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Advice for Instruction | 13. Solving linear equations | Prepare instruction

Goals and objectives Topic overview Resources Print resources

Goals and objectives

In the topic Solving linear equations, studentslearn to solve linear equations to find a value orvalues for a variable, which is a foundational skillfor subsequent topics in this and other mathematicscourses. They also draw connections between themethods for solving linear equations and numerical(tabular), graphical, and symbolic representations of linear functions. Understanding theseconnections gives students additional tools with which to make sense of the situations modeled bylinear equations.

In this topic, students will:

Use linear equations to solve problemsUse inspection, tables, graphs, and algebraic properties to solve linear equationsDetermine whether an equation has one solution, no solution, or many solutionsDetermine the reasonableness of solutions to linear equations in given contextsTransform equations with several variables by solving for one variable in particular

The Staying Sharp problems in Topic 13 are organized around several key ideas:

Problems 1 and 2 (Practicing algebra skills and concepts): Informal solving and gaining facilitywith the forms of linear equationsProblems 3 and 4 (Preparing for upcoming lessons): Working toward graphs of 2-variableinequalitiesProblems 5 and 6 (Reviewing pre-algebra ideas): Measurement and fraction multiplication andcommon denominators

Topic overview

Lesson 13.1: Students examine how a linear equation can be created from a function rule byspecifying a certain output value for the function. Then, they are exposed to different solutionmethods that take advantage of their understanding of multiple ways in which to represent linearfunctions with tables, by graphing, and with symbols. Finally, students make connections betweenthe different solution methods by using them to solve problems that can be modeled by linearequations.

Lesson 13.2: Students continue developing their facility with using tables to solve linear equations.They create tables by hand and come to recognize the benefit of technology for finding anon-integer input corresponding to an integer output.

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Lesson 13.3: Students examine how graphs can be used to solve linear equations. They graph aproblem by hand and recognize the benefits of using technology. Students also learn to defineappropriate intervals and scale when graphing.

Lesson 13.4: Students use a single algebraic or analytical process to solve linear equations. Theyalso continue investigating various methods for solving linear equations.

Lesson 13.5: Students use two algebraic operations to solve linear equations. Then, theyinvestigate how to solve two-step linear equations through an activity in which they build and thensolve equations.

Lesson 13.6: Students use multiple algebraic or analytical processes to solve linear equations.They also continue applying equality and identity properties to solve these equations.

Lesson 13.7: Students explore equations that have no solutions or an infinite number of solutions.Then, they apply what they know about slopes of lines to identify equations with one solution, nosolution, or an infinite number of solutions. During the second half of the class, students will needaccess to a computer lab in order to take an online assessment. In preparation for the onlineassessment, you will need to make an assignment using the Guided assessment for this topic.

Lesson 13.8: Students apply equality and identity properties to solve literal equations. Guidedassessment reports from the online assessments students took last class are also analyzed duringthis lesson. The homework asks students to go online to complete the More practice questions fromthe prior topic as one means of preparing for the mid-unit assessment.

Resources

LESSON RESOURCES

Computer with projection device and Internet connectionGraphing calculatorsOverhead calculatorComputer lab (Lesson 8)Large chart paperMasking tapeChart markersBalance or scalesWhiteboards, dry erase markers, and erasersString

Print resources

Unit 5 Topic 13 Advice for instructionUnit 5 Topic 13 Student Activity Book (Updated 8/6/2014)Unit 5 Topic 13 Student Activity Book with answer key (Updated 8/6/2014)Problem/equation/solution cards


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Equation/solution method cards


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Advice for Instruction | 13. Solving linear equations | Deliver instruction

Lesson materials Lesson preview Lesson activities

Lesson materials

Lesson 13.1 “The connection between linearfunctions and linear equations”Student Activity BookProblem/equation/solution cardsGraphing calculators

Lesson preview

Suggestedtime

Activity Goals

10 min OpenerWrite a function rule to represent a situation and use thisrule to determine an output given a specific input

30 min Core activityConsider four methods you could use to solve a linearequation

10 min Process homework Learn from reviewing the homework due today

25 min Consolidation activityMatch equations with solutions found by using graphs,tables, and algebraic procedures

5 minWrap up and introducehomework

Reflect on the day’s lesson and understand tonight’shomework assignment

Lesson activities

OPENER (10 minutes)Students model a problem situation with a function rule and revisit the familiar notion of findingfunction outputs given particular inputs. The animation on page 3 can be used to debrief theOpener and transition to the core algebraic learning of the lesson—setting up and solvingequations.

Solving a linear equation can be thought of as recovering the input value that produced a particularoutput value under a linear function rule. Connecting the writing and solving of linear equations totabular, graphical, and symbolic representations of linear functions not only gives studentsdifferent strategies for solving equations, but helps develop a deeper understanding of why they


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are solving them. Although you may see that students are already familiar with using algebraicprocedures to solve basic equations, this lesson will help students explore the meaning of asolution to an equation. It will also help them draw connections between their understanding oflinear functions and equation-solving.

Online page 1

Debrief by asking a few students to share their solutions and explain how they arrived at theiranswers. [SAB, questions 1-3] Hold off on validating students’ responses until online page 3,where students will have the opportunity to verify their solutions using an animation.

Online page 2

Preview for students the topic goals addressed in the day’s lesson and the lesson activities.

