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Grade 8Mathematics

High Impact Standards

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Common Core Standards Plus® – Mathematics – Grade 4 Domain: Measurement and Data Focus: Relative Size of Measurement Units Lesson: #4 Standard: 4.MD.1 Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two column table. For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36), ...

Lesson Objective: Students will place customary units for length, volume, mass, and time in order from largest to smallest.

Introduction: “Today we will compare customary units of measurement and place units in order from largest to smallest for length, volume, mass, and time. In the United States, we use a system made up of standard or customary units of measure such as feet, inches, quarts, pounds, minutes, and hours to determine units of length, volume, mass, and time.” Instruction: “The customary system of measurement is commonly used in our everyday life. Look at the top of your page. There is a customary measurement chart displaying units of length, volume, mass, and time. (Review the units for each and stress their relationships. E.g., 1 yd = 3 ft = 36 in, 1 gal = 4 qts, etc. Read chart aloud with students.) Look at the units of length listed below the customary measurement chart. The units are listed from longest to shortest: mile, yard, foot, and inch. In the customary units, mile is the longest unit. Since smaller units make up the largest unit, yard is the next longest unit. The unit after that is foot. One foot is shorter than a yard, but longer than an inch. Therefore, it is listed third in the order. Since an inch is the shortest unit of measurement for length, it is listed last.” Guided Practice: “Look at the example on this page. We are asked to put the units of volume in order from most to least (pint, ounce, cup, quart). Think. Which unit is more? Use the customary measurement chart to help you decide. (Pause.) If you think that quart is the most, you are correct. Write the unit quart on the first blank. Remember, it always takes more of a smaller unit to equal a larger unit. Look at the remaining units. Place them on the blanks from most to least.” Allow students to share their answer with a partner, and then provide the correct order. Elicit volunteers to share why they placed the units in a particular order. Require students to use academic language. Stress the relationships between the units (quart, pint, cup, and ounce). Independent Practice: “Complete problems 1-5 on your own. Use the customary measurement chart to help you list the units in order from largest to smallest.” Review: Review problems 1-5 with students. If time permits, require students to explain their answers. Closure: “Today we reviewed measurement equivalents and relative size within customary units of length, volume, and mass, as well as units of time, and placed units in order from largest to smallest. Can you think of items that you use each day that are measured in length, volume, mass, or time?” Answers: 1. year, day, hour, minute

2. ton, pound, ounce 3. 1 yard, 2 feet, 12 inches 4. 1 year, 1 month, 7 days, 24 hours 5. Answers may vary. A possible answer is 16 ounces, which is equal to 2

cups because it is the most.

Standards Plus Lesson Timeline

Introduction

Instruction

Guided Practice

Independent Practice

Review &Closure

1 min.

2 min.

3 min.

4 min.

5 min.

6 min.

7 min.

8 min.

9 min.

10 min.

11 min.

12 min.

13 min.

14 min.

15 min.

16 min.

17 min.

Introduction

Instruction

Guided Practice

Independent Practice

Review &Closure


5

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Mathematics Grade 8Lesson Index

WHY STANDARDS PLUS INCREASES STUDENT ACHIEVEMENT

Students master grade level standards using the proven effective Standards Plus process

WHY STANDARDS PLUS INCREASES STUDENT ACHIEVEMENT

Students master grade level standards using the proven effective Standards Plus process

Teachers:Teachers are the most important factor in student learning. That’s why every Standards Plus lesson is directly taught by a teacher.

Direct Instruction format:“Direct Instruction is a proven method of instruction that fosters the most significant gains in student achievement and results in deep and enduring understanding of the concept.” (Peladeau, Forget & Gagne, 2003).

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Immediate intervention:For students who need further instruction to master a standard/skill, immediate intervention lessons are available that scaffold below grade level and provide the necessary instruction that allows students to master the standard.

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The lesson index lists every Standards Plus High Impact Standards

Mathematics Grade 8 lesson.

Mathematics Grade 8Lesson Index

Common Core Standards Plus® - Mathematics Grade 8 Lesson Index

CommonCoreStandardsPlus®-Mathematics–Grade8–LessonIndex–PartialYear

Domain Lesson Focus Standard(s) TEPg St.Ed.Pg

Expression

sand

Equ

ations

(Exp

ressions

and

Equ

ations

Stand

ards

:8.EE.1-7,8.EE.7a

-b,8

.EE.8,8.EE.8a

-c)

9 ScientificNotation 8.EE.3:Usenumbersexpressedintheformofasingledigittimesanintegerpowerof10toestimateverylargeorverysmallquantities,andtoexpresshowmanytimesasmuchoneisthantheother.8.EE.4:Performoperationswithnumbersexpressedinscientificnotation,includingproblemswherebothdecimalandscientificnotationareused.Usescientificnotationandchooseunitsofappropriatesizeformeasurementsofverylargeorverysmallquantities.

6 3

10 ScientificNotation 8 4


12 ScientificNotation 12-13 6

E3 Evaluation–ScientificNotation 16 7

13 OperationsUsingScientificNotation

8.EE.4

18 8

14 OperationsUsingScientificNotation 20 9


16 UsingTechnologyw/ScientificNotation 24 11


P3 PerformanceLesson#3–UsingScientificNotation(8.EE.3,8.EE.4) 28 13-15

17 GraphProportionalRelationships&DetermineUnitRate

8.EE.5:Graphproportionalrelationships,interpretingtheunitrateastheslopeofthegraph.Comparetwodifferentproportionalrelationshipsrepresentedindifferentways.

32-33 16-17


36-37 18-19

19 ComparingProportionalRelationships 40 20-21


E5 Evaluation–GraphingandComparingProportionalRelationships 48 24-25

21 SimpleTrianglesandSlope8.EE.6:Usesimilartrianglestoexplainwhytheslopemisthesamebetweenanytwodistinctpointsonanon-verticallineinthecoordinateplane;derivetheequationy=mxforalinethroughtheoriginandtheequationy=mx+bforalineinterceptingtheverticalaxisatb.

