds it took for the game to finish. List your steps, show your work, and write your answer in a complete sentence. c) Rocco tried to solve this problem he took the following steps ! ! = ! = ! = 10! = 7.69 Rocco concluded that the game lasted 7.69 seconds. Explain to Rocco what he did wrong and why it is mathematically illegal. Explain what he should have done to solve the problem. Dear Rocco, Activity CT Algebra I Model Curriculum Version 3.0

63 Name: Date: Page 5 of 5 4. Julie purchased tickets to a Mets baseball game from the Mets website. Each ticket costs $19 and the website charged a convenience fee of $5.75 per ticket. To celebrate going to the game, Julie also bought a new jersey from the website for $60. a) Write an expression that represents the total amount of money she spent on the website based on the number of tickets she bought. First define your variable. b) If her bill came to $159.00, how many tickets did she buy? Write and solve an equation, list your steps, show you work, and then write your answer in a complete sentence. Activity CT Algebra I Model Curriculum Version 3.0

64 Name: Date: Page 1 of 3 Solving Equations with Variables on Both Sides 1. You are looking for a dog spa to care for your dog. Each time you visit a spa, you ask them to give your dog a bath. There are two companies you are considering. Golden Dog Care charges $20 for each visit plus $5 for each bath. Super Dog Delight charges $15 for each visit and a $60 fee for an unlimited number of baths. You need to know how many visits will make the two companies charge the same amount. a) Write an expression for the amount that Golden Dog Care will charge for n visits. b) Write an expression for the amount that Super Dog Delight will charge for n visits. c) Write an equation which determines when these companies charge the same amount. d) We can solve this equation in three different ways. Let s first use a table. Complete the table below. Number of Visits Golden Dog Care Super Dog Delight e) What are some disadvantages to using a table to find the solution of an equation? Activity CT Algebra I Model Curriculum Version 3.0

65 Name: Date: Page 2 of 3 f) Now solve the equation using the properties of equality. List your steps, and show all of your work. Check your solution. g) Now solve the equation by making a graph on your graphing calculator. Set your graphing window using the values in the window below: In the Y= menu, enter the left side expression into Y 1 and the right side expression into Y 2. Press GRAPH and find the point where the two lines intersect. h) What are some disadvantages to using a graph to find the solution? Activity CT Algebra I Model Curriculum Version 3.0

66 Name: Date: Page 3 of 3 2. Dennis is collecting aluminum cans to raise funds for a local animal shelter. He needs to determine whether he should return the cans to the local supermarket or return the cans to the recycling center. If he brings the cans to the supermarket, he receives 5 cents per can. If he brings the cans to the recycling center, he receives 6 cents per can but is forced to pay a $15 recycling fee. How many aluminum cans would he have to return to receive the same amount of money at the supermarket and the recycling center? Write an equation that describes this problem, and then solve it. 3. Create your own problem by filling in the blanks below. A membership to a rock-climbing gym allows you to climb as much as you want for a fee of $ but you also pay $ per day for equipment rental. Nonmembers pay $ per day to use the gym and $ per day for equipment rental. Find the number of days in which the total cost for the members and nonmembers are the same. Write an equation that solves the problem you created above. Then solve the equation and check your solution. Activity CT Algebra I Model Curriculum Version 3.0

67 Name: Date: Page 1 of 1 Practice Solving Equations Solve the following equations. Show your work below each equation. Check your solution. 1. 4c + 8c = c 2. 4f f = w 7 = 2w x + 6x + 49 = 5x y = 20 8y z 1 = 4z z Activity CT Algebra I Model Curriculum Version 3.0