CORE ACTIVITY (30 minutes)Students are introduced to four methods for solving linear equations: graphing, using a table, usingalgebraic operations, and “undoing.” By making the connection between linear functions and linearequations, students build on their previous work in this course with representing functions usingtables, graphs, variables, and words.

Online page 3Students investigate the problem from the Opener and build understanding using an animation.

Remind students of the situation from the Opener. In the animation, panels 1 and 2emphasize the constant rate of change. After playing panel 2, ask:

If the constant rate were 12¢ per mile, how would the function rule change?What do you expect the graph of the function rule r = 24.95 + 0.05m to look like?

Panels 3 and 4 confirm students’ thinking about the graph and remind them of the input-output relationship described by the function. Draw students’ attention to the word “unique”in the captions for these panels. After playing panel 4, ask:

How does the graph show that “each input is associated with one unique output”?Classroom strategy. You may find opportunities as you debrief the Opener to review andreinforce students’ knowledge of linear functions. For example, you could ask:

Why is the graph not very steep?What is the y-intercept of the graph, and how does that relate to the problemsituation?By looking at the graph and the table, what is the cost if you drove 200 miles?How far did you drive if the total cost was $37.45?

However, be sure to manage the time spent discussing the Opener carefully to ensure thatyou have plenty of time for the remaining activities in the lesson.

Online page 4Using the function rule, students write the equation that will help them find the distance drivenfor a given cost.

With the whole class, make the connection between an equation and a function rule.Students will revisit this idea later when they complete a journal describing function rule,equation, and solving an equation. For now, emphasize that when you are using a functionrule to find a specific value, you can do so by solving an equation.


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Students write the equation that will help them find the distance driven for a given cost.[SAB, question 1] To help them make the transition from function rule to specific equation,you could ask:

What value are you looking for?What value is known?How can you use the function rule to find the value you are looking for?

Be sure to reach consensus with the whole class about the equation to be solved: 35.80 =24.95 + 0.05m. This equation sets up the rest of the Core activity.

Online page 5Students work in pairs to solve an equation using each of the four methods they have learned, thenanswer reflection questions.

Have students work with their partners to complete the activity in their activity books usingthe different methods. [SAB, questions 2-4] As you circulate, emphasize the connectionsamong the methods used to solve the equations by asking questions such as:

Does your graph show the same solution as the table? How can you tell?Where is the solution in your table and graph? (Is it a value of m or a value of r?)How is undoing like solving using algebraic operations?

Online pages 6-10Each of the four methods of solving the equation is debriefed.

Ask pairs of students to briefly present to the whole class their solutions for question 2. Makesure each method is presented. Ask the class to check for agreement between each of thesolutions presented. Encourage students to explain their thinking by asking questions such as:

How do you know that value is the solution to the equation?Does that answer make sense in the context of the problem? Does it seem like areasonable number of miles?Why is this answer different from others? (If this is the case)

Use pages 6-10 to review the solution methods and clarify the methods used by students inthe activity. For each method, connect the demonstration in the animation to the studentpresentations that demonstrated that method. Emphasize that students will use and exploremore deeply all four methods presented as they work with solving linear equations in thistopic.Language strategy. Review with students the names of the various methods for solvingequations and alternate names these methods sometimes take, such as graphicsolution/solve graphically, solving with a table/tabular solution, solvinganalytically/algebraically/solving by hand, and undoing/working backward.

Online pages 11-12The four methods are summarized, and the rest of the questions in the activity are debriefed.

Page 11: Use this page to summarize the four methods and debrief question 3. Ask:How can you use the function rule to check the solution to the equation?

Page 12: Use this page to debrief question 4, which helps students formalize the connectionbetween solving by “undoing” and solving using g algebraic operations. Have students look atthe images on the page and ask:

How is the driver’s thinking connected to one of the algebraic operations used to solvethe equation on the right?


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PROCESS HOMEWORK (10 minutes)

Online page 13Students process the homework due today: Homework 12.7 and Staying Sharp 12.7.

Review question 3 with students. It is important to call students’ attention to the differencesbetween the two data sets. By looking at 3a and 3d, you can ask students to compare datathat is linear and approximately linear. Notice that a linear equation can be written torepresent the data for the Railroad Museum. However, for the Town History Museum data,you can only write a linear equation that represents the trend in the data.

CONSOLIDATION ACTIVITY (25 minutes)Students complete a card sort to help them make connections between problem situations,algebraic equations, and the various ways equations can be solved. For this activity, you will needto download the master for the cards from the Lesson materials section. Print one set of cards foreach pair of students. You may want to cut the cards in advance to save class time.

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Have students work with partners to complete the card sort activity described in the activitybook [SAB, question 1]. They will practice using the four methods for solving linear equationsdiscussed in class.In debriefing the card sort, ask students to describe how they matched specific groups ofcards by asking questions such as:

How did you know that equation represented that problem?How did you match the graph to the equation?What value on the graph represents the solution? (Is it the x-value or the y-value?)

The correct card groupings are as follows:1-B-b-g-i2-D-a-h-j3-C-d-f-k4-A-c-e-m

Students complete a journal entry about their understanding of the relationship of function,equation, and a solution to and equations. [SAB, question 2]

WRAP UP AND INTRODUCE HOMEWORK (5 minutes)

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Homework 13.1Students practice solving equations using the various methods. Then they compare thesesolution methods.Staying Sharp 13.1The main concepts and skills in these problems are:

Informally solving a one-step, one-variable equation1.Identifying slope and y-intercept of a line from its equation in slope-intercept form2.Evaluating a possible solution of a two-variable equation3.


