Course Strand Expectation Big Idea
MFM1P NS&A
MFM1P NS&A
MFM1P NS&A
MFM1P NS&A
MFM1P NS&A
MFM1P NS&A
MFM1P NS&A
MFM1P LR
MFM1P LR illustrate equivalent ratios, using a variety of tools (e.g., concrete materials, diagrams, dynamic geometry software) (e.g., show that 4:6 represents the same ratio as 2:3 by showing that a ramp with a height of 4 m and a base of 6 m and a ramp with a height of 2 m and a base of 3 m are equally steep); represent, using equivalent ratios and proportions, directly proportional relationships arising from realistic situations ( Sample problem : You are building a skateboard ramp whose ratio of height to base must be 2:3.Write a proportion that could be used to determine the base if the height is 4.5 m.); solve for the unknown value in a proportion, using a variety of methods
(e.g., concrete materials, algebraic reasoning, equivalent ratios, constant of proportionality) ( Sample problem : Solve = x
4

15
.);
20 make comparisons using unit rates (e.g., if 500 mL of juice costs $2.29, the unit rate is 0.458¢/mL; this unit rate is less than for 750 mL of juice at
$3.59, which has a unit rate of 0.479¢/mL); solve problems involving ratios, rates, and directly proportional relationships in various contexts (e.g., currency conversions, scale drawings, measurement), using a variety of methods (e.g., using algebraic reasoning, equivalent ratios, a constant of proportionality; using dynamic geometry software to construct and measure scale drawings) ( Sample problem : Simple interest is directly proportional to the amount invested. If
Lius invests $84 for one year and earns $1.26 in interest, how much would he earn in interest if he invested $235 for one year?) solve problems requiring the expression of percents, fractions, and decimals in their equivalent forms (e.g., calculating simple interest and sales tax; analysing data) ( Sample problem : Of the 29 students in a
Grade 9 math class, 13 are taking science this semester. If this class is representative of all the Grade 9 students in the school, estimate and calculate the percent of the 236 Grade 9 students who are taking science this semester. Estimate and calculate the number of Grade 9 students this percent represents.). simplify numerical expressions involving integers and rational numbers, with and without the use of technology;* identify, through investigation, some properties of linear relations (i.e., numerically, the first difference is a constant, which represents a constant rate of change; graphically, a straight line represents the relation), and apply these properties to determine whether a relation is linear or non-linear. determine, through investigation, that the rate of change of a linear relation can be found by choosing any two points on the line that represents the relation, finding the vertical change between the points
(i.e., the rise) and the horizontal change between the points (i.e., the run), and writing the ratio rise run
(i.e., rate of change = rise run
);

Course Strand
MFM1P
MFM1P
MFM1P
MFM1P
MFM1P
MPM1D
MPM1D
MPM1D
LR
LR
LR
LR
LR
NS&A
NS&A
LR
Expectation determine, through investigation, connections among the representations of a constant rate of change of a linear relation (e.g., the cost of producing a book of photographs is $50, plus $5 per book, so an equation is C = 50
+ 5 p ; a table of values provides the first difference of 5; the rate of change has a value of 5; and 5 is the coefficient of the independent variable, p , in this equation); compare the properties of direct variation and partial variation in applications, and identify the initial value (e.g., for a relation described in words, or represented as a graph or an equation) ( Sample problem :
Yoga costs $20 for registration, plus $8 per class.Tai chi costs $12 per class. Which situation represents a direct variation, and which represents a partial variation? For each relation, what is the initial value? Explain your answers.); express a linear relation as an equation in two variables, using the rate of change and the initial value (e.g., Mei is raising funds in a charity walkathon; the course measures 25 km, and Mei walks at a steady pace of
4 km/h; the distance she has left to walk can be expressed as d = 25 – 4 t , where t is the number of hours since she started the walk); describe the meaning of the rate of change and the initial value for a linear relation arising from a realistic situation (e.g., the cost to rent the community gym is $40 per evening, plus $2 per person for equipment rental; the vertical intercept, 40, represents the $40 cost of renting the gym; the value of the rate of change, 2, represents the $2 cost per person), and describe a situation that could be modeled by a given linear equation (e.g., the linear equation M = 50 + 6 d could model the mass of a shipping package, including 50 g for the packaging material, plus 6 g per flyer added to the package). describe the effects on a linear graph and make the corresponding changes to the linear equation when the conditions of the situation they represent are varied (e.g., given a partial variation graph and an equation representing the cost of producing a yearbook, describe how the graph changes if the cost per book is altered, describe how the graph changes if the fixed costs are altered, and make the corresponding changes to the equation); simplify numerical expressions involving integers and rational numbers, with and without the use of technology;* solve first-degree equations, including equations with fractional coefficients, using a variety of tools (e.g., computer algebra systems, paper and pencil) and strategies (e.g., the balance analogy, algebraic strategies); identify, through investigation, some properties of linear relations (i.e., numerically, the first difference is a constant, which represents a constant rate of change; graphically, a straight line represents the relation), and apply these properties to determine whether a relation is linear or nonlinear;
Big Idea

