Teaching Algebra With Manipulatives - Sault Schools
Teaching Algebrawith ManipulativesFor use withGlencoe Algebra 1Glencoe Algebra 2
ManipulativesGlencoe offers three types of kits to enhance the use of manipulatives in yourPre-Algebra classroom. The Glencoe Mathematics Overhead Manipulative Resources containstranslucent manipulatives designed for use with an overhead projector. The Glencoe Mathematics Classroom Manipulative Kit contains classroom setsof frequently used manipulatives in algebra, geometry, measurement, probability, andstatistics. The Glencoe Mathematics Student Manipulative Kit contains an individual set ofmanipulatives often used in Student Edition activities.The manipulatives contained in each of these kits are listed on page viii of thisbooklet.Each of these kits can be ordered from Glencoe by calling (800) 334-7344.Glencoe Mathematics Overhead Manipulative ResourcesGlencoe Mathematics Classroom Manipulative KitGlencoe Mathematics Student Manipulative McGraw-HillCopyright by The McGraw-Hill Companies, Inc. All rights reserved.Printed in the United States of America. Permission is granted to reproduce the material containedherein on the condition that such material be reproduced only for classroom use; be provided tostudents, teachers, and families without charge; and be used solely in conjunction with GlencoeAlgebra 1 or Glencoe Algebra 2. Any other reproduction, for use or sale, is prohibited without priorwritten permission of the publisher.Send all inquiries to:Glencoe/McGraw-Hill8787 Orion PlaceColumbus, OH 43240ISBN: 0-07-827755-8Teaching Algebra with Manipulatives1 2 3 4 5 6 7 8 9 10 079 11 10 09 08 07 06 05 04 03 02
ContentsEasy-to-Make ManipulativesPageGrid Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1Centimeter Grid Paper . . . . . . . . . . . . . . . . . . . . .210 10 Centimeter Grids . . . . . . . . . . . . . . . . . . .3Coordinate Planes . . . . . . . . . . . . . . . . . . . . . . . .4Coordinate Planes in Space . . . . . . . . . . . . . . . . .5Coordinate Planes for Trigonometric Functions(degrees) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6Coordinate Planes for Trigonometric Functions(radians) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7Conic Graph Paper — circles and lines . . . . . . . .8Conic Graph Paper — overlapping circles . . . . . .9Algebra Tiles — units . . . . . . . . . . . . . . . . . . . . .10Algebra Tiles — variables . . . . . . . . . . . . . . . . .11Integer Models Summary . . . . . . . . . . . . . . . . . .12Polynomial Models Summary . . . . . . . . . . . . . . .13Equation Models Summary . . . . . . . . . . . . . . . .14Integer Mat . . . . . . . . . . . . . . . . . . . . . . . . . . . .15Equation Mat . . . . . . . . . . . . . . . . . . . . . . . . . . .16Product Mat . . . . . . . . . . . . . . . . . . . . . . . . . . . .17Rectangular Dot Paper . . . . . . . . . . . . . . . . . . . .18Isometric Dot Paper . . . . . . . . . . . . . . . . . . . . . .19Problem Solving Guide . . . . . . . . . . . . . . . . . . .20Spinners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21Die Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . .22Protractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23Rulers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24Normal Distribution Curve . . . . . . . . . . . . . . . . .252-6Algebra Activity RecordingSheet . . . . . . . . . . . . . . . . . . . . . . .49CHAPTER 3Teaching Notes and Overview . . . . . . 503-1Algebra Activity Recording Sheet . . . .563-1Algebra Activity: Formulas . . . . . . . . .573-2Algebra Activity Recording Sheet . . . .593-2Using Overhead Manipulatives:Solving Equations by UsingAddition and Subtraction . . . . . . . .603-2Mini-Project: Using Logic . . . . . . . . . .653-3Using Overhead Manipulatives:Solving Equations by UsingMultiplication and Division . . . . . 