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Differentiation in Mathematics

We know maximum learning occurs at a sweet spot between the overly simple and exceedingly complex, that our students’ natural curiosities are fragile, and that while content prompts interest, curiosity is only maintained under the right cognitive conditions. We know that what is developmentally appropriate is largely contingent on prior learning experiences, and that within diverse international school math classrooms, the teaching challenge can feel daunting.

During this packed experiential workshop, we’ll explore how teachers of mathematics, from elementary through high school, can support the engagement and growth of all students in mixed readiness math classrooms.

Our learning will be guided by the following questions:

• What does it mean to do and learn mathematics?

• What are specific needs of struggling and advanced math learners?

• With these needs in mind, what strategies can we employ to facilitate productive learning experiences for all students?

• How can we assess and report in ways that are compatible with the principles we espouse and supportive of the practices we employ?

We’ll explore these questions through examples and simulations of classroom learning.

Outline:

Session 1 The nature of math learning and the differing needs of diverse learners. Session 2 Characteristics of high quality math instruction Session 3 Tiered instruction and assessment Session 4 Assessment, grading, and reporting in a tiered math class

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From the introduction of How to Teach Now by Bill and Ochan Powell

What Do Teachers Need to Do to Personalize Learning?

In our experience, to effectively personalize learning, teachers need to engage in five ongoing inquiries. We must work to know our students as learners, know ourselves as teachers, know our curriculum, know our assessments, and know our collegial relationships. Knowing our students as learners entails systematically and deliberately exploring our students' cultural identities, linguistic backgrounds, family circumstances, learning styles, intelligence preferences, readiness levels, interests, and many other individual learning traits and then using that information to address specific needs by providing meaningful and appropriately challenging work. Knowing ourselves as teachers includes probing our own cultural biases and assumptions, discovering our preferences in learning style that may have translated into our preferred and dominant teaching style, and recognizing submerged beliefs and expectations that we have about children in general or about students specifically—all of which should help us to more clearly understand and serve our students. Knowing our curriculum at a conceptual level means being able to discriminate between content and transferrable concepts. Concepts are overarching and applicable to many areas of specific content, offering flexibility in choosing access points for students with a variety of cultural backgrounds and learning preferences. Knowing our assessments encompasses selecting and designing tools to match the learning objectives we want to measure, offering students some choice in assessment in order to increase engagement, and bringing students inside the assessment process so that they become the end users of assessment data. Knowing our collegial relationships involves enlisting the help of other professionals with different experiences, backgrounds, skills, and perspectives to support us in planning how to best serve the diverse needs of our students. Education today is a most complex field. As such, it is absurd and counterproductive for teachers to "go it alone." The most enlightened schools are promoting coplanning, coteaching, and the collective analysis of student work. Pursuit of advanced knowledge in all five domains of personalized learning is critical to success. Teachers can fall into focusing on one or two domains, which will limit the effectiveness of instruction. We have created Figure A to show how only exploration of all five domains results in the ability to personalize learning.

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Figure A. Relationship Between Teacher Knowledge and Personalized Learning

Complex Knowledge of Students, Self, and Collegial Relationships

Beginning Knowledge of

Curriculum and Assessment

"Relationship-Oriented Teacher"

• Creates trusting classroom climate.

• Demonstrates extensive empathy.

• Possesses excellent interpersonal skills.

• Does not always identify meaningful or appropriate learning objectives.

• Lesson outcomes ill-defined.

• Assessment may not match learning targets.

• Often drifts off point when teaching.

• Questioning is superficial. • Does not foster complex

thinking.

"Personalizing Teacher" • Creates sense of class

belonging. • Frames meaningful

learning goals. • Matches methodology to

student readiness levels, interests, and learning profiles.

• Uses wide repertoire of instructional strategies and assessments.

• Taps intrinsic student motivation.

• Sets expectation of internal student locus of control and responsibility. Advanced

Knowledge of Curriculum and

Assessment

"Beginning Teacher" • May have subject mastery. • Shows great enthusiasm

and interest. • Possesses limited

knowledge of instructional strategies and assessments.

• Has not yet developed collaborative skills.

• Needs opportunities to develop in all domains.

"Task-Oriented Teacher" • Has content area

expertise/ subject mastery.

• Very effective with highly motivated and capable students.

• Teaches only in own preferred style.

• Does not respond to individual learning needs.

• Does not address varying motivational levels.

• Uses traditional assessments.

Limited Knowledge of Students, Self, and Collegial Relationships Personalized learning does not mean the teacher creates a separate lesson plan for every student. It does, however, presume that the teacher ensures enough flexibility of instruction, activities, and assessment to enable a diverse group of learners to find a good fit most of the time (Tomlinson & Allan, 2000)… Nothing in the classroom can be so rigid that it cannot be adapted to facilitate greater learning. In other words, instructional strategies, use of time, use of materials, approach to content, the grouping of students, and the means of assessment all need to be flexible. The teacher is the architect of that flexibility. ------------------------------------------------------------------------------------------------------------------------------------------ According to Carol Ann Tomlinson differentiated instruction speaks to determining what is best for each learner but within the context of expected learning outcomes. Students still do not necessarily determine these outcomes for themselves. Personalized learning, on the other hand, provides for the possibility that individual learners will meet different and unique outcomes.

(from Learn 231 16 Nov 2011)

TextText

555

SenseMakingClassroomCultures

Movingfrom…to....

Mathematicsisacollectionofproceduresè Mathematicsisawayofthinking

Workingwiththeinexplicableè Workingwiththingsthatmakesense

Significanceofmateriallostonlearner è Materialsignificanttolearner

Studentispassive è Studentisactive

Validatedbyteacher è Validatedbystudent

Truthisaspresented è Truthisasconstructed

Teacherowned è Student-owned

Described/explainedinteacherlanguage è Described/explainedinstudentlanguage

Oftenforgotten/notretrievable è Remembered/retrievable

Popsintoexistence è Growsintobeing

Ignoresstudentreadiness è Considersstudentreadiness

Non-experiential è Experiential

Presentedatbeginningoflesson è Developedatendoflesson

Relianceonmemoryaids è Minimalrelianceonmemoryaids

Isolatedandsuperficial è Connectedandthorough

Followprocedures è Developprocedures

Anxiousaboutmathematics è Senseofpersonalefficacyandconfidence

Deadenthemindandspirit è Enliventhemindandspirit

FromAHandbookonRichLearningTasks:FlewellingandHigginson,2001

666

Response to Intervention in Mathematics

ElementaryandMiddleSchoolMathematics:TeachingDevelopmentally,VandeWalle,Pearson

777

ElementsofHighQualityMathTeachingandLearning1

Dimension TeacherBehaviors

1.Discourse

Buildstowardsasharedunderstandingofmathematicalideas.

Acknowledgesmultipleapproaches/solutionsandhighlightsthemforstudents

Ensuresprogresstowardmathematicalgoalsbymakingexplicitconnectionstostudentapproachesandreasoning

Circulatestheroomduringstudentworktimetomonitorandassessprogress

Plansforpossiblestudentarticulationoftheconcept

2.Questioning

Questionsdrivetowardsgreaterstudentunderstandingthroughadvancingandprobingthinking.

Plansforandasksquestionsthatpromoteorassessstudentsunderstanding,extensionofthinking,and/orprobingofstudentthinkingbutdonottakeoverorfunnelstudentthinking

Allowsforsufficientwaittimesothatmorestudentscanformulateandofferresponses

Canexplainthepurposeforeachquestionasked

3.MakingThinkingVisible

Thelessonrequiresstudentstoshowdeepunderstandingofthecontent.

Carefullyplansquestionsthatwillbeansweredindependentlyinordertogatherdataaboutstudentprogress

Makesinthemomentdecisionsonhowtorespondtostudentswithpromptsthatprobe,scaffold,andextendstudentthinking

Elicitsandgathersevidenceofstudentunderstandingatstrategicpointsduringinstruction

888

4.ProductiveStruggle

Studentsgrapplewithmathematicalideasandrelationships.

