unit of study-2a · students make contextual sense of the various parts of expressions and...

UNIT OF STUDY-2AGSE ALGEBRA IMATHEMATICSTEACHER RESOURCE GUIDE

REASONING WITH LINEAR EQUATIONS AND INEQUALITIES:(August 29 –October 14)By the end of eighth grade students have had a variety of experiences creating equations. Inthis unit, students continue this work by creating equations to describe situations. By the end ofeighth grade, students have learned to solve linear equations in one variable and have appliedgraphical and algebraic methods to analyze and solve systems of linear equations in twovariables. This unit builds on these earlier experiences by asking students to rearrange formulasto highlight a quantity of interest, analyze and explain the process of solving an equation, andto justify the process used in solving a system of equations. Students develop fluency writing,interpreting, and translating between various forms of linear equations and inequalities, andusing them to solve problems. Students explore systems of equations, find, and interpret theirsolutions. Students create and interpret systems of inequalities where applicable. For example,students create a system to define the domain of a particular situation, such as a situationlimited to the first quadrant. The focus is not on solving systems of inequalities. Solving systemsof inequalities can be addressed in extension tasks.

APS Literacy

Page | 1Some of the language used in this document is adapted from the CCSS Progressions, NY, Utah, Arizona, Ohio, & Georgia state resources.

Table of ContentsStandards Addressed ................................................................................................................................... 2

Enduring Understandings.............................................................................................................................6

Lesson One Progression ...............................................................................................................................7

Lesson Two Progression .............................................................................................................................20

Lesson Three Progression...........................................................................................................................34

Sample Fluency Strategies for High School ...............................................................................................45


StandardsAddressed

Duration: maximum of 15 instructional days on an A/B schedule

Georgia Standards of Excellence – Mathematics

Content Standards

(Cluster emphasis is indicated by the following icons. Please note that 70% ofthe time should be focused on the Major Content. ◊ Major Content □Supporting Content)

◊ Create equations that describe numbers or relationships.

MGSE9-12.A.CED.1 Create equations and inequalities in one variable and use them tosolve problems. Include equations arising from linear, quadratic, simple rational, andexponential functions (integer inputs only).

MGSE9-12.A.CED.2 Create linear, quadratic, and exponential equations in two ormore variables to represent relationships between quantities; graph equations oncoordinate axes with labels and scales. (The phrase “in two or more variables” refersto formulas like the compound interest formula, in which A = P(1 + r/n)nt has multiplevariables.)

MGSE9-12.A.CED.3 Represent constraints by equations or inequalities, and bysystems of equations and/or inequalities, and interpret data points as possible (i.e. asolution) or not possible (i.e. a non-solution) under the established constraints.

MGSE9-12.A.CED.4 Rearrange formulas to highlight a quantity of interest using thesame reasoning as in solving equations. Examples: Rearrange Ohm’s law V = IR tohighlight resistance R; Rearrange area of a circle formula A =πr2 to highlight theradius r.

◊Understand solving equations as a process of reasoning and explain the reasoning.

MGSE9-12.A.REI.1 Using algebraic properties and the properties of real numbers,justify the steps of a simple, one-solution equation. Students should justify their ownsteps, or if given two or more steps of an equation, explain the progression from onestep to the next using properties.

◊Solve equations and inequalities in one variable.

MGSE9-12.A.REI.3 Solve linear equations and inequalities in one variable includingequations with coefficients represented by letters. For example, given ax + 3 = 7,solve for x.


□Solve systems of equations.

MGSE9-12.A.REI.5 Show and explain why the elimination method works to solve asystem of two-variable equations.

MGSE9-12.A.REI.6 Solve systems of linear equations exactly and approximately (e.g.,with graphs), focusing on pairs of linear equations in two variables.

◊Represent and solve equations and inequalities graphically.

MGSE9-12.A.REI.10 Understand that the graph of an equation in two variables is theset of all its solutions plotted in the coordinate plane.

MGSE9-12.A.REI.11 Using graphs, tables, or successive approximations, show thatthe solution to the equation f(x) = g(x) is the x-value where the y-values of f(x) andg(x) are the same.

MGSE9-12.A.REI.12 Graph the solution set to a linear inequality in two variables.

Standards for Mathematical Practice

SMP 1. Make sense of problems and persevere in solving them.Students make contextual sense of the various parts of expressions andsituations by representing the context in tactile and/or virtual manipulatives,visual, or algebraic models. They will use estimation skills as a way of makingpredictions from their representations and understand the limitations of acontext. Students analyze systems of equations and utilize various ways to solvethem.

Students: Because Teachers: Read the task carefully. Draw pictures, diagrams, tables, or

use objects to make sense of thetask.

Discuss the meaning of the taskwith classmates.

Make choices about which solutionpath to take.

Try out potential solution paths andmake changes as needed.

Check answers and makes suresolutions are reasonable and makesense.

Explore other ways to solve thetask.

Persist in efforts to solvechallenging tasks, even after

Provide rich tasks aligned to thestandards.

Allow students time to initiate a plan;uses question prompts as needed toassist students in developing apathway.

Continually ask students if their plansand solutions make sense.

Question students to see connectionsto previous solution attempts and/ortasks to make sense of current task.

Consistently ask students to defendand justify their solution by comparingsolution paths.

Provide appropriate time for studentsto engage in the productive struggle ofproblem-solving.


reaching a point of frustration. Differentiate to keep advancedstudents challenged during work time.

SMP2: Reason abstractly and quantitatively.Students represent problems symbolically and make sense of the relationshipbetween quantities expressed contextually. They navigate from abstractrepresentation of mathematical concepts to their concrete real-life basedcounterparts and vice versa. Students make connections between abstract andpractical reasoning as they effectuate mathematical processes.

Students: Because Teachers: Use mathematical symbols to

represent situations Take quantities out of context to

work with them (decontextualizing) Put quantities back in context to

see if they make sense(contextualizing)

Consider units when determining ifthe answer makes sense in termsof the situation

Provide a variety of problems indifferent contexts that allow studentsto arrive at a solution in different ways

Use think aloud strategies as theymodel problem solving

Attentively listen for strategiesstudents are using to solve problems

SMP 3. Construct viable arguments and critique the reasoning of others.Students construct and critique arguments regarding the properties of equalityand inequality. They explain why a generated equation, inequality, or systemmodels a given contextual situation. They make conjectures about the meaningof solutions and investigate their veracity based on the context of the problem.

Students: Because Teachers: Make and tests conjectures. Explain and justifies their thinking

using words, objects, and drawings. Listen to the ideas of others and

decides if they make sense. Ask useful questions. Identify flaws in logic when

responding to the arguments ofothers.

Elaborate with a second sentence(spontaneously or prompted by theteacher or another student) toexplain their thinking and connectit to their first sentence.

Talks about and asks questionsabout each other’s thinking, inorder to clarify or improve their

Encourage students to use provenmathematical understandings,(definitions, properties, conventions,theorems, etc.) to support theirreasoning.

Question students so they can tell thedifference between assumptions andlogical conjectures.

Ask questions that require students tojustify their solution and their solutionpathway.

Prompt students to respectfullyevaluate peer arguments whensolutions are shared.

Ask students to compare and contrastvarious solution methods.

Create various instructional


own mathematical understanding. Revise their work based upon the

justification and explanations ofothers.

opportunities for students to engagein mathematical discussions (wholegroup, small group, partners, etc.).

SMP4: Model with MathematicsStudents translate constraints into equations and inequalities and extractinformation from both the algebraic solution and graphical representation. Theyread and understand a context from a word problem, and decide how torepresent the context, with either a table of values, an equation (in most cases),and/or a graph. They begin to explore the difference between a linear models.