66Algebra Activity Recording Sheet . . . .683-43-4Using Overhead Manipulatives:Solving Equations Using MoreThan One Operation . . . . . . . . . . . .693-4Algebra Activity: Solving Equations . . .13-5Using Overhead Manipulatives:Solving Equations with the Variableon Both Sides . . . . . . . . . . . . . . . . .733-6Using Overhead Manipulatives:Proportions . . . . . . . . . . . . . . . . . . .753-6Using Overhead Manipulatives:Solving Percent Problems . . . . . . . .763-7Using Overhead Manipulatives:Percent of Change . . . . . . . . . . . . .773-7Mini-Project: The Music Business . . . .793-8Algebra Activity: Mixture Problems . .80Algebra 1ActivitiesPageCHAPTER 1Teaching Notes and Overview . . . . . . 271-2Mini-Project: Evaluating Expressions .301-5Using Overhead Manipulatives:Distributive Property . . . . . . . . . . .311-5Algebra Activity Recording Sheet . . . .331-7Algebra Activity: Make a Model . . . . .341-8Algebra Activity Recording Sheet . . . .35CHAPTER 4Teaching Notes and Overview . . . . . . 824-3Using Overhead Manipulatives:Relations . . . . . . . . . . . . . . . . . . . .854-3Algebra Activity Recording Sheet . . . .864-5Using Overhead Manipulatives:Graphing Linear Relations . . . . . . .874-6Using Overhead Manipulatives:Functions . . . . . . . . . . . . . . . . . . . .884-6Mini-Project: Functions or Not? . . . . .894-8Algebra Activity Recording Sheet . . . .904-8Algebra Activity: Functions . . . . . . . .91CHAPTER 2Teaching Notes and Overview . . . . . . 362-2Mini-Project: Adding and SubtractingIntegers . . . . . . . . . . . . . . . . . . . . .392-2Using Overhead Manipulatives:Adding and Subtracting Integers . . .402-3Using Overhead Manipulatives:Multiplying Integers . . . . . . . . . . . .422-4Algebra Activity: Multiplying andDividing Rational Numbers . . . . . .432-5Using Overhead Manipulatives:Measures of Central Tendency . . . .452-5Using Overhead Manipulatives:Statistics and Line Plots . . . . . . . . .462-6Using Overhead Manipulatives:Experimental Probability . . . . . . . .48CHAPTER 5Teaching Notes and Overview . . . . . . 935-1Using Overhead Manipulatives:Find the Slope of a Line . . . . . . . . .965-3Algebra Activity Recording Sheet . . . .985-3Using Overhead Manipulatives: SlopeIntercept Form of Linear Equations 995-5Using Overhead Manipulatives:Linear Equations in Point-Slopeand Standard Form . . . . . . . . . . .100iii
Algebra 1ActivitiesPage5-6Algebra Activity: PerpendicularLines . . . . . . . . . . . . . . . . . . . . . .1025-6Algebra Activity Recording Sheet . . .1045-7Algebra Activity Recording Sheet . . .1055-7Using Overhead Manipulatives:Scatter Plots . . . . . . . . . . . . . . . . .1065-7Mini-Project: Scatter Plots . . . . . . . .1079-49-49-59-59-6CHAPTER 6Teaching Notes and Overview . . . . . 1086-1Using Overhead Manipulatives:Solving Inequalities UsingAddition and Subtraction . . . . . . .1116-1Mini-Project: MultiplyingInequalities by 1 . . . . . . . . . . . .1146-2Algebra Activity Recording Sheet . . .1156-4Algebra Activity: CompoundInequalities . . . . . . . . . . . . . . . . . .116Algebra Activity Recording Sheet . . .1196-56-6Using Overhead Manipulatives:Graphing Inequalities . . . . . . . . . .1209-69-6Using Overhead Manipulatives:Factoring Trinomials . . . . . . . . . . .162Algebra Activity: FactoringTrinomials . . . . . . . . . . . . . . . . . .163Using Overhead Manipulatives:Factoring Differences of Squares .165Algebra Activity Recording Sheet . . .167Using Overhead Manipulatives:Factoring Perfect SquareTrinomials . . . . . . . . . . . . . . . . . .168Mini-Project: Factoring TrinomialSquares . . . . . . . . . . . . . . . . . . . . .170Algebra Activity: FactoringTrinomial Squares . . . . . . . . . . . .171CHAPTER 10Teaching Notes and Overview . . . . . 17310-1Algebra Activity Recording Sheet . . .17610-1Mini-Project: Graphing QuadraticFunctions . . . . . . . . . . . . . . . . . . .177Using Overhead Manipulatives:10-3Solving Quadratic Equationsby Completing the Square . . . . . . .17810-5Algebra Activity: A Fascinating Fold .18010-7Algebra Activity Recording Sheet . . .18110-7Algebra Activity Recording Sheet . . .182CHAPTER 7Teaching Notes and Overview . . . . . 