Course Strand
MPM1D
MPM1D
MPM1D
MPM1D
MPM1D
MPM1D
MPM1D
LR
LR
AG
AG
AG
AG
AG
Expectation compare the properties of direct variation and partial variation in applications, and identify the initial value (e.g., for a relation described in words, or represented as a graph or an equation) ( Sample problem : Yoga costs $20 for registration, plus $8 per class.Tai chi costs $12 per class.
Which situation represents a direct variation, and which represents a partial variation? For each relation, what is the initial value? Explain your answers.); describe the effects on a linear graph and make the corresponding changes to the linear equation when the conditions of the situation they represent are varied (e.g., given a partial variation graph and an equation representing the cost of producing a yearbook, describe how the graph changes if the cost per book is altered, describe how the graph changes if the fixed costs are altered, and make the corresponding changes to the equation).
determine, through investigation, various formulas for the slope of a line segment or a line (e.g., m
 rise run
, m
 thechangeiny thechangeinx
or m

 y
 x
, m
 y x
2
2

 y x
1
1
), and use the formulas to determine the slope of a line segment or a line; identify, through investigation with technology, the geometric significance of m and b in the equation y = mx + b ; determine, through investigation, connections among the representations of a constant rate of change of a linear relation (e.g., the cost of producing a book of photographs is $50, plus $5 per book, so an equation is C = 50
+ 5 p ; a table of values provides the first difference of 5; the rate of change has a value of 5, which is also the slope of the corresponding line; and
5 is the coefficient of the independent variable, p , in this equation); identify, through investigation, properties of the slopes of lines and line segments (e.g., direction, positive or negative rate of change, steepness, parallelism, perpendicularity), using graphing technology to facilitate investigations, where appropriate. describe the meaning of the slope and y -intercept for a linear relation arising from a realistic situation (e.g., the cost to rent the community gym is $40 per evening, plus $2 per person for equipment rental; the vertical intercept, 40, represents the $40 cost of renting the gym; the value of the rate of change, 2, represents the $2 cost per person), and describe a situation that could be modelled by a given linear equation (e.g., the linear equation M = 50 + 6 d could model the mass of a shipping package, including 50 g for the packaging material, plus 6 g per flyer added to the package);
Big Idea

Course Strand
MFM2P
MFM2P
MFM2P
MFM2P
MT
MT
MT
MT
Expectation verify, through investigation (e.g., using dynamic geometry software, concrete materials), properties of similar triangles (e.g., given similar triangles, verify the equality of corresponding angles and the proportionality of corresponding sides); determine the lengths of sides of similar triangles, using proportional reasoning; solve problems involving similar triangles in realistic situations (e.g., shadows, reflections, scale models, surveying) ( Sample problem : Use a metre stick to determine the height of a tree, by means of the similar triangles formed by the tree, the metre stick, and their shadows.).
determine, through investigation (e.g., using dynamic geometry software, concrete materials), the relationship between the ratio of two sides in a right triangle and the ratio of the two corresponding sides in a similar right triangle, and define the sine, cosine, and tangent ratios opposite
(e.g.,sin A = hypotenuse
);
MFM2P MT
MFM2P MT
MFM2P MT
MFM2P MLR
MFM2P MLR
MFM2P MLR determine the measures of the sides and angles in right triangles, using the primary trigonometric ratios and the Pythagorean theorem; solve problems involving the measures of sides and angles in right triangles in real life applications (e.g., in surveying, in navigation, in determining the height of an inaccessible object around the school), using the primary trigonometric ratios and the Pythagorean theorem
( Sample problem : Build a kite, using imperial measurements, create a clinometer to determine the angle of elevation when the kite is flown, and use the tangent ratio to calculate the height attained.); perform everyday conversions between the imperial system and the metric system (e.g., millilitres to cups, centimetres to inches) and within these systems (e.g., cubic metres to cubic centimetres, square feet to square yards), as necessary to solve problems involving measurement ( Sample problem : A vertical post is to be supported by a wooden pole, secured on the ground at an angle of elevation of 60°, and reaching 3 m up the post from its base. If wood is sold by the foot, how many feet of wood are needed to make the pole?); solve first-degree equations involving one variable, including equations with fractional coefficients (e.g. using the balance analogy, computer algebra systems, paper and pencil) ( Sample problem: Solve x
2
 