68 Name: Date: Page 1 of 2 Solving Equations with Balance Scales To explore equations with variable terms on both sides, go to the Algebra Balance Scales Applet on the National Library of Virtual Manipulative (NLVM) website at: g_4_t_2.html Here is an example of an equation being solved with the applet. The rectangles represent x, the squares represent one, and the red balloons represent negative one. The equation is: 3! 3 =! + 5 Step 1: Add 3 to both sides. The equation is now: 3! =! + 8 Step 2: Subtract x from both sides. The equation is now: 2! = 8 Step 3: Divide both sides by 2. (Remove half of each side.) The equation is now:! = 4 This is the solution. Activity CT Algebra I Model Curriculum Version 3.0

69 Name: Date: Page 2 of 2 As you have seen, sometimes you will have variable terms on both sides of the equation. To solve these equations, we must get all the variable terms on one side, get all the constant terms on the other side, and then combine like terms. Solve each equation below. Show your work at each step. You may use Algebra Balance Scales to model and solve the equations. 1. 6! = 2! ! = 2! 3.! + 4! = 3! ! 9 = 8! 4 5.! + 8 = 3! ! + 1 = 4! + 3 Activity CT Algebra I Model Curriculum Version 3.0

70 Name: Date: Page 1 of 2 How Many Solutions? A linear equation can have one solution, no solution, or an infinite number of solutions. Solve each multi-step equation and identify the number of solutions. 1. 4x + 5 x = x x + 8 = x x 3x + 9 = 2 + 8x x 6 = 9x x 5. x + x + 3 = 3 + 2x x 3x = 10 5x Activity CT Algebra I Model Curriculum Version 3.0

71 Name: Date: Page 2 of 2 7. In solving a multi-step equation, a student ended up with 0x = 4. She decided to divide both sides of the equation by 0 and got x = 0. How would you convince her that she has made a mistake? 8. What values of x satisfy the equation 0x = 0? Explain your reasoning. Activity CT Algebra I Model Curriculum Version 3.0

72 Name: Date: Page 1 of 2 Comparing Cab Fares Tonight you and three of your friends plan to go to a movie. But no one has a driver s license and every family member is busy. You need to find transportation. After doing a Google search you find the following three companies. Speedy Cabs $0.70 per mile $5.50 fuel surcharge On Time Cabs $1.50 per mile No fuel surcharge Fast Cabs $0.95 per mile $3.75 fuel surcharge 1. Each of the cab companies lets extra passengers ride free. Write an expression for the cost of using Speedy Cabs for a round trip. Be sure to define your variables. Then do the same for the other two cab companies. 2. Use a website such as to find the distance from your home to the nearest movie theatre. If you don t have access to the internet, estimate the distance. 3. Which of the three companies would be the cheapest for you and your friends to take? Show why. 4. As an alternative, there is a bus route one block from your house with a stop at the movie theater. One way the fare is $1.50. Should you and your friends take the bus? Explain your decision. Activity CT Algebra I Model Curriculum Version 3.0

73 Name: Date: Page 2 of 2 Hasheem lives in Boston, which has the same three cab companies. The fees are the same, as well. Rather than trying to calculate how much it costs to use each company and comparing the prices, Hasheem thought he would just find when they all cost the same amount. 5. Help Hasheem determine how many miles will it take for Fast Cabs and Speedy Cabs to charge the same amount. 6. Using your work from problem 5, explain which cab company, Fast Cabs or Speedy Cabs, is better to use and when. 7. Hasheem now wants to know when Fast Cabs charges the same amount as On Time Cabs. Explain if one company is more economical to use over the other company and when. 8. Make one last calculation for Hasheem. Find when Speedy Cabs charges the same amount as On Time Cabs. Explain if one company is more economical to use over the other company and when. Activity CT Algebra I Model Curriculum Version 3.0

