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Introduction to Higher-Order Algebra forLevel 1 and Level 2 Students7/2018Introduction to HigherOrder Algebra for Level 1and Level 2 StudentsA Workshop by GED Testing Service 2 Copyright GED Testing Service LLC. Allrights reserved.1

Introduction to Higher-Order Algebra forLevel 1 and Level 2 Students7/2018Session Objectives Determine the big ideas ofalgebra Discuss the importance ofteaching the basics ofinequalities and functions(two High Impact Indicators) Connect inequalities andfunctions to real-worldsituations Share resources3The Magic of Algebra Think of a number between 1 and 100. Multiply your number by 4. Add 12. Multiply this number by 2. Add 16. Divide this number by 8. Subtract your original number.Your new number is 5!4 Copyright GED Testing Service LLC. Allrights reserved.2

Introduction to Higher-Order Algebra forLevel 1 and Level 2 Students7/2018Can You Show that Algebraically?Think of a number between 1 & 100.Let's represent this number as n.Multiply your number by 4.We can show this as 4n (4 times n).Add 12.4n 12Multiply this number by 2.2 (4n 12) 8n 24(using the Distributive Property)Add 16.(8n 24) 16 8n (24 16) 8n 40(using the Associative Property)Divide this number by 8.8𝑛 40 8𝑛 88 408 n 5Subtract your original number.n 5–n 0 5(using the Commutative and AssociativeProperties)Your number is 5!0 5 55The Magic of Algebra (What’s the reason?)Think of any number. Multiply it by 2. Add 4. Multiply by 3. Divide by 6. Subtract the number with which you started.You got 2!Explain with algebra why this works.6 Copyright GED Testing Service LLC. Allrights reserved.3

Introduction to Higher-Order Algebra forLevel 1 and Level 2 Students7/2018The answer is . . .Start with the expression that describes theoperations to be performed on your chosennumber, x:and simplify the expression. You'll end up with 2,regardless of the value of x.7Why Use Magic Tricks or Puzzles? They are Fun Non-threatening Motivational Engaging Students begin to use algebraic thinkingwithout knowing that is what they are doing.8 Copyright GED Testing Service LLC. Allrights reserved.4

Introduction to Higher-Order Algebra forLevel 1 and Level 2 Students7/2018Some Big Ideas in Algebra Variable Symbolic Notation Equality Ratio and Proportion Pattern Generalization Equations and Inequalities Multiple Representations of Functions9HoldingAlgebra in YourHandsStarting with the concrete by using algebratiles10 Copyright GED Testing Service LLC. Allrights reserved.5

Introduction to Higher-Order Algebra forLevel 1 and Level 2 Students7/2018When teaching algebraic concepts, modelusing multiple representations Start with the concrete Represent problems using symbols, expressions,and equations, tables, and graphs Model real-world situations Complete problems different ways (flexibility inproblem solving)1112Introduction to Algebra Tiles 𝒙𝟐𝒙𝟐𝟏𝒙 𝟏 𝒙Remember, they could be called x, y, b, t, etc.12 Copyright GED Testing Service LLC. Allrights reserved.6

Introduction to Higher-Order Algebra forLevel 1 and Level 2 Students7/2018What’s My Polynomial?13Big Ideas Using Algebra Tiles Adding and Subtracting Integers; Zero Principle Modeling Linear Expressions Solving Linear Equations Simplifying Polynomials Solving Equations for Unknown Variable Multiplying and Dividing Polynomials Factoring Trinomials Completing the Square Investigations14 Copyright GED Testing Service LLC. Allrights reserved.7

Introduction to Higher-Order Algebra forLevel 1 and Level 2 Students7/2018Look at the Relationships among the Tiles𝟏𝟏𝒙𝒙𝟐15Let’s Use Algebra Tiles to MultiplyPolynomialsProblem: (x 2)(x 3) ?Step 1: Gather all the tiles you need for each term.x 2x 316 Copyright GED Testing Service LLC. Allrights reserved.8