52 26

22 SimpleTrianglesandSlope 54 27

23 DerivetheEquationy=mx 56 28


E6 Evaluation–ProportionalRelationships,Lines,andLinearEquations 60 30-31

P4 PerformanceLesson#4–WhatisSlope?(8.EE.5,8.EE.6) 64 32-33

25 TypesofSolutionstoaLinearEquation8.EE.7a:Giveexamplesoflinearequationsinonevariablewithonesolution,infinitelymanysolutions,ornosolutions.Showwhichofthesepossibilitiesisthecasebysuccessivelytransformingthegivenequationintosimplerforms,untilanequivalentequationoftheformx=a,a=a,ora=bresults(whereaandbaredifferentnumbers).

68 34

26 LinearEquations 70 35

27 Solving1-Stepand2-StepEquations 8.EE.7a,8.EE.7b:Solvelinearequationswithrationalnumbercoefficients,includingequationswhosesolutionsrequireexpandingexpressionsusingthedistributivepropertyandcollectingliketerms.

72 36

28 Solving1-Stepand2-StepEquations 74 37

E7 Evaluation–FindingSolutionsto1-and2-StepLinearEquations 76 38

37 SystemsofEquations

8.EE.8a:Understandthatsolutionstoasystemoftwolinearequationsintwovariablescorrespondtopointsofintersectionoftheirgraphs,becausepointsofintersectionsatisfybothequationssimultaneously.

78 39

38 SystemsofEquations 80 40

39 SystemofEquations 82-83 41

40 SystemsofEquations 86-87 42-44

E10 Evaluation–SystemsofEquations 92 45-46

BOOK 1

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Expression

sand

Equ

ations

(Exp

ressions

and

Equ

ations

Stand

ards

:8.EE.1-7,8.EE.7a

-b,8

.EE.8,8.EE.8a

-c)

9 ScientificNotation 8.EE.3:Usenumbersexpressedintheformofasingledigittimesanintegerpowerof10toestimateverylargeorverysmallquantities,andtoexpresshowmanytimesasmuchoneisthantheother.8.EE.4:Performoperationswithnumbersexpressedinscientificnotation,includingproblemswherebothdecimalandscientificnotationareused.Usescientificnotationandchooseunitsofappropriatesizeformeasurementsofverylargeorverysmallquantities.

6 3



12 ScientificNotation 12-13 6


13 OperationsUsingScientificNotation

8.EE.4

18 8



16 UsingTechnologyw/ScientificNotation 24 11


P3 PerformanceLesson#3–UsingScientificNotation(8.EE.3,8.EE.4) 28 13-15


8.EE.5:Graphproportionalrelationships,interpretingtheunitrateastheslopeofthegraph.Comparetwodifferentproportionalrelationshipsrepresentedindifferentways.

32-33 16-17


36-37 18-19



E5 Evaluation–GraphingandComparingProportionalRelationships 48 24-25

21 SimpleTrianglesandSlope8.EE.6:Usesimilartrianglestoexplainwhytheslopemisthesamebetweenanytwodistinctpointsonanon-verticallineinthecoordinateplane;derivetheequationy=mxforalinethroughtheoriginandtheequationy=mx+bforalineinterceptingtheverticalaxisatb.

52 26

22 SimpleTrianglesandSlope 54 27



E6 Evaluation–ProportionalRelationships,Lines,andLinearEquations 60 30-31

P4 PerformanceLesson#4–WhatisSlope?(8.EE.5,8.EE.6) 64 32-33

25 TypesofSolutionstoaLinearEquation8.EE.7a:Giveexamplesoflinearequationsinonevariablewithonesolution,infinitelymanysolutions,ornosolutions.Showwhichofthesepossibilitiesisthecasebysuccessivelytransformingthegivenequationintosimplerforms,untilanequivalentequationoftheformx=a,a=a,ora=bresults(whereaandbaredifferentnumbers).

68 34

26 LinearEquations 70 35

27 Solving1-Stepand2-StepEquations 8.EE.7a,8.EE.7b:Solvelinearequationswithrationalnumbercoefficients,includingequationswhosesolutionsrequireexpandingexpressionsusingthedistributivepropertyandcollectingliketerms.

72 36

28 Solving1-Stepand2-StepEquations 74 37

E7 Evaluation–FindingSolutionsto1-and2-StepLinearEquations 76 38

37 SystemsofEquations

8.EE.8a:Understandthatsolutionstoasystemoftwolinearequationsintwovariablescorrespondtopointsofintersectionoftheirgraphs,becausepointsofintersectionsatisfybothequationssimultaneously.

78 39

38 SystemsofEquations 80 40

39 SystemofEquations 82-83 41

40 SystemsofEquations 86-87 42-44

E10 Evaluation–SystemsofEquations 92 45-46

BOOK 1


2

Standards Plus - MATH Grade 8 - High Impact Standards

Lesson Index





Func

tions

(Fun

ctions

Stand

ards

:8.F.1-5)

1 DefiningFunctions

8.F.1:Understandthatafunctionisarulethatassignstoeachinputexactlyoneoutput.Thegraphofafunctionisthesetoforderedpairsconsistingofaninputandthecorrespondingoutput.

98 3

2 DefiningFunctions 100 4

3 DefiningFunctions 102 5

4 DefiningFunctions 104 6-7

E1 Evaluation–DefiningFunctions 108 8-9

5 IdentifyingLinearandNon-LinearFunctions 8.F.3:Interprettheequationy=mx+basdefiningalinearfunction,whosegraphisastraightline;giveexamplesoffunctionsthatarenotlinear.

112-113 10-11

6 IdentifyingLinearandNon-LinearFunctions 116 12-13

7 IdentifyingLinearandNon-LinearFunctions 120-121 14-15

8 LinearParentFunction 8.F.3,8.F.2:Comparepropertiesoftwofunctionseachrepresentedinadifferentway(algebraically,graphically,numericallyintables,orbyverbaldescriptions).

124-125 16-17

E2 Evaluation–ComparingFunctions 128 18

Geo

metry

(GeometryStandards:8.G.1-8.G.9)

1 VerifyingProperties 8.G.1:Verifyexperimentallythepropertiesofrotations,reflections,andtranslations.8.G.2 132 19-20

2 ShowingCongruency 8.G.2:Understandthatatwo-dimensionalfigureiscongruenttoanotherifthesecondcanbeobtainedfromthefirstbyasequenceofrotations,reflections,andtranslations;giventwocongruentfigures,describeasequencethatexhibitsthecongruencebetweenthem.