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Evaluating a possible solution of a two-variable inequality4.Determining the side length and area of a square from its perimeter5.Adding and subtracting fractions with like denominators6.


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Lesson materials

Lesson 13.2 “Solving equations with tables”Student Activity BookGraphing calculator

Lesson preview

Suggestedtime

Activity Goals

10 min Opener Solve a linear equation using a table

30 min Core activitySolve linear equations using your graphing calculator’stable utility


25 minReview end-of-unitassessment

Learn from your performance on the end-of-unitassessment



Lesson activities

In today’s lesson, students use tables created by hand to solve equations where the output is notaligned with a single-digit non-negative integer solution. After capitalizing on students’ numbersense to manually and mentally “scroll” up and down a table in search of a solution (input value),technology is used as a tool to expedite finding a solution and solidify the concept of matching thecorrect input to the given output value. Students will then follow a similar developmental processfor solving an equation when the solution is between two integers.

OPENER (10 minutes)Students build a table of values for a function rule by hand. Then they use the table to solve arelated equation.


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Online page 1

Students should build the table by hand. [SAB, questions 1-2] Students use calculator tablesfor the other activities in today’s lesson In debriefing, ask:

What input, or x-values, did you use originally?Did any of your original input values result in an output, or y-value, of 14?What adjustments did you have to make so that you could use a table to solve theequation?

Online page 2


CORE ACTIVITY (30 minutes)Students learn how to use the table feature on the graphing calculator to solve linear equations.

Online page 3Students use the table feature of their calculators to solve the equation from the Opener.

Review the process for using a calculator table to solve a linear equation. Then have studentswork in pairs solve the equation on their calculators. [SAB, question 1]Use the animation on the page to help students solidify their understanding of the tablefeature on the graphing calculator. The animation includes a panel to help students use thedelta-table function on the calculator.

Online pages 4-7Students solve three linear equations using the table feature of their calculators, then compare theprocesses they used.

Page 4: Have students work in pairs to solve the equations and reflect on the processes theyused. [SAB, questions 2-4] Use the following pages to assist with the debrief.Page 5: Debrief question 2a. Ask students to share their processes for solving this equationusing the table feature on the graphing calculator. Then show the animation to validatestudent responses.Page 6: Debrief questions 2b and 3. Pay particular attention to question 3. Ask:

How did you decide what delta-table increment to use?Did you have to change this increment more than once?How did you decide how to refine this increment?

Page 7: Debrief questions 2c and 4. Point out to students that they must enter both sides ofthe equation into the calculator in order to use the table feature to solve. Ask:

How did you use the table to decide what the solution to the equation was?What did you look for in the table to identify the solution to the equation?



Review questions 2, 3a, and 4. Ask students to share their responses to question 4 and helpstudents make connection between the four different methods of solving a linear equation.


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Pay particular attention to the connections students make between “undoing” and usingalgebraic operations. Students should begin to realize that using inverse operations to solve alinear equation is just a formalization of “undoing”.

REVIEW END-OF-UNIT ASSESSMENT (25 minutes)Students use the Assessment Processing Routine to analyze their performance and correct theirwork on the end-of-unit assessment.

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Have students work in partner teams to review the end-of-unit assessment they took in Topic12. As necessary, review and reinforce particular aspects of the Assessment-ProcessingRoutine.Re-collect the assessments after students have had time to respond to comments and makecorrections. Inform students that you will review the corrections and take note of theconcepts and skills for which it seems that individual students and groups of students are stillhaving difficulty. Tell them that you will use the information to make an “action plan” foraddressing areas of difficulty.


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Homework 13.2Students practice solving equations using tables. They solve equations and investigateproblems that can be modeled using linear equations. They also address a commonmisconception about using tables to solve equations.Staying Sharp 13.2The main concepts and skills in these problems are:

Informally solving a multi-step, one-variable equation1.Identifying x- and y-intercepts of a line from its equation in standard form2.Graphing a line from its equation in slope-intercept form3.Shading the upper half-plane of a graph and identifying points in each half-plane4.Comparing the area of a square and its dilation5.Multiplying and reducing fractions6.


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Lesson materials

Lesson 13.3 “Solving equations with graphs”Student Activity BookGraphing calculator

Lesson preview

Suggestedtime

Activity Goals

10 min OpenerSolve a linear equation using a graph that you create on acoordinate grid

30 min Core activitySolve linear equations using your graphing calculator’sgraphing utility


25 min Consolidation activityAnalyze the advantages and disadvantages in using twomethods to find solutions to linear equations



Lesson activities

OPENER (10 minutes)Students solve an equation by graphing by hand.

Online page 1

Allow students several minutes to solve the equation by graphing. [SAB, questions 1-2] Todebrief the Opener, ask students to discuss with their partners how they used a graph to findthe solution to the equation. The solution will be verified in Activity 3.2.

Online page 2


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CORE ACTIVITY (30 minutes)Students investigate how they can solve equations by graphing using their calculator.

Online page 3Students verify their solution to the problem in the Opener.

Use this page to debrief the opener. Students will look at another method for solving thisequation by graphing later in the lesson.

Online pages 4-5Students find the solution to an equation by graphing two linear functions represented in theequation.

Page 4: Show the equation on this page, 5x – 7 = 5 – x. Ask:How many linear functions do you see in this equation?How could you use graphing to solve this equation?

