Numerical And Algebraic Expressions And Equations
2011 Carnegie Learning 2011 Carnegie LearningNumerical andAlgebraic Expressionsand EquationsSometimesit's hard to tellhow a person is feelingwhen you're not talking tothem face to face. People useemoticons in emails and chatmessages to show differentfacial expressions. Eachexpression shows a differentkind of emotion. But youprobably already knewthat. ; )6.1What’s It Really Saying?Evaluating Algebraic Expressions .2956.2 Express MathSimplifying Expressions UsingDistributive Properties .3036.3 Reverse DistributionFactoring Algebraic Expressions . 3096.4 Are They the Same or Different?Verifying That Expressions Are Equivalent . 3176.5 It is time to justify!Simplifying Algebraic Expressions UsingOperations and Their Properties . 325293
Chapter 6 OverviewThis lesson explores the use of algebraicexpressions as efficient representationsfor repeating 17.EE.21This lesson continues the developmentof distributive properties to simplify andfactor nt7.EE.17.EE.21This lesson explores different methods,including the use of graphing calculators,to determine if expressions ingOperationsand TheirProperties7.EE.17.EE.21This lesson requires the use of operationsand properties to justify the steps of thesimplification process to determine ifexpressions are equivalent. Numerical and Algebraic Expressions and Equations293AChapter 61Questions ask students to use tables toorganize values when evaluating algebraicexpressions.XXThis lesson reviews all aspects of thedistributive properties.Questions ask students to analyzemodels and explore various strategies forsimplifying expressions to gain a deepunderstanding of how the properties work.XXXXXX 2011 Carnegie Learning6.EE.67.NS.3TechnologyHighlightsTalk the TalkPacingPeer AnalysisCCSSWorked ExamplesLessonsModelsThis chapter focuses on the use of properties to interpret, simplify, add, subtract, and factor linear expressions.
Skills Practice Correlation for Chapter ingAlgebraicExpressions1–6Define variables and write algebraic expressions7 – 14Evaluate algebraic expressions15 – 20Complete tables of expressions21 – 26Evaluate algebraic expressions for given quantitiesVocabulary6.2SimplifyingExpressionsUsing DistributiveProperties1 – 10Draw models and calculate or simplify expressions11 – 20Use the Distributive Property to rewrite expressions21 – 26Evaluate expressions for given valuesVocabulary6.36.4 2011 Carnegie Learning6.5FactoringAlgebraicExpressionsVerifying ThatExpressions AreEquivalentSimplifyingAlgebraicExpressions UsingOperations andTheir Properties1 – 10Rewrite expressions by factoring out the GCF11 – 20Simplify expressions by combining like terms21 – 26Evaluate expressions for given values by factoring27 – 32Evaluate expressions for given values by combining like terms1–6Determine whether expressions are equivalent by evaluating forgiven values7 – 12Determine whether expressions are equivalent by simplifying13 – 18Determine whether expressions are equivalent by graphing1–89 – 14Complete justification tables to determine if expressions areequivalentGive justification of steps to determine if expressions are equivalentby simplifyingChapter 6Numerical and Algebraic Expressions and Equations 293B
2011 Carnegie Learning294 Chapter 6Numerical and Algebraic Expressions and Equations
What’s It ReallySaying?Evaluating Algebraic ExpressionsLearning GoalKey TermsIn this lesson, you will: variable algebraic expression Evaluate algebraic expressions.Essential Ideas An algebraic expression is a mathematical phraseinvolving at least one variable, and it can containnumbers and operational symbols. 2011 Carnegie Learning An algebraic expression is often used to representsituations in which the same mathematical process isperformed over and over again. To evaluate an expression, you replace each variablein the expression with numbers and then perform allpossible mathematical operations. When you substitute known measures into a formulaand simplify it to determine a new measure, you areevaluating the formula. evaluate an algebraicexpressionCommon Core State Standardsfor Mathematics6.EE Expressions and EquationsReason about and solve one-variable equationsand inequalities.6. Use variables to represent numbers and writeexpressions when solving a real-world ormathematical problem; understand that a variablecan represent an unknown number, or, depending onthe purpose at hand, any number in a specified set.7.NS The Number SystemApply and extend previous understandings ofoperations with fractions to add, subtract, multiply,and divide rational numbers.3. Solve real-world and mathematical problemsinvolving the four operations with rational numbers.6.1Evaluating Algebraic Expressions 295A
