A.CED.1 / ClusterHeading: Create equationsthatdescribenumbers orrelationships.
ContentStandard: Createequationsand inequalitiesin onevariableand usethemtosolveproblems.
Practice Standard: MP2Reason abstractlyand quantitatively.
Problem/TaskSuggestions / Formative AssessmentSuggestions
MagicNumber
A magiciantoldyou topicka numberand rememberit. Add 5.Triplethis answer. Subtract1. Subtractyouroriginal number.Takehalfofthis answer, and subtractyouroriginalnumber again.Themagicianis alwaysabletoguess your finalnumber.
Initially haveall studentspicka number and puttheirfinal answeron a personal whiteboard andshowitatthesametime.Iftherearestudentsthat do nothavearesultof7,do thetaskagain,butthistimehavestudentswrite theiroriginal numberin thecorner and thentheirfinalanswerlargeand inthe middle. Using anymistaken responses,haveseveralstudentsshowtheir thinking processon theboard withsymbols.
If themagician isalwaysabletoguessyour finalnumber,howdoesheor she dothis? Explain. Canyouusenumbersand avariabletoexplain your thinking?
Havestudentsexplain using symbolsand a variableforthenumber chosen. Comparetheseto thenumbersand symbolsthatareon theboard from earlier.
Differentiation
Extensions
•Create yourownMagician’strickin whichyouwill alwaysknowthe answer.
•CreateaMagician’strickwheretheanswerwould alwaysbe the original numbertheaudiencemember picked.
Solution
•[3(?+5)−1−?]− ?=7
2
•Explain theequationsimplifiesto7=7,anequation thatis truefor
any valueof x. / Observation of Students
•Isthestudentabletousementalmath todosimplecomputationor does he/sheneed touseanother tool?MP5
•Doesthestudentimmediatelyattempttowriteaone-variableequation for thesituation?
•Can thestudentexplain theprocessofwriting and solving theequation tootherstudents?MP3
•Doesthestudent’sworkgenerateand solveaone-variable equation to representthesituation?
•Doesthestudent’sworkincludea writtenexplanationofthecreation of a one-variableequation?
QuestionstoGuide Student Thinking
•Whatcould weusein theexpressiontorepresentany number?
•If you dothis again with a differentinput(original number), would you alwayshavethesameoutput(final number)?
Misconceptions
Studentsmay
•Explain thatnomatterwhatnumbertheypick,ityields 7,butwill notbe ableto generateageneralequationwith avariable torepresentthe initial value.
•Dotheorderofoperationsincorrectly,orplaceparenthesesinthewrong spot.
•Distributeorcombinetermsincorrectly.
•Beconvinced thattheproblemreallyismagic.
Vocabulary
•Infinitesolutions,allreal numbers,distribute,combineliketerms
Created by: Illinois StateBoardof Education Content Area Specialist

A.CED.2

VIPDilemma

ClusterHeading: Create equationsthatdescribenumbers orrelationships

ContentStandard:Createequationsintwoormore variablestorepresentrelationshipsbetweenquantities;graph equationson coordinate axeswith labelsand scales.

Practice Standard(s): MP1Makesenseof problemsand persevereinsolving them,MP4Model with mathematics.

Problem/TaskSuggestionsFormative AssessmentSuggestions

Myphonebeepswith a textfrom myfriendwho isan event coordinator:

“Myproducersent me only50feet ofredvelvetrobeand 4poles! I don’tknow what hewas thinking,howcan I fitall oftheVIPs in this section?“

a.Whatis themathematicalquestion that is beingasked?

b. Preciselyanswerthequestion thatwewrote.(Ideally“What is the maximum area you can haveforarectanglewith a constantperimeterof

50feet? Whatwould thesidelengths be?)

c.Writea text to theevent coordinatorto explain theanswerand to explain howhecould alwaysdo similarproblems in thefuture.

Differentiation

Supports

•Askwhat mathwordsrelate to the event coordinators problem.

•Provideyarnand pushpins orgraph paperfor thestudent to construct the figure.