74 Page 1 of 7 Unit 2: Investigation 4 (4 Days) Solving Equations Using the Distributive Property CCSS: 8EE 7, A-SSE 3, A-CED 1, A-REI 1, A-REI 3 Overview Students write and solve multi-step linear equations using the distributive property in a number of different contexts. The investigation begins with an introduction to the distributive property and how it can be used in equation solving. Activities focus on solving equations containing fractions and solving multi-step equations in context, and reinforce student understanding of contradictions and identities. Assessment Activities Evidence of Success: What Will Students Be Able to Do? Solve multi-step equations in a variety of contexts using the distributive property and combining like terms. Assessment Strategies: How Will They Show What They Know? Exit Slips & ask students to solve non-contextual equations using the distributive property. Exit Slip asks students to write an equation to model a situation and use the distributive property and combining like terms to solve the equation. Journal Prompt asks students to reflect on what they find difficult about solving multi-step equations and how they can overcome the difficulty. Launch Notes 1. Introduce students to a real world problem which can be modeled by an equation that involves parenthesis. Activity begins with the problem You and three friends go to the local fair. You each buy a $3 food ticket and a stamp for unlimited rides. If the total cost for the four of you is $32, how much does the stamp for unlimited rides cost? Model this problem by the equation 4! + 3 = 32 and then use this equation to discuss the distributive property. Closure Notes The majority of the investigation involves contextual situations in which students may work in cooperative groups to explore different situations. Students should be asking questions and reporting answers and strategies to the class often. Therefore, the closure activity may consist of a summary of different strategies to ensure that all students are comfortable with the act of solving multi-step equations. Unit 2 Investigation 4 Overview CT Algebra I Model Curriculum Version 3.0

75 Page 2 of 7 Teaching Strategies I. In Activity Solving Problems using the Distributive Property, students write expressions and equations to model real world situations and then solve the equations. Encourage students to write equations using parenthesis so they can apply the distributive property when solving the equations. In the first problem, the equation has the form 4! + 3 = 32 where u represents the cost of an unlimited ride stamp. At this point you may formally introduce the distributive property or break students into groups and ask the class to develop their own definition. A formal statement of the distributive property is that if a, b, and c are any real numbers, then!! +! =!" +!". Depending on your student s backgrounds you, may choose to show this property with algebra tiles or an area model as shown below. A fuller description of the property is the distributive property of multiplication over addition. Group Activity Break students into groups of two or three and ask them to come up with a statement for the distributive property in their own words. Then, combine two groups, and have the two groups compare and agree upon a statement for the distributive property. Finally, have the groups share their definition and have the class agree on a statement for the distributive property. Note: If you still have students who use the flow-chart method for solving equations, that is fine. Make sure to advise them they can continue to use that method, but they must first simplify the equation (distribute, combine like terms, and get the variable on only one side of the equation) before they put the equation into the flowchart. Differentiated Instruction (Enrichment) The distributive property with which we are familiar relates multiplication and division. Students may want to explore whether there are other pairs of operations for which there is a distributive property? For example does multiplication distribute over subtraction? Does division distribute over addition? Does addition distribute over multiplication? II. In Activity Distributive Property with Algebra Tiles, students use algebra tiles to model expressions and equations, and to understand how the distributive property Unit 2 Investigation 4 Overview CT Algebra I Model Curriculum Version 3.0

76 Page 3 of 7 works. The last part of the activity asks students to identify whether or not the distributive property was used correctly to rewrite expressions. Differentiated Instruction (For Learners Needing More Help) Algebra tiles (Activity 2.4.2) provide students a visual understanding of the distributive property. III. In Activity Using the Distributive Property, students use the distributive property to rewrite algebraic expressions. Differentiated Instruction (For Learners Needing More Help) Activity provides students an opportunity to practice using the distributive property. IV. You may pose a problem to the class which leads to an equation with parentheses on both sides, such as: The menu at the food stand said that French fries cost $2 and onion rings cost $4. You couldn t tell how much hot dogs cost because there were ketchup stains all over the menu. You were both very hungry after going on so many rides. You ordered three hot dogs, each with a serving of fries. Your friend ordered two hot dogs and two servings of onion rings. You each were charged the exact same price. How much did each hot dog cost? Allow students to think about this question for a while. Students should arrive at the equation 3! + 2 = 2(! + 4). You may use a diagram similar to the one below to model this equation. The diagram displays 3 groups of (! + 2) on the left side and 2 groups of (! + 4) on the right side. Students should recognize that two x s on the left side balance out with the two x s on the right side. Also, the six ones on the left side balance with six of the ones on the right side. By removing what is equal from both sides, we are left with one x on the left side and two ones on the right side, meaning that x must equal two. Students can check that this solution by evaluating each expression when x = 2. Once students are comfortable with the fact that x equals two, ask them to solve the equation by applying the distributive property. Unit 2 Investigation 4 Overview CT Algebra I Model Curriculum Version 3.0