OverviewStudents are introduced to algebraic expressions and variables as an efficient method to representsituations where the same mathematical process is repeated over and over again. In the first twoproblems, students write and use algebraic expressions to represent given contexts. Students areintroduced to the terminology “evaluating an algebraic expression” and will practice this skill in theremaining two problems. Students complete tables that record the results when evaluating the same 2011 Carnegie Learningexpression with multiple values.295B Chapter 6Numerical and Algebraic Expressions and Equations
Warm Up1. Simplify each numeric expression.a. 8 1 9 226b. 8 1 9 335c. 8 1 9 444d. 8 1 9 553e. 8 1 9 662f. 8 1 9 771g. 8 1 9 880h. 8 1 9 9892. Explain the rule you used to simplify these problems? 2011 Carnegie LearningI use the order of operations rules and multiplied before I added.3. What was the same in each problem?Every problem has 9 times a number, and then 8 added to it.4. What was different in each problem?The number 9 was multiplied by a different value every time.5. What patterns did you notice in your solutions?The solutions increased by 9 every time. Except for the last solution, the tens-digit increasedby one every time and the ones-digit decreased by one every time.6.1Evaluating Algebraic Expressions 295C
2011 Carnegie Learning295D Chapter 6Numerical and Algebraic Expressions and Equations
What’s It ReallySaying?Evaluating Algebraic ExpressionsLearning GoalKey TermsIn this lesson, you will: variable algebraic expression Evaluate algebraic expressions. evaluate an algebraicexpressionDo you have all your ducks in a row? That’s just a drop in the bucket!That’s a piece of cake!What do each of these statements have in common? Well, they are all idioms.Idioms are expressions that have meanings which are completely different fromtheir literal meanings. For example, the “ducks in a row” idiom refers to asking ifsomeone is organized and ready to start a task. A person who uses this idiom isnot literally asking if you have ducks that are all lined up.For people just learning a language, idioms can be very challenging to understand.Usually if someone struggles with an idiom’s meaning, a person will say “that’sjust an expression,” and explain its meaning in a different way. Can you think of 2011 Carnegie Learning 2011 Carnegie Learningother idioms? What does your idiom mean?6.16.1Evaluating Algebraic ExpressionsEvaluating Algebraic Expressions 295295
Problem 1A context is given andstudents complete a tableusing repeated multiplicationto determine a total cost. Therepeated calculations providethe background to introducealgebraic expressions andvariables as an efficient methodto represent situations wherethe same mathematical processis repeated over and over again.Students will write and use thealgebraic expression for thegiven context.Problem 1Game Day SpecialYou volunteer to help out in the concession stand at your middle-school football game.You must create a poster to display the Game Day Special: a hot dog, a bag of chips, anda drink for 3.75.1. Complete the poster by multiplying the number of specials by the cost of the special.Game Day Special1 hot dog, 1 bag of chips, and a drinkfor 3.75Number of SpecialsCost123456 3.75 7.50 11.25 15.00 18.75 22.50GroupingHave students completeQuestions 1 through 3 witha partner. Then share theresponses as a class.In algebra, a variable is a letter or symbol that is used to represent a quantity. Analgebraic expression is a mathematical phrase that has at least one variable, and it cancontain numbers and operation symbols.Share Phase,Questions 1 through 3 What is the advantage ofWhenever you perform the same mathematical process over and over again, an algebraicexpression is often used to represent the situation.2. What algebraic expression did you use to represent the total cost on your poster? What is the advantage ofhaving a Game Day specialfor the sellers?3. The cheerleading coach wants to purchase a Game Day Special for every student onthe squad. Use your algebraic expression to calculate the total cost of purchasingWhy was it suggested thatyou multiply to determine thecosts?Could you have gottenthose same results byusing another mathematicaloperation? Explain.How is the term variablerelated to its root word“vary”?Which of these are algebraic3 1 8,3x 1 8,expressions:44x, xy 2 z and 9?296 Chapter 618 Game Day Specials.3.75(18) 67.50The total cost of 18 Game Day Specials is 67.50. 296 Chapter 6Numerical and Algebraic Expressions and EquationsHow did your algebraic expression help you solve for the total cost ofpurchasing 18 Game Day Specials?Is there another way to write the algebraic expression?NoteEncourage students to use parentheses when substituting values in algebraicexpressions. It makes the substitution more visible.Numerical and Algebraic Expressions and Equations 2011 Carnegie Learning The algebraic expression I used was 3.75s. 2011 Carnegie LearningLet s represent the number of Game Day Specials.having a Game Day specialfor the customers?
