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I. UNIT OVERVIEW & PURPOSE:

This unit specifically addresses the concept of transformational graphing. Absolute value, polynomial, and square root functions will be examined.

II. UNIT AUTHOR:

Donna Deplazes

Craig County Public Schools

III. COURSE:

Mathematical Modeling: Capstone Course

IV. CONTENT STRAND:

Algebra

V. OBJECTIVES:

To recognize and translate the graph of an absolute value function. To recognize and translate the graph of polynomial (specifically quadratic) functions. To establish a pattern for easy recognition of higher order polynomials. To recognize and translate the graph of square root function. To expose students to possible real world situations involving transformational graphing.

VI. MATHEMATICS PERFORMANCE EXPECTATION(s):

MPE.2: Collect and analyze data, determine the equation of the curve of best fit, make predictions, and solve real-world problems, using mathematical models. Mathematical models will include polynomial, exponential, and logarithmic functions. MPE.12: Transfer between and analyze multiple representations of functions, including algebraic formulas, graphs, tables, and words. Select and use appropriate representations for analysis, interpretation, and prediction. MPE.14: Recognize the general shape of function (absolute value, square root, cube root, rational, polynomial, exponential, and logarithmic) families and convert between graphic and symbolic forms of functions. A transformational approach to graphing will be employed. Graphing calculators will be used as a tool to investigate the shapes and behaviors or these functions.

VII. CONTENT:

This unit specifically addresses the topic of transformation graphing and general recognition of absolute value, polynomial (specifically quadratic), and exponential functions. All lessons are discussed in the context of a real world application.

VIII. REFERENCE/RESOURCE MATERIALS:

Graphing calculators will be required. Student Exploration Worksheets and Exit Slip Assessments will be needed for all three lessons. Computers to access GeoGebra would also be beneficial when presenting if you find or make your own animation applets to help students visualize the concepts.

IX. PRIMARY ASSESSMENT STRATEGIES:

Please see Exit Slip Assessments attached to each lesson. Upon completion of more functions a unit assessment on the transformation of functions will be given.

X. EVALUATION CRITERIA:

Students will complete a five question exit slip at the end of the lesson. Documents will be attached to each individual lesson. Students will earn a classwork grade out of 10 points - 5 of these points are from the answers to the exit slip, 2 points are for each group's participation in the class discussion, and 3 points are for each student's participation in their group.

XI. INSTRUCTIONAL TIME:

Three 90-minute class periods.

Strand

Algebra

Mathematical Objective(s)

Functions, Quadratic Functions, Transformational Graphing In WUiV leVVon VWuTenWV will TiVcuVV WranVformaWional grapUing by examining WUe conVWrucWion of VuVpenVion briTgeV. SWuTenWV will explore mulWiple repreVenWaWionV of quaTraWic funcWionV. TUey will Volve given problemV by WranVforming WUe grapU of a quaTraWic funcWion. TableVH grapUVH anT equaWionV will be uVeT by VWuTenWV Wo aiTe in finTing WUe neceVVary VoluWionV.

Mathematics Performance Expectation(s)

MPE.2: Collect and analyze data, determine the equation of the curve of best fit, make preTicWionVH anT Volve real-worlT problemVH uVing maWUemaWical moTelV. ÓaWUemaWical moTelV will incluTe polynomialH exponenWialH anT logariWUmic funcWionV. ÓPN.12J TranVfer beWween anT analyYe mulWiple repreVenWaWionV of funcWionVH incluTing algebraic formulaVH grapUVH WableVH anT worTV. SelecW anT uVe appropriaWe repreVenWaWionV for analyViVH inWerpreWaWionH anT preTicWion. ÓPN.14J RecogniYe WUe general VUape of funcWion (abVoluWe valueH Vquare rooWH cube rooWH raWionalH polynomialH exponenWialH anT logariWUmic) familieV anT converW beWween grapUic anT Vymbolic formV of funcWionV. A WranVformaWional approacU Wo grapUing will be employeT. GrapUing calculaWorV will be uVeT aV a Wool Wo inveVWigaWe WUe VUapeV anT beUaviorV of WUeVe funcWionV.