136 21

3 MappingFigures 138 22

4 MappingFigures 140 23

E1 Evaluation–UsingRotations,Reflections,andTranslations

8.G.1,8.G.2 142 24

5 DilatingFigures 8.G.3:Describetheeffectofdilations,translations,rotations,andreflectionsontwo-dimensionalfiguresusingcoordinates.8.G.4:Understandthatatwo-dimensionalfigureissimilartoanotherifthesecondcanbeobtainedfromthefirstbyasequenceofrotations,reflections,translations,anddilations;giventwosimilartwo-dimensionalfigures,describeasequencethatexhibitsthesimilaritybetweenthem.

144-145 25

6 TransformingFigures 148-149 26-27

7 TransformingFigures 152 28

8 TransformingFigures 154 29

E2 TransformingFigures 156 30

9 DescribeaSequenceofTransformations 8.G.4 158 31

10 AngleSumandExteriorAngleTheorems 8.G.5:Useinformalargumentstoestablishfactsabouttheanglesumandexteriorangleoftriangles,abouttheanglescreatedwhenparallellinesarecutbyatransversal,andtheangle-anglecriterionforsimilarityoftriangles.

160 32

11 ApplyingtheAngleSumofaTriangle 162 33

12 ApplytheAngleSumandExteriorAngleofTriangles 164 34

E3 Evaluation–TheAngleSumandExteriorAngleofTriangles

8.G.4,8.G.5 166 35

21 ProofofthePythagoreanTheorem

8.G.6:ExplainaproofofthePythagoreanTheoremanditsconverse.

168 36-37

22 ProofofthePythagoreanTheorem 172 38-39

23 ProofofthePythagoreanTheorem 176 40-41

24 ConverseofthePythagoreanTheorem 180 42-43

E6 Evaluation–ProofsofthePythagoreanTheoremandItsConverse 184 44

BOOK 2


3

Standards Plus - MATH Grade 8 - High Impact StandardsLesson Index





Geo

metry

(GeometryStandards:8.G.1-8.G.9)

25 ApplyingthePythagoreanTheorem

8.G.7:ApplythePythagoreanTheoremtodetermineunknownsidelengthsinrighttrianglesinreal-worldandmathematicalproblemsintwoandthreedimensions.

186 45

26 ApplyingthePythagoreanTheorem 188 46



E7 Evaluation–ApplythePythagoreanTheorem 194 49

33 FindingtheDistanceBetweenPointsonaCoordinatePlane

8.G.8:ApplythePythagoreanTheoremtofindthedistancebetweentwopointsinacoordinatesystem.

196 50

34 DistanceFormula 198 51-52

35 ApplyingtheDistanceFormula 202 53

36 DistanceFormulaandtheConverseofthePythagoreanTheorem 204 54

E9 Evaluation–PythagoreanTheorem 206 55-56

P11 PerformanceLesson#11–PythagoreanTheorem(8.G.6,8.G.7,8.G.8) 210 57

BOOK 2


4

Standards Plus - MATH Grade 8 - High Impact Standards

Lesson Index


Implementation Models

14 Week Model Teach one lesson a day

7 Week Model Teach two lessons a day

Saturday School/Bootcamp Model Teach five lessons per session



Sample Lessons Included in this Booklet

Common Core Standards Plus® ‐ Mathematics – Grade 8 – Lesson Index

Domain Lesson Focus Standard(s) Student Page

DOK Level

Functio

ns

(Fun

ctions Stand

ards: 8

.F.1‐5)

1 Defining Functions 8.F.1: Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.

118

1‐22 Defining Functions 119 3 Defining Functions 120 4 Defining Functions 121‐122

E1 Evaluation – Defining Functions 123, 125

5 Identifying Linear and Non‐Linear Functions 8.F.3: Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.

127‐128

1‐2

6 Identifying Linear and Non‐Linear Functions 129‐130


8 Linear Parent Function 8.F.3, 8.F.2: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

133‐134

E2 Evaluation – Comparing Functions 135

9 Linear Functions in y = k Form 8.F.2, 8.F.3 137

1‐2

10 Rate of Change 8.F.4: Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

138

11 Rate of Change 139

12 Rate of Change 140‐141

E3 Evaluation – Comparing Functions 8.F.3, 8.F.4 142‐143

13 Rewrite Linear Equation into Slope‐Intercept Form

8.F.2

145

1‐214 Comparing Properties of Two Functions 146

15 Comparing Properties of Two Functions 147



P6 Performance Lesson #6 – Linear Functions and Relationships (8.F.1, 8.F.2, 8.F.3, 8.F.4) 151‐152 3

17 Comparing the Properties of Two Functions 8.F.2 153

1‐2

18 Construct/Interpret a Function to Model a Linear Relationship

8.F.4

154

19 Construct/Interpret a Function to Model a Linear Relationship 155

20 Construct/Interpret a Function to Solve Problems 156‐157

E5 Evaluation – Constructing and Interpreting Functions

8.F.2, 8.F.4 158

21 Sketch a Function Graph 8.F.5: Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

159

1‐222 Describe Functional Relationships 160

23 Describe Functional Relationships

8.F.5

161‐162

24 Describe Functional Relationships 163‐164

E6 Evaluation – Use Functions to Model Relationships 165

P7 Performance Lesson #7 – Functional Relationships (8.F.2, 8.F.4, 8.F.5) 167 3



DOK Level

Functio

ns

(Fun

ctions Stand

ards: 8

.F.1‐5)

1 Defining Functions 8.F.1: Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.

118

1‐22 Defining Functions 119 3 Defining Functions 120 4 Defining Functions 121‐122