3 x

1 and verify.); connect the rate of change of a linear relation to the slope of the line, and define the slope as the ratio m = rise run
; identify, through investigation, properties of the slopes of lines and line segments (e.g., direction, positive or negative rate of change, steepness, parallelism), using graphing technology to facilitate investigations, where appropriate;
Big Idea

Course Strand
MFM2P QR
Expectation determine, through investigation using technology, that a quadratic relation of the form y = ax 2 + bx + c ( a  
0) can be graphically represented as a parabola, and determine that the table of values yields a constant second difference (
: Graph the quadratic relation y = x 2
– 4, using technology. Observe the shape of the graph.
Consider the corresponding table of values, and calculate the first and second differences. Repeat for a different quadratic relation. Describe your observations and make conclusions.);
MPM2D QR
MPM2D T
MPM2D T determine, through investigation using technology, that a quadratic relation of the form y = ax 2 + bx + c ( a  
0) can be graphically represented as a parabola, and determine that the table of values yields a constant second difference (
: Graph the quadratic relation y = x 2
–
4 x , using technology. Observe the shape of the graph.
Consider the corresponding table of values, and calculate the first and second differences. Repeat for a different quadratic relation. Describe your observations and make conclusions.); verify, through investigation (e.g., using dynamic geometry software, concrete materials), the properties of similar triangles (e.g., given similar triangles, verify the equality of corresponding angles and the proportionality of corresponding sides); describe and compare the concepts of similarity and congruence;
MPM2D T
MPM2D T
MPM2D T
MPM2D T
MPM2D T solve problems involving similar triangles in realistic situations (e.g., shadows, reflections, scale models, surveying) ( Sample problem: Use a metre stick to determine the height of a tree, by means of the similar triangles formed by the tree, the metre stick, and their shadows.). determine, through investigation (e.g., using dynamic geometry software, concrete materials), the relationship between the ratio of two sides in a right triangle and the ratio of the two corresponding sides in a similar right triangle, and define the sine, cosine, and tangent ratios opposite
(e.g.,sin A = ); hypotenuse determine the measures of the sides and angles in right triangles, using the primary trigonometric ratios and the Pythagorean theorem; solve problems involving the measures of sides and angles in right triangles in real life applications (e.g., in surveying, in navigating, in determining the height of an inaccessible object around the school), using the primary trigonometric ratios and the Pythagorean theorem. explore the development of the sine law within acute triangles (e.g., use dynamic geometry software to determine that the ratio of the side lengths equals the ratio of the sines of the opposite angles; follow the algebraic development of the sine law and identify the application of solving systems of equations [student reproduction of the development of the formula is not required]);
Big Idea