77 Page 4 of 7 Activity Walk-A-Thon poses several problems that lead to equations with parentheses. The last two problems in the activity incorporate the use of graphing calculators. V. Activity Epic Win, Epic Fail introduces students to the Pan Balance Expressions applet at NCTM Illuminations. Students use the pan balance to solve equations and develop a graphical understanding of the solution of a linear equation. Activity 2.4.5a Practice Solving Equations provides students practice solving a variety of multi-step equations. To solve the equations, students must combine like terms, use the distributive property, and execute multiple steps. The equations include identities and contradictions. Differentiated Instruction (For Learners Needing More Help) Activity 2.4.5a Solving Equations Skill Practice may be particularly helpful to students who need sequential step-by-step instruction. Group Work You can use Activity 2.4.5a Solving Equations Skill Practice as a group activity. Have the students break into pairs. They can each solve different problems, then swap papers and check their partners work by substituting the solution back into the equation. Doing the activity this way cuts down on the number of problems each student has to do themselves and provides an opportunity for students to help each other out and to analyze and discuss errors in each other s work. VI. In Activity 2.4.6a Pizza Party, students construct and solve equations which involve parenthesis. The activity begins with the problem: For Raul s birthday, his girlfriend Jessica invited some of their friends to the fair. They got a picnic table and she ordered 6 large pizzas. Luckily, she had a coupon for $3 off each pizza. If the bill came to $38.94, how much was each large pizza? The activity contains scaffolding to assist students develop the equation. Once the equation 6! 3 = is developed, ask students what happens if we first divide both sides by 6 instead of distributing first. This may lead to an interesting discussion regarding the different ways to undo an equation involving parenthesis. The second problem asks students to create an equation that involves parenthesis and combining like terms. The problem states: What if Jessica bought small pizzas instead? She decides to buy 7 small pizzas, but she only has four coupons. Each coupon reduces the cost by $2. She bought four small pizzas at the discounted price and paid full price for the other three. If the bill came to $44.50, how much was each small pizza? If p equals the price of a small pizza, the resulting equation is 4! 2 + 3! = Unit 2 Investigation 4 Overview CT Algebra I Model Curriculum Version 3.0

78 Page 5 of 7 Differentiated Instruction (Enrichment) Activity 2.4.6b Pizza Party does not contain scaffolding. This version could be used with students who are more comfortable constructing equations. Activity Multi-Step Equation Challenge provides students additional practice solving multi-step equations, some of which are identities and contradictions. You may want to complete the first problem with the whole class to model the steps needed to solve the equation. For additional practice, students may go to the site ml and solve the multi-step equations listed there. They can check their answers and see how multi-step equations are solved. You may ask students to print out their answers and bring them into class to show you. VII. VIII. In Activity Fraction Busters, students learn how to remove fractions from an equation by multiplying both sides of the equation by the lowest common denominator of any fractions in the equation. In Activity Geometry and Sports, students create and solve equations that model sports related problems and geometrical figures. Differentiated Instruction (Enrichment) Activity Geometry and Sports is an activity for students needing more of a challenge. The problems provide strong independent learners an opportunity to use their problem-solving skills. Use some of this material with all students if time permits. In Activity Arithmetic Sequences Revisited, students use the formula for the n th term of an arithmetic sequence to solve problems. Differentiated Instruction (Enrichment) Activity Arithmetic Sequences Revisited can be used for students ready for a bigger challenge. This activity revisits arithmetic sequences, this time using the formula for finding the nth term. Differentiated Instruction (For Learners Needing More Help) Encourage students who are struggling solving multi-step linear equations to visit the following websites to see examples and videos. Unit 2 Investigation 4 Overview CT Algebra I Model Curriculum Version 3.0