Students: Because Teachers: Use mathematical models (i.e.

formulas, equations, symbols) tosolve problems in the world

Use appropriate tools such asobjects, drawings, and tables tocreate mathematical models

Make connections betweendifferent mathematicalrepresentations (concrete,verbal, algebraic, numerical,graphical, pictorial, etc.)

Check to see if an answer makessense within the context of asituation and changing the modelas needed

Provide opportunities for students tosolve problems in real life contexts

Identify problem solving contextsconnected to student interests

SMP5: Use Appropriate Tools StrategicallyStudents create and use tables. They use a number line to represent solutions tolinear inequalities in one variable. Students use the coordinate plane and thenumber line to help them understand what the solution(s) means in terms ofthe context of the problem.

Students: Because Teachers: Use technological tools to explore

and deepen understanding ofconcepts

Decide which tool will best helpsolve the problem. Examples mayinclude:

o Calculatoro Concrete modelso Digital Technologyo Pencil/papero Ruler, compass, protractor

Make a variety of tools readilyaccessible to students and allowingthem to select appropriate tools forthemselves

Help students understand the benefitsand limitations of a variety of mathtools


Estimate solutions before using atool

Compare estimates to solutions tosee if the tool was effective

SMP6: Attend to precision.Students work on algebraic and graphical representation of solutions. They paycareful attention to precision especially in labeling axes and clarifying thecorrespondence with quantities in a problem. They understand that obtainingaccurate solutions is key in ensuring that algebraic and graphical solutionsagree. Students provide carefully articulated explanations using academicvocabulary involving linear equations and inequalities.

Students: Because Teachers: Communicate precisely using clear

language and accuratemathematics vocabulary

Decide when to estimate or give anexact answer

Calculate accurately and efficiently,expressing answers with anappropriate degree of precision

Use appropriate units;appropriately labeling diagramsand graphs

Explicitly teach mathematicsvocabulary

Insist on accurate use of academiclanguage from students

Model precise communication Require students to answer problems

with complete sentences, includingunits

Provide opportunities for students tocheck the accuracy of their work

Note: All of the Standards for Mathematical Practice (SMPs) are critical to students fullyand appropriately attending to the content. Not all SMPs will occur in every lesson,however SMPs 1, 3, and 6 should be regularly apparent. All SMPs should be taught intandem with the content standards.

EnduringUnderstandings

In order to support deep conceptual learning it is important that student leave this unitexperience with the following understandings:

Linear equations and inequalities in one variable can be used in a contextualsituation to solve problems.

Real world situations can be represented symbolically and graphically. Algebraic expressions and equations generalize relationships from specific

cases.


Lesson One ProgressionDuration 3 days

Focus Standard(s)

Create equations that describe numbers or relationships.

MGSE9-12.A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Includeequations arising from linear, quadratic, simple rational, and exponential functions (integer inputs only).

MGSE9-12.A.CED.4 Rearrange formulas to highlight a quantity of interest using the same reasoning as insolving equations. Examples: Rearrange Ohm’s law V = IR to highlight resistance R; Rearrange area of a circleformula A =πr2 to highlight the radius r.

Understand solving equations as a process of reasoning and explain the reasoning.

MGSE9-12.A.REI.1 Using algebraic properties and the properties of real numbers, justify the steps of a simple,one-solution equation. Students should justify their own steps, or if given two or more steps of an equation,explain the progression from one step to the next using properties.

Solve equations and inequalities in one variable.

MGSE9-12.A.REI.3 Solve linear equations and inequalities in one variable including equations with coefficientsrepresented by letters. For example, given ax + 3 = 7, solve for x.

Performance ObjectivesAs a result of their engagement with this unit…

*MGSE9-12.A.CED.4 – SWBAT rearrange formulas IOT manipulate and solve contextual problems for specifiedvariables.

MGSE9-12.A.CED.1 – SWBAT create linear equations and inequalities in one variable IOT represent and solveequations and inequalities.

*MGSE9-12.A.REI.3 – SWBAT use the properties of equality IOT to solve linear equations and inequalities inone variable.

*MGSE9-12.A.REI.1 – SWBAT apply algebraic properties and properties of real numbers IOT construct viablearguments justifying their solutions.

Note: Performance objectives that are indicated with a (*) are either implicit or will be interwoven throughoutthis progression and the unit in total.


Building CoherenceAcross Grades:

Within Grades:

Terms and DefinitionsAlgebra: The branch of mathematics that deals withrelationships between numbers, utilizing letters andother symbols to represent specific sets of numbers,or to describe a pattern of relationships betweennumbers.

Algebraic Expression: A mathematical phraseinvolving at least one variable and sometimesnumbers and operation symbols and an absence ofequality.

Coefficient: A number multiplied by a variable in analgebraic expression.

Inequality: Any mathematical sentence that containsthe symbols > (greater than), < (less than), ≤ (less thanor equal to), or ≥ (greater than or equal to).

Property of Inequalities: Here a, b and c stand for arbitrarynumbers in the rational or real number systems.

Exactly one of the following is true: a < b, a = b, a > b.

If a > b and b > c then a > c.

If a > b, then b < a.

If a > b, then –a < –b.

•Student workwith expressionsand equationsincludingfomulas

7thGrade

8thGrade

Students will consider linear equations as functions in the next unit andexplore implications for the domain and range to fit contextual situations.



Within Grades:










•Student usevariables torepresentquantities inequations andinequalities

8thGrade

9thGrade




Within Grades:










Studentsconsiderconstraints,and useequations andinequalities tomodel real-world context

9thGrade



Continuous: Describes a connected set of numbers,such as an interval.

Discrete: A set with elements that are disconnected.

Equation: A number sentence that contains an“equal” symbol.

If a > b, then a ± c > b ± c.

If a > b and c > 0, then a × c > b × c.

If a > b and c < 0, then a × c < b × c.

If a > b and c > 0, then a ÷ c > b ÷ c.

If a > b and c < 0, then a ÷ c < b ÷ c.

Substitution: To replace one element of amathematical equation or expression with another ofequal value.

Term: A value in a sequence--the first value in asequence is the 1st term, the second value is the 2ndterm, and so on; a term is also any of the monomialsthat make up a polynomial.

Guiding Question(s) How do I justify the solution to an equation? How do I solve an equation or inequality in one variable?

Interpretations and Reminders Provide examples of real-world problems that can be modeled by writing an equation or inequality in one

variable. Discuss the importance of using appropriate labels and scales on the axes when representing functions with

graphs. Embed discussions around restriction on the domain in contextual problems. Note when data is discrete

versus continuous. Provide visual and/or hands on analogies like comparing the sides of an equation to the arms of a balance

scale, and students will have a stronger feel for what 'balance' and 'equality' really mean in math.

The Properties of OperationsHere a, b and c stand for arbitrary numbers in a given number system. The properties of operations apply to

the rational number system, the real number system, and the complex number system.

Associative property of addition: (a + b) + c = a + (b + c)Commutative property of addition: a + b = b + aAdditive identity property of 0: a + 0 = 0 + a = a

Existence of additive inverses: For every a there exists –a that a + (–a) = (–a) + a = 0.Associative property of multiplication: (a × b) × c = a × (b × c)

Commutative property of multiplication: a × b = b × aMultiplicative identity property of 1: a × 1 = 1 × a = a

Existence of multiplicative inverses: For every a ≠ 0 there exists 1/a so that a × 1/a = 1/a × a = 1.Distributive property of multiplication over addition: a × (b + c) = a × b + a × c
















































The Properties of EqualityHere a, b and c stand for arbitrary numbers in the rational, real, or complex number systems.

Reflexive property of equality a = aSymmetric property of equality If a = b, then b = a.