Course Strand
MPM2D T
MPM2D T
MPM2D T
Expectation explore the development of the cosine law within acute triangles (e.g., use dynamic geometry software to verify the cosine law; follow the algebraic development of the cosine law and identify its relationship to the
Pythagorean theorem and the cosine ratio [student reproduction of the development of the formula is not required]); determine the measures of sides and angles in acute triangles, using the sine law and the cosine law ( Sample problem : In triangle ABC,
 A = 35°,
 B = 65°, and AC = 18 cm. Determine BC. Check your result using dynamic geometry software.); solve problems involving the measures of sides and angles in acute triangles.
Big Idea
MEL3E EP3.2
MEL3E EP3.8
MEL3E SI2.1
MEL3E SI2.4
MEL3E TT2.1 estimate the sale price before taxes when making a purchase (e.g., est imate 25% off of $38.99 as 25% or ¼ off of $40, giving a discount of about $10 and a sale price of approximately $30; alternatively, estimate the same sale price as about ¾ of $40) compare the unit prices of related items to help determine the best buy
Sample problem: Investigate whether or not purchasing larger quantities always results in a lower unit price. determine, through investigation using technology (e.g., calculator, spreadsheet), the effect on simple interest of changes in the principal, interest rate, or time, and solve problems involving applications of simple interest determine, through investigation using technology, the effect on the future value of a compound interest investment of changing the total length of time, the interest rate, or the compounding period
Sample problem: Compare the results at age 40 of making a deposit of
$1000 at age 20 or a deposit of $2000 at age 30, if both investments pay
6% interest per annum, compounded monthly. determine distances represented on maps (e.g., provincial road map, local street map, web-based maps), using given scales
MBF3C
MBF3C
MBF3C
MM2.6
PF1.6
DM2.2 distinguish exponential relations from linear and quadratic relations by making comparisons in a variety of ways (e.g., comparing rates of change using finite differences in tables of values; inspecting graphs; comparing equations), within the same context when possible (e.g., simple interest and compound interest, population growth) determine, through investigation using technology (e.g., a TVM Solver on a graphing calculator or on a website), the effect on the future value of a compound interest investment or loan of changing the total length of time, the interest rate, or the compounding period Sample problem:
Investigate whether doubling the interest rate will halve the time it takes for an investment to double. determine the theoretical probability of an event (i.e., the ratio of the number of favourable outcomes to the total number of possible outcomes, where all outcomes are equally likely), and represent the probability in a variety of ways (e.g., as a fraction, as a percent, as a decimal in the range 0 to 1)

Course
MBF3C
MBF3C
MCF3M
MCF3M
Strand
GT1.2
GT2.1
EF1.6
EF3.1
Expectation represent three-dimensional objects, using concrete materials and design or drawing software, in a variety of ways (e.g., orthographic projections, perspective isometric drawings, scale models) solve problems, including those that arise from real-world applications by determining the measures of the sides and angles of right triangles using primary trigonometric ratios distinguish exponential functions from linear and quadratic functions by making comparisons in a variety of ways (e.g., comparing rates of change using finite differences in tables of values; identifying a constant ratio in a table of values; inspecting graphs; comparing equations), within the same context when possible (e.g., simple interest and compound interest, population growth) compare, using a table of values and graphs, the simple and compound interest earned for a given principal and a fixed interest rate over time
Big Idea
MCF3M
MCF3M
TF1.1
TF1.3 solve problems, including those that arise from real-world applications, by determining the measures of the sides and angles of right triangles using the primary trigonometric ratios verify, through investigation using technology the sine law and the cosine law
MCR3U CF3.3 simplify rational expressions by adding, subtracting, multiplying, and dividing, and state the restrictions on the variable values
MCR3U
MCR3U
EF2.1
DF2.1 distinguish exponential functions from linear and quadratic functions by making comparisons in a variety of ways (e.g., comparing rates of change using finite differences in tables of values; identifying a constant ratio in a table of values; inspecting graphs; comparing equations) identify sequences as arithmetic, geometric, or neither, given a numeric or algebraic representations
MCR3U TF1.1 determine the exact values of the sine, cosine, and tangent of the special angles: 0, 30, 45, 60, and 90
MCR3U
MCR3U
TF1.6
TF1.7 pose problems involving right triangles and oblique triangles in two dimensional settings, and solve these and other such problems using the primary trigonometric ratios, the cosine law and the sine law pose problems involving right triangles and oblique triangles in three dimensional settings, and solve these and other such problems using the primary trigonometric ratios, the cosine law and the sine law