Give students a few minutes to work with their partner to graph the two linear functionsrepresented in the equation by hand and using the calculator. [SAB, question 1]Technology tip. Make sure students understand that, because there are two distinctfunctions represented in this equation, each must be a separate entry in the equation editorfor their graphing technology. In other words, they will be entering and graphing two distinctfunctions with the technology.Page 5: Use this page to debrief question 1. Ask:

Where on the graph is the solution to the equation? How do you know?Use the first check reveal on the page to verify the graphs that students constructed and tohelp students answer the question posed above. Ask:

What does the x-coordinate of the intersection point tell you?What does the y-coordinate of the intersection point tell you? [SAB, question 2]

Use the second check revel on the page to help students solidify their understanding of themeaning of the intersection point.Technology tip. Demonstrate how to use a calculator to find graphical solutions by:

tracing to estimate the coordinates of the point of intersection;zooming in or our and then tracing to refine the estimate; andusing the calculator's built-in feature to find a more precise intersection

Online page 6-8Students practice solving equations using the graphing feature of their calculators.

Page 6: Have students work with their partners to complete the activity in their activity book.[SAB, questions 3-4]Page 7: Use this page to debrief question 3. Before revealing the solution, ask students howthey used a graph to solve the equation 3x + 6 = -6. Ask students to identify the intersectionpoint and discuss the meaning of the coordinates of that point. Then, use the reveal to verifystudents’ responses.Discuss how this method of solving by graphing compares with the graphing method used tosolve this equation in the opener.Page 8: Use this page to debrief question 4. Ask students how the solution to this equation isdifferent from the solutions of the other equations they have looked at. They should notice


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that the solution to this equation is not an integer, so solving by graphing by hand wouldmake it difficult to find the solution. Use the check reveal on this page to discuss whygraphing by hand would not be an appropriate method for solving the equation 8x + 10 = 5.



Go over question 6 as a class. A common misconception that students have is that the outputvalue given by a table is the solution to the equation. Use the problem to remind studentsthat the y-value gives the output while the x-value will be the input value that would makethe equation true. You can relate this easily to the method being discussed in today’s lesson,solving by graphing, by pointing out that if they were to solve this equation by graphing, theintersection point would be (9,13).

CONSOLIDATION ACTIVITY (25 minutes)

Online page 10Students solve a series of equations using different solution methods. Then, they compare anddiscuss which method worked best for each equation.

Organize students in pairs, with one student being Student A and the other Student B.Student A should use tables to find solutions to the equations in parts a, c, and e, whileStudent B uses graphs to solve those three equations. Then students should switch methodsfor solving the other three equations. [SAB, question 1]Ask students to sketch their actual calculator display in the space provided.

Classroom strategy. Monitor student work. Asking assessing and advancing questions isparticularly important when students are learning new capabilities of their graphingcalculators. Asking the right questions at the right time, instead of telling students whatbutton to push next, will empower students to be independent users of technology to findsolutions to equations more quickly.Ask students to reflect on these two solution methods. [SAB, question 2] Then, have studentsshare their responses to this question.


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Homework 13.3Students practice solving equations and interpreting solutions using graphs.Staying Sharp 13.3The main concepts and skills in these problems are:

Informally solving a two-step, one-variable equation1.Comparing the equation of a line in slope-intercept and standard form2.Evaluating a possible solution of a two-variable equation3.Evaluating a possible solution of a two-variable inequality4.Comparing the perimeter and area of a rectangle and its dilation5.Finding a common denominator and adding fractions with unlike denominators6.


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Lesson materials

Lesson 13.4 “Solving one-step equations”Student Activity BookAlgebra tilesEquation/solution method cardsGraphing calculator

Lesson preview

Suggestedtime

Activity Goals

10 min Opener Compare solving with tables and solving with graphs

30 min Core activity Use algebraic operations to solve one-step equations


25 min Consolidation activityPractice using algebraic operations to solve one-stepequations



Lesson activities

OPENER (10 minutes)Students solve a one-step linear equation using a table or graph and compare the two methods.

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Allow students several minutes to solve the equation and answer the reflection question.[SAB, questions 1-2]Debrief by having one student who solved the equation using tables and one who solved itusing graphs present their solutions. Then engage the class in a discussion about thechallenges involved with each method.

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CORE ACTIVITY (30 minutes)Students investigate how algebraic operations are used to solve one-step linear equations.

Online pages 3-6Use these pages to launch the Core activity with the whole class. Although many students mayalready be familiar with using algebraic procedures to solve basic equations, these pages helpestablish the mathematical concepts behind those procedures.

Page 3: Use the animation to build intuition for the balance scale model. As the animationmoves from 3 = 3 to 3 + 2 = 5 and students experiment with the balance scale, they willobserve that, in order to keep the scale balanced, equal amounts must be added to orsubtracted from both sides of the scale. Use this observation to link to the idea of “keepingequations balanced”. Ask students to reflect on this in the SAB. [SAB, question 1] Ask thefollowing questions to guide the discussion:

How is solving an equation like using a balance scale?How can you keep the scale balanced?