Transitive property of equality If a = b and b = c, then a = c.Addition property of equality If a = b, then a + c = b + c.

Subtraction property of equality If a = b, then a – c = b – c.Multiplication property of equality If a = b, then a × c = b × c.

Division property of equality If a = b and c ≠ 0, then a ÷ c = b ÷ c.Substitution property of equality If a = b, then b may be substituted for a in any expression containing a.

Misconceptions

Students may not understand that an equation is a mathematical statement of two equivalent expressionsjoined by an equals sign. Sometimes students will try to solve “x + 6” thinking it is “x + 6 = 0”

When solving an equation such as 6 + 3x + 2 = 4x + 3, students may realize that they need to subtract 2, butoften do so from all the constants rather than from each side of the equation with the appropriate liketerm.

Many students fail to treat expressions in parentheses as a unit.

Students may confuse the rule of changing a sign of an inequality when multiplying or dividing by anegative number with changing the sign of an inequality when one or two sides of the inequality becomenegative (for ex., 3x > -15 or x < - 5).

Some students may have trouble translating equations and inequalities from contextual situations.

Some students may believe that for equations containing fractions only on one side, it requires “clearingfractions” (the use of multiplication) only on that side of the equation. When addressing themisconception, start by demonstrating the solution methods for equations similar tox + x + x + 46 = x, and stress that the multiplication Property of Equality is applied to both sides, by

multiplying by 60, which is the LCD.

Some students may believe that subscripts can be combined as b1 + b2 = b3 and the sum of differentvariables d and D is d +D = 2D.


Learning Progression (Suggested Learning Experiences)Procedural Fluency: (Recommended for 5 - 10 minutes each day: Fluency strategies are useful to activatestudent voice, solicit prior knowledge and develop fluency based on conceptual understandings.) For additionalfluency practice strategies see the table at the end of this document.

Using students’ prior knowledge about perfect square numbers, spend the first 10 minutesallowing for reflection and effortless retrieval of these facts.

Provide students with the following statements b = 5. Have students work in pairs to come up with a numberstory whose solution can be illustrated by the statement.

---------------------------------------------------------------------------------------------------------------------------------------------------Graduated Measure (The graduated measure is a quick opportunity to diagnose students’ level of comfort withthe material before you begin the progression.)

Student Directions: Select a question/problem from the table below that you feel best equipped to answersuccessfully.

Beginning Developing ProficientSolve for x:

3(x – 2) = 12Solve for b:

3(23 – 2b) + 3b = 48Solve for n:

−6n − 20 = −2n + 4(1 − 3n)

Opening Lesson

A deep exploration of contextual situations will require practice. The following progression is a guide, andnot an exhaustive experience with the content.

When readdressing solving simple equations, it may be beneficial to remind students of the most commonsolving techniques, such as converting fractions from one form to another, removing parentheses in thesentences, or multiplying both sides of an equation or inequality by the common denominator of the fractions.Students must be aware of what it means to check an inequality’s solution. The substitution of the end pointsof the solution set in the original inequality should give equality regardless of the presence or the absence ofan equal sign in the original sentence. The substitution of any value from the rest of the solution set shouldgive a correct inequality.

Careful selection of examples and exercises is needed to provide students with meaningful review and tointroduce other important concepts, such as the use of properties and applications of solving linear equationsand inequalities. Stress the idea that the application of properties is also appropriate when working withequations or inequalities that include more than one variable, fractions and decimals. Regardless of the type of









3(23 – 2b) + 3b = 48Solve for n:

−6n − 20 = −2n + 4(1 − 3n)

Opening Lesson












3(23 – 2b) + 3b = 48Solve for n:

−6n − 20 = −2n + 4(1 − 3n)

Opening Lesson





numbers or variables in the equation or inequality, students have to examine the validity of each step in thesolution process.

Gradual Release of ResponsibilityFocus LessonProvide students with the following problem:Tomatoes cost $0.50 each and avocados cost $2.00 each. Anne buys six more tomatoes than avocados. Hertotal bill is $8. How many tomatoes and how many avocados did Anne buy?

As students begin to think about this problem ask? What are our “knowns” and “unknowns”? Elicit steps tohelp students define their variable

Let a = the number of avocados Anne buys. (Discuss why you have only identified one variable when there aretwo quantities in the problem.) Then help students begin to further translate the information given in theproblem. This is an area of struggle for many students. Make certain students have the connection betweenthe words and selected model used for the problem.

Anne buys six more tomatoes than avocados. This means that a + 6 = the number of tomatoes. Tomatoes cost$0.50 each and avocados cost $2.00 each. Her total bill is $8. This means that .50 times the number oftomatoes plus 2 times the number of avocados equals 8. Why isn’t there a t for tomatoes in the problem?

Another Example:

Your little brother starts a new business selling lemonade. He spends a total of $15.82 for supplies and sellslemonade on the first day for $0.50 a glass. He made a total profit of $6.18. Write an equation that can besolved to find the number of glasses he sold.

Explain to students the significance of each value in the problem:

-$15.82: The amount of money spent on supplies. Emphasize to students that the value is negativebecause it is money that was spent and not earned.

Probe students by asking them tomake a statement about what it

means for a to equal 2.


0.50: The cost for one glass of lemonade $6.18: The amount of money made by the end of the day. g: the number of glasses sold

The resulting equation would be -15.82 + .50g = 6.18.

Guided PracticeDuring this phase of gradual release provide students with the following graphic organizer as a strategy tohelp students make sense of a context when writing an equation:

Given a real-world context that represents a linear equation in one variable, complete the table below. Havestudents notice patterns from their picture, table or equation, and make predictions about the solution.

Contextual Example: Lashawn has saved $600 so far to buy a new flat screen TV. If the TV she wants costs$1800 and she continues to save $20 a week, how long will it take her to buy the TV?

Diagram/ VisualRepresentation

What units willyou use?

What are yourknown anunknown

quantities?

Write theequation

Solve the problem

Use the following problems to provide students with additional supervised practice.

Creating and solving an equation in one variable.

Paying the Rent.pdf

The perimeter of two gardens are equal. The measures of each garden is shown below. Find the perimeters of thegardens.

4x – 3

x – 3

3x + 1 3x + 1


Collaborative Practice

Place students into groups of 2-3 homogenously to work the following task.

Basketball (1).docx

Independent Practice

BuyingACar (1).docx

Inequalities in one variable

In grades 6-8, students solve and graph linear equations and inequalities. There are two major reasons fordiscussing the topic of inequalities along with equations: First, there are analogies between solving equationsand inequalities that help students understand them both. Second, the applications that lead to equationsalmost always lead in the same way to inequalities. Graphing experience with inequalities should be limited tographing on a number line diagram.

A discussion of linear inequalities in one variable should logically follow the previously described engagementwith linear equations in one variable. Students should understand that inequalities, like equations, arerelationship statements that compare values. Use the table below to address the expected language andnotation around inequalities.

Notation Verbal Description Examplea < b “ a is less than b” -4 < 10a < b “a is less than or equal to b” Tammie is at most the same

age as Franka > b “a is greater than b” Mike makes more money

that Jamala > b “a is greater than or equal to b” 52 > the number of states in

the US

Use some of the examples in the chart above to point out the solution sets involved in inequalities and howthat differs from the single solution for most equations.

Consider the following example:

3x


In 7 years, Roderick will be old enough to vote in an election. (You must be at least 18 years old to vote.) Whatcan you say about how old she is now?

Ask students to generate possible solutions to illustrate the variance. Use students’ responses to generate thefollowing inequality.

Roderick’s age + 7 > 18Remind students that solutions sets can be illustrated using set notation or graphed on the number line.