Course
Strand/
Number
Expectation
MEL4E RD2.1
MEL4E AM1.4
MEL4E AM3.1
MEL4E AM3.2 determine the theoretical probability of an event (i.e., the ratio of the number of favourable outcomes to the total number of possible outcomes, where all outcomes are equally likely), and represent the probability in a variety of ways (e.g., as a fraction, as a percent, as a decimal in the range 0 to 1) convert measures within systems (e.g., centimeters and metres, kilograms and grams, litres and milliliters, feet and inches, ounces and pounds), as required within applications that arise from familiar contexts. identify and describe applications of ratio and rate, and recognize and represent equivalent ratios (e.g., show that 4:6 represents the same ratio as 2:3 by showing that a ramp with a height of 4m and a base of 6m and a ramp with a height of 2m and a base of 3m are equally steep) and equivalent rates (e.g., recognize that paying $1.25 for 250mL of tomato sauce is equivalent to paying $3.75 for 750mL of the same sauce), using a variety of tools identify situations in which it is useful to make comparisons using unit rates, and solve problems that involve comparisons of unit rates
Big Idea
MEL4E AM3.3
MEL4E AM3.4
identify and describe real-world applications of proportional reasoning
(e.g., mixing concrete; calculating dosages; converting units; painting walls; calculating fuel consumption; calculating pay; enlarging patterns), distinguish between a situation involving a proportional relationship and a situation involving a non-proportional relationship in a personal and/or workplace context, and explain their reasoning identify and describe the possible consequences (e.g., overdoses of medication; seized engines; ruined clothing; cracked or crumbling concrete) of errors in proportional reasoning (e.g., not recognizing the importance of maintaining proportionality; not correctly calculating the amount of each component in a mixture)
MAP4C MM2.6
MAP4C MM2.4
MAP4C GT1.1 recognize that a linear model corresponds to a constant increase or decrease over equal intervals and that an exponential model corresponds to a constant percentage increase or decrease over equal intervals, select a model (i.e., linear, quadratic, exponential) to represent the relationship between numerical data graphically and algebraically, using a variety of tools (e.g., graphing technology) and strategies (e.g., finite differences,regression), and solve related problems identify when the rate of change is zero, constant or changing, given a table of values or a graph of a relations, and compare two graphs by describing rate of change perform required conversions between the imperial systems and the metric systems using a variety of tools (e.g. tables, calculators, online conversion tools), as necessary within applications

Course Strand Expectation
MAP4C GT2.1
MAP4C DM1.5
MAP4C DM1.6 solve problems in two dimensions using metric or imperial measurements, including problems that arise from real-world applications by determine the measures of the sides and angles of right triangles using the primary trigonometric ratios, and of acute triangles using the sine law and the cosine law determine an algebraic summary of the relationship between two variables that appear to be linearly related, using a variety of tools and strategies and solve related problems describe possible interpretations of the line of best fit of a scatter plot
(e.g. the variables are linearly related) and reasons for misinterpretations
(e.g., using too small a sample, failing to consider the effect of outliers; interpolating from a weak correlation; extrapolating non-linearly related data)
MCT4C TF1.1 determine the exact values of the sine, cosine, and tangent of the special angles: 0°, 30°, 45°, 60°, and 90°
MCT4C TF1.4
MCT4C AG2.2 solve multi-step problems in two and three dimensions, including those that arise from real-world applications (e.g. surveying, navigation), by determining the measures of the sides and angles of right triangles using the primary trigonometric ratios perform required conversions between the imperial system and the metric system using a variety of tools as necessary within applications
Big Idea
MHF4U CF1.2
MHF4U CF3.1 recognize that the rate of change for a function is a comparison of changes in the dependent variable to changes in the independent variable, and distinguish situations in which the rate of change is zero, constant, or changing by examining applications, including those arising from real-world situations (e.g., rate of change of the area of a circle as the radius increases, inflation rates, the rising trend in graduation rates among Aboriginal youth, speed of a cruising aircraft, speed of a cyclist climbing a hill, infection rates) Sample problem: The population of bacteria in a sample is 250 000 at 1:00 p.m., 500 000 at 3:00 p.m., and 1
000 000 at 5:00 p.m. Compare methods used to calculate the change in the population and the rate of change in the population between 1:00 p.m. to 5:00 p.m. Is the rate of change constant? Explain your reasoning. compare, through investigation using a variety of tools and strategies the characteristics of various functions
MHF4U TF1.1 recognize the radian as an alternative unit to the degree for angle measurement, define the radian measure of an angle as the length of the arc that subtends this angle at the centre of a unit circle, and develop and apply the relationship between radian and degree measure