1227-1Mini-Project: Comparing Options . . .1247-2Algebra Activity Recording Sheet . . .1257-3Using Overhead Manipulatives:Elimination Using Addition andSubtraction . . . . . . . . . . . . . . . . . .1267-5Algebra Activity: Graphing Systemsof Inequalities . . . . . . . . . . . . . . . .127CHAPTER 11Teaching Notes and Overview . . . . . 18311-2Mini-Project: Matching Radicals . . . .18511-4Using Overhead Manipulatives: ThePythagorean Theorem . . . . . . . . . .18611-4Algebra Activity: The PythagoreanTheorem . . . . . . . . . . . . . . . . . . . .18711-4Using Overhead Manipulatives:Graphing Irrational Numbers . . . .18911-6Using Overhead Manipulatives:Similar Triangles . . . . . . . . . . . . .19011-7Algebra Activity Recording Sheet . . .19111-7Algebra Activity Recording Sheet . . .192CHAPTER 8Teaching Notes and Overview . . . . . 1298-1Using Overhead Manipulatives:Multiplying Monomials . . . . . . . .1348-1Algebra Activity Recording Sheet . . .1358-4Algebra Activity Recording Sheet . . .1368-5Algebra Activity Recording Sheet . . .137Using Overhead Manipulatives: Adding8-5and Subtracting Polynomials . . . . .1388-5Mini-Project: Multiplying Binomials .1418-6Using Overhead Manipulatives:Multiplying a Polynomial by aMonomial . . . . . . . . . . . . . . . . . . .1428-7Algebra Activity Recording Sheet . . .1448-7Using Overhead Manipulatives:Multiplying Polynomials . . . . . . . .1458-7Algebra Activity: MultiplyingPolynomials . . . . . . . . . . . . . . . . .1478-8Using Overhead Manipulatives:Three Special Products . . . . . . . . .149CHAPTER 9Teaching Notes and Overview . . . . . 1529-2Algebra Activity Recording Sheet . . .1569-3Algebra Activity Recording Sheet . . .1599-3Using Overhead Manipulatives:Factoring Trinomials . . . . . . . . . . .160CHAPTER 12Teaching Notes and Overview . . . . . 19312-5Using Overhead Manipulatives:Dividing Polynomials byMonomials and by Binomials . . . .19512-5Algebra Activity Recording Sheet . . .19712-9Mini-Project: Rational Roundup . . . .198CHAPTER 13Teaching Notes and Overview . . . . . 19913-3Algebra Activity: Modeling Data . . . .20013-5Algebra Activity Recording Sheet . . .203CHAPTER 14Teaching Notes and Overview . . . . . 20414-1Algebra Activity Recording Sheet . . .20614-3Algebra Activity: Probability andCompound Events . . . . . . . . . . . .207iv
Algebra 1ActivitiesPage14-5Algebra Activity: Theoretical andExperimental Probability . . . . . . .20814-5Algebra Activity Recording Sheet . . .2097-87-8Algebra Activity: Inverse Relations . .256Algebra Activity Recording Sheet . . .258CHAPTER 8Teaching Notes and Overview . . . . . .2598-1Algebra Activity Recording Sheet . . .2638-2Algebra Activity Recording Sheet . . .2648-2Mini-Project: Conics . . . . . . . . . . . . .265Algebra Activity Recording Sheet . . .2668-48-4Algebra Activity Recording Sheet . . .2678-6Algebra Activity: Conic Sections . . . .2688-6Algebra Activity Recording Sheet . . .270Algebra 2ActivitiesPageCHAPTER 1Teaching Notes and Overview . . . . . 2101-2Algebra Activity Recording Sheet . . .2121-2Algebra Activity Recording Sheet . . .2131-3Algebra Activity: Solving Equations .2141-6Mini-Project: Solving Absolute ValueInequalities . . . . . . . . . . . . . . . . . .216CHAPTER 2Teaching Notes and Overview . . . . . 2172-5Algebra Activity Recording Sheet . . .2182-6Mini-Project: Graphing AbsoluteValue Functions . . . . . . . . . . . . . .219CHAPTER 9Teaching Notes and Overview . . . . . .2719-1Mini-Project: InterpretingPowers of a BinomialGeometrically . . . . . . . . . . . . . . . .2729-3Algebra Activity Recording Sheet . . .273CHAPTER 3Teaching Notes and Overview . . . . . 