Problem 2Students write an algebraicexpression including bothmultiplication and addition torepresent a context. They willuse the algebraic expression tocalculate the results for multiplevalues of the variable; this timethe results are not representedin a table.Problem 2Planning a Graduation PartyYour aunt is planning toDeluxe Graduation Receptionhost your cousin’s highIncludes:One salad (chef or Cæsar)One entree (chicken, beef, or seafood)Two side dishesOne dessertFee: 105 for the reception hall plus 40 per guestschool graduation party atLattanzi’s Restaurant andReception Hall. Lattanzi’shas a flyer that describesthe Deluxe GraduationReception.GroupingHave students completeQuestions 1 and 2 with apartner. Then share theresponses as a class.1. Write an algebraic expression to determine the cost ofthe graduation party. Let g represent the number ofguests attending the party.105 1 40gShare Phase,Questions 1 and 2 Is there another way to write2. Determine the cost of the party for each number of attendees.Show your work.a. 8 guests attendthe algebraic expression? 2011 Carnegie LearningThe party would cost 425 if 8 guests attended.How is this algebraicexpression different from theone in Problem 1? How did your algebraicexpression help you solve forthe cost of the party for eachnumber of guests? What makes more sense inthe context of this situation,the expression (105 1 40)g orthe expression (105 1 40g)?Explain.What makes more sense inthe context of this situation,the expression (105 1 40g)or the expression(40g 1 105)? Explain.105 1 40(8) 5 425b. 10 guests attendThe party would cost 505 if 10 guests attended.105 1 40(10) 5 505 2011 Carnegie Learning So, an equationhas an equals sign.An expressiondoes not.c. 12 guests attendThe party would cost 585 if 12 guests attended.105 1 40(12) 5 585Note6.1Evaluating Algebraic Expressions 297The expression 105 1 40g will probably make more sense to students than40g 1 105 because it follows the context chronologically; 105 is the initial feeand then you pay for the guests. Follow what makes sense to students, it is notnecessary to guide students to y 5 mx 1 b form at this time.6.1Evaluating Algebraic Expressions 297
Problem 3Students are introducedformally to the terminology“evaluate an algebraicexpression”. They will evaluatealgebraic expressionsrepresented both with andwithout tables.Problem 3Evaluating ExpressionsIn Problems 1 and 2, you worked with two expressions, 3.75s and (105 1 40g). Youevaluated those expressions for different values of the variable. To evaluate an algebraicexpression, you replace each variable in the expression with a number or numericalexpression and then perform all possible mathematical operations.1. Evaluate each algebraic expression.a. x 2 7GroupingHave students completeQuestion 1 with a partner. Thenshare the responses as a class. for x 5 28x 7 8 7 15 for x 5 211x 7 11 7 18 for x 5 16x 7 16 7 9Use parenthesesto showmultiplicationlike -6(-3).b. 26yShare Phase,Question 1 What does it mean to for y 5 23 6y 6( 3) 18 for y 5 0 6y 6(0) 0 for y 5 7 6y 6(7) 42evaluate an expression? c. 3b 2 5What integer rule did you useto evaluate this expression?How is this question similarto those in the previousproblems?298 Chapter 6for b 5 223b 5 3( 2) 5 6 5 11 for b 5 33b 5 3(3) 5 9 5 4 for b 5 93b 5 3(9) 5 27 5 22d. 21.6 1 5.3nHow is this question differentfrom those in previousproblems?298 Chapter 6 for n 5 25 1.6 1 5.3n 1.6 1 5.3( 5) 1.6 1 ( 26.5) 28.1 for n 5 0 1.6 1 5.3n 1.6 1 5.3(0) 1.6 1 0 1.6 for n 5 4 1.6 1 5.3n 1.6 1 5.3(4) 1.6 1 21.2 19.6Numerical and Algebraic Expressions and EquationsNumerical and Algebraic Expressions and Equations 2011 Carnegie Learning How is the term evaluaterelated to its root word“value”? 2011 Carnegie Learning