Plansforandanticipateswhatstudentsmightstrugglewithduringalessonandispreparedtosupportthemproductivelythroughthestruggle

Givesstudentsadequateuninterruptedtimetostruggleandasksquestionsthatscaffoldstudentthinkingwithoutsteppingintodotheworkforthem

Reiteratesthaterrorsandconfusionarepartofthelearningprocessbyfacilitatingdiscussionsonmistakes,misconceptions,andstruggles

Praisesstudentsfortheireffortsinmakingsenseofmathematicalideasandfortheirperseveranceinreasoningthroughproblems

5.Worthytasks2

Tasksmovestudentsforwardintheprogressionofconceptualunderstanding.

• Alignwithworthwhilemathematicscontent• Havearelevantorinterestingmathematicalorreal-worldcontext• Provideopportunitiesforstudentstodevelopanddemonstratemathematicalhabitsofmind• Requirestudentstodemonstrateunderstandingormaketheirthinkingvisibleandpromotediscourse• Allowfortheuseofdifferentrepresentations,approaches,andentrypoints• Involvestudentsinaninquiry-orientedorexploratoryapproach• Connectpreviousknowledgetonewlearning• Arecognitivelydemanding

1.ThistableisbasedonPrinciplestoActions:EnsuringMathematicalSuccessforAllpublishedbytheNationalCouncilofTeachersforMathematics,2014andtheKIPPMathematicalUnderstandingEvidenceGuide.

2.Thislistisaguide,notachecklist.Aworthytaskmaynothaveallofthesecharacteristics.

999

5/1/17

1

3a.i.Howmanysquareswilltherebensecondslater?Labelasketchthatshowswhyyourexpression makessense.

1secondlater

2secondslater

3secondslater

ii.Howmanysecondswillittakeforthefigure tohave100squares?

3c.Thencreateaninterestingstorythatincorporatesyourpattern.Whatquestionmightbeaskedtoprovokea

problemsolver togeneralizethepattern?

3b.Createauniquepatternofyourownwhichisgrowingataconstantrate.Howmanysquareswillbeinthenth figure?

1010

Unit1PerformanceAssessment Name:_________________________________

TheVirusProblem Block:_________________________________Part1:First,answerthefollowingquestions.Show,explain,orinsomewaydemonstrateyourthoughtprocess.Writeortypethisupneatlyandcompletely.Or,ifyouprefer,youmaycreateavideoexplanationinstead.

All:1. Whenwillthevirushittheborder?2. Whichcolorwillhittheborder?3. Howbigwillthevirusbewhenithits?

AdditionalforBlueLevel:1. Whereexactlywillthevirushittheborder?

AdditionalforBlackLevel:1. Ifthepixelpatterngrowsforever,whatwillitsaspectratiobe?

Part2:Afteryouhavecheckedyouranswersinclass,youwillreflectonyourmathunderstanding.Thoughtfullyrespondtothequestionsbelowthatapplytoyou.Thiswillbeaddedtowhatyou'vealreadycompletedforpart1.NOTE:Youmayneedtoscan/takeapictureofyourworkifityoudiditbyhand,sothatitisavailabletoyouwhileyouworkonthisreflection.

• Ifyourperformanceassessmentanswerswerecorrect;

o howmightyouhavegoneaboutsolvingtheproblemsinamoreefficientoreffectiveway?

o howdidyoursolution(s)comparetothesolutionsyousawpresentedduringclass?Whatsimilaritiesanddifferencesdidyounotice?

• Ifyoudiscoveredthatyoumadeamistake;

o doyouknowwhereyourreasoningwentwrongandhowthaterroraffectedyoursolution?Explaininawaythatrevealsyournewlevelofunderstanding.

• Ifyoufeltextremelyconfusedandarrivedtoclasswithoutasolution;

o canyouidentifywhataspectoftheproblemoritssolutionconfusedyou?Explainyourthoughts,andalsoincludeanexplanationthatshowswhatyounowunderstandabouttheproblem(s).

PART1and2EVALUATION

SuccessfullysolvestheproblemAll1. Yes/No2. Yes/No3. Yes/NoAdd.Blue1. Yes/NoAdd.Black1. Yes/No

REASONING

Yourmethodisillogical,unclear,andshowslittleornounderstanding.

Yourmethodissomewhatlogical,accurate,andshowssomeunderstanding.

Yourmethodismostlylogical,accurate,andshowsconsiderableunderstanding.

Yourmethodisverylogical,accurate,andshowssophisticatedunderstanding.

COMMUNICATION

Yourexplanationisunclearwithlittleornodetails.

Yourexplanationissomewhatcleardetailedandprecise.

Yourexplanationismostlyclear,detailedandprecise.

Yourexplanationisveryclear,detailedandprecise.

Howmanyhintcardsdidyouuse:0 1 2 3 111111

Whatdoesitmeantobeveryclear?Thethinkingisveryeasyforthereadertounderstandandinterpret.Whatdoesitmeantobeverydetailed?Thereaderdoesn’tneedtoinferhowandwhydecisionsweremade.Whatdoesitmeantobeveryprecise?Thewritercarefullyusesvocabularytomakethewritingasconciseandmeaningfulaspossible.Part3:Reflectiononthislearningprocess.Thisperformanceassessmentgaveyouanopportunitytoemployahighlevelofresilienceandrelating.ReflectonhowyouappliedBOTHoftheselearningdispositionsinyourapproachtotheseproblems.First,giveexamplesandexplainhowyouusedeachdisposition.Then,explainhowyoumighttrytoimproveinthefuture.PART3EVALUATION

Yourreflectionisminimallythoughtful

andclear.

Yourreflectionissomewhatthoughtful

andclear.

Yourreflectionismostlythoughtfuland

clear.

Yourreflectionisverythoughtfulandclear.

Noexamplesprovidedfromtheperformanceassessmentexperience.

Itispartiallysupportedwith

examplesfromtheperformance

assessmentexperience.

Itismostlysupportedwithexamplesfromtheperformanceassessmentexperience.

Itishighlysupportedwithexamplesfromtheperformanceassessmentexperience.

LearningDispositions&Capacities

Resilient

isbeingready,willingandabletolockontolearning-knowinghowtoworkthroughdifficultieswhenthepressuremountsorthegoinggetstough.

Inquisitive:hasaquestioningandpositiveattitudetolearning

Focused:observant,concentrateswell,ignoresdistractions,becomesengrossed

Adventurous:willingtoriskand“haveago”.Upforanewchallenge.

Persistent:staysdetermined,positiveandpatientinthefaceofdifficultyormistakes

Relating

isbeingready,willingandabletolearnaloneorwithotherpeople-usingasenseofindependentjudgmenttogetherwithskillsincommunicationandempathy.

Independent:ableto“standtheirground”;showsinitiative.

Collaborative:agoodteam-player;helpsgroupstoworkwelltogether.

Empathic:understandsothers;offershelpfulfeedbackandsuggestions;receptiveandimitative.

OpenMinded:asksfor,listenstoandmakesgooduseofinformation,feedbackandadvice.

121212

HINTS

1. How long was the virus from side to side before any time passed by?

2. Each second, how much longer did the virus grow sideways?

3. At this rate, how long will it take the virus to reach the side borders?

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ElementsofHighQualityMathTeachingandLearning1

Dimension TeacherBehaviors StudentBehaviors

1.Discourse

Buildstowardsasharedunderstandingofmathematicalideas.

Acknowledgesmultipleapproaches/solutionsandhighlightsthemforstudents

Ensuresprogresstowardmathematicalgoalsbymakingexplicitconnectionstostudentapproachesandreasoning

Circulatestheroomduringstudentworktimetomonitorandassessprogress

Plansforpossiblestudentarticulationoftheconcept

Focusesonthewhybehindtheirapproachtothetaskorproblem,notthehow

Justifiesorexplainstheirthinking

Makesconjecturesorpredictions

Buildsuponandcritiquesthereasoningofothers

Activelylistenswhenothersarespeaking

2.Questioning

Questionsdrivetowardsgreaterstudentunderstandingthroughadvancingandprobingthinking.