Related SOL

AII.6 TUe VWuTenW will recogniYe WUe general VUape of funcWion (quaTraWic) familieV anT will converW beWween grapUic anT Vymbolic formV of funcWionV. A WranVformaWional approacU Wo grapUing will be employeT. GrapUing calculaWorV will be uVeT aV a Wool Wo inveVWigaWe

WUe VUapeV anT beUaviorV of WUeVe funcWionV.

AFDA.3 TUe VWuTenW will collecW TaWa anT generaWe an equaWion for WUe curve (quaTraWic) of beVW fiW Wo moTel real-worlT problemV or applicaWionV. SWuTenWV will uVe WUe beVW fiW equaWion Wo inWerpolaWe funcWion valueVH make TeciVionVH anT juVWify concluVionV wiWU algebraic anTIor grapUical moTelV. AŃMA.4 TUe VWuTenW will WranVfer beWween anT analyYe mulWiple repreVenWaWionV of funcWionVH incluTing algebraic formulaVH grapUVH WableVH anT worTV. SWuTenWV will VelecW anT uVe appropriaWe repreVenWaWionV for analyViVH inWerpreWaWionH anT preTicWion. Please note that transformation graphing will be applied in all lessons within the unit. The type of funcWion aTTreVVeT will cUange Taily. IW iV inWenTeT for VWuTenWV Wo uVe baVic WranVformaWion grapUing WecUniqueV Wo Uelp WUem on a Tay-Wo-Tay baViV.

NCTÓ SWanTarTV

Understand relations and functions and select, convert flexibly among, and use various repreVenWaWionV for WUem. Interpret representations of functions of two variableV. Use symbolic algebra to represent and explain mathematical relationships.

ÓaWerialVIReVourceV

Classroom set of graphing calculators.

Students need to be familiar with the structure of suspension bridges. Below are a couple of WUe famouV American VuVpenVion briTgeV. o Golden Gate Bridge o Brooklyn Bridge

SmartBoard or LCD Projector

Building Bridges - SWuTenW NxploraWion PS #1

Building Bridges - SWuTenW NxploraWion PS #2

Building Bridges - NxiW Slip AVVeVVmenW

Assumption of Prior Knowledge

Students should have completed Algebra II.

Students should have experience using a graphing calculator, specifically finding a

Vpecific viewing winTow.

Students might find it difficult to realiYe WUaW UoriYonWal WranVformaWionV are repreVenWeT ͞reǀersely" from graphic to symbolic representation. The relevant real life context in this problem iV relaWeT Wo WUe conVWrucWion of a

VuVpenVion briTge.

Introduction: Setting Up the Mathematical Task

In this lesson, students will investigate how vertical and horizontal translations affect the

Vymbolic repreVenWaWion of a quaTraWic funcWion.

InWroTucWion - 20 minuWeV

SWuTenW NxploraWion #1 - 20 minuWeV

MiVcuVVion of SN #1 - 10 minuWeV

SWuTenW NxploraWion #2 -20 minutes

MiVcuVVion of SN #2 - 10 minuWeV

CP AVVignmenWJ NxiW Slip AVVeVVmenW - 10 minuWeV

Have students discuss the similarities and differenceV beWween WUe grapUV of abVoluWe value funcWionV anT quaTraWic funcWionV. SWuTenWV will be reminTeT abouW WUe VymmeWric cUaracWeriVWicV of WUe grapU of a quaTraWic funcWion. IW iV imporWanW Wo noWe WUaW only WUree poinWV are neeTeT Wo moTel a quaTraWic funcWion. TUe TI-84+ will give a Vymbolic repreVenWaWion founT WUrougU regreVVion WUaW iV in VWanTarT form. HoweverH if you uVe WUe VymmeWry of WUe parabola you can finT WUe Vymbolic repreVenWaWion in verWex form. o First use the parent function ݕ funcWionV. o Then use ݕ Lt:T VymmeWry (cenWereT arounT WUe verWex) in WUe Wable WUaW repreVenWV WUe funcWion. o Have students find the quadratic regression using their calculaWorV anT WUeVe o Then have the VWuTenWV verify WUaW ݕ Lt:T sxT Eu{.