Solving An Inequality. Consider the following problem:

Sam and Alex play in the same soccer team. Last Saturday Alex scored 3 more goals than Sam, but togetherthey scored less than 9 goals. What are the possible number of goals Alex scored?

A = 3 + SS + A < 9S + (3 + S) < 93 + 2S < 92S < 6S < 3

Guided Practice

Use the following problems to guide students through writing and solving linear inequalities. Work theproblems with students in a whole group and follow up with discussion questions designed to assess studentsunderstanding of the standard. Note: Have students to not only create and solve the inequality but to alsograph the inequality and write an explanation of their reasoning.


Inequality Problems.pdfHerman.docx

Collaborative PracticeUse the following flipchart practice to provide students with additional supervised practice.

Creating and solving an inequalities in one variable.

InequalityWordProblems.flipchart

One-variableInequality Word Problems.pdf

Solving Literal Equations:Solving equations for the specified letter with coefficients represented by letters (e.g., A = ½(B + b) whensolving for b) is similar to solving an equation with one variable. Provide students with an opportunity toabstract from particular numbers and apply the same kind of manipulations to formulas as they did toequations. One of the purposes of doing abstraction is to learn how to evaluate the formulas in easier waysand use the techniques across mathematics and science.

Ask students:How are formulas used in the real-world?Why is it that a formula can have more than one variable?What if I want to find the value of a variable in the formula other than the isolated variable?

Ask students to consider the equation for finding the area of a square/rectangle: A = l(w). Then ask studentsto use the equation to solve for the length of a rectangle with an area of 40 m2 and a width is 5 m. Moststudents will more than likely arrive at an answer by substituting 40 and 5 into the equation in the followingmanner:

A = l x w40 = l (5)40/5 = l8 = l

There may be some students who struggle to solve for a variable in an equation if the variable is not isolated.


At this time, show students how they can isolate any variable in the equation by using the same reasoning.

A = l (w)

=

= l

In order to isolate (l), show studentshow to divide on both sides of theequation by the width in order tosolve for length. If students substitutethe values into the result, they will beable to verify the balance of theequation.

A = l (w)

=

= w

In order to isolate (w), show studentshow to divide on both sides of theequation by the length in order tosolve for width. If students substitutethe values into the result, they will beable to verify the balance of theequation.

Contextual Example: The length of a rectangle is 1 centimeter (cm) less than 3 times the width. If theperimeter is 54 cm, find the dimensions of the rectangle.

Diagram/ VisualRepresentation


What are yourknown anunknown

quantities?

Write theequation

Solve the problem

Draw students’ attention to equations containing variables with subscripts. The same variables with differentsubscripts (e.g., x1 and x2) should be viewed as different variables that cannot be combined as like terms. Avariable with a variable subscript, such as an, must be treated as a single variable – the nth term, wherevariables a and n have different meaning.

Example:

Acceleration is the measure of how fast a velocity is changing. The formula for acceleration ist

vva if

.

Where, a represents the acceleration rate, fv is the final velocity, iv is the initial velocity, and t represents the time in

seconds. Ask students to solve the formula for fv . Use this example to illustrate the difference between variables that

have unique subscripts.Given that the following trapezoid has area 54 cm2,

set up an equation to find the length of theunknown base, and solve the equation.


Additional Unit Assessments –Assessment

Name

Assessment Assessment Type StandardsAddressed

Cognitive Rigor

Acting Out

Acting Out.docx

Scaffolding Task A.CED.1A.CED.3N.Q.1N.Q.2N.Q.3

DOK 2/3

Lucy’s LinearEquations &Inequalities

LucysLinearEquation&Inequalities.docx

Practice Task A.CED.1A.SSE.1A.SSE.1aA.SSE.1b

DOK 3

Jayden Phone Plan

JaydenPhonePlan.docx

Scaffolding Task A.REI.1A.REI.3

DOK 2/3

Differentiated SupportsLearningDifficulty

• Jigsaw strategy: Students are divided into four groups to write, strategize, simplify andevaluate their own solutions to their equations.

• Allow students to use the TRACE feature on the graphing calculator to solve equations.Students will see the solution is the x value that corresponds to the same two y-values inthe table.

• Have students use algebra tiles to model solving equations.

• Additional opportunity to re-engage students with this Map Shell lesson.

•linear equations inone variable r1.pdf

High-Achieving

Challenge high-achieving students with the following Formative Assessment Lesson


skeleton_tower.pdf

EnglishLanguageLearners

Have students write their understandings in a math journal. You may want to considerallowing EL students to write notes in their first language and annotate by identifying mathspecific words they do not have a translation for.

Allow students to create pictorial flash cards for new math words. Create a graphic organizer that outlines the properties of rational numbers

Online/Print ResourcesDigitalResources Same Solution Task

http://www.illustrativemathematics.org/illustrations/613This task gives students 6 different equations and asks them to show whether they havethe same solution.

Balancing Equations Applethttp://illuminations.nctm.org/ActivityDetail.aspx?id=10This applet allows you to demonstrate how a point of intersection of two lines is the x–value where the two expressions have the same y–value. The applet shows the graphalongside a scale with algebraic expressions as the ‘weights.’

PrintResources

PayingRent.docx BernardoAndSylviaPlayAGame (1).docx

solvingmulitstepinequalities.ppt

Solving Equationswith Variables on Both Sides.ppt

Rally RobinCooperative Learning Strategy.docx

Algebra Tiles Equation MatsTextbook AlignmentAlgebra I, McGraw-Hill Education pp. 75 – 102, 155 – 170, 126 – 131, 285 - 300


Lesson Two ProgressionDuration 6 days

Focus Standard(s)

Create equations that describe numbers or relationships.

MGSE9-12.A.CED.2 Create linear, quadratic, and exponential equations in two or more variables to representrelationships between quantities; graph equations on coordinate axes with labels and scales. (The phrase “intwo or more variables” refers to formulas like the compound interest formula, in which A = P(1 + r/n)nt hasmultiple variables.)

MGSE9-12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/orinequalities, and interpret data points as possible (i.e. a solution) or not possible (i.e. a non-solution) under theestablished constraints.

Represent and solve equations and inequalities graphically.

MGSE9-12.A.REI.10 Understand that the graph of an equation in two variables is the set of all its solutionsplotted in the coordinate plane.

MGSE9-12.A.REI.12 Graph the solution set to a linear inequality in two variables.

Performance ObjectivesAs a result of their engagement with this unit…MGSE9-12.A.CED.2 – SWBAT to create linear equations in two or more variables IOT model relationshipsbetween two quantities.

MGSE9-12.A.REI.10 – SWBAT evaluate and plot values of a two variable equation IOT analyze and verify thesolutions graphically.

MGSE9-12.A.REI.12 – SWBAT Graph the solution set to a liner inequality in two variables IOT model real-worldsituations with constraints.


MGSE9-12.A.CED.3 – SWBAT represent constraints of equations IOT interpret solutions as viable or nonviablein a modeling context.

Note: Performance objectives are written in the suggested order that they should be taught. Objectives thatare indicated with a (*) are either implicit or will be interwoven throughout the progression.


Within Grades:

Terms and DefinitionsAlgebra: The branch of mathematics that deals with Linear Inequality: An inequality in two variables such

•Student workwith expressionsand equationsincludingfomulas

7thGrade

8thGrade

Students will engage in creating and rewriting quadtraticand exponential equtions in upcoming units.





Within Grades:


•Student usevariables torepresentquantities inequations andinequalities

8thGrade

9thGrade






Within Grades:


Studentsconsiderconstraints,and useequations andinequalities tomodel real-world context

9thGrade



relationships between numbers, utilizing letters andother symbols to represent specific sets of numbers,or to describe a pattern of relationships betweennumbers.