Plansforandasksquestionsthatpromoteorassessstudentsunderstanding,extensionofthinking,and/orprobingofstudentthinkingbutdonottakeoverorfunnelstudentthinking

Allowsforsufficientwaittimesothatmorestudentscanformulateandofferresponses

Canexplainthepurposeforeachquestionasked

Expectstobeaskedtoexplain,clarify,orelaborateontheirthinking

Listensto,commentson,andquestionsthecontributionsoftheirclassmates

Thinkscarefullyabouthowtopresenttheirresponsestoquestionsclearly,withoutrushingtorespondtoquickly

3.MakingThinkingVisible

Thelessonrequiresstudentstoshowdeepunderstandingofthecontent.

Carefullyplansquestionsthatwillbeansweredindependentlyinordertogatherdataaboutstudentprogress

Makesinthemomentdecisionsonhowtorespondtostudentswithpromptsthatprobe,scaffold,andextendstudentthinking

Elicitsandgathersevidenceofstudentunderstandingatstrategicpointsduringinstruction

Sharesresponsesverballytoquestions(inpairsorwholegroup)onlyafterthey’vehadachancetoanswerthemindividuallysothatteacherhasanaccurategaugeofindividualprogress.

Reflectsonmistakesandmisconceptionstoimprovemathematicalunderstanding

Asksquestionsof,respondsto,andgivessuggestionstosupportthelearningoftheirclassmates

Assessesandmonitorstheirownprogresstowardsmathematicslearninggoalsandidentifiesareasinwhichtheyneedtoimprove

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4.ProductiveStruggle

Studentsgrapplewithmathematicalideasandrelationships.

Plansforandanticipateswhatstudentsmightstrugglewithduringalessonandispreparedtosupportthemproductivelythroughthestruggle

Givesstudentsadequateuninterruptedtimetostruggleandasksquestionsthatscaffoldstudentthinkingwithoutsteppingintodotheworkforthem

Reiteratesthaterrorsandconfusionarepartofthelearningprocessbyfacilitatingdiscussionsonmistakes,misconceptions,andstruggles

Praisesstudentsfortheireffortsinmakingsenseofmathematicalideasandfortheirperseveranceinreasoningthroughproblems

Self-regulatesandself-monitorstheirownthinkingthroughoutthedurationofthetask

Looksforpatternstogainadeeperunderstandingofthematerial

Looksformultiplesolutionstrategieswhenapproachingataskorproblem

Asksquestionsthatarerelatedtothesourceoftheirstrugglesandwhichwillhelpthemmakeprogressinunderstandingconceptsandsolvingtasks

Makessenseofproblemsandperseveresinsolvingthem

5.Worthytasks2

Tasksmovestudentsforwardintheprogressionofconceptualunderstanding.

• Alignwithworthwhilemathematicscontent• Havearelevantorinterestingmathematicalorreal-worldcontext• Provideopportunitiesforstudentstodevelopanddemonstratemathematicalhabitsofmind• Requirestudentstodemonstrateunderstandingormaketheirthinkingvisibleandpromotediscourse• Allowfortheuseofdifferentrepresentations,approaches,andentrypoints• Involvestudentsinaninquiry-orientedorexploratoryapproach• Connectpreviousknowledgetonewlearning• Arecognitivelydemanding

1.ThistableisbasedonPrinciplestoActions:EnsuringMathematicalSuccessforAllpublishedbytheNationalCouncilofTeachersforMathematics,2014andtheKIPPMathematicalUnderstandingEvidenceGuide.

2.Thislistisaguide,notachecklist.Aworthytaskmaynothaveallofthesecharacteristics.

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Teaching Math to Students with Low Working Memory

1. Teach(dyscalculic(students(by(explicitly(scaffolding(working(memory(processes.((

((((

4

Teaching students to add fractions with different denominators (Munro, 2003).

1. encode the task in working memory; 2. stimulate what students already know about this type of task; this provides the existing knowledge

base for encoding and representing the new ideas; 3. guide students to link the new task with ones they already can do; this assists them to focus on

the particular features of the new task; 4. guide students to encode the type of problem in working memory for later storage in long term

memory; 5. stimulate what students already know about key components of the new task; this assists them to

retrieve additional relevant aspects of their existing knowledge; 6. stimulate students to use the new links to complete a specific task; this assists them to encode a

particular example in context in working memory; 7. repeat with similar particular tasks; this guides students to encode the type of task in working

memory; 8. guide the students to identify and describe the new procedure and to practise applying it; this

guides students to encode the new procedure in working memory; 9. guide the students to identify when to use the new procedure this guides students to encode in

working memory the types of contexts in which they will use the new procedure; 10. guide the students to automatize what they know about how to add two fractions.

Focus of the teaching How teacher scaffolds student to use of the components of working memory:

Encode the task in working memory

scaffold students to interpret the task : Read the task and say what it says. Make a picture of what it says.

Stimulate what students already know about this type of task.

scaffold students to say what they know about the type of task : What types of fractions can you add? Can you add 2/6+ 3/6 =? Write down some other tasks you can add.

Guide students to link the new task with ones they can do.

scaffold students to say: How do these tasks differ from the one we are working on?

Guide students to encode the type of problem

scaffold students to ask: Can I make task I can’t do like the ones I can do ?

Stimulate what students already know about key components of the new task.

scaffold students to recall how they can say each fraction in other ways: What are other fractions that say the same as 1/3 ? Write down some other tasks you can add. Repeat for ½. What do these two sets of pictures show ? How could you use what they tell us ?

Stimulate students to use the new links to complete the particular task.

Scaffold students to see how they can use the alternative names for 1/2 and 1/3 to solve the task 1/2 + 1/3 =. Ask : Remember we want to find two fractions that are the same as 1/2 and 1/3. Can you find another fraction for 1/2 and 1/3 that have the same denominator ? Cue the students to rehearse the new links

Repeat with similar tasks. Scaffold students to use the procedure with similar tasks using the worked example as a model, for example, 2/3 + 1/4 =. Cue them to say what they will do before they begin using the earlier task. Scaffold each step.

Guide the students to identify and describe the new procedure and to

Cue the students to review the tasks they have completed and identify the procedure. Guide them to recognize key aspects of the procedure.

5

A teaching regime that includes explicit scaffolding of working memory processes across topics. As well as scaffolding students to meet the working memory demands while teaching a particular multistep procedure in mathematics, teachers can teach students to use working memory strategies across all topics in mathematics. Key teaching procedures to assist students who have mathematics learning difficulties to encode and manipulate their knowledge in working memory include the following. You can teach students to 1. stimulate explicitly what they already know about the task they will learn. Your

teaching can include the relevant mathematics procedures, concepts, the mathematics symbolism and the factual knowledge they will need to use. Students can learn to link the mathematics procedures you will teach with the procedures they already know.

2. say and paraphrase relevant mathematics information such as tasks, number sentences and mathematics ideas. This helps them to ‘read’ the number sentences into working memory. Have them ‘tell themselves’ what concrete patterns, pictures of mathematics ideas and mathematics actions show.

3. visualize mathematics ideas, for example, number sentences. Scaffold them to use

these strategies and give them time to do so. 4. learn new ideas first through 2 or 3 specific examples and then to extract the

procedure. Ask them to talk about what the three examples show.

5. learn each mental action, for example, adding or subtracting as physical actions first and gradually internalize them.

6. use a sequence of self-instructional strategies to guide their way through any task. When they are doing mathematics tasks, teach them to

say what a task says; visualize it; say what the solution will be like; categorise the task;; say what they will do first, second, …;; plan what they will do the task.