Students will work the exploration activities and then the class will discuss their conclusions.

NacU group will be aVkeT Wo anVwer TiVcuVVion queVWionV on eacU exploraWion workVUeeW anT commenW on WUe anVwerV proviTeT by oWUer groupV. All VWuTenWV will Uave WUe correcW anVwerV before WUe enT of claVV. Students are asked to explore transformational graphing of quadratic functionV in WUe Wwo exploraWion workVUeeWV aWWacUeT. The figure given in the exploration activities is designed to assist students who struggle to

͞picture" the actual situation.

Student Exploration 1:

Group Work (groups of 2 or 3)

SWuTenWITeacUer AcWionVJ

Students should use the information from the introduction to complete a worksheet UoriYonWal WranVlaWionV uVing WUe verWex form of a quaTraWic funcWion. Teacher will be guiTing VWuTenWV aV neeTeT if queVWionVIproblemV ariVeH buW will noW anVwer

WUe queVWionV for WUe VWuTenWV.

Explore Learning activity gizmo allows students to change ܽ cUangeV aV well. TUiV webViWe iV noW free - a VubVcripWion iV requireT. You may view free for five 50

Monitoring Student Responses

o Students are expected to discuss the exploration activities togeWUer in WUeir groupV anT WUen TiVcuVV WUe queVWionV aV a claVV aW WUe enT of WUe acWiviWy. o The teacher will aVViVW VWuTenWV wUo Uave TifficulWieV anT exWenT WUe maWerial (aTT a VWep of TifficulWy) for VWuTenWV WUaW are reaTy Wo move forwarT.

Student Exploration 2:

Group Work (groups of 2 or 3)

SWuTenWITeacUer AcWionVJ

Students should use the information from the introduction and the discussion of the first WUaW aTTreVVeV UoriYonWal anT verWical WranVlaWionV aV well aV VWreWcUeV anT compreVVionV of a given quaTraWic funcWion. Teacher will be guiding students as needed if questions/problems arise, but will not answer

WUe queVWionV for WUe VWuTenWV.

o Students sUoulT realiYe WUaW WUey only neeT Wo cUange WUe locaWion of WUe Wwo WowerV (leaving WUe TiVWance beWween WUe Wwo WowerV anT WUe UeigUW of WUe

WowerV WUe Vame).

o Students should realize that they need to change the height of the towers and WUe UeigUW of WUe cable above WUe briTge Vurface Wo form a verWical WranVlaWion. o Changing the distance between the towers or just the height of the towers woulT reVulW in a VWreWcU or compreVVion. Nxplore Learning acWiviWy giYmo allowV VWuTenWV Wo cUange ܽ cUangeV aV well. TUiV webViWe iV noW free - a VubVcripWion iV requireT. You may view free for five minuWeV.

Monitoring Student Responses

o Students are expected to discuss the exploraWion acWiviWieV WogeWUer in WUeir groupV anT WUen TiVcuVV WUe queVWionV aV a claVV aW WUe enT of WUe acWiviWy. o At the end of the second exploration students are asked to make generalizations abouW Uow UoriYonWal anT verWical VUifWV (WranVlaWionV) affecW WUe funcWion rule. o The teacher will aVViVW VWuTenWV wUo Uave TifficulWieV anT exWenT WUe maWerial (aTT a VWep of TifficulWy) for VWuTenWV WUaW are reaTy Wo move forwarT.

Assessment

Students will complete a five question exit slip at the end of the lesson. PleaVe Vee aWWacUeT TocumenW. SWuTenWV will earn a CP graTe ouW of 10 poinWV - 5 of WUeVe poinWV are from WUe

answers to the eǆit slip, 2 points are for each group's participation in the class discussion, and 3

Extensions and Connections (for all students)

The concept of vertical and horizontal stretching and compressing was brought up in the last exploraWion queVWion anT will be TiVcuVVeT in a claVVroom VeWWing. TUe Wopic of WranVformaWional

Strategies for Differentiation

For ELL learners, teachers should work with the ELL teacher to provide bridges between mathematics ǀocabulary and the student's primary language. Learning disabled students may benefit if the teacher provides multiple choice anVwerV Wo

WUe VWuTenW exploraWionV.