E1 Evaluation – Defining Functions 123, 125

5 Identifying Linear and Non‐Linear Functions 8.F.3: Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.

127‐128

1‐2



8 Linear Parent Function 8.F.3, 8.F.2: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

133‐134


9 Linear Functions in y = k Form 8.F.2, 8.F.3 137

1‐2

10 Rate of Change 8.F.4: Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

138

11 Rate of Change 139

12 Rate of Change 140‐141

E3 Evaluation – Comparing Functions 8.F.3, 8.F.4 142‐143

13 Rewrite Linear Equation into Slope‐Intercept Form

8.F.2

145

1‐214 Comparing Properties of Two Functions 146




P6 Performance Lesson #6 – Linear Functions and Relationships (8.F.1, 8.F.2, 8.F.3, 8.F.4) 151‐152 3

17 Comparing the Properties of Two Functions 8.F.2 153

1‐2

18 Construct/Interpret a Function to Model a Linear Relationship

8.F.4

154

19 Construct/Interpret a Function to Model a Linear Relationship 155

20 Construct/Interpret a Function to Solve Problems 156‐157

E5 Evaluation – Constructing and Interpreting Functions

8.F.2, 8.F.4 158

21 Sketch a Function Graph 8.F.5: Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

159

1‐222 Describe Functional Relationships 160

23 Describe Functional Relationships

8.F.5

161‐162

24 Describe Functional Relationships 163‐164

E6 Evaluation – Use Functions to Model Relationships 165

P7 Performance Lesson #7 – Functional Relationships (8.F.2, 8.F.4, 8.F.5) 167 3

Func

tions



DOK Level

Geom

etry

(Geo

metry Stand

ards: 8.G.1‐8.G.9)

1 Verifying Properties 8.G.1: Verify experimentally the properties of rotations, reflections, and translations. 8.G.2: See Below 197‐198

1‐2

2 Showing Congruency 8.G.2: Understand that a two‐dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

199 3 Mapping Figures 200 4 Mapping Figures 201

E1 Evaluation – Using Rotations, Reflections, and Translations

8.G.1, 8.G.2 202

5 Dilating Figures 8.G.3: Describe the effect of dilations, translations, rotations, and reflections on two‐dimensional figures using coordinates. 8.G.4: Understand that a two‐dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two‐dimensional figures, describe a sequence that exhibits the similarity between them.

203

1‐26 Transforming Figures 204‐2057 Transforming Figures 206 8 Transforming Figures 207 E2 Transforming Figures 208

9 Describe a Sequence of Transformations 8.G.4 209

1‐2

10 Angle Sum and Exterior Angle Theorems 8.G.5: Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle‐angle criterion for similarity of triangles.

210 11 Applying the Angle Sum of a Triangle 211

12 Apply the Angle Sum and Exterior Angle of Triangles 212

E3 Evaluation – The Angle Sum and Exterior Angle of Triangles 8.G.4, 8.G.5 213

13 Defining Angles Made by a Transversal

8.G.5

215

1‐214 Measuring the Angles Formed by a Transversal 216 15 Measuring Angles Formed by a Transversal 217 16 Measuring Angles Formed by a Transversal 218 E4 Evaluation – Parallel Lines Cut by a Transversal 219 17 Parallel Lines Cut by a Transversal

8.G.5

221

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18 Use Transversals to Find the Angle Sum of a Triangle 222

19 Properties and Criteria for Similar Triangles 223 20 Criteria for Similar Triangles 224

E5 Evaluation – Transformations, Triangles, and Parallel Lines Cut by Transversals 225

P10 Performance Lesson #10 – 2‐D Figures & Transformations (8.G.1, 8.G.2, 8.G.3, 8.G.4, 8.G.5) 227‐228 3 21 Proof of the Pythagorean Theorem

8.G.6: Explain a proof of the Pythagorean Theorem and its converse.

229‐230

1‐2

22 Proof of the Pythagorean Theorem 231‐23223 Proof of the Pythagorean Theorem 233‐23424 Converse of the Pythagorean Theorem 235‐236

E6 Evaluation – Proofs of the Pythagorean Theorem and It’s Converse 237

25 Applying the Pythagorean Theorem

8.G.7: Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real‐world and mathematical problems in two and three dimensions.

239

1‐226 Applying the Pythagorean Theorem 240 27 Applying the Pythagorean Theorem 241 28 Applying the Pythagorean Theorem 242 E7 Evaluation – Apply the Pythagorean Theorem 243

Geo

met

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Sample lessons continue on the following page.


Common Core Standards Plus® – Mathematics – Grade 8Domain: Functions Focus: Defining Functions Lesson: #1Standard: 8.F.1: Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.

Lesson Objective: Students will understand the difference between a relation and a function and identify a function.

Introduction: “Today you will distinguish between a relation and a function.”

Instruction: “A relation is a set of ordered pairs. A function is a subset or special type of relation. For a set of ordered pairs to be a function, they must follow a rule. A function is a relation where the value of one variable depends on the value of the other variable. In a function, each input (x-coordinate) may be paired with exactly one output (y-coordinate). The y-coordinate value is dependent on the x-coordinate value. In a function any x-value number can be substituted into the equation and yield exactly one y-value out of the equation. Given a set of ordered pairs, a table of x- and y-values, an equation, or a graph, you should be able to determine if the given is a function or just a relation.”

Guided Practice: “Today we will only look at a set of coordinate pairs and tables of x-yvalues. Let’s complete the example together. We are to determine if the relation is a function and explain. The ordered pairs are: ( 2, 5),(7, 2),(3, 3),(8, 5),(7, 8) . Is this a function or not? It’s not a function because for the same x-value of 7, there are two different y-values: a -2 and an 8. Remember, each x-value in a pair can only be paired with exactly one unique y-value.”

Independent Practice: “It’s your turn to determine if the set of ordered pairs or the table of x-y values represent a function or not. You must explain your reasoning in writing.”

Review: When the students are finished, go over the problems.

Closure: “Think about what you practiced today. Using your own words, tell your elbow partner what a function is.” After a minute, call on volunteers to share their definition out loud to the class. Answers: 1. Function. Even though the y-value repeats, the unique x-values are

paired with their own y-value. 2. Not a function. -2 has two different y-value outputs: 0 and 5 3. Not a function. 3 has 5 different y-value outputs. 4. Function. Each x-value is paired with it’s own unique y-value 5. Answers will vary. The x-value should be -5, 4, or -8 with a different y-

value that is already given in the table.