7. organise new mathematics ideas into categories. When they have learnt a new

mathematics idea, have them recognize instances of it, teach it as a category and teach them a name for the category.

practise applying it. Have them work through a set of practice tasks. Before they begin each task, ask them to say what they will do.

Guide the students to identify when to use the new procedure

Cue the students to note how these tasks differ from the ones they could already do. Guide them to give a name to the new type of task so that it is distinguished from the earlier type. Ask them to make up tasks that are not ready and ones that are ready. classify tasks. say what they know about the two types of tasks

Guide the students to automatize what they know about how to add two fractions

Cue students to decide rapidly whether an addition of fractions is ready to add and if so to work out the new denominator, numerators and add them.

161616

2. Teach(working(memory(strategies((As#well#as#scaffolding#students#to#meet#the#working#memory#demands#while#teaching#a#particular#multistep#procedure#in#mathematics,#teachers#can#teach#students#to#use#working#memory#strategies#across#all#topics#in#mathematics.##Teach students with math learning difficulties to:

(

(From(“The(Role(of(Working(Memory(in(Mathematics(Learning(and(Numeracy”(by(John(Munro,(University(of(Melbourne(

5

A teaching regime that includes explicit scaffolding of working memory processes across topics. As well as scaffolding students to meet the working memory demands while teaching a particular multistep procedure in mathematics, teachers can teach students to use working memory strategies across all topics in mathematics. Key teaching procedures to assist students who have mathematics learning difficulties to encode and manipulate their knowledge in working memory include the following. You can teach students to 1. stimulate explicitly what they already know about the task they will learn. Your

teaching can include the relevant mathematics procedures, concepts, the mathematics symbolism and the factual knowledge they will need to use. Students can learn to link the mathematics procedures you will teach with the procedures they already know.

2. say and paraphrase relevant mathematics information such as tasks, number sentences and mathematics ideas. This helps them to ‘read’ the number sentences into working memory. Have them ‘tell themselves’ what concrete patterns, pictures of mathematics ideas and mathematics actions show.

3. visualize mathematics ideas, for example, number sentences. Scaffold them to use

these strategies and give them time to do so. 4. learn new ideas first through 2 or 3 specific examples and then to extract the

procedure. Ask them to talk about what the three examples show.

5. learn each mental action, for example, adding or subtracting as physical actions first and gradually internalize them.

6. use a sequence of self-instructional strategies to guide their way through any task. When they are doing mathematics tasks, teach them to

say what a task says; visualize it; say what the solution will be like; categorise the task;; say what they will do first, second, …;; plan what they will do the task.

7. organise new mathematics ideas into categories. When they have learnt a new

mathematics idea, have them recognize instances of it, teach it as a category and teach them a name for the category.

practise applying it. Have them work through a set of practice tasks. Before they begin each task, ask them to say what they will do.

Guide the students to identify when to use the new procedure

Cue the students to note how these tasks differ from the ones they could already do. Guide them to give a name to the new type of task so that it is distinguished from the earlier type. Ask them to make up tasks that are not ready and ones that are ready. classify tasks. say what they know about the two types of tasks

Guide the students to automatize what they know about how to add two fractions

Cue students to decide rapidly whether an addition of fractions is ready to add and if so to work out the new denominator, numerators and add them.

6

8. review regularly how they learn the ideas and the thinking strategies they used. This helps them learn a repertoire of working memory rehearsal and transformational strategies that they can use on later occasions.

9. say what they have learnt when they have learnt a new mathematics idea. Ask them to say how the new idea is similar to and different from what they already knew. Teach them explicitly to link it with what they knew. This helps them to store the new mathematics knowledge in memory.

10. automatise the new knowledge by teaching independent use of the idea initially.

Scaffold them to do more of the idea and gradually remove the scaffolding as they do more of the idea by themselves. Have them practise doing it.

An instructional sequence for teaching these working memory strategies is described in Munro (2011). Conclusion In its conclusion the review of working memory and mathematics by Raghubar, et al., (2010) draws attention to our lack of knowledge of the relationship. The authors note the need for a theory of mathematical processing that integrates strategy discovery and selection, the use of mathematical knowledge and specific aspects of working memory. Contemporary developments in neuropsychology could well contribute to such a theory. Teachers and schools could make an invaluable contribution to such a model. Teachers every day observe how students in their classes respond to the mathematics teaching information. They develop at least an intuitive awareness of the influence of working memory processes on students’ on-going mathematical understandings in a way that is denied most researchers and investigators from other disciplines. Insights from how students respond to regular classroom teaching is a potential mine of useful information. References Baddeley, A.D. and Logie, R.H. (1999). Working memory: The multiple-component model. In: Miyake, A. and Shah, P. (Eds.). Models of working memory: Mechanisms of active maintenance and executive control. Cambridge University Press, Cambridge (1999), pp. 28–61. De Rammelaere, S., Stuyven, E. and Vandierendonck, A. (2001), Verifying simple arithmetic sums and products: Are the phonological loop and the central executive involved? Memory and Cognition, 29 pp. 267–273. Fletcher, J.M. (2005). Predicting math outcomes: Reading predictors and comorbidity, Journal of Learning Disabilities 38 pp. 308–312. Imbo, I. and Vandierendonck, A. (2007). The role of phonological and executive working memory resources in simple arithmetic strategies, European Journal of Cognitive Psychology 19, 910–933. Munro, J. (2011). The high reliability literacy teaching procedures to building students’ literacy knowledge. In Hopkins, D., Munro, J. and Craig, W. (Eds.) Powerful Learning: A Strategy for Systemic Educational Improvement. Camberwell, Aus: Australian Council for Educational Research, pp 67-81.

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Teachstrugglingstudentsbyexplicitlyscaffoldingworkingmemoryprocesses.

FocusoftheTeaching Howteacherscaffoldsstudenttousethecomponentsofworkingmemory

1.Encodethetaskinworkingmemory.

2.Stimulatewhatstudentsalreadyknowaboutthistypeoftask.

3.Guidestudentstolinkthenewtaskwithonestheycando.

4.Guidestudentstoencodethetypeofproblem.

5.Stimulatewhatstudentsalreadyknowaboutkeycomponentsofthenewtask.

181818

6.Stimulatestudentstousethenewlinkstocompletetheparticulartask.

7.Repeatwithsimilartasks.

8.Guidethestudentstoidentifyanddescribethenewprocedureandtopracticeapplyingit.

9.Guidethestudenttoidentifywhentousethenewprocedure.

10.Guidethestudentstoautomatizewhattheyknowabouthowtocompletethistypeofproblem.