GroupingHave students completeQuestion 2 with a partner. Thenshare the responses as a class.Sometimes, it is more convenient to use a table to record the results when evaluating thesame expression with multiple values.2. Complete each table.Share Phase,Question 2 What mathematicala.operations are involved inthis question? Where is the substitutionrepresented in your work? What integer rules didyou use to evaluate thisexpression? What makes a table moreconvenient than a series ofquestions?h 2h 72 1121 58 23277b.aa416 12 10 4033.556 2011 Carnegie Learning 2011 Carnegie Learningc.xx2 51 43463122 1d.y 5 1015122 y 1 35524533523526.16.1Evaluating Algebraic ExpressionsEvaluating Algebraic Expressions5 299299
Problem 4Students use integers,decimals, and mixed numbersto evaluate algebraicexpressions.Problem 4Evaluating Algebraic Expressions Using Given Values1.1. Evaluate each algebraic expression for x 5 2, 23, 0.5, and 223a. 23xGrouping 3(2) 6Have students completeQuestions 1 through 3 witha partner. Then share theresponses as a class. 3(0.5) 1.51 7 3 23 3( 3) 9(b. 5x 1 10Share Phase,Question 1 When evaluating the5(2) 1 10 205( 3) 1 10 55(0.5) 1 10 12.51 1 10 112 1 10 125 2333(expression, which value waseasiest to work with? Why?When evaluating theexpression, which value wasthe most difficult to workwith? Why?)c. 6 2 3x6 3(2) 06 3( 3) 156 3(0.5) 4.51 6 1 7 136 3 23)d. 8x 1 758(2) 1 75 918( 3) 1 75 518(0.5) 1 75 791 1 75 5618 233(300300 Chapter 6 Chapter 6)Numerical and Algebraic Expressions and EquationsNumerical and Algebraic Expressions and Equations 2011 Carnegie Learning(Using tables may helpyou evaluate theseexpressions. 2011 Carnegie Learning )
Share Phase,Question 2 When evaluating the1.2. Evaluate each algebraic expression for x 5 27, 5, 1.5, and 216a. 5xexpression, which value waseasiest to work with? Why? 5( 7) 35When evaluating theexpression, which value wasthe most difficult to workwith? Why?5(5) 255(1.5) 7.551 55 166()b. 2x 1 3x2( 7) 1 3( 7) 352(5) 1 3(5) 252(1.5) 1 3(1.5) 7.551 1 3 11 52 1666() ()c. 8x 2 3xI'm noticingsomethingsimilar about all ofthese expressions.What is it?8( 7) 3( 7) 358(5) 3(5) 258(1.5) 3(1.5) 7.551 3 11 58 1666) () 2011 Carnegie Learning 2011 Carnegie Learning(6.16.1Evaluating Algebraic ExpressionsEvaluating Algebraic Expressions 301301
Share Phase,Question 3 When evaluating the5.3. Evaluate each algebraic expression for x 5 23.76 and 2216expression, which value waseasiest to work with? Why? a. 2.67x 2 31.852.67(23.76) 31.85 31.5892(33 x 1 56b. 1148)5 31.85 90.1452.67 216When evaluating theexpression, which value wasthe most difficult to workwith? Why?3 335.5553 (23.76) 1 561148()5 1 5633 21114681313 47 1 56648()6157 13531 1 20024246Talk the TalkTalk the TalkStudents describe a strategyfor evaluating an algebraicexpression.1. Describe your basic strategy for evaluating any algebraic expression.I substitute a value for the variable and then follow the order of operation rules.Grouping2. How are tables helpful when evaluating expressions?Have students completeQuestions 1 and 2independently. Then share theresponses as a class.Be prepared to share your solutions and methods.302302 Chapter 6 Chapter 6Numerical and Algebraic Expressions and EquationsNumerical and Algebraic Expressions and Equations 2011 Carnegie Learning 2011 Carnegie LearningTables help me organize the values when I evaluate an expression.