2203-2Mini-Project: Solving Systems ofEquations by Elimination . . . . . . .2223-3Algebra Activity: Graphing Systemsof Inequalities . . . . . . . . . . . . . . . .2233-5Algebra Activity Recording Sheet . . .2253-5Algebra Activity: Equations of Planes 226CHAPTER 10Teaching Notes and Overview . . . . . .27410-1Algebra Activity Recording Sheet . . .27510-1Algebra Activity: ExponentialFunctions . . . . . . . . . . . . . . . . . . .27610-6Mini-Project: Graceful Numberingof Graphs . . . . . . . . . . . . . . . . . . .278CHAPTER 4Teaching Notes and Overview . . . . . .2284-1Algebra Activity: Matrices . . . . . . . .2294-3Mini-Project: Determinants . . . . . . . .231CHAPTER 11Teaching Notes and Overview . . . . . .27911-1Mini-Project: Sequences . . . . . . . . . .28211-1Algebra Activity Recording Sheet . . .28311-6Algebra Activity Recording Sheet . . .28411-6Algebra Activity Recording Sheet . . .28511-6Algebra Activity: InvestigatingFractal Cut Outs . . . . . . . . . . . . . .286CHAPTER 5Teaching Notes and Overview . . . . . .2325-2Algebra Activity Recording Sheet . . .2345-4Algebra Activity Recording Sheet . . .2355-5Algebra Activity: Radical Roots ofPolynomials . . . . . . . . . . . . . . . . .2365-6Algebra Activity Recording Sheet . . .2385-9Mini-Project: nth Roots of NegativeNumbers . . . . . . . . . . . . . . . . . . . .2395-9Algebra Activity Recording Sheet . . .240CHAPTER 12Teaching Notes and Overview . . . . . .28812-4Algebra Activity Recording Sheet . . .29112-4Algebra Activity: Probability . . . . . . .29212-8Mini-Project: Probability . . . . . . . . . .29412-8Algebra Activity RecordingSheet . . . . . . . . . . . . . . . . . . . . . .29512-9Algebra Activity RecordingSheet . . . . . . . . . . . . . . . . . . . . . .296CHAPTER 6Teaching Notes and Overview . . . . . .2416-1Mini-Project: Parabolas . . . . . . . . . . .2436-4Algebra Activity Recording Sheet . . .2446-4Algebra Activity: Completing theSquare . . . . . . . . . . . . . . . . . . . . .2456-6Algebra Activity: QuadraticFunctions . . . . . . . . . . . . . . . . . . .247CHAPTER 13Teaching Notes and Overview . . . . . .29713-1Algebra Activity: IndirectMeasurement . . . . . . . . . . . . . . . .29913-2Algebra Activity Recording Sheet . . .30113-6Mini-Project: Linear Interpolationby Graphing . . . . . . . . . . . . . . . . .302CHAPTER 7Teaching Notes and Overview . . . . . .2497-4Mini-Project: Synthetic Substitutionand Zeros . . . . . . . . . . . . . . . . . . .2527-7Algebra Activity: ExploringIteration . . . . . . . . . . . . . . . . . . . .253CHAPTER 14Teaching Notes and Overview . . . . . .30314-7Mini-Project: Trigonomessage . . . . . .304v
Teacher’s Guide to UsingTeaching Algebra with Manipulativeshe book contains three sections of masters—Easy-to-Make Manipulatives, Algebra 1Activities, and Algebra 2 Activities. Tabs helpyou locate the chapter resources in each section. Acomplete list of manipulatives available in each of thethree types of Glencoe Mathematics Manipulative Kitsappears on the next page.Mini-Projects are short projects that enable students towork cooperatively in small groups to investigate mathematical concepts.TUsing Overhead Manipulatives provides instructionsfor the teacher to demonstrate an alternate approach tothe concepts of the lesson by using manipulatives onthe overhead projector.Easy-to-Make ManipulativesThe first section of this book contains masters for making your own manipulatives. To make more durablemanipulatives, consider using card stock. To makealgebra tiles similar to those shown in the StudentEdition, have students use markers to color the tilesappropriately or use colored card stock.Student Recording Sheets accompany the AlgebraActivities found in the Student Edition. Students caneasily record the results of the activity on preparedgrids, charts, and figures.Algebra Activities provide additional activities toenrich the students’ experiences. These masters ofteninclude a transparency master to accompany theactivity.You can also make transparencies of frequently useditems such as grid paper and number lines.Activity MastersEach chapter begins with Teaching Notes andOverview that summarizes the activities for the chapterand includes sample answers. There are four types ofmasters.vii