191919

202020

DifferentiatingforStrugglingMathLearnersDifferentiatingteachersdon’tlimitstudents’exposuretoadvancedorsophisticatedmaterialjustbecausetheyhaven’tyetmasteredthefoundations.Theadvancedideastowhichweexposestudents,withorwithoutfoundations,providecontextandmotivationforlearningbasicideas(Wormeli).However,fastlearnersareusuallyfastretrieversbecausetheyhaven’tclutteredtheirmemorywithnetworksoftrivia.They’velearnedaconceptinlessthantheallottedtimeandhaveselectedthecriticalattributesforstoragewhilediscardingwhattheydecideisunimportant.Prematurelyincreasingcomplexityforslowerlearnersislikeaskingthemtotakefivebigsuitcasesonanovernighttrip,whereasthefastlearnersaretakingasmallbagpackedwiththeessentials.Slowerlearnersneedmoretimetosortaconcept’ssub-learningsintoimportantandunimportantcategories(Sousa).Strugglinglearnersneedmoredirection.Whilemanylearnersbenefitfrominquirybased,constructivisttechniques,explicitinstructionwithstrugglingstudentshasshownconsistentlypositiveeffects.Strugglingstudentsaremoresuccessfulwhenteachersprovideclearmodelsforsolvingaproblem,extensivepractice,opportunitiestothinkaloud,andextensivefeedback(NationalAdvisoryPanelFinalReport).Specialneedsstudentsneedtobecomestrategiclearnersandnothaphazardlyusewhateverstrategiesortechniquestheyhavedevelopedontheirown.Tobeabletodecidewhichstrategiestouse,forexample,studentsneedtoobservehowothersthinkoractwhenusingvariousstrategies.Learningskillsdevelopwhenstudentsreceiveopportunitiestodiscuss,reflectupon,andpracticepersonalstrategieswithclassroommaterialsandappropriateskills(Sousa).Effectiveinterventionsaddressmotivationalfactorsasmuchastheyaddressmathcontent(Forbringer).Whenstrugglerswalkthroughthedoor,theyarrivewithacognitivebeliefsystemconsistingofalltheirassumptions,beliefs,andknowledgeaboutmathlearning.Thisinfluencestheirself-concept,alltheyknowandbelieveaboutthemselves,whichdeterminestheiropennesstoinputsfromthelearningenvironment.Often,studentswithspecialneedsrequirealotofreassurancefromteachersthatthey’reontherighttrack.Becauseofpastexperiences,thesestudentsbelievetheycannotlearnorthattheworkissimplytoodifficult.Asaresult,theymaybelievetheycannotachievesuccessthroughtheirownefforts.Teacherswhoaddressself-conceptshutdownandwithdrawalbyre-teachingmaterial(oftenmoreslowlyorloudly)areattackingtheproblemfromthefrontendoftheinformationprocessingsystem.ItistheequivalentofputtingabrighterlightoutsidetheclosedVenetianblinds,hopingthelightwillpenetrate.Iftheblindsarefullyclosedandeffective,nolightwillgetthrough,regardlessofhowbrightitmaybe.Toopentheblinds,thelearnermustbelievethatparticipatingwillproducenewsuccessesratherthanrepeatpastfailures(SousaandPowells).Successfulteachersunderstandthedestructiveimpactofanxiety,andtheycontinuouslystrivetolessenanxietytriggers(Sousa).Maintaininghighexpectationsforstudentgrowthiscrucial.Teachersholdinganinnateviewofmathabilityaremorelikelytoexpresssupportandencouragementforstrugglingmathlearnersinunproductiveways.“Kind”strategies,suchasassigninglesshomeworkandnotcallingonstudentsduringclass,failtomotivatestrugglingstudentstoimprove.Studentswhoreceivecomfort-orientedfeedback—asopposedtomorestrategy-focusedfeedback—oftenassumetheirteachershavelowexpectationsforwhattheymightaccomplish,andconsequently,havelowerexpectationsandmotivationconcerningtheirownabilitiesandperformance(Rattan,Good,andDweck).Successfulteachersworkto

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buildself-esteeminordertohelpstudentsacceptresponsibilityandseeaconnectionbetweentheirachievementandeffort.Theseteachersavoidbeingoverprotective;theyallowstudentstosuffertheconsequencesoftheirbehavior(Sousa).Lowachieversareoftenthoughtofaslackingabilityorbeingslow,butthey’reoftenlowachievingbecausetheydon’tknowwhattheyshouldbepayingattentionto(Boaler).Nomatterwhatwedecidestudentsneedtolearn,notmuchwillhappenuntilstudentsunderstandwhattheyaresupposedtolearnandsettheirsightsonlearningit.Unlessstudentssee,recognize,andunderstandthelearningtargetfromtheverybeginningofalesson,onefactorwillremainconstant:theteacherwillalwaysbetheonlyoneprovidingthedirection.Thestudents,ontheotherhand,willfocusondoingwhattheteachersays,ratherthanonlearning.Thisfliesinthefaceofwhatweknowaboutnurturingmotivated,self-regulated,andintentionallearners(BrookhartandLong).Differentiatingteachersunderstandthatself-assessmentisalearnedskill,whichmusttobedevelopedfromtheearliestyears.Theircurriculumknowledgeprovidesinvaluableclarityaboutwhatevidenceconstitutesmastery,andthisclarityhelpsthemprovideampleself-assessmentopportunities.Differentiatingteacherspromptself-assessmentbeforeofferingoutsideevaluationbecausetheyknowstudentswillbecomedependentlearnersiftheydon’tlearnhowtoself-assess(Betts).Recommendations

• Helpstrugglersimprovetheirself-worthbyprovidingnon-contingentacceptance;tellthemthattheiracceptanceisnotcontingentontheirlevelsofperformanceintheclassroom.

• Trytogettoknowaboutthestudent’slifeoutsideofschool.• Showstudentsthatyougenuinelycareabouttheirwellbeing.Fosteracaring

atmospherebyworkingwithstudentstoidentifypressuresandburdens.• Exhibitanunderstandingofyourstudents’problemsandawillingnesstolisten

andofferadvice.• Helpstudentsdealwiththeirfrustrationsandreassurethemthattheyhavethe

abilitytoovercomedifficultchallenges.• Shakehandswithstudentswhenyougreetthemandusestudents’nameswhen

addressingthem• Avoidcriticizingastudent’squestion.• Allowstudentstomakedecisionsaboutsomeaspectsofclasswork.• Spendextratimewithstrugglingstudents.• Haveconversationswitheverystudent.• Pointoutpositiveaspectsofyourstudents’work.• Helpstudentsturnfailureintoapositivelearningexperience.• Celebrateyourstudents’achievements,nomatterhowsmall.• Increaseengagement;stimulateinterestwithhumor,unexpectedintroductions,

andotherattentiongrabbers.• Givestudentssomeleadershipintheirlearning–helpstudentsdevelopattainable,

personalgoalsfortheirlearning.Successisakeyfactorinmaintaininginvolvement.

• Coachstudentstotakenotes.Writingisnotonlyagoodmemorytoolbutalsohelpsstudentsorganizetheirthoughtsandfocusonwhatisimportant.

• Increaseretention;useclosurestrategiessuchasjournalwritingandgroupprocessing.

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• Provideopportunitiesforstudentstoworkinproductivegroups.Theopportunitytodiscusswhatthey’relearningkeepsthesestudentsengagedandhelpsthempracticeinterpersonalskills.

• Nurturetraitsofresilience:socialcompetence,problem-solvingskills,autonomy,andasenseofpurposeandfuture.

• Makemathirresistible–teacherscommunicateapassionforthecontent.Beenthusiasticandshowthatmathcanmakeadifference.

• Resisttheoverwhelmingtemptationto“justdrill‘em.’”Aconceptualapproachisthebestwaytohelpstudentswhostruggle.Achildwhohasdifficultieshascertainlybeendrilledinthepast.Althoughdrillmayprovidesomeveryshort-termsuccess,drillwillhavelittleeffectinthelongrun.

• Goforthebigpictureandshowhowconceptsareconnectedtosatisfythepatternseekingbrain.

• Findoutwhattheyalreadyknowtobuildonpriorknowledge.• Model.Showstudentshowtodoitusingmodelsandexamples.• Thinkandtalkaloudtomodelthestepsincognitiveprocessing.• Supportthedevelopmentofself-regulationbyhelpingstudentstounderstandthe

learningtargetfromthelesson’soutset.• Provideexposuretoadvancedandsophisticatedmaterialascontextfor

motivation.• Coachstudentssotheyaskyouquestionsinordertoclarifymisconceptions.This

hastheaddedbenefitofforcingstudentstorehearsetheinformationintheirheadsinordertocomposethequestion.

• Coachstudentstorereadwhattheydon’tunderstand–thisstrategygivesstudentsanotheropportunitytocomprehendwhattheyarereading.

• Coachstudentstoaskotherstochecktheirwork–peerfeedbackisavaluableandnon-threateningtooltoverifytheaccuracyandcompletenessoftheassignment.

• Coachstudentstosearchforandcorrecterrorsintheirwork–studentsarelesslikelytorepeaterrorstheydiscoverontheirown.

• Coachstudentstotakenotes–addingthekinestheticactofwritingincreasesattentionandthelikelihoodthatthestudentwillretainthenewinformation.