Average Rate of Change: The change in the value of aquantity by the elapsed time. For a function, this isthe change in the y-value divided by the change in thex-value for two distinct points on the graph.

Constant Rate of Change: With respect to thevariable x of a linear function y = f(x), the constantrate of change is the slope of its graph.



Domain: The set of x-coordinates of the set of pointson a graph; the set of x-coordinates of a given set ofordered pairs; The value that is the input in a functionor relation.


X-intercept: The point where a line meets or crossesthe x-axis

Y-intercept: The point where a line meets or crossesthe y-axis

as Ax + By > C, where the solution set consist ofordered pairs (x, y) that produce a true statementwhen the values of x and y are substituted into theinequality.











Range: The set of all possible outputs of a function.

Slope: The ratio of the vertical and horizontal changesbetween two points on a surface or a line.


Term: A value in a sequence--the first value in asequence is the 1st term, the second value is the 2ndterm, and so on; a term is also any of the monomials






















































that make up a polynomial.

Guiding Question(s) How can you recognize a linear equation in two variables? How does the graph of a linear equation or inequality relate to the solution(s)?

Interpretations and Reminders Provide examples of real-world problems that can be modeled by writing an equation or inequality in two

variables.

Discuss the importance of using appropriate labels and scales on the axes when representing functions withgraphs. Examine real-world graphs in terms of constraints that are necessary to balance a mathematicalmodel with the real-world context. For example, a student writing an equation to model the maximumperimeter of a rectangular space when the length is given by the equation l = 2w should understand that wis only reasonable when w>0

This restriction on the domain is necessary because the side of a rectangle under these conditions cannotbe less than or equal to 0.

Provide visual and/or hands on analogies like comparing the sides of an equation to the arms of a balancescale, and students will have a stronger feel for what 'balance' and 'equality' really mean in math.







The Properties of EqualityHere a, b and c stand for arbitrary numbers in the rational, real, or complex number systems.

Reflexive property of equality a = aSymmetric property of equality If a = b, then b = a.

Transitive property of equality If a = b and b = c, then a = c.Addition property of equality If a = b, then a + c = b + c.




Misconceptions Students may interchange slope and y-intercept when creating equations. For example, a taxi cab costs $4

for a dropped flag and charges $2 per mile. Students may fail to see that $2 is a rate of change and is slopewhile the $4 is the starting cost and incorrectly write the equation as y = 4x + 2 instead of y = 2x + 4.

Given a graph of a line, students use the x-intercept for b instead of the y-intercept. Given a graph, students incorrectly compute slope as run over rise rather than rise over run. For example,

they will compute slope with the change in x over the change in y.

Students do not know when to include the “or equal to” bar when translating the graph of an inequality.







Using students’ prior knowledge about perfect square numbers, spend the first 10 minutesallowing for reflection and effortless retrieval of these facts. Ask students to consider the following scenario
































and share out their reasoning. Encourage the use of mental math in their discussions.

You and your friend have a coupon for $25 towards a dinner for two. The dinner does not include tax orgratuity; however, you and your friend want to try to spend at most $25. It is okay if you do not use the entirevalue of the coupon. List some possible combinations of prices for your mean and your friends’ meal.---------------------------------------------------------------------------------------------------------------------------------------------------Graduated Measure (The graduated measure is a quick opportunity to diagnose students’ level of comfort withthe material before you begin the progression.)


Beginning Developing ProficientIdentify the slope and y-interceptof the equation y = -7x + 12

Describe the steps needed tograph the equation y = 2x - 15

Write the following equation inslope intercept form: 3x + 4y = 20

Gradual Release of ResponsibilityFocus Lesson

Equations in two variables:

Recall this problem from the first progression: “Your little brother starts a new business selling lemonade. Hespends a total of $15.82 for supplies and sells lemonade on the first day for $0.50 a glass. He made a totalprofit of $6.18. Write an equation that can be solved to find the number of glasses he sold.”

This was an example of an equation in one variable. However, you can write a linear equation in twovariables as well. Consider the variation of the earlier problem:

Your little brother starts a new business selling lemonade. He spends a total of $15.82 for supplies and sellslemonade on the first day for $0.50 a glass. Write an equation that shows the relationship between thenumber of glasses sold and the total profit.

Again, emphasize to students the following:

-$15.82: The amount of money spent on supplies. Emphasize to students that the value is negativebecause it is money that was spent and not earned.

0.50: The cost for one glass of lemonade g: the number of glasses sold


P: the total profit

The one difference is that one less variable has been identified, therefore the resulting equations would be:The resulting equation would be -15.82 + .50x = P.

This equation cannot be solved for one variable however the equation has a solution set (a group of possiblesolutions). To review:

Ask students to generate a table of possible solutions to this equation. Allow pair share work and then askseveral pairs to contribute an additional solution set to the table.(Sample Table)

Total Number of Glasses Total Profit1 glass

10 glasses20 glasses30 glasses

Ask Students: Do you see a relationship between the total glasses of lemonade sold and the total profit. Havethem come up with a statement that illustrates the relationship.

At this point, show students how they can take the equation in two variables (from the example) and plot iton a graph. Take some time to discuss where g and P fall on the coordinate plane based on which quantity isdependent on the other.

g P

1 -15.32

10 -10.82

20 -5.82

30 -0.82

40 4.18

50 9.18-20

-15

-10

-5

0

5

10

15

0 10 20 30 40 50 60

Prof

it

# of glasses sold

Lemonade Stand Profit


Show students how to analyze and understand the graph by asking the following probing questions andproviding an opportunity for discussion:

Are the values in the table the only possible number of glasses that could have been sold? Why or whynot? What are some other options for the value of g? Can the value of g be a decimal? Why or whynot?

Is $9.18 the greatest profit that can be made? Why or why not? What does it mean to have a negativeprofit?

Should the dots indicating (glasses, profit) be connected? Why or why not? Is there anything wrongwith the graph above?

Teacher Note: Engage students in deep conversations around the analysis of the equation, graph and table asit relates to the context.

Guided Practice

Strategies to help students make sense of a context when writing an equation in two variables:Given a real-world context that represents a linear equation, complete the table below. Have students noticepatterns from their picture, table or equation, and make predictions about the solution.

Contextual Example: An office manager is buying new file cabinets. He has only 60 square feet of floorspace. Cabinet A requires 3 square feet and Cabinet B requires 10 square feet.Picture/Visual


What are your knownan unknown quantities?

Write theequation

Solve the problem

Ask students to generate an equation that represents the situation.Solution: Let x = the number of Cabinet A and let y = the number of Cabinet B. Each Cabinet A requires 3square feet and each Cabinet B requires 10 square feet for a total of 60. The equation is 3x + 10y = 60.

Part b: If the office manager decides to purchase 3 of Cabinet B, how many of Cabinet A should he purchase?Solution: substitute 3 for y and solve for x.3x + 10y = 603x + 10(3) = 603x + 30 = 603x = 30x = 10


He should buy 10 of Cabinet A

Use the following additional problem to provide students with supervised practice.

Creating and solving an equation in twovariables. CAB COMPANY.docx

Collaborative PracticePlace students into groups of 2-3 homogenously. Have students to complete linear equation performance task.Additional problems are also being provided for which students can create a table and a graph to accompanythe linear equation.

Linear Equation Collaborative Practice.pdf Linear Word Problems.doc

Inequalities in two variables


It is important to point out the ways students may encounter linear inequalities. This will provide a model forthe types of equations they can create to model their contextual situations.

cbyax cbyax cbyax cbyax

Point out that a, b, and c are constant values while x and y are variables where the ordered pair (x, y) is thesolutions to the linear inequality if it makes the sentence true.