Visual learners will benefit from the graphical representations and the ability to dynamic exploraWion alloweT wiWUin Nxplore Learning aV well aV WUe grapUing calculaWor. Auditory learners will benefit from the classroom anT group TiVcuVVionV. Kinesthetic learners will benefit from movement from individual work to group work and

WUe involvemenW in claVVroom preVenWaWion.

High ability students may start to begin to compare groups for similarities or differences anT offer opinions to lead into tomorrow's lessons. TUe figure VUowV a VuVpenVion briTge. Pe are going Wo finT WUe quaTraWic funcWion Wo VymboliYe WUe paWU of WUe VupporW cable beWween WUe Wwo WowerV WUaW are 100 feeW Wall. TUere iV a TiVWance of 400 feeW beWween WUe Wwo WowerV. TUe cable reacUeV iWV loweVW poinW aW WUe miTTle of the span at a height of 15 feet aboǀe the bridge's surface. Below is a scale Trawing on a coorTinaWe plane of wUaW WUe ViWuaWion lookV like WUaW VUoulT Uelp you.

ViWuaWion Wo anVwer WUe following queVWionV.

1. What number or numbers in the siWuaWion woulT you cUange Wo move WUe quaTraWic

funcWion UoriYonWally only (no VWreWcUing or compreVVing WUe acWual VUape)? Nxplain wUy or Uow your cUange(V) woulT move WUe funcWion UoriYonWally.

2. What number or numbers in the situation would you change Wo move WUe quaTraWic

funcWion verWically only (no VWreWcUing or compreVVing WUe acWual VUape)? Nxplain wUy or Uow your cUange(V) woulT move WUe funcWion UoriYonWally.

3. What number or numberV in WUe ViWuaWion woulT you cUange Wo VWreWcU WUe funcWion

UoriYonWally? Nxplain wUy or Uow your cUange(V) woulT VWreWcU WUe funcWion UoriYonWally.

4. What number or numberV in WUe ViWuaWion woulT you cUange Wo compreVV WUe funcWion

UoriYonWally? Nxplain wUy or Uow your cUange(V) woulT compreVV WUe funcWion

UoriYonWally.

5. What number or numberV in WUe ViWuaWion woulT you cUange Wo VWreWcU WUe funcWion

verWically? Nxplain wUy or Uow your cUange(V) woulT VWreWcU WUe funcWion verWically.

6. What number or numberV in WUe ViWuaWion woulT you cUange Wo compreVV WUe funcWion

verWically? Nxplain wUy or Uow your cUange(V) woulT compreVV WUe funcWion verWically.

1. Draw a graph on a coorTinaWe plane Wo VymboliYe WUe paWU of WUe VupporW cable of a

VuVpenVion briTge. TUe UeigUW of WUe Wwo WowerV iV 80 feeW. TUere iV a TiVWance of 350 feeW beWween WUe Wwo WowerV. TUe cable reacUeV iWV loweVW poinW aW WUe miTTle of WUe Vpan aW a UeigUW of 22 feet aboǀe the bridge's surface. Be sure to label and diǀide your axis using an appropriate scale. Give the coordinates of three keys points of interest on

WUe grapU.

2. Using the three points iTenWifieT in WUe grapU uVe your grapUing calculaWor Wo finT WUe

quaTraWic regreVVion Wo repreVenW WUe ViWuaWion.

3. Now using the symmetry of the graph, develop the quadratic function that represents

WUe ViWuaWion in verWex form.

x y

4. Verify that the regreVVion equaWion in queVWion 2 iV equivalenW Wo WUe funcWion you founT

in queVWion 3.