Thecommentsandrecommendationsabovearebasedontheworkof:

• DavidSousa–HowtheSpecialNeedsBrainLearns• RickWormeli–FairIsn’tAlwaysEqual• JohnVandeWalle–ElementaryandMiddleSchoolMathematics:Teaching

Developmentally• TheNationalMathAdvisoryCouncilFinalReport• BillandOchanPowell–MakingtheDifference• JoBoaler• BambiBetts• Moss,C.Brookhart,S.&Long,B.(2011)KnowingYourLearningTarget,

EducationalLeadership:WhatStudentsNeedtoLearn,68.• AneetaRattan,CatherineGood,andCarolDweck.It’sOK–NotEveryoneCanBe

GoodatMath:InstructorswithanEntityTheoryComfort(andDemotivate)Students.”JournalofExperimentalSocialPsychology,April2012.

• LindaForbringerandWendyFuchs,RtIinMath:Evidence-BasedInterventionsforStrugglingStudents

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Mathperformanceimproveswhenanxietyisalleviated.Anxietyisoftenconnectedto:

Teacherattitudes:Teacherattitudesinfluencemathanxietyandarethemostdominatingfactorinmoldingstudentattitudesaboutmath.Successfulteachers…

• Presentanagreeabledispositionthatshowsmathtobeagreathumaninvention.

• Strivetoengagestudentswithworkthat’schallengingenoughtodemandeffortbuteasyenoughtoexpectsuccess.

• Displayconfidenceinyourteaching.Teacherswhoareanxioustransmitthisfeartostudents.

• Demonstrateawarenessofandunderstandstudentconfusionandfrustration.Curriculum:Primarystudentsusuallyratemathasoneofthesubjectstheylikemost.Studentsbelievetheyhavethecompetencetodomathematicsandthathardworkwillbringsuccess.Mathanxietyoftensurfacesby4thgradewhenthecurriculumshiftstomoreabstractthinking.Studentsstarttobelievesuccessisduetoinnateabilityandeffortmatterslittle.Thematerialgetsevenmoreabstractinhighschool,andstudentsrealizethatmemorizationisnotsufficienttosucceed.Successfulteachers…

• Provideopportunitiesforstudentstousemathasatoolfordiscoveringnewideas.

• Helpstudentstransitiontoabstractthinking.InstructionalStrategies:Teachingtechniquesthatcenteron“explain-practice-memorize”areamongthemainsourcesofmathanxietybecausethefocusisonmemorizationratherthanonunderstanding.Successfulteachers…

• Posequestionsinanefforttohelpstudentscontinuouslylearn.• Limitthefrequencyofmemorizinganddoingrotepractice.• Developmeaningthroughpracticalapplications.

ClassroomCulture:Sense-makingclassroomsgeneratelessanxietythanprocedurefocusedclassrooms.Emphasizingspeeddoesnotencouragestudentstoreflectontheirthinkingprocessesortoanalyzetheirresults.Successfulteachers…

• Createaculturewherestudentscanaskquestions,discoverlearning,exploreideas,feelsecureintakingrisks,andnotfeelembarrassedforgivingwronganswers.

• Discouragevaluingspeedovertimeforreflection.• Encouragesensemakingovermemorizingstepsorprocedures

Assessment:Testsoftendiminishconfidencebecausethere'snoflexibilityintheprocess.Successfulteachers…

• Limittimedtests,whichdisruptprocessinginbothworkingandlong-termmemories.

• Incorporatemultiplemethodsofassessmentsuchasoral,written,ordemonstrationformats.

(Sousa,HowtheBrainLearnsMath,172-78)

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AdvancedLearnerCharacteristics,Needs,andRecommendationsAdvancedmathlearners:

• graspmathematicalideasveryquickly.Advancedmathstudentswilllosemotivationifinstructionisshallow,repetitiveor,tooslow.Giftedstudentsaremorelikelytoretainscienceandmathcontentaccuratelywhentaught2-3timesfasterthan“normal”classpace(Rogers,1991)

• operateeasilywithsymbolsandspatialconceptsandquicklyrecognizesimilarities,differences,andpatterns.

• sustainconcentrationandshowtenacityinpursuingsolutions.

(Sousa,HowtheGiftedBrainLearnsMathematics,170-171)Advancedmathlearnersneed:

• acurriculumthatisdifferentiated(bylevel,complexity,breadth,anddepth)andconductedatamorerapidrate.Combinedaccelerationandenrichmentshouldbetheinterventionofchoice.

NationalMathAdvisoryPanelFinalReport

• assessmentsthatallowfordifferencesinunderstanding,creativity,andaccomplishment;theyneedachancetoshowwhattheyhavelearned.

(DanaJohnson:TeachingMathematicstoGiftedStudentsinaMixed-AbilityClassroom)

Ifthereneedsaren’tmet,advancedlearners:

• canbecomementallylazy.Abrainlosescapacityand“tone”withoutvigoroususe.

• maybecome“hooked”onthetrappingsofsuccess,doingwhatis“safe”ratherthanwhatcouldresultinlearning.

• maybecomeperfectionists.Externalpraiseforbeingthebestcausesthemto

attachmuchoftheirself-worthtotherewardsofschooling,andfailurebecomessomethingtoavoidatallcosts.Creativeproductiontypicallyhasahighfailure-to-successratioandgiftedperfectionistsareunlikelytoseetheirproductivecapacityrealized.

• mayfailtodevelopasenseofself-efficacywhichcomesfromstretchingyourselfto

achieveagoal.Thesestudentsoftengothroughlifefeelinglikeimpostors,fearfullyawaitingthedaytheworlddiscoverstheyaren’tsocapableafterall.

• mayfailtodevelopstudyandcopingskillsbecausetheycoastthroughschoolwith

onlymodesteffort.Successinlifetypicallyfollowspersistence,hardwork,andrisk.

(Tomlinson,HowtoDifferentiateInstructioninMixedAbilityClassrooms:11-12)

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Researchsupportsaccelerationthroughcompactingandenrichment.Specifically:

• Provideanoverview,assessproficiency,andthenallowstudentstomoveontomoreadvancedandcomplexcontentthatinvolveshigher-levelthinking,morecomplexorabstractideas,anddeeperlevelsofunderstanding(e.g.,writing,proofs,solutionstochallengingproblems).

• Provideactivitiesthatgobeyondthetopicofstudytocontentthatisnot

specificallyapartofthegrade-levelcurriculumbutisalignedwiththelessonobjectives.

(DanaJohnson:TeachingMathematicstoGiftedStudentsinaMixed-AbilityClassroom)

Animportant,additionalfinding:

Amajorresearchfindingisthatwhatisdevelopmentallyappropriateislargelycontingentonprioropportunitiestolearn.Claimsbasedontheoriesthatchildrenofparticularagescannotlearncertaincontentbecausetheyare“tooyoung,”“notintheappropriatestage,”or“notready”haveconsistentlybeenshowntobewrong.Norareclaimsjustifiedthatchildrencannotlearnparticularideasbecausetheirbrainsareinsufficientlydeveloped,eveniftheypossesstheprerequisiteknowledgeforlearningtheideas.

(NationalMathAdvisoryPanelReport,2008)

StrategiesandGroupingFormats-ResearchFindingsCurriculumCompacting(replacingmathcurriculumwithadvancedmathatthestudent’struelearninglevelandatanacceleratedpace) Effectsize=.83

ClusterGrouping(3-6students) Effectsize=.62

Pull-outsthatareadirectextensionoftheregularschoolcurriculum Effectsize=.65

Full-TimeAbilityGrouping(SecondaryStudentsGrades7-12) Effectsize=.33

Full-TimeAbilityGrouping(ElementaryStudentsinAllAcademicAreas) Effectsize=.49

*Inclassroomterms,aneffectsizeof1.00isapproximatelyequaltooneschoolyear.Aneffectivesizeof.25iseducationallysignificant.Sotheimpactofcompactingisabout8

grade-equivalentschoolmonthsofadditionalachievement.