Page 4: Use panels 1 through 5 to discuss the "balance scale" model for solving equations.Then, use panels 6 and 7 to introduce the algebra tile model to help students cement theirunderstanding of using algebraic operations to solve one-step linear equations. [SAB,questions 2-3]Page 5: Formalize the concept of keeping equations balanced by performing the sameoperations on each side of the equation.Page 6: Show the page and ask:

What are two ways to “undo” addition?Give student pairs a few minutes to compare the two methods (using subtraction and usingthe additive inverse) for solving the equation x + 2 = 7. Then, play the animation to supportstudents’ discussion. [SAB, question 4] Ask:

How would you “undo” subtraction?Classroom strategy. Have one student in each pair model the subtraction strategy for solvingthe equation in the animation using algebra tiles, while the other student in the pair modelsthe additive inverse strategy. Then, have the two compare their approaches and theiranswers. This can help reinforce for many students the connection between subtraction andaddition of the opposite.

Online page 7Students practice solving one-step equations. They should use algebra tiles and sketch their tilesolutions for at least the first two equations, and for the other equations as needed to connect tothe numerical representation.

Use this page to remind students about zero pairs and connect the concept of zero pairs tosolving equations using algebra tiles. Students will need to be able to identify zero pairs inorder to use the algebra tiles to model equations and solving equations.Give students sets of algebra tiles and have them complete the activity in their activitybooks. [SAB, questions 5-9] Have students use their tiles as they solve the first equations.After that, students are only asked to write down their solutions numerically. The importantidea is that students can move flexibly between the physical tiles, sketches of tiles, andnumerical representations.Circulate as students work. A helpful question is


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How does the algebra tile model connect to keeping a scale balanced?Classroom strategy. Some students may need to rely on algebra tiles longer than others. Itmay be helpful to have the tiles available over the next few lessons for any students thatneed them.

Online page 8-10Use these pages to debrief students' work with questions 5-9.

Page 8: Discuss the use of inverse operations as a way to “undo” operations and isolate thevariable in an equation. Emphasize the application of the same operation to each side of theequation to keep it “in balance” and relate this idea back to the tile and balance scalemodels. Also, connect the idea of inverse operations (addition and subtraction) to thecreation of zero pairs.Page 9: Debrief questions 6-8 using the animation. Invite students to the class computer toshow the tile solution for the different equations. Be sure the students explain the propertiesbeing used in each step.

The equation 5 – x = 7 requires students to deal with the opposite of the variable. While theanimation does not support the use of tiles to model the final algebraic step, students shouldbe able to algebraically justify how to deal with the situation to solve the equation.

Page 10: Use this page to debrief question 9. As you display the page, ask:What are two ways to “undo” multiplication?

Give students a few minutes to discuss the two methods. Then, play the animation to supportstudents’ discussion.Reinforce the connection between division and multiplication by the reciprocal. Ask:

For what kinds of equations does it make more sense to divide to undo themultiplication of a number and variable?For what kinds of equations might it make more sense to multiply by the reciprocal toundo the multiplication of a number and a variable?

Online page 11Students summarize the process for solving a simple linear equation.

Use the questions on the page as needed to support the summary. Then, display the checkreveal to allow students to verify their understanding.



Discuss question 3 as a class. Students should realize that they can find the equation of eachline on the graph in order to determine the equation for which the solution is shown. Then,ask students to identify the intersection point of each graph and interpret the meaning ofeach coordinate of the intersection point.

CONSOLIDATION ACTIVITY (25 minutes)

Online page 13Students practice with partners solving equations using the methods they have been working with


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in the Core activity. For this activity, you will need to download the master for the cards from theLesson materials section. Print one set of cards for each pair of students. You may want to cut thecards in advance to save class time.

Equations to be solved and methods to be used are determined randomly by drawing cards.Solution methods include using algebra tiles, and solving numerically using the addition,subtraction, multiplication, and division properties of equality.After each partner solves the equation on the drawn card using the methods on their drawncards, pairs should compare their solutions and address any discrepancies.Pay special attention to whether students are distinguishing between the addition andsubtraction properties (and the multiplication and division properties) when solving equationsaccording to the method on their card.Debrief the activity by asking a few students to share their solutions to equations you select.You may want to pick students whose work will highlight particular misconceptions orimportant ideas.


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Homework 13.4Students practice solving one-step equations using algebraic operations and analyze acommon error related to equation solving.Staying Sharp 13.4The main concepts and skills in these problems are:

Solving a two-step, one-variable equation using a table1.Identifying slope and y-intercept of a line from its equation in slope-intercept form2.Graphing a line from its equation in standard form3.Shading the lower half-plane of a graph and identifying points in each half-plane4.Determining the volume of a cube5.Finding a common denominator and subtracting fractions with unlike denominators6.


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Lesson materials

Lesson 13.5 “Solving two-step equations”Student Activity BookAlgebra tilesStudent whiteboards, markers, and erasersGraphing calculator

Lesson preview

Suggestedtime

Activity Goals

10 min OpenerReason how to solve equations requiring more than one“undoing”

25 min Core activity Use algebraic operations to solve two-step equations


30 min Consolidation activity Use algebraic operations to solve two-step equations



Lesson activities

OPENER (10 minutes)

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Allow students a few minutes to think about the equation and answer the questions. [SAB,questions 1-3] When debriefing, pay particular attention to question 3 where students areasked to identify the operation that they would “undo” first.Classroom strategy. Relate the idea of “undoing” to solve an equation to the idea offollowing the order of operations to “do” in order to evaluate expressions. Thinking aboutsolving equations as “undoing” the order of operations may help students who are unsure ofwhat do to first when solving an equation.

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CORE ACTIVITY (25 minutes)Students build intuition for solving two-step linear equations using “undoing” and by looking atalgebra tile models. Then, they investigate solving two-step equations using algebraic operations.