Functions - Teacher Lesson Plan - Sample


15


Example: Determine if the following relation is a function. Explain your reasoning.( 2, 5),(7, 2),(3, 3),(8, 5),(7, 8)

Directions: Complete the following.

Determine if the following relations are functions. Explain your reasoning.

1. 1 , 4 ,(3, 4),( 4, 4),(2,10)4

2.

3.

4. ( 1, 0),(0,1),(1, 2),(2, 3)

5. Complete the table below to make a relation that is not a function.

x y -2 0 0 -2 3 5 -2 5 9 11

x 3 3 3 3 3 y -0.5 1.3 2.5 4 10

x y - 5 3 4 2 - 8 7

Functions - Student Page - Sample


Common Core Standards Plus® – Mathematics – Grade 8Domain: Functions Focus: Defining Functions Lesson: #2Standard: 8.F.1: Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Lesson Objective: Students will identify if a graph of ordered pairs represents a function.

Introduction: “Yesterday you determined if a set of ordered pairs and a table of x-y values represent a function. Today you will determine if a graph of a set of ordered pairs represents a function.”

Instruction: “The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. You can determine if the graph is a function if the output for a given input is unique. In other words, only one y-coordinate output exists for a given x-coordinate input. Visually, this means that no two points are lined up vertically on the graph.”

Guided Practice: “Let’s complete the example together. Is the graph a function? (It is a function.) How do you know? For each x-value input, there is exactly one y-value output. No two points are lined up vertically.”

Independent Practice: “It’s your turn to determine if the graphs of a set of ordered pairs are functions or not.”


Closure: “Today you evaluated graphs of ordered pairs to determine if the graphs represented functions. How can you visually determine if a graph of ordered pairs is a function?” Call on a student for the answer and confirm answer.

Answers: 1. Cici is not correct. The given graph is not a function. She would have to remove the point (2, 1) or (2, -2).

2. Yes. The graph is a function.


Lesson Objective: Students will determine if a relation curve is a function using the vertical line test.

Introduction: “Today you will determine if a relation curve is a function using the vertical line test.”

Instruction: “We have looked at a set of ordered pairs to determine if the ordered pairs represent a function. A set of ordered pairs is a function when the input value of x has a rule or equation applied to it and when a unique corresponding output value of y is the result. We have looked at ordered pairs written as a list, in a table, and plotted on a coordinate plane or grid. Today you will view the graph of the actual rule or equation using a set of infinite points.This means the graphs will be continuous and not discrete points. We learned from analyzing the graphs in the previous lesson, that if two or more points are vertically lined up on the graph, the graph is not a function. You will use the vertical line test to help you determine if the relation curve, or graph, is a function. A relation is a function if no vertical lines intersect the graph at more than one point. You may be asking yourself if there is such a thing as a horizontal line test. You will hear about that in high school. We are only focusing on the vertical line test as a short-cut method for determining and confirming if a relation curve is a function.”

Guided Practice: “Let’s complete the examples together. Is the graph shown below a function? Draw a few vertical lines to see. The graph is not a function. A vertical line intersects the curve in more than one place. This means that for a given x input value, there is more than one y output value, so the relation is not a function. The vertical line test works because you can test to see if the vertical line, through any and all x-values, intersects the curve in one, or more than one, place. If all possible vertical lines intersect in only one place for each line, then there is one and only one y output value for an input x-value and the relation curve is a function. The test can be applied incorrectly if the vertical line is not drawn accurately. For example, only one vertical line is drawn at an x-value where it would intersect the relation curve in only one place. If drawn in a different location, it would intersect the relation curve in more than one place.”

Independent Practice: “Determine if the relation curves are functions by using the vertical line test. You can draw more than one vertical line to test for a function.”


Closure: “Today you evaluated relation curves, including lines, to identify functions. You used the vertical line test.” Answers: 1. No

2. Yes 3. Yes 4. No


Lesson Objective: Students will identify a function from a scenario and an equation.

Introduction: “Today you will be given scenarios with tables of values and equations. You will determine if the scenarios and the equations represent functions.”

Instruction: “To determine if an equation represents a function, we must see if there is one unique output value y for an input value x. We will do this by substituting an input value into the equation and finding the output value. To determine if variable a is a function of variable b given a table of values, we must see if there is one unique value a for each b. To determine if variable b is a function of variable a, we must see if there is one unique value b for each a. If we find different output numbers for the same input number, the scenario does not represent a function.”

Guided Practice: “Let’s complete the examples. We are given the equation x2 + y2 = 9in the first problem, and we must determine if it represents a function. Let’s pick any number for x and solve for y. If it is a function, we will find that any input value x will give us only one output value y. We’ll substitute 2 for x to get 4 + y2 = 9. Solving for y we get y2 = 9 – 4 = 5, so y2 = 5. To solve for y, we take the square root of both sides of the equation to get 5.y We have two answers for y: 5and 5. Since we do not have one unique output value y for our input value of 2 for x, the equation does not represent a function. The table shows the number of employees needed at Paws Doggy Day Care for the number of dogs being cared for. In question 2, is the number of dogs a function of the number of employees? This means for each number of employees, is there one unique value of the number of dogs being cared for? Look at the numbers. For 9 employees present, there are both 89 and 87 number of dogs being cared for. For 7 employees present, there are both 65 and 68 dogs being cared for. In question 3, is the number of employees a function of the number of dogs? Look at the numbers from the other direction. For 94 dogs being cared for, there are both 10 and 12 employees present. For question 4, is the number of dogs a function of the day? The days do not repeat, so each input and corresponding output is unique and therefore a function.”

Independent Practice: “Apply the same line of thinking for the practice problems as we did in the example problems.”


Closure: “Today you were given equations and scenarios, and you determined if each was a function.”

Answers: 1. Function (substitute x = 2) 2

2 2

42

y x

y

yy

2. Function (substitute x = 2) 2

2

22 24 26

y xyyy

3. Not a function (substitute x = 5) 2

2

2

25 23

3 (Two different outputs for one input)

x yyy

y

4. Function (substitute x = 5) 2

5 23

x yy

y

5. Yes, the score is a function of the number

correct. For each number correct, there is a unique score.

6. No, the number of customers is not a function of the number of cars. There is a repeat of 40 for the number of cars on two different days and the number of customers is different for the 40 cars (40 and 53).