Graphing A Linear Inequality. Consider the following:

In the problem above, the line is solid because the inequality sign used contained a portion of the equal sign (≤or ≥). The region above the line is shaded and therefore represents all possible solutions excluding anysolution that fall on the line y = -3x + 4.

Also, consider the following:


In the problem above, the line is dotted because the inequality sign used is less than or equal to (< or >) aportion of the equal sign (≤ or ≥). The region above the line is shaded and therefore represents all possiblesolutions including any solution that fall on the line y = -x – 5.

Have students consider the following example:Zakyiah is starting her own company making bracelets. Each beaded bracelet takes 3 hours to make and eachleather bracelet takes 5 hours to make. Zakyiah can work no more than 20 hours per week making bracelets.

Part a: Write an inequality to represent Zakyiah's situation.Solution: Let b = the number of beaded bracelets and let l = the number leather bracelets. Zakyiah can work nomore than 20 hours, use less than or equal to, <. (Engage students in a discussion about the selection of the“less than or equal to” symbol rather than the “less than” symbol.)3s + 5p < 20

Ask students “Can Zakyiah make 2 beaded bracelets and 3 leather bracelets?”Solution: Substitute 1 for s and 3 for p.3s + 5p < 203(2) + 5(3) < 206 + 15 < 20

21 < 20, is a false statement so the answer is no.

Now ask students to generate various sample solutions that will work under Zakyiah’s constraints. Havestudents place their solutions in a table. We can then graph the equation and plot all of the possible solutionsand have student make conjectures about the graph of an inequality in two variables.


Example – A focus on constraints:

You are going to a street fair where they are selling popcorn for $2.50 and sodas for $2. You and your friendshave $20 altogether, which you can spend all or part of.

Part a: Write an inequality that represents the possible number of sodas and snacks your group can buy.Differentiate for students by defining variable for struggling students.Let x = the number of bags of popcorn boughtLet y = the number of sodas bought.

Assume the graph below represents the inequality.

Solution: Each bag of popcorn (x) costs $2.50 and each soda (y) costs $2.2.5x + 2y < 20

A focus on constraints:

All of students’ solutionpoints should fall within theshaded region including onthe line itself. Discuss this

with students.


Ask students to name two additional constraints. Provide opportunities for various pairs to justify theirreasoning.Solution: The numbers of bags of popcorn and snacks cannot be negative, two additional constraints are x > 0and y > 0. This will further restrict the solution to Quadrant I (positive x and y values only).

Based on the graph of the inequality ask for two possible solutions.Solution: There are many possible solutions, choose any two points within the shaded region in Quadrant I withinteger coordinates. Make certain that students identify each ordered pair in context.

Possible answers:(0, 0), no bags of popcorn and no sodas(4, 4), 4 bags of popcorn and 4 sodas

Explore non-examples by asking students to name two possible non-solutions. Students should justify theirreasoning.Solution: There are many possible non-solutions, choose any two points that are not within the shaded region.(10, 0), 10 bags of popcorn and no sodas(6, 4), 6 bags of popcorns and 4 sodas

Independent Practice

Lebron James scored 38 points in the series against Detroit and will scoreanother 15 points for every quarter he plays in the playoffs. To set a record,he needs to score at least 400 points.

1) Write an equation for how many points (p) Lebron will score after qquarters.

2) If he wants to score at least 400, he must score________ 400. (use aninequality symbol!)

3) Write an inequality for how many quarters he has to play to score at least400.


Name

Assessment Assessment Type Standards Addressed Cognitive Rigor

Pizza Promotion

PizzaPromotion.docx

Performance TaskIndividual/Partner

MGSE9-12.A.CED.2MGSE9-12.A.CED.3MGSE9-12.A.REI.12

DOK 2/3


Speed Zones

Speed Zones.docx

Performance TaskIndividual/Partner

MGSE9-12.A.CED.2MGSE9-12.A.CED.3MGS.9-12.A.REI. 10

DOK 2/3

Forget The Formula

ForgettheFormula.docx

Scaffolding Task MGSE9-12.A.CED.2MGSE9-12.A.CED.3MGSE9-12.A.CED.4

DOK 2/3

Differentiated SupportsLearningDifficulty

• Jigsaw strategy: Students are divided into four groups to write, strategize, simplify andevaluate their own solutions to their equations.

• Allow students to use the TRACE feature on the graphing calculator to solve equations.Students will see the solution is the x value that corresponds to the same two y-values inthe table.

• Have students use algebra tiles to model solving equations.

• Provide students a graphic organizer for solving equations and inequalities.• Create a “Modeling Contexts” Choice Board that allows students to attempt problems

based on their comfort level. Include scenarios that can be modeled by various types ofequations. See template below:

•Modeling Choice

Board.doc

High-Achieving

Challenge high-achieving students with the following Formative Assessment Lesson

G9.U1.FAL.TheLargest Loser N.Q.1,3 A.CED.2.docx

EnglishLanguageLearners

Have students write their understandings in a math journal. You may want to considerallowing EL students to write notes in their first language and annotate by identifying mathspecific words they do not have a translation for.

Allow students to create pictorial flash cards for new math words. Create a graphic organizer that outlines the properties of rational numbers

Online/Print ResourcesPrintResources

Escalator.pdf best_buy_tickets.pdf

The Race Task.pdf


Algebra Tiles DesmosTextbook AlignmentAlgebra I, McGraw-Hill Education pp.317 – 323

Lesson Three ProgressionDuration 6 days

Focus Standard(s)Solve systems of equations.

MGSE9-12.A.REI.5 Show and explain why the elimination method works to solve a system of two-variableequations.

MGSE9-12.A.REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing onpairs of linear equations in two variables.

MGSE9-12.A.REI.11 Using graphs, tables, or successive approximations, show that the solution to the equationf(x) = g(x) is the x-value where the y-values of f(x) and g(x) are the same.

Performance ObjectivesAs a result of their engagement with this unit…MGSE9-12.A.REI.5 – SWBAT use the properties of equality IOT justify solution methods when solving a systemof two-variable equations.

MGSE9-12.A.REI.6 – SWBAT solve linear equations IOT model and approximate solutions in algebraic orcontextual problems.

MGSE9-12.A.REI.11 – SWBAT evaluate equations using graphs, tables, or approximations IOT prove that the


solution to the equation f(x) = g(x) is the x-value where the y-values of f(x) and g(x) are the same.


Within Grades:

Terms and DefinitionsSystem of Linear Equations: a collection of two ormore linear equations with a same set of unknowns.

Consistent System of Equations: A system ofequations that has at least one solution.

Independent: A consistent system with exactly onesolution.

Elimination: The process of eliminating one of thevariables in a system of equations using addition orsubtraction in conjunction with multiplication ordivision and solving the system of equations.

Linear Model: A linear function representing real-world phenomena. The model also representspatterns found in graphs and/or data.

Parameter: The independent variable or variables in asystem of equations with more than one dependentvariable.


Inconsistent System of Equations: A system ofequations that has no solutions. (Attempts to solveinconsistent systems typically result in false

Students uselinear

functions tomodel

relationshipsbetween

quantities

8thGrade

9thGrade

With an understanding of solving equations as a reasoning process, students can organize the various methods for solving differenttypes of equations into a coherent picture in preparation for modeling quadratic and exponential functions later in the couse.




Within Grades:









Studentsanalyze

systems oflinear

equations bothanalytically

and graphically

9thGrade

11thGrade





Within Grades:









•Studentsanalyzesystems ofnon-linearequations

11thGrade



Dependent: A consistent system with infinitely manysolutions.

statements such as 0 = 3)

Guiding Questions How can you find exact solutions to a system of equations without using a graphing calculator? What does it mean to be a solution to a system of linear equations? Does solving a system algebraically always result in a unique system? Why? How do you decide whether to solve a system using elimination by substitution, elimination by

multiplication, and elimination by addition? How do I prove that a system of two equations in two variables can be solved by multiplying and adding to

produce a system with the same solutions?

