Bound for Success Scope and Sequence Statements

MATHEMATICSPatterns and AlgebraConcept In Year 1the student:In Year 2the student:In Year 3the student:In Year 4the student:In Year 5the student:In Year 6the student:In Year 7the student:In Year 8the student:In Year 9the student:only 2 on the other side sowe need to put anotherperson on that side tomake it balanced”)(including 2 + 3 + 3 isgreater than 4 + 1 + 1) or‘unequal’ (including 6 – 4does not equal 10 – 5)equivalent, leaving it atthat)being told that the value ofa is 5 writes ‘3 + a = 4 + a– 1 = 12 – a – 2 = a – 3 +6’)to solve simple non-linearequations (e.g. finds thetime taken to travel 300km at an average speed of85 km/h, finds the value ofx for which 2x 2 + 3 = 53and determines theapproximate dimensions ofa rectangle with area100 cm 2 , width w andlength 3 cm greater than itswidth)Patterns,sequences andgeneralisationsPatterns provide thepower to think beyondspecific cases togeneral ones• Recognises patterns intheir daily lives (e.g.sunrise and sunset, goingto school for 5 days andhaving 2 days off)• Identifies a repeatingpattern made of actions orsounds or objects andperforms these in the orderidentified (e.g. hearssomeone stamp, click andclap, stamp, click, clap,and can repeat that pattern)• Recognises, continues anddescribes repeatingpatterns, identifying theelements of the patterncycle (e.g. says “there arethree colours in thepattern; red, blue andgreen and it goes red,blue, green, red, blue,green, red,…”)• Identifies and describespatterns as growing orrepeating and whatchanges from one elementof the pattern to the next(e.g. for the pattern, [ring],[ring, stamp, ring], [ring,stamp, ring, stamp, ring]says “there is another‘ring, stamp’ each time”)• Creates and continuespatterns and makes generalstatements of predictionabout what comes next(e.g. says ”the sixth onewill be green” andcontinues the pattern to seeif they are right)• Uses materials to representnumber and spatialpatterns for both growingand repeating patterns anddescribes what is changingfrom one element to thenext• Given the first 5 elementsof a pattern predicts thenext five and explains thebasis for their prediction,particularly for growingpatterns, and continues thepattern showing theirprediction to prove theyare right; Shows what iscommon (or generalisable)in a repeating pattern byusing a different form (e.g.shows the colour patterngreen, blue, green, blueusing actions such as clap,jump, clap, jump by firstrecognising there are twoelements in the pattern andthey repeat alternately)• Uses materials to representnumber and spatialpatterns (e.g. uses countersto model and identifytriangular and squarenumbers, and matchsticksto make growth patterns)• Understands therelationship between anelement in a pattern cycleand its position in thatcycle and can predict whatcomes next in the pattern(e.g. for the patternF E F E F E1 2 3 4 5 6 7they are able to see that ‘E’is in the ‘even’ positionsand can predict that the16th position will be evenand hence ‘E’ if the patterncontinues• Identifies and continuesnumber patterns anddescribes what is changingin words (e.g. continues 3,6, 9, 12…. and says “thepattern is ‘add three’ eachtime”, and continues 2, 5,10, 17…saying “thedifference between the firsttwo numbers is 3, the nexttwo 5, and the next two 7:this makes a new numberpattern where thedifference is two eachtime”); Writes the pointsgenerated as ordered pairs[e.g. (1,3),(2, 6), (3, 9) …and (1,2), (2,5), (3,10) ]• Identifies number patternsthat are ‘linear’ byexamining the differencepattern and recognisingthat linear patterns have aconstant difference termbetween the elements ofthe pattern, and that whenthese number patterns aregraphed in the firstquadrant the points form astraight line [e.g. (1,3),(2,6),(3, 9) … will form alinear pattern while (1,2),(2,5), (3,10) will not] andplots these points by handand with a graphingcalculator• Expands numbers andvariables represented byindex powers (e.g. 3 4 = 3 x3 x 3 x 3 and y 3 = y x y xy); Generalises numberpatterns and describeswhat they have noticed aschanging in the pattern in ageneral sense (e.g. onexamining the pattern2,4,8,16,… can describethe pattern as “eachnumber in the pattern isdoubled to get the nextterm”, for the sequence )• Explores and establisheslinear equivalences [e.g.2(4x + 8) = 4(2x + 4) = 8(x+ 2) ] and explores andestablishes simple nonlinearequivalences (e.g.generalises numberpatterns including 27 2 =(20 + 7) 2 = 20 2 + 2 x 20 x7 + 7 2 = 400 + 280 + 49 =72920 +7400 140 20140 49 +7• Uses a diagram to show (a+ 2) x (a + 3) = a 2 + 5a +6 and (2x) 2 = 4x 4 ;Generalises from numberexamples including 4 3 x 4 2= 4 5 and 4 5 ÷ 4 2 = 4 3 toshow that x 3 x 2 = x 5 andx 5 ÷ x 2 = x 3• Identifies where letters areused in mathematics torepresent words (e.g. ‘3c’represents ‘three cents’ and‘20 m’ represents ‘twentymetres’)• Identifies where letters areused in formulae torepresent differentquantities (e.g. l = 2 x wcan be used to representthe formula “length equalstwo times the width” andthat the two quantities‘length’ and ‘width’ canvary depending on the sizeof the quadrilateral) andcan tell the differencebetween two statementsusing letters: one wherethe letter represents a word(e.g. 5c) and one where theletter represents a quantitythat varies• Uses a letter/variable torepresent a variablequantity and explains whythe letter does not stand forthe word but for a number(e.g. given the statement‘Mary is three years olderthan John’ is representedby M = J + 3, can explainwhy ‘M’ does notrepresent ‘Mary’ but infact represents Mary’s age- a number, and in theformula P = 2 x l + 2 x w,l represents the numbermeasurement of the lengthand not the word ‘length’,reading this as “Perimeter• Understands concatenation(i.e. that ‘3x’ represents ‘3lots of whatever number xrepresents’ since x is avariable quantity; if ‘c’represents ‘the number ofcats’ then ‘4c’ does notrepresent four cats but‘four lots of the number ofcats’) and can complete atable such as the followingshowing differentrepresentations; Usesconcatenation to expand4(2x + 8) by reading thisas “four lots of, two lots ofwhatever number xrepresents, plus 8”;• Re-arranges linear andsome simple non-linearalgebraic expressions (e.g.p = 3q – 2 to obtain q =(p-2)/3; given A = πr 2 anda specific value of A, findsthe corresponding value ofr and obtains the generalcase r = √A/π noting that rmust be positive in therelated practical context;Uses words and symbols torepresent variables andconstants and interpretsalgebraic expressions forrelationships developed incontext; Constructsexpressions that involveBound for Success Scope and Sequence Statements V2 Page 44 Working Document Semester One 2007