Online page 3Students are introduced to the concept of equivalent expressions. They solve a two-step linearequation by investigating how the equation was built and then “undoing” each operation that wasdone to build the operation. Then they review the undoing process using an animation.

Have students work in pairs to solve the equation 2x + 5 = 11 . Then have students describe inwords the process they used to solve the equation. [SAB, questions 1-2]Play panels 1 and 2 of the animation to verify student responses and conceptualize theprocesses of building and solving two-step equations. Ask students to compare theseprocesses.Panel 3 reinforces the idea of equivalent equations by looking at the graphs of each equationin the equation column of question 1. Make sure to call students’ attention to the idea that,for each equation being graphed, the solution, x = 3, is always the same.

Online page 4Students use algebra tiles to solve a two-step equation. They review the process using an animationand compare the algebra tile solution with the algebraic solution.

Before showing the page, ask students to create an algebra tile model that they could use tosolve the equation 2x + 5 = 11. Then, play panel 1 of the animation on page 4 to showstudents the algebra tile model. [SAB, question 3]Play panel 2 and review the concept of inverse operations. Play panel 2 again, pausingbetween each step of the solution. Between each step, ask:

How are inverse operations used in this step to solve the equation?

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Have students work with their partners to complete the activity in their activity books. [SAB,questions 4-5] When students have had enough time to complete the activity, pick one ortwo pairs of students to share their solutions with the whole class for any of the equationsthat require further discussion. Use the check reveals on page 5 to provide another point ofcomparison. Use the discussion questions to debrief question 5, formalizing some of theprocedures they have been using.



Go over the solutions for question 1 as a class. Ask students to write their solution to theequation in part a on their whiteboard and hold it above their heads. Tell students to lookaround and make sure that the class agrees on a solution. Use this as an opportunity forformative assessment of student understanding. Repeat for the eight additional equations. Ifyou notice that students are having trouble with a particular equation, you may want tospend some time discussing that equation.


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Classroom strategy. Ensure that students are using the color-coding in the Homework-Processing Routine. This will help them to identify strengths and weaknesses. Let studentsknow that self-assessment and reflection are important skills that they will be developing asthey analyze not only their homework, but also their assessments.

CONSOLIDATION ACTIVITY (30 minutes)Students build intuition for solving two-step equations by building equations and then solving themby “undoing”.

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Have students work in pairs on this activity. [SAB, questions 1-4] Walk through theinstructions. You may want to use the example to demonstrate to students what they aresupposed to do during the activity. The example provided shows the steps that the studentshould go through before handing their paper over to their partner. You can then show thepartner’s role in solving the equation.Classroom strategy: In this activity, students will be working with their partners as they buildequations and then solve them by “undoing”. Students will have to be very careful whenwriting their equations. As you walk around the class, look for students making the commonmistake of saying that they multiplied both sides of the equation by a number but incorrectlyapplying the multiplications. For example, a student may say that they multiplied both sidesof the equation x + 2 = 3 by 2 but then write the equation 2x + 2 = 6 instead of 2(x + 2) = 6.Once students have completed the activity, ask them to compare the steps taken to “build”the equations to the steps needed to “undo” or solve the equation. [SAB, question 5] Fromthis discussion, you can reinforce the idea of inverse operations.


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Homework 13.5Students practice solving two-step equations. They continue investigating the concept ofequivalent equations by looking at the graphs of equivalent equations.Staying Sharp 13.5The main concepts and skills in these problems are:

Solving a two-step, one-variable equation using a graph1.Identifying x- and y-intercepts of a line from its equation in standard form2.Evaluating possible solutions of a two-variable inequality3.Plotting solutions of a two-variable inequality on a plane to recognize patterns4.Comparing the volume of a cube and its dilation5.Finding the reciprocal of a fraction6.


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Lesson materials

Lesson 13.6 “Solving multi-step equations”Student Activity BookGraphing calculator

Lesson preview

Suggestedtime

Activity Goals

10 min OpenerIdentify algebraic properties used to solve a linearequation

25 min Core activity Use algebraic operations to sole multi-step equations


30 min Consolidation activityAddress some common errors in using algebraicoperations to solve multi-step equations



Lesson activities

OPENER (10 minutes)Students review various algebraic properties as they investigate the solution process for a linearequation.

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Students identify the algebraic properties used to solve an equation. [SAB]To debrief, ask students for alternate solution methods. For example, in the second step ofthe process demonstrated, the Addition Property of Equality and additive inverses are used.However, students should realize that they could have use the subtraction property ofequality as an inverse operation to get the same resulting equation of 2x = -10. Similarcomments could be made about the fourth step in the process demonstrated.


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CORE ACTIVITY (25 minutes)Students expand their investigation of solving linear equations to equations that contain a variableterm on both sides of the equation. They also look at the properties of equality that they will usein order to manipulate and solve multi-step equations.

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Page 3: The table of properties introduces students to a more formal way of stating theproperties of equality with mathematical symbols.If students are having difficulty interpreting the symbolic statements, have them replace theletters with consistent numbers (e.g., a = 2, b = 2, and c = 3) to show how the properties worknumerically.Students complete a journal activity to help them check their own understanding about theproperties of equality and how they are used. They should compare their journal entries withtheir partner’s, then work in pairs to use the properties of equality in solving an equationwith several steps. [SAB, questions 1-2]Circulate as students work to solve the equations. These questions can help students moveforward:

How is this equation similar to other equations you have solved?How is it different?What form would you like this equation to have so that you can solve for the variable?How could you apply properties of equality to get it into that form?