Interpretations and Reminders

The focus of standard REI.5 is to provide mathematics justification for the addition (elimination) andsubstitution methods of solving systems of equations that transform a given system of two equations into asimpler equivalent system that has the same solutions as the original.

Build on student experiences in graphing and solving systems of linear equations from middle school tofocus on justification of the methods used. Include cases where the two equations describe the same line(yielding infinitely many solutions) and cases where two equations describe parallel lines (yielding nosolution).


Systems of linear equations can have one solution, infinitely many solutions or no solutions. Students willdiscover these cases as they graph systems of linear equations and solve them algebraically.

A system of linear equations whose graphs meet at one point (intersecting lines) has only one solution, theordered pair representing the point of intersection. Use the academic vocabulary of “consistentindependent system” to describe this case.

A system of linear equations whose graphs do not meet (parallel lines) has no solutions and the slopes ofthese lines are the same. A system of linear equations whose graphs are coincident (the same line) hasinfinitely many solutions, the set of ordered pairs representing all the points on the line. Use the academicvocabulary of “inconsistent system” to describe this case.

By making connections between algebraic and graphical solutions and the context of the system of linearequations, students are able to make sense of their solutions.

Students need opportunities to work with equations and context that include whole number and/ordecimals/fractions.

The system solution methods can include but are not limited to graphical, elimination/linear combination,substitution, and modeling.

Systems can be written algebraically or can be represented in context. Students may use graphing calculators, programs, or applets to model and find approximate solutions for

systems of equations.Some of the language is adapted from the MGSE High School Flipbook

Misconceptions• Some students may neglect to check their answers by substituting into both equations in the system.• Students may not understand the case in which a system of linear equations has infinitely many solutions, if

this is not introduced in a real-world context.• Algebraically, students arriving at 0=0 may think the solution is at point (0, 0) rather than a system with

infinite solutions. Students arriving at -5=0 may think the solution is at point (-5, 0) rather than a systemwith no solutions.

• When students begin to use elimination using multiplication, they sometimes forget to multiply each termon both sides of the equation. You may ask students to show this step when they are learning to apply it tohelp them remember to multiply each term


This progression is all about interpreting contextual situations. Students should be able to transferthe context to the abstracted math and vice versa. Use the fluency strategy below to incite their thinking

Directions:

Write a scenario that could be modeled by thegraph. Be prepared to share your story and justifyyour reasoning.
















Directions:

















Directions:



around interpreting contexts.

Graduated Measure (The graduated measure is a quick opportunity to diagnose students’ level of comfort withthe material before you begin the progression.)


Beginning Developing ProficientWhat is the solution to the systemof equations below:

(-2. 1)

Verify (2, 1) is a solution to thesystem of equations:

y = 6x − 11−2x − 3y = −7

Yes, (2, 1) is a solution bysubstitution.

Solve the system of equations:

−2x + 6y = 6−7x + 8y = −5

(3, 2)

Teaching Notes

A system of linear equations can have one solution, no solution or infinite solutions.

A system of linear equations has one solution if the graphs of the two lines intersect at one point. Theintersection is the solution.

A system of linear equations has no solution if the graphs of the two lines never intersect (and aretherefore parallel).

A system of linear equations has infinite equations has infinite solutions if the two lines coincide (are thesame line).

The Addition and Multiplication Properties of Equality allow finding solutions to certain systems of equations.In general, any linear combination, m(Ax + By) + n(Cx + Dy) = mE +nF, of two linear equations

Ax + By = E andCx + Dy = F


intersecting in a single point contains that point. The multipliers m and n can be chosen so that the resultingcombination has only an x-term or only a y-term in it. That is, the combination will be a horizontal or verticalline containing the point of intersection.

A Simple Proof

In the proof of a system of two equations in two variables, where one equation is replaced by the sum of thatequation and a multiple of the other equation, produces a system that has the same solutions, let point (x1, y1)be a solution of both equations:

Ax1 + By1 = E (true)Cx1 + Dy1 = F (true)

Replace the equation Ax + By = E with Ax + By + k(Cx + Dy) on its left side and with E + kF on its right side.The new equation is Ax + By + k(Cx + Dy) = E + kF.

Show that the ordered pair of numbers (x1, y1) is a solution of this equation by replacing (x1, y1) in the left sideof this equation and verifying that the right side really equals E + kF:Ax1 + By1 + k(Cx1 + Dy1) = E + kF (true)

Classifying Systems

Systems of equations are classified into two groups, consistent or inconsistent, depending on whether or notsolutions exist. The solution set of a system of equations is the intersection of the solution sets for theindividual equations. Stress the benefit of making the appropriate selection of a method for solving systems(graphing vs. elimination). This depends on the type of equations and combination of coefficients forcorresponding variables, without giving a preference to either method.

Opening

The graphing method can be the first step in solving systems of equations. A set of points representingsolutions of each equation is found by graphing these equations. Even though the graphing method is limited infinding exact solutions and often yields approximate values, the use of it helps to discover whether solutionsexist and, if so, how many are there. The next step is to turn to algebraic methods, elimination or substitution,to allow students to find exact solutions. For any method, stress the importance of having a well-organizedformat for writing solutions.

Students have had previous experience with solving systems of equations using graphing in middle school.Prior to this focus lesson, you can review the following with students:

A system of equations is a group of two or more equations that have the same set of unknowns. For thepurposes of this lesson, we are only speaking of systems with two linear equations.


Gradual Release of ResponsibilityFocus LessonBegin with simple, real-world examples and help students to recognize a graph as a set of solutions to an equation. Forexample, if the equation y = 6x + 5 represents the amount of money paid to a babysitter (i.e., $5 for gas to drive to thejob and $6/hour to do the work), then every point on the line represents an amount of money paid, given the amount oftime worked.

Next, remind student of graphing techniques for linear equations:

Point out that more that one equation can be graphed on the same corrdinate plane, and if we consider howthose equations work together then we are looking at a “system of equaitons”

Explore visual ways to solve an equation such as -3x + 4 =3 x – 2 by graphing the functions y = -3x + 4 and y =3 x– 2. Students should recognize that the intersection point of the lines is at (1, 1). They should be able toverbalize that the intersection point means that when x = 1 is substituted into both sides of the equation, eachside simplifies to a value of 1.

Emphasize to students that although the solution was found graphically and is precise, it is not always efficientto construct a graph. Solutions to systems can be generated algebraically by using two methods: substitutionand elimination. This progression will focus specifically on the elimination method.

Elimination Method

Example: The elimination method eliminates one variable by adding or subtracting the equations to get anequation in one variable. It does not matter which variable is eliminated however sometimes a particularvariable may be easier to eliminate than the other. Consider the problem given above:

y = -3x + 4y = 3x – 2If we add both equations together the result will be:

2y = 0 + 2

Think aloud as you graph thissystem for students.

Discuss what the ordered pair atthe intersection point means toeach equation.


2y = 2Y = 1Once the value of y is determined, we can determine the value of x by substituting the value of y into one ofthe original equations:

1 = -3x + 4-3 = -3x1 = xTherefore the intersection of the two lines is (1,1) – the same solution that was generated from graphing bothlines.

Example: Have students to keep in mind that elimination works best if the equations are manipulated so thatall like terms and equal signs are in the same column. At this time, demonstrate the second example tostudents:

In this example, the equations can be subtracted to eliminate the variable y. The result will be:

11x = -11x = -1Again, once the x value is determined, substitute the value into one of the original equations to calculate the yvalue.