7. Yes, the number of cars is a function of the number of customers. There is not a repeat of the number of customers.


Lesson Objective: Students will understand the difference between a relation and a function and identify a function.

Introduction: “Today you will distinguish between a relation and a function.”

Instruction: “A relation is a set of ordered pairs. A function is a subset or special type of relation. For a set of ordered pairs to be a function, they must follow a rule. A function is a relation where the value of one variable depends on the value of the other variable. In a function, each input (x-coordinate) may be paired with exactly one output (y-coordinate). The y-coordinate value is dependent on the x-coordinate value. In a function any x-value number can be substituted into the equation and yield exactly one y-value out of the equation. Given a set of ordered pairs, a table of x- and y-values, an equation, or a graph, you should be able to determine if the given is a function or just a relation.”

Guided Practice: “Today we will only look at a set of coordinate pairs and tables of x-yvalues. Let’s complete the example together. We are to determine if the relation is a function and explain. The ordered pairs are: ( 2, 5),(7, 2),(3, 3),(8, 5),(7, 8) . Is this a function or not? It’s not a function because for the same x-value of 7, there are two different y-values: a -2 and an 8. Remember, each x-value in a pair can only be paired with exactly one unique y-value.”

Independent Practice: “It’s your turn to determine if the set of ordered pairs or the table of x-y values represent a function or not. You must explain your reasoning in writing.”


Closure: “Think about what you practiced today. Using your own words, tell your elbow partner what a function is.” After a minute, call on volunteers to share their definition out loud to the class. Answers: 1. Function. Even though the y-value repeats, the unique x-values are

paired with their own y-value. 2. Not a function. -2 has two different y-value outputs: 0 and 5 3. Not a function. 3 has 5 different y-value outputs. 4. Function. Each x-value is paired with it’s own unique y-value 5. Answers will vary. The x-value should be -5, 4, or -8 with a different y-

value that is already given in the table.

Lesson Set at a Glance

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Common Core Standards Plus® – Mathematics – Grade 8Domain: Functions Focus: Defining Functions

Evaluation: #1

Directions: Complete the following problems independently. Determine if the graphs are functions. Justify your answer.

1.

2.

3.

Functions Assessment Sample



1 2 3 4 1 +

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Performance Lesson - Teacher Lesson Plan - Sample

Common Core Standards Plus® – Mathematics – Grade 8Performance Lesson #7 – Domain: Functions

Lesson Objective: The students will analyze and describe a functional relationship.

Overview: Students will use their knowledge of functional relationships as addressed in Common Core Standards Plus Functions Daily Lessons 17-24, E5-E6.

Students will:• Analyze a given graph to write a scenario that would result in the same graph.• Analyze and describe the functional relationship displayed in the graph.

Guided Practice: (Required Student Materials: St. Ed. Pg. 167)• Review vocabulary.• Review functional relationships.

Independent Practice: (Required Student Materials: St. Ed. Pg. 167)Have the students:

• Analyze a given graph.• Write a scenario that would result in the given graph.• Analyze and describe the functional relationship displayed in the graph.

Review & Evaluation:• Have students share their scenarios in small groups. • Share a few examples of the students’ scenarios with the class. • Discuss how there could be different scenarios that result in the same graph.

Standard Reference: 8.F.2: Compare properties of two functions each represented in a different way (algebraically, graphi-cally, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. 8.F.4: Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. 8.F.5: Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

Required Student Materials: • Student Pages: St. Ed. Pg. 167 (Student Worksheet) • Lined paper

St. Ed. Pg. 167


Teacher Lesson Plan

346


19

Performance Lesson - Student Page - Sample

Common Core Standards Plus® – Mathematics – Grade 8Performance Lesson #7 – Domain: Functions


Student Page

347

Vocabulary: Relation: A set of ordered pairs.

Ordered pairs: Corresponding numbers in a table that are used to locate a point on a coordinate plane.

Function: A subset or special type of relation; a function is a relation where the value of one variable depends on the value of the other variable. The y-‐coordinate value is dependent on the x-‐coordinate value. Any x-‐value can be substituted into the equation and yield exactly one y-‐value out of the equation.

Relation curve: The graph of a function.

Vertical line test: A relation is a function if no vertical lines intersect the graph at more than one point.

Linear function: The graph of a linear function results in a straight line; all the points described by the function lie on the same line.

Slope (Rate of Change): The measure of how steep a line is, or vertical change divided by horizontal change; slope may be positive, moving up from left to right; negative, moving downward from left to right; or zero, moving neither up nor down from left to right.

Coefficient: The number that is multiplied by the variable or an algebraic expression in an algebraic term.

Constant: A number without a variable; a value that does not change.

Directions: Write a scenario that could be shown using the graph below. Describe the functional relationship between the two quantities.

15

14

13

12

11

10

9

8

7

6

5

4

3

2

1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15


Common Core Standards Plus® – Mathematics – Grade 8Domain: Geometry Focus: Verifying Properties Lesson: #1Standard: 8.G.1: Verify experimentally the properties of rotations, reflections, and translations. 8.G.2: Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

Lesson Objective: Students will verify experimentally the properties of rotations, reflections, and translations on the length of line segments, the measure of angles, and parallelism.

Introduction: “Today we will investigate the properties of rotations, reflections, and translations on geometric figures.”

Instruction: “Today we will verify experimentally the properties of rotations, reflections, and translations on the length of line segments, the measure of angles, and parallelism.”

Guided Practice: “Let’s complete the example problem together. Go ahead and plot triangle ABC on the coordinate plane below as you see me do. (Model all graphing and transformations for the students. The completed graph is shown below.) Be sure to label the vertices and sides. Use your transparency to trace triangle ABC and label the transparency as well. Next we reflect the triangle over the line y = x which is already drawn for us. If we folded our paper along the line of reflection, the triangles would exactly line up. To reflect across the line y = x, our ordered pairs of (x, y) become (y, x) for each vertex. For example, the point (-4, -1) becomes (-1, -4). Go ahead and sketch in the reflected triangle and label the corresponding vertices as A’B’C’. Next we translate the triangle up two units. What do you notice as you move your transparency up two units? It overlaps exactly triangle DEF. Triangle ABC and triangle DEF are congruent. This means that the corresponding line segments of the sides of the triangle and the angle measures of the vertices are equal after a reflection and translation are applied. To be specific, the corresponding angles A and D, B and E, C and F are the congruent. Corresponding line segment lengths a and d, b and e, and c and f are also congruent.”