(Sousa,HowtheGiftedBrainLearns,51–52)

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Unit 5 Summative Assessment 1

Jakarta Intercultural School

8th Grade – AG1 Summative Assessment

Name: ___________________ Block: ______________________ Date: _______________________

Unit 5: Exponents and Exponential Functions

Standard Grade: Advanced Highly Advanced

Mastered Proficient Approaching Emerging No Proficiency

Meeting Learning Goals State of Your Work Higher Order Thinking

A All learning goals are met within the topic of study.

Work is accurate, clear and organized, showing attention to detail at all times.

Sophisticated understanding shown through communication of higher order thinking

B Most of the learning goals are met within the topic of study.

Work is accurate, clear and organized, showing attention to detail most of the time.

Considerable understanding shown through communication of higher order thinking.

C Some of the learning goals are met within the topic of study.

Work is accurate, clear and organized, showing attention to detail some of the time.

Some understanding shown through communication of higher order thinking.

D Few of the learning goals are met within the topic of study.

Work is rarely accurate, clear or organized, and rarely shows attention to detail.

Limited understanding shown through communication of higher order thinking.

I None of the learning goals are met within the topic of study.

Work is never accurate, clear or organized, and rarely shows attention to detail.

No understanding shown through communication of higher order thinking.

Unit 5 Learning Goals Green Blue Black

Model and solve word problems involving real-life exponential growth and decay scenarios.

Apply Properties of Exponents involving: • Multiplication • Division • Negative and Zero Exponents

Black: Including understanding of logarithms.

Apply scientific notation.

Graph exponential functions, identifying domain and range. Black: Including understanding of logarithms.

Simplify radicals.

Unit 5 Exponential Functions PSA Project

27


CALCULATORSECTION

Readallinstructionsandchooseproblemscarefully.Youmustdoanentireproblemtoreceivecredit.Showallstepsofyourworkandcheckworkcarefully. Part1–ModelRealWorldProblems 1(ALL)Youbuyausedcarthatis8yearsold.Therateofdepreciationforthismodelcarisshown

tobeabout18%peryear.Ifyoupaid$6,500forthecar,whatwasthevaluewhenitwasnew?

2(GREEN)Abusinesshasa$5,000profitin1990.Thisprofitincreasesby15%peryear.

a.Writeafunctionthatmodelsthissituation.Don’tforgettodefineyourvariables.

b.Findtheannualprofitafter10years.

c.Inwhatcalendaryearwillthisbusinesshaveprofitsexceeding$50,000?2(BLUE)Abusinesshasaprofitof$5,000in1990.Fiveyearslaterithasaprofitof$10,502.

a. Findtheannualgrowthrateforthisbusinessandwriteanequationthatmodelsthisbusiness’profitovertime.

b. Findtheannualprofitintheyear2000.

c. Inwhatcalendaryearwillthisbusinesshaveprofitsexceeding$100,000?

2(BLACK)Abusinesshasaprofitof$5,000in1990.Tenyearslater,theirannualprofitshavegrownto$24,034.14peryear.

a. Findtherateofgrowthforthisbusiness.

b. Findoutwhattheirprofitwouldhavebeenin1988.

c. Tothenearesthundredth,howmanyyearsafterthestartwillthebusiness’profitsexceed$100,000?

WorkSpaceGreenBlueBlack

28


3(GREENIn2007,therewereapproximately7,300SumatranOrangutansinthewild.Thispopulationisdecreasingbyabout2.5%peryear.Approximatelyhowmanyorangutansweretherein1970?

3(BLUE)Gilberthasaplantomakehisownsunglasses.Hetakessomefilterfilmandattachesittoapairofregularglasses.Hehaslearnedthatthefilterfilmabsorbs20%ofthelightpassingthroughit.Healsolearnsthatagoodpairofsunglassesabsorbs80%ofthelight.Hefigureshe’lluse4layersoffilterfilmandthat’llbethesame.

a. IsGilbertcorrect?Defendyouranswerwithspecificnumbers.

b. Writeaformulatocalculatetheamountoflightpassingthroughxlayers.

3(BLACK)Marytakesoutaloanfor$25,000ata7%interestrate,compoundedannually.Attheendofeveryyear,shemakesapaymentonthesamedaythattheinterestiscalculatedforthefollowingyear.Sheisthenchargedinterestontheamountthatshestillowes.Shewantstopaybacktheloaninthreeequal,annualpayments.Howmuchshouldeachpaymentbetocoverthecostoftheoriginalloanandallinterestdue?


4(GREEN)Iboughtastockfor$40persharein2014.In2015,mystockwasworth$80pershare.

a. Whatisthepercentofincreaseformystock?

b. Writeafunctiontomodelmystock’svalue.

c. Ifitcontinuesatthisrate,whatwillmystockbeworth,pershare,in2018?

29


4(BLUE/BLACK)Thehalf-lifeofaradioactiveelementistheamountoftimethatittakesforhalfofanyamountoftheelementtodecayaway.

Baliniumisanartificiallyproducedelementwithahalf-lifeof10minutes.

a. WriteafunctionthatrelatesthenumberofgramsofBaliniumremainingaftertminutesfroma40-gramsample.

b. Howmuchremainsafter½hourpasses? c. (BLUE)Howlongwillittakebeforeabout1/32oftheoriginalsampleremains? c.(BLACK)Calculatethetimeittakesforonly1/1000oftheoriginalsampletoremainto3decimal

places.5(ALL)ThefeesatJISintheyear2016were$30,000.Theboardagreedthatatuitionincreasewasnecessary.Oneboardmemberproposesthatthefeesshouldberaisedeveryyearby$2,500.Anotherboardmemberproposesthatthefeesshouldbeincreasedby7%everyyear.Makearecommendationconsideringboththeshorttermandthelongtermchanges.Explaintheadvantagesanddisadvantagesofeachproposaltosupportyourchoice.

Includeequations,tables,andgraphsforeachproposalinyourdiscussion.

Workspaceonnextpage…

30


5(ALL)Workspace

31


Unit5:ExponentsandExponentialFunctions Name: ________________________SummativeAssessment

Block: ________________________

NON-CALCULATORSECTION Date: ________________________

Readallinstructionsandchooseproblemscarefully.Youmustdoanentireproblemtoreceivecredit.Showallstepsofyourworkandcheckworkcarefully. PartII–PropertiesofExponents1(ALL)Usethepropertiesofexponentstosolveforx.Showyourwork.3x=92•3•2732(ALL)Simplify

a. (3x-2)(-3x)(x0)b. !"#$% &'

()$%

3(GREEN)Comparethetwoexpressionsineachstatementusinggreaterthan(>),lessthan(<)orequalto(=).Determinetheanswerwithoutmultiplyinganynumbersandshowyourreasoning.

a. (5• 3)3 ___5• 33 b. 84 • 83 ___812 3(BLUE)Simplifyorevaluatethefollowingexpressions.Writeanswersinsimplestform.

a. −1,-./ % −1,%.- -b.(7)$/ 33-

$%

3(BLACK)Evaluate,showingallsteps.

a.