Page 4: Use the animation on this page to debrief students’ solutions to the equation inquestion 2. Highlight the use of the Distributive Property for solving this equation.Emphasize that the Distributive Property works in both directions. Applying the property inone direction is often called “collecting like terms,” as with re-writing 2x + 3x as (2 + 3)x, or5x. Applying it in the other direction might involve re-writing the expression 2(x + 3) by"distributing the 2" as 2x + 6.

Page 5: Ask students work in pairs to solve the two equations on this page, 8 – 3y = 7y – 2 – 5yand 4(k –3) + 7 = 2k – (k + 8). [SAB, questions 3-4] Once students have had time to work,invite different students to share their solution strategies. Then, use the check reveals toallow students to check their answers. Ask students to identify the properties of equality usedin the check reveal to solve the two equations.



Discuss question 3 as a class. Ask students to share their responses to the three questionsposed.

For part a, students should notice that the x-coordinate of each intersection point is thesame.For part b, make sure students understand that this means that the three equationshave the same solution and are, therefore, equivalent equations.Part c asks students to make connections between the three equations. Students should


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notice that these three equations outline the equations that would be produced whensolving the original equation, 2x – 10 = 4, using algebraic operations.

CONSOLIDATION ACTIVITY (30 minutes)Students explore some common procedures and mistakes involved with solving multi-stepequations. Students then practice solving multi-step equations and justifying their procedures.

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Launch the activity by reading through the first problem with the whole class. [SAB, question1] Before looking over Marcos’ steps, ask:

Is Marcos' final answer reasonable for the situation? Why or why not?Have students work in pairs to complete the activity. They first correct errors in otherstudents’ work, then they solve equations, justifying their steps. [SAB, questions 2-5]Debrief questions 2 and 3 by asking students to explain the errors they find. Probe theirunderstanding with questions such as:

Which property did Marcos apply incorrectly? What about David and Elise?Can you find more than one error in any of the solutions?

Emphasize that there is often more than one correct way to solve a multi-step equation.Operations can be undone in different orders, as long as the properties are applied correctly.Debrief questions 4 and 5 by asking two different students to show two different ways tosolve one of the equations.


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Homework 13.6Students practice solving multi-step equations using algebraic (inverse) operations andidentifying the properties of equality used in doing so. They verify solutions by graphing.Staying Sharp 13.6The main concepts and skills in these problems are:

Informally solving a one-step, one-variable equation with a fractional solution1.Identifying slope and y-intercept of a line from its equation in standard form2.Generating and verifying a solution of a two-variable inequality3.Generating and verifying a non-solution of a two-variable inequality4.Determining the radius of a circle from its circumference5.Finding a common denominator of three fractions with unlike denominators6.


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Lesson materials

Lesson 13.7 “How many solutions?”Student Activity BookGuided assessmentGraphing calculatorComputer lab or other 1-1 computing environment

Lesson preview

Suggestedtime

Activity Goals

10 min OpenerUse number sense to find solution sets that do notcontain one unique number

20 min Core activityIdentify and solve linear equations that do not have onenumber as a solution


35 min Online assessmentAssess your understanding of key ideas and skills fromthis unit



Lesson activities

OPENER (10 minutes)Students use their number sense to begin investigating the two equations, one of which has nosolution and the other of which has infinitely many solutions, that will be solved in the core of thelesson.

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Allow students several minutes to think about possible solutions for the equations. [SAB,questions 1-2] Debrief by asking students to share their ideas and explanations. Hold off onvalidating student responses; this will be covered later in the lesson.


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CORE ACTIVITY (20 minutes)Students use graphs and algebraic operations to investigate linear equations that have one solution,no solutions, or infinitely many solutions.

Online pages 3-5Launch the activity using these pages. The use of algebra tiles gives students an intuitive sense ofwhat it means when an equation has no solutions or many solutions.

Page 3: Have students think about solving the problem using algebra tiles. Emphasize themeaning of the solution: the value for x that makes the equation true. In this case, there isonly one value for x that makes the equation true.Pages 4 and 5: Again, have students think about solving these equations using algebra tiles.These pages should provide the intuitive sense of what it means for an equation to have nosolution and many solutions. Have students work with partners to solve the equations in their activity books using algebraicproperties and graphs. [SAB, question 1] Students have already solved the first threeequations in parts a-c using number sense and algebra tile representations. This activityprovides a way to connect their intuitive understanding of the number of solutions tographical and algebraic representations. Then parts d-f provide more practice with usingalgebraic properties to solve equations and interpret the meaning of the solutions. Importantunderstandings to look for as students work include:

Solutions represent values of x. To clarify this idea, students should indicate solutions onthe x-axis, rather than on the equation-line itself.Algebraic and graphical solutions should show agreement between them.The meaning of a solution of x = 0 is different from the meaning of “no solutions.”

Online pages 6-8Use these pages to debrief the activity with the whole class. The check-reveals in these pages showsolutions for parts a-c. After discussing these solutions, have several pairs of students present theirsolutions for parts d-f.

Language strategy. As on page 7, make the connections between the vocabulary of “nosolutions” and “empty set” and the symbolic notation { } and Ø. [Note: The empty set symbolwas inspired by the 28th letter Ø in the Danish and Norwegian alphabet. It is not related inany way to the Greek letter Φ.]



