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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.
444
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?
131313
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
141414
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.
151515
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
232323
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
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Unit 5 Summative Assessment 2
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
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Unit 5 Summative Assessment 3
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?
WorkSpaceGreenBlueBlack
4(GREEN)Iboughtastockfor$40persharein2014.In2015,mystockwasworth$80pershare.
a. Whatisthepercentofincreaseformystock?
b. Writeafunctiontomodelmystock’svalue.
c. Ifitcontinuesatthisrate,whatwillmystockbeworth,pershare,in2018?
29
Unit 5 Summative Assessment 4
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$2,500.Anotherboardmemberproposesthatthefeesshouldbeincreasedby7%everyyear.Makearecommendationconsideringboththeshorttermandthelongtermchanges.Explaintheadvantagesanddisadvantagesofeachproposaltosupportyourchoice.
Includeequations,tables,andgraphsforeachproposalinyourdiscussion.
Workspaceonnextpage…
30
Unit 5 Summative Assessment 5
5(ALL)Workspace
31
Unit 5 Summative Assessment 6
Unit5:ExponentsandExponentialFunctions Name: ________________________SummativeAssessment
Block: ________________________
NON-CALCULATORSECTION Date: ________________________
Readallinstructionsandchooseproblemscarefully.Youmustdoanentireproblemtoreceivecredit.Showallstepsofyourworkandcheckworkcarefully. PartII–PropertiesofExponents1(ALL)Usethepropertiesofexponentstosolveforx.Showyourwork.3x=92•3•2732(ALL)Simplify
a. (3x-2)(-3x)(x0)b. !"#$% &'
()$%
3(GREEN)Comparethetwoexpressionsineachstatementusinggreaterthan(>),lessthan(<)orequalto(=).Determinetheanswerwithoutmultiplyinganynumbersandshowyourreasoning.
a. (5• 3)3 ___5• 33 b. 84 • 83 ___812 3(BLUE)Simplifyorevaluatethefollowingexpressions.Writeanswersinsimplestform.
a. −1,-./ % −1,%.- -b.(7)$/ 33-
$%
3(BLACK)Evaluate,showingallsteps.
a.
932
⎛
⎝⎜⎞
⎠⎟8
23
⎛
⎝⎜⎞
⎠⎟b.log322
32
Unit 5 Summative Assessment 7
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
Unit 5 Summative Assessment 8
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
Unit 5 Summative Assessment 9
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.
WorkSpaceGreenBlueBlack
35
Unit 5 Summative Assessment 10
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
Unit 5 Summative Assessment 11
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?
WorkSpaceGreenBlueBlack
Domain:Range:
x y
PartV–SimplifyingRadicals1(ALL)Simplify,showallwork. − 404-7%
37
Unit 5 Summative Assessment 12
2(GREEN)Simplify. 25!3"
2(BLUE)Evaluate.Showallwork.25
)'
2(BLACK)Whatisthelargestintegerksuchthat 8 125k < ?
WorkSpaceGreenBlueBlack
3(GREEN)Simplify. 200
3(BLUE)Whatistheleastpossiblepositiveintegervalueofxsuchthat 15 ∙ ! ∙ 35isaninteger?
3(BLACK)Solve.Showallwork.
3! + 3! + 3! +⋯ = 15
WorkSpaceGreenBlueBlack
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
notreally Kindof Forthe
mostpartYes,
definitely!
Haveyoushown/explainedhowyouarrivedatyourequation(s)?Noor
notreally Kindof Forthe
mostpartYes,
definitely!
Haveyougivenaclearandfactual(orplausible)explanationfor
thechangeyoumadetoyourgrowth/decayfactor?
Noor
notreally Kindof Forthe
mostpart
Yes,
definitely!
Haveyouclearlyshown(withgraphs,tables,andequations)and
explainedhowthechangeyoumadetothegrowth/decayfactor
impactedyourscenario?
Noor
notreally Kindof Forthe
mostpartYes,
definitely!
Isyourworkclearlyorganizedandnicelypresented?Noor
notreally Kindof Forthe
mostpartYes,
definitely!
BlueandBlackLevelOnly:Haveyouthoughtofascenariothatillustratesthedifference
betweenexponentialandlinearfunctions?
Noor
notreally Kindof Forthe
mostpartYes,
definitely!
BlackLevelOnly:Haveyouexpandedyourscenariotodemonstratehowlogarithms
canbeusedtohelpunderstandrealworldproblems?
Noor
notreally Kindof Forthe
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