Independent Practice: “It’s your turn to complete the practice problems. By the end of the lesson, you will have verified experimentally the properties of various combinations of rotations, reflections, and translations on geometric figures.” You may ask the students to draw the intermediate transformation on the grid if time permits. They are shown in the answer section.

Review: When the students are finished, review the problems together.

Closure: “Today we investigated the properties of rotations, reflections, and translations on geometric figures. We saw that figures remain congruent when any combination of these three transformations are applied.”

Answers: 1. The graph shows the 90-degree counterclockwise rotation as triangle A’B’C’. Triangle DEF is the result of the translation of 6 units up and 2 units to the left. The lengths of line segments a and d, b and e, and c and f are congruent. Anglemeasures A and D, B and E, and C and F are also congruent.

2. The graph shows the reflection of trapezoid DEFG over the line x = 2 as trapezoid D’E’F’G’. Trapezoid WXYZ shows the final transformation of rotating 90 degrees clockwise around the origin.

, , , .DG WZ GF ZY FE YX ED XW Corresponding angles are congruent. Parallel line segments DE and FG remain parallel after the transformations are applied and are taken to parallel line segments WX and YZ.

Geometry - Teacher Lesson Plan - Sample


21


Example: On the coordinate plane below, plot and connect the points A(-4, 5), B(-1, 5), C(-4, -1) to form a triangle. Label the line segment across from angle A as a, across from angle B as b and across from angle C as c. Use tracing paper or a transparency to trace triangle ABC. Reflect triangle ABC over the line y = x and translate the reflected triangle 2 units up.

What do you notice about the line segments and angle measures of triangle ABC and triangle DEF? Be specific and precise in your explanation.

Directions: Complete the following problems. 1. On the coordinate plane below, plot and connect the points A(1, 2), B(6, 2), C(4, 6) to

form a triangle. Label the line segment across from angle A as a, across from angle B as b, and across from angle C as c. Use tracing paper or a transparency to trace the triangle. Rotate the triangle 90 degrees counterclockwise around the bottom left vertex, then translate the triangle six units up and two units to the left.

What do you notice about the lengths of the line segments and angle measures of the triangle you drew ABC and triangle DEF? Be specific and precise in your explanation.

Geometry - Student Page - Sample


Common Core Standards Plus® – Mathematics – Grade 8Domain: Geometry Focus: Showing Congruency Lesson: #2Standard: 8.G.2: Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibitsthe congruence between them. Lesson Objective: Students will show congruency through rotations, reflections, and translations.

Introduction: “Today we will show congruency through determining transformations to carry one figure to another.”

Instruction: “Today we will work with the three transformations of rotations, reflections, and translations. These three transformations carry a figure onto another. The three transformations of rotations, reflections, and translations also preserve distance and angle. Sometimes, more than one transformation is required to show congruency between two figures. You will determine what those transformations could be to show congruency.”

Guided Practice: “Let’s complete the example together. Rectangles 1 and 2 are congruent. We need to show that the rectangles are congruent by stating a series of translations followed by a rotation that maps one rectangle to the other. Let’s transform rectangle 1 to rectangle 2. Label the bottom right vertex of rectangle 1 as A and the corresponding vertex of rectangle 2 as A’. To map rectangle 1 to 2, first translate rectangle 1 five units to the right then three units up. We can draw the translated rectangle and label the angle between the corresponding vertices with a symbol. This shows we need to rotate rectangle 1 clockwise by the same angle to map to rectangle 2. When we started, notice that the rectangles vertices were not labeled for us. The angle was not labeled either. We don’t know what the angle is, but that doesn’t mean we can’t talk about the angle. We define unknowns with symbols or letters in algebra. Labeling figures gives you a specific way to talk about the figure. (The rectangles and angle are not labeled, which gives students the opportunity to label as they see fit to be able to describe the transformations.) The second question asks if it is possible to show congruence with only one reflection. What do you think? What did you find? You can’t show congruence with only one reflection. The line of reflection (shown as a dashed line) would be the perpendicular bisector of the line segment between the points A and A’. Reflecting either one of the rectangles across the line, would not result in the other rectangle.”

Independent Practice: “It’s your turn to apply the same process to the practice problems.” You may provide transparencies or tracing paper for the students to use.


Closure: “Today you showed congruency through rotations, reflections, and translations.”

Answers: 1. Yes. Rotate 180 degrees clockwise or counterclockwise around the origin. 2. Yes. Reflect across the x-axis then the y-axis OR reflect across the y-axis then

the x-axis. 3. No. Translating moves the figure horizontally and vertically but does not change

the orientation. A reflection or rotation is needed to change the orientation.

Common Core Standards Plus® – Mathematics – Grade 8Domain: Geometry Focus: Mapping Figures Lesson: #3Standard: 8.G.2: Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibitsthe congruence between them. Lesson Objective: Students will describe a sequence of transformations that exhibit the congruence between figures.

Introduction: “Today you will be given congruent figures on a coordinate plane and you will describe one or more transformations to map one figure to the other.”

Instruction: “There is usually more than one sequence to describe the transformation of one figure to another. In today’s lesson, you will determine two different sequences for each transformation to exhibit the congruency between the figures. For each case, you will have to determine what one transformation will map one figure to another.”

Guided Practice: “Let’s complete the example together. We are given two congruent quadrilaterals on a coordinate plane. We need to map one of the quadrilaterals to the other using a sequence of two transformations. Any suggestions of what transformations we can use? We can use a reflection and translation. Let’s label the quadrilateral on the left as 1 and the one on the right as 2. We can translate quadrilateral 1, 14 units to the right then reflect the figure over the line x = 9. What is another sequence we can apply? We could reflect quadrilateral 1 over the y-axis and then translate 4 units to the right. There are numerous sequences that will map quadrilateral 1 to quadrilateral 2. Now we need to determine one transformation that will map quadrilateral 1 to quadrilateral 2. We know we will need to do a reflection. The line of reflection will always be the perpendicular bisector of the line segment between any two corresponding points of the figures. In this case, the line of reflection will be the line x = 2.”