8(-1) + y = -16-8 + y = -16y = -8The solution to the system (the point where the lines intersect) is (-1,-8)

Example: Now show students a third example, where more manipulation will be needed to solve the problem:

Start by simplifying the top equation:2x – y = -3-3 – 7y = 10x

Now attempt to align all like terms and the equal sign in the same column:2x – y = -3-10x -7y = 3


In order to be able to add or subtract these equations in a manner where one variable will be eliminated, weneed to decide which variable we want to eliminate. The left side shows a possible process for elimination ofthe x variable while the right side shows a possible process for the elimination for the y variable:

5 (2x – y = -3)-10x – 7y = 3

10x – 5y = -15-10x – 7y = 3(add the two equations)

-12y = -12y = 1

Now substitute y into one of the originalequations:

2(1) – y = -32 – y = -3-y = -5y = 5

The solution to the system (the point wherethe lines intersect) is (1,5)

-7 (2x – y = -3)-10x – 7y = 3

-14x + 7y = 21-10x – 7y = 3(add the two equations)

-24x = 24x = -1

Now substitute x into one of the originalequations:

2 (-1) – y = -3-2 – y = -3-y = -1y = 1

The solution to the system (the point wherethe lines intersect) is (1,-1)

Guided Practice

Use the following problem/activity to guide students through solving systems of equations. Even thoughstudents have learned the substitution method in 8th grade, encourage students to use the elimination methodwith the problem because that is aligned to the standard and objective.

CashRegisters.docx

Collaborative Practice

Place students into groups of 2-3 homogenously. Have students to complete The Cycle Shop performance task.Please note the following:

Most students will probably complete the task in 20-25 minutes. The document contains a grading rubric and annotated student work. Upon completion, students can either present their work to the class or conduct a gallery walk of their


classmates’ work.

TheCycleShop.docx

Using Technology

Using technology, have students graph a function and use the trace function to move the cursor along thecurve. Discuss the meaning of the ordered pairs that appear at the bottom of the calculator, emphasizing thatevery point on the curve represents a solution to the equation.

Use the table function on a graphing calculator to explore some systems of equations. For example, to solve:

5

204

xy

xystudents can examine the equations y = 4x – 20 and y = -x – 5 and determine that they intersect at

the point (3, -8) by examining the graph and/or table to find where the y-values are the same.


Name

Assessment Assessment Type StandardsAddressed

Cognitive Rigor

Cashbox

Cashbox.pdf

Problem SolvingTask (Groups)

MGSE9-12.A.CED.2MGSE9-12.A.CED.3MGSE9-12.A.REI.5

DOK 3

Linear Equations inTwo Variables

linear equations intwo variables r1.pdf

FormativeAssessment Lesson

MGSE9-12.A.REI.5

Lisa’s Problem

Lisa's Problem.pdf

Learning Task MGSE9-12.A.REI.5 DOK 3


AccuratelyWeighing Pennies

Accurately weightingpennies Part 1.pdf

Problem SolvingTask (Pairs/Groups)

MGSE9-12.A.REI.6MGSE9-12.A.REI.11

DOK 2/3

Boomarangs FAL

maximizing profit -selling boomerangs r1.pdf

FormativeAssessment Lesson

MGSE9-12.A.REI.5MGSE9-12.A.REI.6MGSE9-12.A.REI.12

DOK 3

Differentiated SupportsLearning Difficulty Use Foldables for making study guides from notes, examples, graphs, or other

representations regarding systems of equations. They can also serve as anadditional tool for identifying similarities and differences through comparing thedifferent methods for solving a system, thereby organizing their knowledge.

Create a classroom blog where students can write entries, explaining how theydecide when to use each of the different methods for solving systems ofequations. One thing they can do is write a pros and cons list for each method.

Use Algebra tiles as a physical model to help visualize the addition andsubtraction properties in the elimination method.

High-Achieving Students Challenge high achieving students with an opportunity to decide which is the bestdeal on the following task:

CALS_DINNER_CARD_PLAN Task.pdf

English LanguageLearners

It is important to ensure that ELLs understand content presented in their secondlanguage. This can be accomplished by clarifying key concepts; through modelingand teacher explanation; and cueing.

To support students in quickly accessing the background information necessaryto understand contexts, the use of short videos and photographs can also beuseful.

Online/Print ResourcesDigital Resources http://www.youtube.com/watch?v=1qHTmxlaZWQ

why-does-the-elimination-method-work-161153 Video to explain why theelimination method works.

http://www.purplemath.com/modules/systlin5.htm Guided notes on solvingsystems of equations

Coordinate Grid Paper http://math.about.com/library/Worksheets/Coordinate-Grid-w.pdf

Systems of Linear Equations: Graphinghttp://www.purplemath.com/modules/systlin2.htm


Print Resources

system of EquationWord Problems.ppt

Systems ofEquations Word Problems.pdf

Solving Systems ByGraphing.ppt

Textbook AlignmentAlgebra I, McGraw-Hill pp.335 - 370

Sample Fluency Strategies for High SchoolExpression of the Week Choose an expression for the week. Allow students to consider the given expression.

Students will then think of different ways to rewrite that expression in order to createequivalent expressions. As course concepts progress ask students to incorporateadditional concepts in the creation of their expression. Provide students an opportunityto explain their thinking.

Algebraic Talks Extending from the idea of Number Talks. Teachers can help students develop flexibilitythrough Algebraic Talks.


Print Resources








systems of equations(Notes and homework).doc


Print Resources









Algebraic Talks (The Norms): No Pencils No calculators Mental Math Only Consider the problem individually during the first few minutes

Place a problem on the board and allow students to consider possible solutionpathways. Next, allow 5 or 6 student to explain their strategy and record the variousmethods on the board for all students to see. Name the strategy (preferably after thestudent who made the conjecture). Allow other students to ask questions about thestrategies while the proposing student is allowed to defend his/her work.

Also see: Jo Boaler Number Talks Videos

Picture This Provide students with a graph, or chart and ask them to come up with a contextualsituation that makes the picture relevant

Find the odd man out


Note 1: This is an open ended question where students can choose any one of thechoices A-D as the odd man as long as they are able to defend their choice. Forexample,A is the odd equation because it is in the only equation with coefficients as 1 or -1B is the odd equation because it is in the only equation with negative slopeC is the odd equation because it is in the only equation whose line passes through theoriginD is the odd equation because it is in the only equation which is not represented in thestandard form.

Note 2: Teacher can provide 4 equations/ 4 triangles / 4 transformations / 4 trig ratioswhere there is more than one way of classifying. Students can choose the odd man bymere inspection (surface level) or by examining the four components with a deeperunderstanding of the concept.

Example 2:

A is the odd equation because it has all positive coefficients or because it has twonegative roots.

B is the odd equation because it has two positive roots.

C is the odd equation because it has one positive and one negative root.

D is the odd equation because the parabola opens downwards

A

x – y = 5

B

2x + 3y = 6

C

2x – 3y = 0

D

3y = 2x – 4

A

y = x2+5x+6

B

y = x2- 5x+6

Cy = x2- x-6

Dy = - x2 + 6


Generate examples andnon-examples Example 1: Draw as many triangles as possible that are similar to

Note: Teachers can ask students to generate examples and non-examples for a varietyof topics to improve fluency. A few more possibilities are listed here:1: Create quadratic equations with imaginary roots2. Create quadratic equations with complex roots3. Create non-examples of complementary angles.4. Create examples of circles whose center is (3, -2)5. Draw triangle ABC where Sin A= 0.3

Locating the error Choose a student’s work to be displayed on board after removing the name and ask theclass to locate the error and also work the solution correctly. This can be done in anytopic where the problem requires multiple steps to solve.

unit of study-2a · students make contextual sense of the various parts of expressions and...

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