79 Page 6 of 7 Journal Prompt What do you find most difficult about solving multi-step equations? How do you plan on overcoming this difficulty? Group Activity Activity Big Brain Contest is a power point presentation which contains 15 multi-step equations. Each slide displays one equation that awards a certain number of points. You can divide the class into two teams, having each team sit on opposite sides of the room. Have both teams solve the equation for a set amount of time. Set up a timer. (3 minutes per problem may be a good place to start, depending on the class and the equation s level of difficulty.) Tell students to stand up when they ve completed the problem and have an answer, but wait for the timer to expire before asking students for an answer (allowing more students to finish). You may want to extend the time if there are only a few students standing. You can award points either by issuing them to the team with the first correct answer or by issuing them to each person who was able to get the correct answer (the more students with the correct answer, the more points for the team.) Reserve 10 minutes at the end of class to review any problems the students want to see worked out. You can also have the students who get the correct answer put their work on the board. This activity is intended as a review activity. Resources and Materials Activity Solving Problems Using the Distributive Property Activity Distributive Property with Algebra Tiles Activity Using the Distributive Property Activity Walk-A-Thon Activity Epic Win, Epic Fail Activity 2.4.5a Practice Solving Equations Activity 2.4.6a Pizza Party Activity 2.4.6b Pizza Party Activity Multi-Step Equation Challenge Activity Fraction Busters Activity Geometry and Sports Activity Arithmetic Sequences Revisited Activity Big Brain Contest Exit Slip Distribute! Exit Slip Let s Solve! Exit Slip Saving for College Student Journals Graphing Calculators Algebra Tiles Unit 2 Investigation 4 Overview CT Algebra I Model Curriculum Version 3.0

80 Page 7 of 7 Pan Balance Expressions applet at NCTM Illuminations. Link to online practice for solving multi-step linear equations Link to online videos and examples of solving multi-step 1-variable equations Unit 2 Investigation 4 Overview CT Algebra I Model Curriculum Version 3.0

81 Name: Date: Page 1 of 4 Solving Problems using the Distributive Property Class description of the Distributive Property: 1. You and three friends go to the local fair. You each buy a $3 food ticket and a stamp for unlimited rides. If the total cost for the four of you is $32, how much does the stamp for unlimited rides cost? a) What is the unknown cost in the problem? b) What is the known cost in the problem? c) Write an expression showing the cost for one person. d) Write an expression showing the total cost for everyone. e) The total cost equals $32. Write an equation that models this situation. f) Now we want to solve this equation to find the cost of the stamp. If you followed the steps correctly, your equation should have a pair or parentheses in it. Up until now, we have not solved any equations with parentheses. How we are going to get rid of the parentheses? Explain below. Activity CT Algebra I Model Curriculum Version 3.0

82 Name: Date: Page 2 of 4 g) Use the distributive property to eliminate the parentheses in the equation. h) Now solve the equation. Show your work and circle your answer. 2. Mr. Matthews organized a field trip to Lake Compounce for his ninth grade students who had perfect attendance. He bought the admission ticket for each student, plus a gift for each student. The gifts for the girls cost $5 each and the gifts for the boys cost $4 each. 12 boys and 15 girls attended the field trip. The total cost of the tickets and gifts was $798. After Mr. Matthews spends the money, the Principal demands to know much each admission ticket cost. Mr. Matthews has never been good at math, so he needs your help. How much did each ticket cost? a) Write an expression modeling the cost for just the girls. b) Write an expression modeling the cost for just the boys. c) Using the two expressions, write an expression modeling the total cost. d) Use the distributive property to simplify the expression. e) Write an equation that you can use to find the ticket cost. Solve the equation and check your solution. Activity CT Algebra I Model Curriculum Version 3.0