Introduction to Higher-Order Algebra forLevel 1 and Level 2 Students7/2018Multiplying Polynomials(x 2)(x 3) ?Step 2: Lay out the tiles on opposite axes of the algebra tilemat as shown below. 3x 2x17Multiplying Polynomials(x 2)(x 3) ?Step 3: Using other tiles, form a perfect rectangle using thetiles on axes of the product mat as the indicators for thedimensions of the rectangle as shown. 3Notice that no othercombination of tiles wouldmake a perfect rectanglematching the dimension oftiles on the axes of theproduct mat.x 2x18 Copyright GED Testing Service LLC. Allrights reserved.9

Introduction to Higher-Order Algebra forLevel 1 and Level 2 Students7/2018Resources Working With Algebra Tiles – AlgebraTiles.htm Factoring Polynomials Using Algebra Tiles - Del tActivity.pdf Multiplying Polynomials Using Algebra Tiles – Virginia Dept.of arch/sol/math/A/mess a-2b 2.pdf19Algebra Tile Apps Illuminations (National Council for Teachers of d 3482 Michigan Virtual Universityhttp://media.mivu.org/mvu pd/a4a/homework/index.html National Library of Virtual tmlDon’t forget phone apps!20 Copyright GED Testing Service LLC. Allrights reserved.10

Introduction to Higher-Order Algebra forLevel 1 and Level 2 Students7/2018Inequalities –No Problem!A High Impact Indicator21Focusing on High Impact Indicators InequalitiesA.3 Write,manipulate,solve, andgraph linearinequalitiesA.3.a Solvelinearinequalities inone variablewith rationalnumbercoefficients.A.3.b Identifyor graph thesolution to aone variablelinear inequalityon a numberline.A.3.c Solvereal-worldproblemsinvolvinginequalities.A.3.d Writelinearinequalities inone variable torepresentcontext.22 Copyright GED Testing Service LLC. Allrights reserved.11

Introduction to Higher-Order Algebra forLevel 1 and Level 2 Students7/2018Can your students . . .23What is an inequality?An inequality is a mathematical sentence that usessymbols such as , , , or to compare twoquantities.AB24 Copyright GED Testing Service LLC. Allrights reserved.12

Introduction to Higher-Order Algebra forLevel 1 and Level 2 Students7/2018Inequalities Are EverywhereSituationMathematical InequalitySpeed limitLegal speed on the highway 65 miles perhourCredit cardMonthly payment 10% of your balance inthat billing cycleTextmessagingAllowable number of text messages per month 250Travel timeTime needed to drive from home toschool/work 18 minutes25Recognize the Symbols and the VocabularyPhraseInequality“a is more than b”a b“a is at least b”a b“a is less than b”a b“a is at most b;” or“a is no more than b”a bInequality tells what is “allowable” or “possible.” An inequalityplaces conditions on the value of the variable .26 Copyright GED Testing Service LLC. Allrights reserved.13

Introduction to Higher-Order Algebra forLevel 1 and Level 2 Students7/2018Recognize the Symbols and the VocabularyTermInequalityCoefficient4a 8Boundary PointA solution that makes theinequality trueSolution SetThe range of values thatmake the inequality trueInclusivea 6Exclusivea 6An inequality tells what is “allowable” or “possible.”27Rules for Solving Inequalities1.2.3.4.Make the same changes to both sides of the inequalityIsolate the variableCombine like termsUse the inverse operation to remove clutter from thevariable5. If your inverse operation is multiplication or division bya negative number, reverse the inequality sign becomes becomes becomes becomes 28 Copyright GED Testing Service LLC. Allrights reserved.14

Introduction to Higher-Order Algebra forLevel 1 and Level 2 Students7/2018Properties of InequalitiesAddition and SubtractionIf a b, then a c b cIf a b, then a - c b – cReal-life situationBecky is older than Janet: b jAdd 10 years: b 10 j 10Subtract 10 years: b – 10 j - 1029Properties of InequalitiesMultiplication and DivisionIf a b, then ac bc, if c 0If a b, then ac bc, if c 0Real-life situationBecky is older than Janet: b jWhen they are twice their current age:b(2) j(2)When they were half the age they are now:𝑏𝑗 2230 Copyright GED Testing Service LLC. Allrights reserved.15