Independent Practice: “It’s your turn to apply the same process to the practice problems. There can be more than one correct answer.” You can have the students exchange their sequences with a classmate to check each other’s work.


Closure: “Today you described a transformation or a sequence of transformations that mapped one figure to another. You realized that there could be many correct answers.” Answers: 1. Answers will vary. Students can label the quadrilaterals as 1 and 2. An

example is reflect quadrilateral 1 over the line x = 9 then reflect over the line y = 2.

2. Rotate clockwise 180˚ around the point (9, 2).

Common Core Standards Plus® – Mathematics – Grade 8Domain: Geometry Focus: Mapping Figures Lesson: #4Standard: 8.G.2: Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

Lesson Objective: Students will describe a sequence of transformations that exhibit the congruence between figures.

Introduction: “Today you will be given congruent figures on a coordinate plane and you will describe a sequence of transformations to map one figure to the other.”

Instruction: “There is usually more than one sequence to describe the transformation of one figure to another. In today’s lesson, you will determine different sequences for each transformation to exhibit the congruency between the figures.”

Guided Practice: “Let’s complete the example together. We need to describe the sequence of transformations that exhibits the congruence between the figures shown on the coordinate plane below. It appears that the triangle is translated and rotated. We will start with the top triangle. Let’s label the vertices as EFG, so we can describe the process precisely. Let’s translate triangle EFG so that vertex F is at the same location of the corresponding vertex of the second figure. To do this we need to translate triangle EFG 9 units down and 5 units to the right. Write this sequence and draw triangle E’F’G’. Finally, we need to rotate the triangle clockwise. Let’s label the other triangle as E”F”G”. We use double prime to represent the same triangle that is

transformed again. If possible, we can give the degree of rotation. If it is not simple to determine the degree of rotation, then we can define the angle with a symbol such as theta. In our case, it is not hard to determine the angle of rotation. Notice that line segment E”F” goes through the points (0, 0) and (7, 7). This means the line segment has a slope of -1. A slope of 1 or -1 is a 45 degree angle. Line segment E’F’ is horizontal which is a 0 slope. To rotate the line segment E’F’, and thus the triangle, from a zero slope to a -1 slope, we need to rotate the triangle 45 degrees clockwise. We have just mapped triangle EFG to triangle E”F”G” with a sequence of translations and a rotation.”

Independent Practice: “It’s your turn to apply the same process on the practice problem. Be sure to notice that the corresponding vertices of the transformed figures are labeled. This is important to write an accurate transformation. There are many correct answers.” You can have the students exchange their sequences with classmates.


Closure: “Today you described a transformation or a sequence of transformations that mapped one figure to another. You confirmed that there could be many correct answers.”

Answers: Answers will vary. Possible answers: 1. Rotate clockwise 180 degree around

the origin. Reflect across the x-axis then reflect across the y-axis Reflect across the x-axis then translate 10 units to the right.

2. Rotate 90 degree clockwise around (-5, 5). Translate 3 units left and 3 units up, then rotate 90 degrees clockwise around point (-8, 5).


Lesson Objective: Students will verify experimentally the properties of rotations, reflections, and translations on the length of line segments, the measure of angles, and parallelism.

Introduction: “Today we will investigate the properties of rotations, reflections, and translations on geometric figures.”

Instruction: “Today we will verify experimentally the properties of rotations, reflections, and translations on the length of line segments, the measure of angles, and parallelism.”

Guided Practice: “Let’s complete the example problem together. Go ahead and plot triangle ABC on the coordinate plane below as you see me do. (Model all graphing and transformations for the students. The completed graph is shown below.) Be sure to label the vertices and sides. Use your transparency to trace triangle ABC and label the transparency as well. Next we reflect the triangle over the line y = x which is already drawn for us. If we folded our paper along the line of reflection, the triangles would exactly line up. To reflect across the line y = x, our ordered pairs of (x, y) become (y, x) for each vertex. For example, the point (-4, -1) becomes (-1, -4). Go ahead and sketch in the reflected triangle and label the corresponding vertices as A’B’C’. Next we translate the triangle up two units. What do you notice as you move your transparency up two units? It overlaps exactly triangle DEF. Triangle ABC and triangle DEF are congruent. This means that the corresponding line segments of the sides of the triangle and the angle measures of the vertices are equal after a reflection and translation are applied. To be specific, the corresponding angles A and D, B and E, C and F are the congruent. Corresponding line segment lengths a and d, b and e, and c and f are also congruent.”

Independent Practice: “It’s your turn to complete the practice problems. By the end of the lesson, you will have verified experimentally the properties of various combinations of rotations, reflections, and translations on geometric figures.” You may ask the students to draw the intermediate transformation on the grid if time permits. They are shown in the answer section.


Closure: “Today we investigated the properties of rotations, reflections, and translations on geometric figures. We saw that figures remain congruent when any combination of these three transformations are applied.”

Answers: 1. The graph shows the 90-degree counterclockwise rotation as triangle A’B’C’. Triangle DEF is the result of the translation of 6 units up and 2 units to the left. The lengths of line segments a and d, b and e, and c and f are congruent. Anglemeasures A and D, B and E, and C and F are also congruent.

2. The graph shows the reflection of trapezoid DEFG over the line x = 2 as trapezoid D’E’F’G’. Trapezoid WXYZ shows the final transformation of rotating 90 degrees clockwise around the origin.

, , , .DG WZ GF ZY FE YX ED XW Corresponding angles are congruent. Parallel line segments DE and FG remain parallel after the transformations are applied and are taken to parallel line segments WX and YZ.

Lesson Set at a Glance

Lesson Sets build student competence



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Common Core Standards Plus® – Mathematics – Grade 8Domain: Geometry Focus: Map One Figure to Another Using Rotations, Reflections, and Translations

Evaluation: #1

Directions: Complete the following problems independently using the congruent figures on the coordinate plane shown below.

1. Describe a transformation to map figure 1 to figure 2.



Geometry Assessment Sample



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