83 Name: Date: Page 3 of 4 3. Your bank charges a monthly fee of $2.25 for your checking account and an additional $1.25 for each transaction you make with your debit card. Your May bill is for $ How many transactions did you make with your debit card in May? a) Assign a variable for the number of transactions that you made in May. b) Write an equation to find the number of transactions you made in May. c) Solve the equation. Does your answer seem reasonable? Why or why not? 4. Because you always pay your monthly bill on time, the bank says that in June, the first two transactions that you make with your debit card will not be charged a $1.25 fee. a) Let x equal the number of transactions you made in June. Write an expression for the number of transactions that you will be charged for in June. b) Now write an expression for the cost of the transactions in June. c) Suppose that your June bill is $ Write an equation to find the number of transactions you made in June. d) Solve the equation. Does your answer seem reasonable? Why or why not? Activity CT Algebra I Model Curriculum Version 3.0

84 Name: Date: Page 4 of 4 5. Something went wrong in July! Your monthly fee increased to $2.45 and you are again being charged for each transaction. Your July bill is $8.95, and you know that you only made five transactions. What were you charged for each transaction? Write an equation and solve. 6. The bank wants to keep your business, so they change their policy. In August, your first two transactions are not charged a fee. You suspect that the transaction fee changed. Your monthly fee is $2.45, and your August bill is $ You know that you made 10 transactions. What were you charged for each transaction? Write an equation and solve. Activity CT Algebra I Model Curriculum Version 3.0

85 Name: Date: Page 1 of 4 Distributive Property with Algebra Tiles KEY: = x = 1 = 1 Write the expression represented by each group of tiles Represent each expression using tiles. Draw your tiles below each expression. 4. 2! ! 1 6.! 4 The expression 2(2! + 3) indicates two groups of tiles as shown below on the left. You can combine like terms to simplify the expression and you get what is shown below on the right.!!" +! = 2! ! + 3 = 4! + 6 The tiles show that the product of 2 and (2! + 3) is 4! + 6. This illustrates the Distributive Property. The model above represents the equation 2 2! + 3 = 4! + 6. Write an equation for each model. (Use the example above as a guide.) 7. Activity CT Algebra I Model Curriculum Version 3.0

86 Name: Date: Page 2 of Use tiles to represent the expression. Then rewrite the expression without parenthesis (! + 1) 11. 4(2! 3) 12. 2(3! 1) 13. Does 4(2! 5) equal 8! 5 or 8! 20? Explain. Activity CT Algebra I Model Curriculum Version 3.0

87 Name: Date: Page 3 of 4 Write an equation to model the picture. 14. = 15. = 16. = Draw a picture to model the equation ! 7 = ! + 1 = ! 3 = 3 Activity CT Algebra I Model Curriculum Version 3.0

88 Name: Date: Page 4 of 4 In problems 20 23, decide if the distributive property was applied correctly. Explain your answer ! + 1 = 8! + 1 YES or NO ! 2 = 15! + 10 YES or NO ! 1 = 21! 7 YES or NO ! = 18! +6 YES or NO 24. Suppose one of our classmates was absent from class today. She will need to know what the distributive property means. Look over your work on this activity and briefly describe the distributive property below. Activity CT Algebra I Model Curriculum Version 3.0