932

⎛

⎝⎜⎞

⎠⎟8

23

⎛

⎝⎜⎞

⎠⎟b.log322

32


4(GREEN)Simplify.

a.45 ∙ 4%75 -b. 5&'(

%&('-

4(BLUE)a.Findthevalueofkintheequation:

35 ∙ 35 ∙ 2- ∙ 6! = >!b.Findthevalueofxif…

3% ∙ 9% ∙ 27% ∙ … ∙ 2187% = 3&

4(BLACK)Evaluate,showingallsteps.

a.Solvetheequationforx:3 BC = 27 %C b.Supposethat2x=10.Evaluate24x–2

5(GREEN)Solveforx.Checkyouranswer.

a.D ∙ D& ∙ D&E% = D3Fb.3" = 35 &

33


5(BLUE)Solveforx.

a.7& = 495Fb.125 ∙ 5/ = 5& + 5& + 5& + 5& + 5&c.235 = 3I

&

5(BLACK)

a. Solve for a, b and c if: 5J ∙ 5K = 5L 3J K = 81 and 5M5N =

33"

b. Solve for x: 35O =

%PC∙B

RSC

%TC

34


PartIII–ScientificNotation1(ALL)Eachofthefollowingnumbersiscurrentlywritteninscientificnotation.Pleaserewrite

themindecimalnotation.

a. Theuniverseisestimatedtobenearly1.5 × 1010yearsold.

b. Thetimetakenbylighttotravelonemeterisroughly3×10-9seconds.2(GREEN)Eachofthefollowingnumbersiscurrentlywrittenindecimalnotation.Pleasewritetheminscientificnotation.

a. Thediameterofahydrogenatomis0.00000000005meterslong.b. Thenumberofneuralconnectionsinthehumanbrainisabout102,500,000,000,000.2(BLUE)Imaginethateachofyourbrain’sneuralconnectionswerethelengthofahydrogenatomsdiameter.Ifyouunwoundallofyourbrainsneuralconnectionstoformasingle,chain-linkstringofconnections,howlongwouldthissinglestringbeinscientificnotation?Usetheinformationprovidedinthegreenlevelproblem.

2(BLACK)Thenumberofgrainsofsandinaonecubiccentimetersamplewas9.37x103.Usingthisinformation,predicthowmanygrainsofsandwouldbeinacubicmeterofsand.


35


PartIV–GraphingExponentialFunctions1(ALL)Matchthefollowingequationstothegraphsbelow.Explainhowyoucanbesurethatyour

selectionistheonlycorrectchoice.

A. y=10(¾)x B. y=-3x+10 C. y=10(1.3)x D. y=3x+10 E. y=10(3)x F. y=10(0.1)x

i. Equation ii. Equation

Explanation: Explanation:

iii. Equation iv. Equation

Explanation: Explanation:

36


2(GREEN)a.Makeagraphofthefollowingfunctionforx-valuesfrom-2to+2:# = 2 3

%&

b.Whatisthedomain?c.Whatistherange?2(BLUE)a.Makeagraphofthefollowingfunctionforx-valuesfrom-2to+2:# = − 3

5&

b.Whatisthedomain?c.Whatistherange?

2(BLACK) a.Makeagraphofthefollowingfunction:

# = log35!

b.Whatisthedomain?c.Whatistherange?


Domain:Range:

x y

PartV–SimplifyingRadicals1(ALL)Simplify,showallwork. − 404-7%

37


2(GREEN)Simplify. 25!3"

2(BLUE)Evaluate.Showallwork.25

)'

2(BLACK)Whatisthelargestintegerksuchthat 8 125k < ?


3(GREEN)Simplify. 200

3(BLUE)Whatistheleastpossiblepositiveintegervalueofxsuchthat 15 ∙ ! ∙ 35isaninteger?

3(BLACK)Solve.Showallwork.

3! + 3! + 3! +⋯ = 15


38

ExponentialFunctionsMultimediaProject

Afterspendingourfirstsemesterlearningaboutlinearfunctions,we’vespentthisunitstudying

exponentialfunctions,whichbehaveverydifferently.Thegoalsofthisprojectareforyouto:

1. Learnaboutaninterestingreallifesituationthatrelatestothisunit.2. Demonstrateyourunderstandingofexponentialfunctions.3. Creativelyapplymultimediaskillswhileeducatingandmotivatingothers.

GREEN-EVERYONE

Part1Goal Createavideoorawrittenreportthateducatesyouraudienceaboutthetopicyou’veselected.Ifyoumakeavideo,itshouldbeinterestingandamaximumlength

of4minutes.Followthesesteps:

1. Describeaninteresting,reallifesituationthatillustrateshowexponentialfunctionsareuniqueandpowerful.Representyourscenariowithtables,graphs,

andequations.Explainhowyouarrivedatyourequations.Also,citeyoursources!

2. Showhowachangeinthegrowthordecayfactorcanhaveasignificantimpacton

yoursituationbydoingthefollowing:

• First,changethegrowth/decayfactorduetoareallifeoccurrence.Ifpossible,

makethischangebasedonsomethingthatactuallyhappened.Ifthatistoo

difficult,createarealistic“whatif”scenario.

• Explainyourreasoningforchangingthefactorandmakesureyoushowthe

impactofthischangewith“overlapping/parallel”tables,graphs,andequations.

Part2GoalCreatea30secondvideothatiseducational,engagingandmotivating.Inspireyour

audiencetolearnmoreaboutyourtopic(studentswhodopart1asavideodonot

needtocompletepart2).Keepthefollowingrequirementsinmind:

1. Videosmustbe30secondslong.Writeyourscriptandplancarefully.

2. Usevisuals,soundeffects,and/ormusictohighlightyourmessage.

3. Previewyourvideotoensuregoodimageandsoundquality.

**NoteaboutGroupWork:Youmayworkonthisprojectwithapartnerifyouwish,whichmeansyoucanusethesametopicandcollaborateonyourresearchandanalysis.

However,eachpersonneedstoturnintheirownuniqueproducts.AdditionalBlueLevelRequirements

Inadditiontomeetingthegreenlevelrequirements,yourtopicmustalsoilluminatethedifferencesbetweenexponentialandlinearfunctionsintables,graphs,andequations.

AdditionalBlackLevelRequirements

Inadditiontomeetingthegreenandbluelevelrequirements,yourworkmustalso

demonstratethatyouunderstandhowlogarithmscanbeusedtohelpunderstandreal

worldproblems.

39

Name:_________________________________________ ProjectColorLevel:____________________Whenyoufinishyourproject,carefullycompletethisrubricandsubmititwithyourproject.

STUDENTSELF-EVALUATION ColorLevel___________________

Part1.WrittenReportorMultimediaExplanation CircleOne

Haveyouclearlydescribedaninteresting,reallifesituationthat

illustrateshowexponentialfunctionsareuniqueandpowerful?

Noor

notreallyKindof

Forthe

mostpart

Yes,

definitely!

Haveyouclearlycitedthesourceofyourinformation?Noor

notreally Kindof Forthe

mostpartYes,

definitely!

Haveyoushownbyusinggraphs,tables,andequationshow

exponentialfunctionsareuniqueandpowerful?

Noor


mostpartYes,

definitely!

Haveyoushown/explainedhowyouarrivedatyourequation(s)?Noor


mostpartYes,

definitely!

Haveyougivenaclearandfactual(orplausible)explanationfor

thechangeyoumadetoyourgrowth/decayfactor?

Noor


mostpart

Yes,

definitely!

Haveyouclearlyshown(withgraphs,tables,andequations)and

explainedhowthechangeyoumadetothegrowth/decayfactor

impactedyourscenario?

Noor


mostpartYes,

definitely!

Isyourworkclearlyorganizedandnicelypresented?Noor


mostpartYes,

definitely!

BlueandBlackLevelOnly:Haveyouthoughtofascenariothatillustratesthedifference

betweenexponentialandlinearfunctions?

Noor


mostpartYes,

definitely!

BlackLevelOnly:Haveyouexpandedyourscenariotodemonstratehowlogarithms

canbeusedtohelpunderstandrealworldproblems?

Noor


mostpartYes,

definitely!

Part2Video(onlyifyoudidawrittenreportforPart1) CircleOne

Doesitshowthatyou’veputalotofeffortintomakingahigh

qualityvideothatdemonstratesthepowerofexponential

functionsandmotivatestheviewertolearnmoreaboutthetopic?

Noor

notreally KindofForthe

most

partYes,

definitely!

TEACHEREVALUATION ColorLevel:GreenBlueBlack

Part1DepthofUnderstanding Limited Partial Considerable Sophisticated

LevelofQuality Low Medium High

Part2 LevelofQuality Low Medium High

Qualityofstudentself-assessment Low Medium High

OVERALLLEARNINGGOAL Emerging Approaching Proficient Mastered

40

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