•Askwhat theperimeterofthefigureneeds to be.

Extension

•Howwould theproblem changeifyou could useawall as oneofthesidesof theVIP section?Would anothershapeprovidemorespace?

Solutions

a.Explain that perimeter must be50feetandtheyaretrying to find the maximum area. Answers could varyifstudents considerthelooping of rope orthink about onecontinuousrope.

b. Writetheequation y=x (25-x),createatableofvalues,or graph thepossible values to find the maximumvalue(vertex)to get two sidelengths of12.5feet and an area of 156.25squarefeet.

c.Explain thata squarewouldbethebestpossibilitybecauseit maximizesarea.

Explain to theevent coordinator,thatanytimehehasfourpoles,hecould

dividetheropelength by4to find thebest length foreach side.Hecould also writea quadratic equation and find themaximum value.

Created by: Illinois StateBoardof Education Content Area Specialists

Observation of Students

•Dothestudentsrecognizethemath in thesituation?

•Arethestudentsableto makesenseof thedilemma and comeupwith a mathematical questioneither alone,in theirgrouporduring full class discussion?

•Isthestudentabletorecognizewhattheproblemisasking?(MP1)

•Dostudentsrecognizethatthesituation canberepresented asa quadraticfunction?

•Arethestudentsabletoexplain in wordsthemethodstheyareusing?

QuestionstoGuide Student Thinking

•Whatarethekeywordsthatweneedtocreateamathematical question?

•Howcan you becertain? Howcouldyoucheck?

•Howcouldyoumathematicallyconvinceaskepticthatthereisnoother answerthatis better and thatyou havecheckedeverypossibility?

Misconceptions

Studentsmay:

•Arguethat12x13isthebestanswer becausetheycreated a tableof whole numbers.

•Arguethat12x13isjustasgood as12.5x12.5becausethedifference between 156and 156.25squarefeetis negligibleinaVIPsection.

•Believethatthetwosideshavetoadd upto100instead of all four sides.

Vocabulary

•Parabola,quadratic,vertex,perimeter,area,maximum

A.REI.3 / ClusterHeading: Solve equationsand inequalitiesinone variable.
ContentStandard:Solvelinear equationsand inequalitiesinonevariable,including equationswith coefficientsrepresented byletters.
Practice Standard: MP3Constructviableargumentsand critiquethereasoningofothers.
Problem/TaskSuggestions / Formative AssessmentSuggestions
Compare and Contrast
Compareand contrastthefollowing twoequations.Solveeach forx, explaining eachofyoursteps. (Assumeu,tand q are non-zeroreal numbers).
a) b)
Differentiation
Supports
•Havethestudentfirstsolve parta and then usetheexactsamesteps tosolvepart b.
•Givethestudenta numberforoneormoreofu,tandq.
Extension
•Writea word problemthatwouldyield thefirstequation. Howwould you havetochangethatword problemtoyield thesecond?Have studentspair and sharetheir wordproblemsand discussthevalidity
of each problemin relationtotheoriginal equation.
Solution
•Parta:x=3.
•Partb:?=?+?or?=?+?
???
•Students explanation should include:Foreachequation,firstadd
eitherthe1orthettoboth sides,then dividebothsides ofthe equation bythecoefficientofx, either7or q. Theonlydifferenceis thatthefirstusesnumbercoefficientsand thesecond usesletter coefficients. / Observation of Students
•Doesthestudentrecognizethatsolving anumericone-variableequation usesthesameprocessas solving a literalequation?MP7
•Isthestudentabletoexplain theprocessofsolving aone-variable equation togroupmembers?(Consider usingThink,Pair,Share). MP3
•Doesthestudentincludeawrittenexplanationof each of thestepsto solveaone-variableequation?
QuestionstoGuide Student Thinking
•Whatoperations“undo”the operationsof subtractionand multiplication?
•Whatoperation is impliedwhen a number isnexttoavariable?What doesqxmean?
Misconceptions
Studentsmay
•Notrecognizesimilaritiesbetween thetwoequations.
•Arguethereis nowaytosolvethesecondequation.
•Believethatqxcannot bedivided.
Vocabulary
•Coefficient,Variable, Equivalence
Created by: Illinois StateBoardof Education Content Area Specialist