Course Strand Expectation
MHF4U TF1.4
MFH4U TF3.3 determine, without technology, the exact values of the primary trigonometric ratios and the reciprocal trigonometric ratios for the special angles and their multiples less than or equal to 2π recognize that trigonometric identities are equations that are true for every value in the domain (i.e., a counter-example can be used to show that an equation is not an identity), prove trigonometric identities through the application of reasoning skills, using a variety of relationships (e.g., tan x = sin x/cos x; sin 2 x + cos 2 x = 1; the reciprocal identities; the compound angle formulas), and verify identities using technology
Sample problem: Use the compound angle formulas to prove the double angle formulas.
MDM4U CP1.3
MDM4U CP2.2
MDM4U CP2.3
MDM4U
MDM4U
SA3.1
SA3.2 determine the theoretical probability, P i
of each outcome of a discrete sample space, recognize that the sum of the probabilities of the outcomes is 1, recognize that the probabilities P i
form the probability distribution associated with the sample space, and solve related problems solve simple problems using techniques for counting permutations and combinations, where all objects are distinct, and express the solutions using standard combinatorial notation [e.g., n!, P(n, r), (n r)]
Sample problem: In many Aboriginal communities, it is common practice for people to shake hands when they gather. Use combinations to determine the total number of handshakes when 7 people gather, and verify using a different strategy. solve introductory counting problems involving the additive counting principle (e.g., determining the number of ways of selecting 2 boys or 2 girls from a group of 4 boys and 5 girls) and the multiplicative counting principle (e.g., determining the number of ways of selecting 2 boys and 2 girls from a group of 4 boys and 5 girls) interpret statistics presented in the media (e.g., the UN's finding that 2% of the world's population has more than half the world's wealth, whereas half the world's population has only 1% of the world's wealth), and explain how the media, the advertising industry, and others (e.g., marketers, pollsters) use and misuse statistics (e.g., as represented in graphs) to promote a certain point of view (e.g., by making a general statement based on a weak correlation or an assumed cause-and-effect relationship; by starting the vertical scale at a value other than zero; by making statements using general population statistics without reference to data specific to minority groups) assess the validity of conclusions presented in the media by examining sources of data, including Internet sources, methods of data collection, and possible sources of bias, and by questioning the analysis of the data and conclusions drawn from the data
Big Idea

Course Strand Expectation
MCV4U RC2.5
MCV4U RC2.8 determine, through investigation using technology, the graph of the derivative f'(x)or dy/dx of a given exponential function [i.e., f(x) = a x > 0, a
≠ 1)] [e.g., by generating a table of values showing the instantaneous rate of change of the function for various values of x and graphing the ordered pairs; by using dynamic geometry software to verify that when f(x) = a x , f'(x) = kf(x)], and make connections between the graphs of f(x) and f'(x) or y and dy/dx [e.g., f(x) and f'(x) are both exponential; the ratio f'(x)/f(x) is constant, or f'(x) = kf(x); f'(x) is a vertical stretch from the x-axis of f(x)]
Sample problem: Graph, with technology, f(x) = a(x) (a > 0, a≠ 1) and f'(x) on the same set of axes for various values of a (e.g., 1.7, 2.0, 2.3, 3.0,
3.5). For each value of a, investigate the ratio f'(x)/f(x) for various values of x, and explain how you can use this ratio to determine the slopes of tangents to f(x). verify, using technology (e.g., calculator, graphing technology), that the derivative of the exponential function f(x) = a x is f'(x) = a x ln a for various values of a [e.g., verifying numerically for f(x) = 2 x that f'(x) = 2 x ln 2 by using a calculator to show that is ln 2 or by graphing f(x) =
2 x , determining the value of the slope and the value of the function for specific x-values, and comparing the ratio f'(x)/f(x) with ln 2] Sample problem: Given f(x) = e x , verify numerically with technology using
that f'(x) = f(x)ln e.
MCV4U GA1.2
MCV4U GA2.1 represent a vector in two-space geometrically as a directed line segment, with directions expressed in different ways, and algebraically, and recognize vectors with the same magnitude and direction but different positions as equal vectors perform the operations of addition, subtraction, and scalar multiplication on vectors represented as directed line segments in two-space, and on vectors represented in Cartesian form in two-space and three-space
Big Idea