5. Discuss how this quadratic function has been transformed (horizontally or vertically

WranVlaWeTH compreVVeT or VWreWcUeT) compareT Wo WUe quaTraWic funcWion from exploraWion acWiviWy #1 - ݕ

Lଵ଻

quotesdbs_dbs6.pdfusesText_11

[PDF] [PDF] Transformational Graphing in the Real World

I. UNIT OVERVIEW & PURPOSE:

II. UNIT AUTHOR:

Donna Deplazes

Craig County Public Schools

III. COURSE:

Mathematical Modeling: Capstone Course

IV. CONTENT STRAND:

Algebra

V. OBJECTIVES:

VI. MATHEMATICS PERFORMANCE EXPECTATION(s):

VII. CONTENT:

VIII. REFERENCE/RESOURCE MATERIALS:

IX. PRIMARY ASSESSMENT STRATEGIES:

X. EVALUATION CRITERIA:

XI. INSTRUCTIONAL TIME:

Three 90-minute class periods.

Strand

Algebra

Mathematical Objective(s)

Mathematics Performance Expectation(s)

Related SOL

WUe VUapeV anT beUaviorV of WUeVe funcWionV.

NCTÓ SWanTarTV

ÓaWerialVIReVourceV

 Classroom set of graphing calculators.

 SmartBoard or LCD Projector

 Building Bridges - SWuTenW NxploraWion PS #1

 Building Bridges - SWuTenW NxploraWion PS #2

 Building Bridges - NxiW Slip AVVeVVmenW

Assumption of Prior Knowledge

 Students should have completed Algebra II.

Vpecific viewing winTow.

VuVpenVion briTge.

Introduction: Setting Up the Mathematical Task

Vymbolic repreVenWaWion of a quaTraWic funcWion.

InWroTucWion - 20 minuWeV

SWuTenW NxploraWion #1 - 20 minuWeV

MiVcuVVion of SN #1 - 10 minuWeV

SWuTenW NxploraWion #2 -20 minutes

MiVcuVVion of SN #2 - 10 minuWeV

CP AVVignmenWJ NxiW Slip AVVeVVmenW - 10 minuWeV

͞picture" the actual situation.

Student Exploration 1:

Group Work (groups of 2 or 3)

SWuTenWITeacUer AcWionVJ

WUe queVWionV for WUe VWuTenWV.

Monitoring Student Responses

Student Exploration 2:

Group Work (groups of 2 or 3)

SWuTenWITeacUer AcWionVJ

WUe queVWionV for WUe VWuTenWV.

WowerV WUe Vame).

Monitoring Student Responses

Assessment

Extensions and Connections (for all students)

Strategies for Differentiation

WUe VWuTenW exploraWionV.

WUe involvemenW in claVVroom preVenWaWion.

ViWuaWion Wo anVwer WUe following queVWionV.

1. What number or numbers in the siWuaWion woulT you cUange Wo move WUe quaTraWic

2. What number or numbers in the situation would you change Wo move WUe quaTraWic

3. What number or numberV in WUe ViWuaWion woulT you cUange Wo VWreWcU WUe funcWion

4. What number or numberV in WUe ViWuaWion woulT you cUange Wo compreVV WUe funcWion

UoriYonWally.

5. What number or numberV in WUe ViWuaWion woulT you cUange Wo VWreWcU WUe funcWion

6. What number or numberV in WUe ViWuaWion woulT you cUange Wo compreVV WUe funcWion

1. Draw a graph on a coorTinaWe plane Wo VymboliYe WUe paWU of WUe VupporW cable of a

WUe grapU.

2. Using the three points iTenWifieT in WUe grapU uVe your grapUing calculaWor Wo finT WUe

3. Now using the symmetry of the graph, develop the quadratic function that represents

WUe ViWuaWion in verWex form.

4. Verify that the regreVVion equaWion in queVWion 2 iV equivalenW Wo WUe funcWion you founT

5. Discuss how this quadratic function has been transformed (horizontally or vertically

Lଵ଻

Classroom set of graphing calculators.

SmartBoard or LCD Projector

Building Bridges - SWuTenW NxploraWion PS #1

Building Bridges - SWuTenW NxploraWion PS #2

Building Bridges - NxiW Slip AVVeVVmenW

Students should have completed Algebra II.