For question 2, you may want to ask a few students to explain the steps they took to solvethe equation 7x + 10 = 24 as well as justify each step by identifying the algebraic propertyused. Discuss questions 3 and 4 as a class. Students most likely did not justify the steps theytook to complete question 4.

ONLINE ASSESSMENT (35 minutes)


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Online page 10 and Guided assessment pages 1-9Students complete the online Guided assessment for this topic. You will need to reserve acomputer lab for this activity. Completion of the items in Guided assessment will provide you andyour students with formative assessment data related to student understanding of the key ideas inthis topic, and will help students prepare for the upcoming mid-unit assessment. To view andanalyze data from this assessment, you will need to create an assignment for your class, and thenview the assignment report.


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Homework 13.7Students will continue investigating equations with many or no solutions. They will usealgebraic operations to solve equations, but will also look at tables and graphs to determine asolution set for an equation.Staying Sharp 13.7The main concepts and skills in these problems are:

Informally solving a two-step, one-variable equation with the variable on both sides1.Identifying x- and y-intercepts of a line from its equation in slope-intercept form2.Evaluating possible solutions of a two-variable inequality3.Examining possible solutions of a two-variable inequality on the plane to recognizepatterns

4.

Determining the height of a cylinder from its volume and radius5.Finding the opposite reciprocal of a fraction6.


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Lesson materials

Lesson 13.8 “Rearranging formulas”Student Activity BookGraphing calculator

Lesson preview

Suggestedtime

Activity Goals

10 min Opener Review the formula for the circumference of a circle

40 min Core activity Solve literal equations for a specified variable


20 min Review online assessmentAnalyze and learn from performance on the onlineassessment



Lesson activities

OPENER (10 minutes)Students review the formula for the circumference of a circle. Students find the circumferencegiven the radius and are asked to find the radius given the circumference.

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Allow several minutes for students to solve the problems. [SAB, questions 1-2] Debrief byasking for volunteers to share their answers with the class. Question 1 is fairlystraightforward; however, students will have to draw upon what they have learned aboutsolving one-step equations to answer question 2.

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CORE ACTIVITY (40 minutes)Students connect solving equations in a specific context to basic measurement concepts frommiddle school geometry. Students practice solving various literal equations for a specified variable.Then, they practice solving equation for a specific variable by transforming the equations of line instandard form to slope-intercept form.

Online page 3Students solve the circumference formula for r in two ways.

Ask students to answer questions 1 and 2 in their activity books. [SAB, questions 1-2] Then,use the animation on page 3 to debrief the questions.When debriefing ask:

What algebraic step did you perform on the first task to solve for r?How did solving for r help you to determine the accuracy of your measurements?

Online pages 4-6Students work in pairs to complete the activity.

Page 4: This page cues students to work with their partner on the remainder of the activity.[SAB, questions 3-4]Page 5: Use this page to debrief question 4a.Page 6: This page allows students to check their work for question 4b. Ask for volunteers tocome to the class computer to help fill in the puzzle.Classroom strategy. Remind students of the algebraic properties used when solving linearequations by asking students to identify the algebraic properties used to transform eachequation in questions 3 and 4.



Discuss question 4 as a class. Ask:How many solutions does the equation have?How can you tell by looking at the equation that it will have exactly one solution?

Call students’ attention to the fact that the slopes of the graphs of the lines y = -10x + 3 and y= 5x – 27 are different. Remind students that an equation has no solution when the graphs ofthe lines are parallel, or when the slopes of the graphs of the line are the same and they-intercepts of the graphs of the line are different. Also, an equation has an infinite numberof solutions when the graphs of the lines are the same line, or when the slope of the graph ofthe lines is the same and the y-intercept of the graph of the line is the same. For thisproblem, since the slopes of the two lines are different, we know that the lines can beneither parallel nor the same line. So, the equation cannot have an infinite number ofsolutions and cannot have no solution. Thus, the equation must have exactly one solution.

REVIEW ONLINE ASSESSMENT (20 minutes)

Students analyze class performance on the Guided assessment.


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Display the class report for the Guided assessment questions. Have students identify theitems the class did well on and those the class did not do well on. Instruct students to writethe question numbers in the table provided in the activity book. [SAB]Work as a class to identify the mathematics of problems the class did well on, and discussthese concepts and skills. Be prepared to project particular Guided assessment questions.Then identify and discuss the mathematics involved for problems on which the class did notperform well. Be prepared to project particular questions (perhaps two questions) from theGuided assessment. Allow students several minutes to work with their partners to re-do theseproblems in their activity books, and then discuss as a class.Share with students their individual score reports. Tell students they should take some timeoutside of class to revisit problems from the Guided assessment, in particular those for whichthey did not get the correct answers.


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Homework 13.8This homework assignment is designed to help students prepare for the mid-unit assessment.Students will be asked to review course materials related to the key skills and ideas coveredso far in the unit. Students will also be taking the online More practice assignment from thetopic Solving linear equations. Students will need to bring their Student Activity Books homeand will also need online access in order to complete Parts I and II of the homework. Remindstudents where and when they can get online access in school or in the community.Staying Sharp 13.8The main concepts and skills in these problems are:

Informally solving a two-step, one-variable equation with a negative solution1.Rewriting the equation of a line from standard form into slope-intercept form2.Evaluating a possible solution of a two-variable inequality3.Generating a solution of a two-variable inequality to recognize patterns on the plane4.Determining a length of the side of a trapezoid using similarity5.Finding a common denominator of two fractions and checking whether it is the LCD6.


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