Introduction to Higher-Order Algebra forLevel 1 and Level 2 Students7/2018But there is one exceptionWhen you multiply or divide each side of an inequality by anegative number, you must reverse the inequality symbol!WHY?Multiplying or dividing both sides of an equation by anegative number changes the sign of each side of theequation. On both sides, what was positive becomesnegative, and what was negative becomes positive.31Testing the Property3 2Multiply by -1(-1)(3) 2 (-1)-3 -2FALSE-3 -2TRUEMultiplying by anegative flipped theinequality sign from“greater than” to “lessthan.”3232 Copyright GED Testing Service LLC. Allrights reserved.16

Introduction to Higher-Order Algebra forLevel 1 and Level 2 Students7/2018Solve the Inequality4 x 124 x 12(draw wall down inequality)4 x 12(box in variable)4 x 12(minus 4 both sides)-4-4x 83333Graph the Solutionx 81. Draw a number line. Just need a few numbers on eitherside of the solution number.2. Decide if open circle or closed circle. Place it above thesolution number.3. Determine which way your arrow goes by substituting anumber in for the variable to make the statement true.Then draw the arrow pointing in that direction.3434 Copyright GED Testing Service LLC. Allrights reserved.17

Introduction to Higher-Order Algebra forLevel 1 and Level 2 Students7/2018Solve the Inequality48 8m48 8m886 m(draw wall, box variable)(divide 8 both sides)3535Graph the Solution6 m1. Draw a number line. Just need a few numbers on eitherside of the solution number.2. Decide if open circle or closed circle. Place it above thesolution number.3. Determine which way your arrow goes by substituting anumber in for the variable to make the statement true.Then draw the arrow pointing in that direction.3636 Copyright GED Testing Service LLC. Allrights reserved.18

Introduction to Higher-Order Algebra forLevel 1 and Level 2 Students7/2018Your Turn!Coach told us weneeded to completemore than 18repetitions.Which one iscorrect?1.N 182.N 183.N 184.N 1837Your Turn!Children under 12 pay adiscounted price at themovie theatre.Which one is correct?1.N 122.N 123.N 124.N 1238 Copyright GED Testing Service LLC. Allrights reserved.19

Introduction to Higher-Order Algebra forLevel 1 and Level 2 Students7/2018Your Turn!Marie has purchasedparty supplies for amaximum of 12 people.Which one is correct?1.N 122.N 123.N 124.N 1239Sample Question from GEDTS40 Copyright GED Testing Service LLC. Allrights reserved.20

Introduction to Higher-Order Algebra forLevel 1 and Level 2 Students7/2018ResourcesOne-Variable Inequalities – a/one-variable-linear-inequalitiesVirtual Nerds: What is an Inequality?https://www.youtube.com/watch?v wcBwdz-ZBaMVery Basics of Graphing Inequalities(on a number line)https://www.youtube.com/watch?v nif2PKA9bXA41ResourcesMath is Fun – Solving uality-solving.htmlSolving and Graphing tch?v EE2qWIyjKD0Math Dude Unit 1-4 – SolvingInequalitieshttps://www.youtube.com/watch?v 8hhewFQ K0w42 Copyright GED Testing Service LLC. Allrights reserved.21

Introduction to Higher-Order Algebra forLevel 1 and Level 2 Students7/2018FunctionsAnother High Impact Indicator43Focusing on High Impact Indicators FunctionsA.7 Compare,represent, andevaluate functions.A.7.a Compare twodifferent proportionalrelationshipsrepresented indifferent ways.A.7.b Represent oridentify a function ina table or graph ashaving exactly oneoutput (one elementin the range) foreach input (eachelement in thedomain).A.7.c Evaluatelinear and quadraticfunctions for valuesin their domain whenrepresented usingfunction notation.A.7.d Compareproperties of twolinear or quadraticfunctions eachrepresented in adifferent way(algebraically,numerically intables, graphically orby verbaldescriptions).44 Copyright GED Testing Service LLC. Allrights reserved.22