Follow UpAssignmentUse the Assignment for Lesson 6.1 in the Student Assignments book. See the Teacher’s Resourcesand Assessments book for answers.Skills PracticeRefer to the Skills Practice worksheet for Lesson 6.1 in the Student Assignments book for additionalresources. See the Teacher’s Resources and Assessments book for answers.AssessmentSee the Assessments provided in the Teacher’s Resources and Assessments book for Chapter 6.Check for Students’ UnderstandingUse a variable to write an algebraic expression to represent each series of numerical expressions.After each algebraic expression, list the values for the variable.1. 5 3 1 73. 3 8 1 95 4173 8 1 115 5173 8165 6173 8 1 255 x173 81zx 5 3, 4, 5 and 6z 5 9, 11, 6 and 254. 11(5 1 9)6 81711(9 1 8)9 817(12 1 9)118 81711 (9 1 9)y 81711(x 1 9)y 5 2, 6, 9 and 8x 5 5, 8, 12, and 9 2011 Carnegie Learning2. 2 8 1 76.1Evaluating Algebraic Expressions 302A
2011 Carnegie Learning302B Chapter 6Numerical and Algebraic Expressions and Equations
Express MathSimplifying Expressions UsingDistributive PropertiesLearning GoalsKey TermsIn this lesson, you will: Write and use thedistributive properties. Use distributive propertiesDistributive Property of Multiplication over AdditionDistributive Property of Multiplication over SubtractionDistributive Property of Division over AdditionDistributive Property of Division over Subtractionto simplify expressions.Essential Ideas The Distributive Property provides ways to write 2011 Carnegie Learningnumerical and algebraic expressions in equivalentforms.Common Core State Standardsfor Mathematics7.NS The Number SystemApply and extend previous understandings ofoperations with fractions to add, subtract, multiply,and divide rational numbers. When the Distributive Property is applied to numericalexpressions only, it is a helpful tool in computingmath mentally. The area of a rectangle model is useful indemonstrating the Distributive Property. There are four versions of the distributive property:7.EE Expressions and Equations Distributive Property of Multiplication over Addition:If a, b and c are any real numbers, thena (b 1 c) 5 a b 1 a c.Use properties of operations to generate e
1 – 10 Draw models and calculate or simplify expressions 11 – 20 Use the Distributive Property to rewrite expressions 21 – 26 Evaluate expressions for given values 6.3 Factoring Algebraic Expressions Vocabulary 1 – 10 Rewrite expressions by factoring out the GCF
b. Perform operations on rational algebraic expressions correctly. c. Present creatively the solution on real – life problems involving rational algebraic expression. d.Create and present manpower plan for house construction that demonstrates understanding of rational algebraic expressions and algebraic expressions with integral exponents. 64
Unit 2: Algebraic Expressions Media Lesson Section 2.4: Simplifying Algebraic Expressions Steps for Simplifying Algebraic Expressions Step 1: Simplify within parentheses Step 2: Use distributive property to eliminate parentheses Step 3: Combine like terms. Example 1: Simplify the following algebraic expressions. Show all possible steps.
Jun 12, 2017 · Station 2 Independent Practice: Algebraic and Numerical Expressions Distribute Handout 7.3: Algebraic and Numerical Expressions. Tell students they will write an algebraic expression for 5 situations and write numerical expressions for the volume of two cubes and the area of three squares.File Size: 1MB
Teacher guide Interpreting Algebraic Expressions T-1 Interpreting Algebraic Expressions MATHEMATICAL GOALS This lesson unit is intended to help you assess how well students are able to translate between words, symbols, tables, and area representations of algebraic expressions. It will help you to identify and support students who have .
9.5 Addition and Subtraction of Algebraic Expressions In the earlier classes, we have also learnt how to add and subtract algebraic expressions. For example, to add 7x 2 – 4x 5 and 9x – 10, we do 7x2 – 4x 5
Simplifying Algebraic Expressions 1-9 LESSON A number, a variable, or a product of numbers and variables that are separated by plus and minus signs. The number that is multiplied by the variable in an algebraic expression. Lesson Objectives Simplify algebraic expressions Vocabulary term (p. 42) coefficient (p. 42) Additional Examples Example 1
Explain 1 Interpreting Algebraic Expressions in Context In many cases, real-world situations and algebraic expressions can be related. The coefficients, variables, and operations represent the given real-world context. Interpret the algebraic expression corresponding to the given context. Example 1 Curtis is buying supplies for his school.
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