Glencoe Mathematics ManipulativesGlencoe Mathematics Overhead Manipulative ResourcesISBN: 0-07-830593-4Transparenciesinteger matequation matproduct matinequality matdot paperisometric dot papercoordinate gridsOverhead Manipulativescentimeter gridnumber lineslined paperregular polygonspolynomial modelsinteger modelsequation modelsalgebra tilesspinnerstwo-dimensional cupsred and yellow countersdecimal models (base-ten blocks)compassprotractorgeoboard/geobandsgeometric shapestransparency pens in 4 colorsGlencoe Mathematics Classroom Manipulative KitISBN: 0-02-833116-8Measurement, Probability,and StatisticsAlgebraalgebra tilescounterscupscentimeter cubesequation mat/product matcoordinate grid stamp andink padbase-ten modelsmarblesmeasuring cupsnumber ape rsisometric dot grid stamppattern blockstangramsGlencoe Mathematics Student Manipulative KitISBN: 0-02-833654-2algebra tilesred and yellow counterscupsequation /product tape measureviii
Grid Paper Glencoe/McGraw-Hill1Teaching Algebra with Manipulatives
Centimeter Grid Paper Glencoe/McGraw-Hill2Teaching Algebra with Manipulatives
10 10 Centimeter Grids Glencoe/McGraw-Hill3Teaching Algebra with Manipulatives
Coordinate Planes Glencoe/McGraw-Hill4Teaching Algebra with Manipulatives
Coordinate Planes in SpacezOyxzOyx Glencoe/McGraw-Hill5Teaching Algebra with Manipulatives
Coordinate Planes for Trigonometric Functions(degrees)y1O90 180 270 360 450 540 630 720 90 180 270 360 450 540 630 720 x 14y321O 1x 2 3 4y1 360 270 180 90 O90 180 270 360 x 1 Glencoe/McGraw-Hill6Teaching Algebra with Manipulatives
Coordinate Planes for Trigonometric Functions(radians)y1O 2 3 22 5 23 7 24 2 3 22 5 23 7 24 x 14y321O 1x 2 3 4y1 2 3 2 2O 2 3 22 x 1 Glencoe/McGraw-Hill7Teaching Algebra with Manipulatives
Conic Graph Paper(circles and lines) Glencoe/McGraw-Hill8Teaching Algebra with Manipulatives
Conic Graph Paper(overlapping circles) Glencoe/McGraw-Hill9Teaching Algebra with Manipulatives
Algebra 11111111111111111111111111111111 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Glencoe/McGraw-Hill10Teaching Algebra with Manipulatives
Algebra Tiles(variables)xxxxxxxxxxxxxxxxxxxx x x x x x x x x x x x x x x x x x x x xx2 x x22 xGlencoe/McGraw-Hillx22 xx2112 xx22 x2Teaching Algebra with Manipulatives
Integer Models SummaryThere are two types of integers tiles. 11A zero pair is formed by pairing one positive integer tile and one negativeinteger tile. 110You can remove or add zero pairs to a set without changing the value ofthe set.Remove 4 zero pairs.1111 1 1 1 115 ( 4) 1 1 1 1 1 ( 1) ( 1) ( 1) ( 1)1111 1 1 1 111 [ 1 ( 1)] [ 1 ( 1)] [ 1 ( 1)] [ 1 ( 1)] 1Add 2 zero pairs. 1 111 1 1 1 1 3 Glencoe/McGraw-Hill 1 1 3 [ 1 ( 1)] [ 1 ( 1)] 312Teaching Algebra with Manipulatives
Polynomial Models SummaryThere are three basic tiles used for modeling a polynomial.xxx12x x x11 1x x21 1 1x 1 xEach tile has an opposite. x 12 xA zero pair results when a tile and its opposite are paired. 11x2 x20 xx00You can add or remove zero pairs to any polynomial without changingits value.Like terms are represented by tiles that are the same shape and size. Glencoe/McGraw-Hill13Teaching Algebra with Manipulatives
Equation Models SummaryA zero pair is formed by pairing one positive tile and one negative tile ofthe same type. 11x2 x2 xx000You can remove or add the same number of like tiles to each side of theequation mat without changing its value.Add 2 negative ones to each side.1x11 x11x 2 11 1 1x 2 ( 2)3 111 3 ( 2) 1 1You can remove or add zero pairs to either side of the equation matwithout
v Algebra 1 Activities Page 14-5 Algebra Activity: Theoretical and Experimental Probability . . . . . . .208 14-5 Algebra Activity Recording Sheet . . .209 Algebra 2 .
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