Mathematics:Functions

F.IF.4 / ClusterHeading:Interpretfunctionsthatarisein applicationsintermsofthecontext.
ContentStandard: For a function thatmodels a relationship betweentwoquantities,interpretkeyfeaturesof graphsandtablesintermsof thequantities,and sketchgraphsshowing keyfeaturesgiven averbal description.
Practice Standards: MP4Modelwithmathematics,MP6Attend toprecision.
Problem/TaskSuggestions / Formative AssessmentSuggestions
BallToss
A ball wasthrown intotheair and data wasrecorded tomodel itspath,with x representing horizontaldistancefromthestartpoint,and y representing the heightin feet.
Describethe pathof the ball in mathematicalterms. Includeall criticalpoints.
Differentiation
Supports
•Askthestudentsspecificsaboutthe problem,orprovidea worksheet thathasthesequestionson it:Howhigh did theball reach?Where did theball land if itspathcontinues?(Considermaking a graph or extending thetable)Whatshapewould a graphof the data take? Whatarethecriticalpointson thisgraph?
Extensions
•Writetheequationthat modelsthedata.(y=-.5x2+ 5x+5.5)
Solution
•Theball began at 5.5feetin theair.On a graph,thiswould bethey- intercept.Theballtraveled in a parabolicpath,landing 11feetaway fromthestart.Graphically,thisisthex-intercept.The highestpoint, thevertexof theparabola,is when theball is5feet horizontallyaway and 18feet high. / Observation of Students
•Isthestudentabletomodel parabolicdata?
•Doesthestudentcontextualizethevertexasthehighestpointreached?
•Doesthestudentcontextualizethex-interceptasthefarthestpoint traveled in a real-lifesituation?
•Doesthestudentcorrectlylabelthex- and y-variablescontextuallyand abstractly?
•Isthestudentabletoextrapolatea table byeithercreating a graph,rule, or by using thepatterntocontinuethetable?
QuestionstoGuide Student Thinking
•Whatdoesthepoint(0,5.5)represent?
•Whatdo you noticeaboutthedata andthepathoftheball?
•When you throwa ball howcanyou predictwhereitwill land?
Misconceptions
Studentsmay
•Believethatthe ballhitthe ground beforethestarting pointbecause thatis anotherlocationwherethey-value equals0.
•Believetheballneverhittheground becausethetableis incomplete.
•Believethatthe maximum heightis5 yardsbecausethatisthe horizontal distancewhentheballreachesitshighest point.
•Call theshapean upsidedown “U”or“V”,instead ofusing precise language.
•Saythattheball landsat7or16feet becausethis is wherethetable ends.
Vocabulary
•Parabola,Vertex,x-intercept, y-intercept
Created by: IllinoisStateBoardof Education Content Area Specialist