89 Name: Date: Page 1 of 2 Using the Distributive Property If three people buy the same laptop computer and same phone (from the same store at the same time), the total cost for the three people can be represented in two different ways. Since the total cost is unique, the two representations must be equal.!"#$%!"#$ =!"#$!" 3!"#$%#& +!"#$!" 3!h!"#$ or!"#$%!"#$ = 3 (!"#$!"!!"#$%# +!"#$!"!!h!"#) This idea can be applied to algebraic expressions. Recall that multiplication can be thought of as repeated addition, so 5(! + 3) means there are five groups of (! + 3). So, there are five x s and five 3 s, so all together, there is 5! This is represented below. Write equivalent expressions without parentheses: 1. 3(a + 6) 2. 5(n 4) 3. 4(c + d) Do you notice a pattern? In each case, you end up with the products of the outside number with each of the inside numbers. This pattern is called the distributive property and it can be expressed with the following rules:!! +! =!" +!"!!! =!"!" Sometimes it s helpful to draw arrows from the outside factor to the inside factors like this: 7 ( a + 4 ) = 7a (2v + 3 ) = 16v + ( 24) = 16v 24 Activity CT Algebra I Model Curriculum Version 3.0

90 Name: Date: Page 2 of 2 Simplify the following expressions using the distributive property. 4. 9(5 + y) = 5. 5( p + 6) = 6. 11(3c 4) = 7. 2(4m 7) = 8. 3(10x + 4) = 9. 1(p + 8) = 10. 2(3a + 4c 5d) = (8m 4n) = 12. 6(1 5m) = 13. 2(1 5v) = 14. 3(4 + 3r) = 15. 3(6r + 8) = 16. 6(7k + 11) = 17. 3(7n + 1) = 18. 6(1 + 11b) = (a 5) = 20. (3 7k)( 2) = (8x + 20) = 22. (7 + 19b)( 15) = 23. (x +1)(14) = Activity CT Algebra I Model Curriculum Version 3.0

91 Name: Date: Page 1 of 2 AWalk-a-Thon Raul s school decided to participate in a walk-a-thon to raise money for a local charity. His homework last night was to get pledges and bring them back to class. Three of Raul s aunts said they would each pledge $4.50 for his participation and then $1.50 for every mile that Raul walked. His classmate Casey got her grandfather to pledge $200 for the walk-a-thon. His friend Thanoj got his mother and sister to each pledge $8.00 for every mile that he completed. 1. If we let m = miles walked, then the expression: 3( m) describes the amount of money pledged to Raul. Briefly explain why this expression models his pledge amount. 2. Write expressions for Casey s and Thanoj s pledges. 3. In your opinion, which of the three students received the best pledge? Briefly justify your answer. 4. Anyone who earns more than $100 receives an itunes gift card. How many miles would Raul need to walk to earn the gift card? Explain how your answer. 5. How many miles would Thanoj need to walk to earn the gift card? Does either Raul or Thanoj have a realistic chance? Activity CT Algebra I Model Curriculum Version 3.0

92 Name: Date: Page 2 of 2 6. How many miles would Raul need to walk to collect the same amount in pledges as Casey? Briefly explain your strategy. 7. How many miles would Thanoj and Raul have to walk to both earn the same amount of money? 8. The graphing calculator can also be used to explore the three students pledges. Put Raul s pledge expression into Y 1, Casey s into Y 2, and Thanoj s into Y 3. Use the table feature of the graphing calculator to solve problems 3 7. Copy values from the tables in the graphing calculator into the tables below. Then explain how the values in the tables may be used to answer these problems. Raul Casey Thanoj x y x y x y Graph the three equations together in the same window. Use the window settings below. Does the graph verify your predictions? How is using a graph similar to using a table? How is it different? Activity CT Algebra I Model Curriculum Version 3.0

93 Name: Date: Page 1 of 3 1. Solve the following equation. 5! + 2 = 3! + 4 Epic Fail, Epic Win 2. Using the Pan Balance Applet at NCTM Illuminations, I entered one side of the equation in the left pan and the other side in the right pan. What does the image above tell you about the value(s) of x that makes the pans balance? 3. Solve the following equations. (a) 2! 2 = 3 + 2! (b) 3! + 7 = (4 3!) 4. What happened? We get false statements and cannot find a real number that makes the statements true. Activity CT Algebra I Model Curriculum Version 3.0

