Mathematics

C. DISCRETE FUNCTIONSTHE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics Grade 11, University PreparationOVERALL EXPECTATIONSBy the end of this course, students will:1. demonstrate an understanding of recursive sequences, represent recursive sequences in a variety ofways, and make connections to Pascal’s triangle;2. demonstrate an understanding of the relationships involved in arithmetic and geometric sequencesand series, and solve related problems;3. make connections between sequences, series, and financial applications, and solve problems involvingcompound interest and ordinary annuities.SPECIFIC EXPECTATIONS1. Representing SequencesBy the end of this course, students will:1.1 make connections between sequences anddiscrete functions, represent sequences usingfunction notation, and distinguish between adiscrete function and a continuous function[e.g., f(x) = 2x, where the domain is the set ofnatural numbers, is a discrete linear functionand its graph is a set of equally spaced points;f(x) = 2x, where the domain is the set of realnumbers, is a continuous linear function andits graph is a straight line]1.2 determine and describe (e.g., in words; usingflow charts) a recursive procedure for generatinga sequence, given the initial terms(e.g., 1, 3, 6, 10, 15, 21, …), and representsequences as discrete functions in a varietyof ways (e.g., tables of values, graphs)1.3 connect the formula for the nth term of asequence to the representation in functionnotation, and write terms of a sequence givenone of these representations or a recursionformula1.4 represent a sequence algebraically using arecursion formula, function notation, or theformula for the nth term [e.g., represent 2, 4,8, 16, 32, 64, … as t 1 = 2; t n = 2t n – 1 , asnn1 2 3f(n) = 2 , or as t n = 2 , or represent , , ,2 3 44 5 611, , , … as t 1 = ; t n = t n – 1 + ,5 6 72n(n + 1)nnas f(n) = , or as t n = , where nn + 1n + 1is a natural number], and describe the informationthat can be obtained by inspectingeach representation (e.g., function notationor the formula for the nth term may showthe type of function; a recursion formulashows the relationship between terms)Sample problem: Represent the sequence0, 3, 8, 15, 24, 35, … using a recursionformula, function notation, and the formulafor the nth term. Explain why this sequencecan be described as a discrete quadraticfunction. Explore how to identify a sequenceas a discrete quadratic function by inspectingthe recursion formula.1.5 determine, through investigation, recursivepatterns in the Fibonacci sequence, in relatedsequences, and in Pascal’s triangle, andrepresent the patterns in a variety of ways(e.g., tables of values, algebraic notation)1.6 determine, through investigation, anddescribe the relationship between Pascal’striangle and the expansion of binomials,and apply the relationship to expand binomialsraised to whole-number exponents4 5 6 2 5[e.g., (1 + x) ,(2x–1) , (2x – y), (x + 1) ]50