Mathematics:Geometry

G.CO.5
G.SRT.8 / ClusterHeading:Experimentwithtransformationsintheplane.
ContentStandard: Given ageometricfigureand arotation,reflection,ortranslation,drawthetransformed figureusing, e.g.,graph paper, tracing paper,or geometrysoftware.Specifya sequenceof transformationsthatwill carrya given figureontoanother.
ClusterHeading:Definetrigonometricratiosand solveproblemsinvolving righttriangles.
ContentStandard:Usetrigonometricratiosand the Pythagorean Theoremtosolverighttrianglesinapplied problems.
Practice Standards: MP1Makesenseof the problemand perseverein solving it,MP4Model withmathematics.
Problem/TaskSuggestions / Formative AssessmentSuggestions
EscalatorProblem
Atthelocalmall,an escalator willcarrypeoplefromtheground floor tothe basement. Thebasementis 3.6m belowtheground floor.Eachstepon the escalator needstomeasure24cmwideand18cmhigh.
-Howmuch horizontaldistanceisneeded toallowroomfor the escalator?
-Whatwillbethemeasureof theangleformed bytheescalator and basementfloor? Justifyyour response.
-Describethetransformation for a person startingatthetop ofthe escalator and riding tothe bottom. Justify your response.
Differentiation
Support
•Supplya drawing that hasthe vertical and horizontal distances labeled,along withthefirststep.
Extension
•Havethestudentsdescribethetransformation thatwould takeplace if insteadof anescalator, therewerestairs.
Solution
•Thenecessaryhorizontal distancewould be480cmor4.8 meters.
•Theanglewould beapproximately53.13 degreesor 0.927radians becauseyouwould needtodotheinversetangentofthehorizontal distance/vertical distance.
•Thetranslationwould bethata person would translate600cm downwardatan angleof 53.13 degrees. An alternativewould befor theperson totranslate360cm down and 480cmhorizontally, however,thisdoesnot describethe path the person traveled. / Observation of Students
•Can thestudentvisualizetheproblem,eithermentally orbyusing paper and pencil?
•Doesthestudentrecognize he/shewill haveto useaninverse trigonometricfunction tofind theangle?
•Can thestudentexplain theprocesshe/sheusedtofind theangleto their classmatesand inwriting? MP3
•Doesthestudentrecognizethedifferencebetween adiagonal translationand acombination ofa horizontaland vertical translation?
Questionsto Guide Student Thinking
•Could you drawa diagramof thesituation tohelp you?
•Howmanystepswillyou need toreach the basement?
•If you knowtwosidelengths,howcanyou findtheangleinvolved?
•Which transformationwillneed tooccurto movethe person?
Misconceptions
Studentsmay
•Forgettoconvertallofthe numberstoeithermetersor centimeters.
Vocabulary
Tangent,InverseTangent,Angle,Transformation,Translation
Adapted from: Bauer,et.al. (2006).SimmsIntegrated MathematicsLevel2,Dubuque,Iowa:Kendall/HuntPublishing. ISBN#978-0-7575-2030-3

Mathematics: Statistics and Probability

S.ID.3 / Cluster Heading: Summarize, represent, and interpret data on a single count or measurement variable.
Content Standard: Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).
Possible Practice Standard: MP3 Construct viable arguments and critique the reasoning of others.
Problem/Task Suggestions / Formative Assessment Suggestions
Clerk’s Wages
What measure of central tendency would most closely represent the potential pay of a new clerk if there are ten clerks who all make about the same amount and a manager who makes ten times as much? In your group, create a presentation report justifying why this would be the best choice.
Differentiation
Supports
  • Give the student examples of possible wages, or have them come up with possible wages, calculate the measures of central tendency and then decide which most closely represents the new clerks pay.
  • First have the student define or look up the meanings of the measures of central tendency.
Extension
  • If posting this job, what measure of central tendency would the company want to display?
  • Create a situation where another measure of central tendency would be more pertinent.
Solution
  • The new pay would most closely be represented by the median. The manager is pulling the mean heavily, making the wage look about double of what it would be. The mode may not be helpful because of the phrase “about the same amount”. The median would show the mid-range of the numbers, which would be one of the middle paid clerks.
/ Observation of Students
  • Is the student able to define and use correct vocabulary? (MP6)
  • Is the student able to see the value in the different measures of central tendency and use it to evaluate within the context?
Examining students’ presented work
Does the group
  • Address the median as the most appropriate answer?
  • Explain why they chose this measure of central tendency and use appropriate vocabulary? (MP6)
  • Communicate to the whole class?
  • Include full participation of all members of the group?
  • Answer questions from the class as needed?
Questions to Guide Student Thinking
  • What are the measures of central tendency?
  • Pick numbers that would model the situation and decide if the measure of central tendency that you picked previously still appears to most closely represent the new clerk’s wages.
Misconceptions
Students may
  • Explain that the answer is the mean because that is the average.
  • Explain that all of the measures of central tendency represent the same data, so the answer could be any of them.

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