94 Name: Date: Page 2 of 3 5. EPIC FAIL! What do the following images tell you about the value(s) of x that make the pans balance? 6. Solve the following equations. (a) 2! 2 = 4 + 2! (b) 0.2! 4 =! (! 20)! 7. What happened? We get true statements. The expressions on both sides are identical. EPIC WIN! 8. What do the following images tell you about the value(s) of x that make the pans balance? Activity CT Algebra I Model Curriculum Version 3.0

95 Name: Date: Page 3 of 3 Vocabulary: Equations where no value of x can make them true are called contradictions (EPIC FAILS!), and equations where any value of x can make them true are called identities (EPIC WINS!). 9. Write down everything you see about the equations, the graphs and the solutions for the above problems. Activity CT Algebra I Model Curriculum Version 3.0

96 Name: Date: Page 1 of 2 Pizza Party 1. For Raul s birthday, Jessica invited Raul and some of their friends to a fair. They got a picnic table and she ordered 6 large pizzas. Luckily, she had a coupon for $3 off each pizza. If the bill came to $38.94, what was the price of a large pizza? a) What is the unknown in this problem? b) What is known about the actual cost of one pizza? c) Write an expression for the cost of one pizza. d) Write an expression for the cost of all six pizzas. e) The total cost for all the pizzas was $ Write an equation that models this situation. f) Solve the equation. What was the price of each large pizza before the discount? 2. What if Jessica bought small pizzas instead? She decides to buy 7 small pizzas, but she only has four coupons. Each coupon reduces the cost by $2. She bought four small pizzas at the discounted price and paid full price for the other three. If the bill came to $44.50, how much was each small pizza? a) Write an expression for the cost of the 3 small pizzas that Jessica bought at full price. b) Write an expression for the cost of the 4 small pizzas that Jessica bought at a discounted price. Activity 2.4.6a CT Algebra I Model Curriculum Version 3.0

97 Name: Date: Page 2 of 2 c) Write an equation that models this situation. d) Solve the equation. What was the price of each small pizza before the discount? 3. Raul, Jessica, and their friends enjoyed eating the pizza on a rectangular picnic table. The dimensions of the picnic table are presented in the figure below. The perimeter of the rectangular table is 96 in. Find the length of each side. 3(x + 5) 2(3x + 3) a) Write an equation modeling the perimeter of the rectangle. b) Solve the equation. c) List the lengths of each side: Side 1: Side 2: Side 3: Side 4: d) Examine your answers in question c. What is this kind of rectangle called? Activity 2.4.6a CT Algebra I Model Curriculum Version 3.0

98 Name: Date: Page 1 of 2 Pizza Party 1. For Raul s birthday, Jessica invited Raul and some of their friends to a fair. They got a picnic table and she ordered 6 large pizzas. Luckily, she had a coupon for $3 off each pizza. If the bill came to $38.94, what was the price for a large pizza? 2. What if Jessica bought small pizzas instead? She decides to buy 7 small pizzas, but she only has four coupons. Each coupon reduces the cost by $2. She bought four small pizzas at the discounted price and paid full price for the other three. If the bill came to $44.50, how much was each small pizza? Activity 2.4.6b CT Algebra I Model Curriculum Version 3.0

99 Name: Date: Page 2 of 2 3. Raul, Jessica, and their friends enjoyed eating the pizza on a rectangular picnic table. The dimensions of the picnic table are presented in the figure below. The perimeter of the rectangular table is 96 in. Find the length of each side. 3(x + 5) 2(3x + 3) a) List the lengths of each side: Side 1: Side 2: Side 3: Side 4: b) Examine your answers in question a. What is this kind of rectangle called? Activity 2.4.6b CT Algebra I Model Curriculum Version 3.0