Grade Eleven Math Curriculum
Introduction
Grade 11 mathematics builds on the elementary program, relying on the same fundamental principles on which that program was based. It is founded on the premise that students learn mathematics most effectively when they have a thorough understanding of mathematical concepts and procedures, and when they build that understanding through an investigative approach, as reflected in the inquiry model of learning. This curriculum is designed to help students build a solid conceptual foundation in mathematics that will enable them to apply their knowledge and skills and further their learning successfully. Grade 11 is also an extremely competitive year where students tend to feel the pressure of college applications and extra credits for admissions. Mathematics at this point becomes even more critical as a mandatory subject, making its understanding imperative. The information provided in this page are identical to the Official Grade Eleven Math Curriculum established by the Ministry of Education.
Table of Contents
Overall and Specific Expectations
Four types of courses are offered in the senior mathematics program of grade 11: university preparation, university/college preparation, college preparation, and workplace preparation. Students choose course types on the basis of their interests, achievement, and postsecondary goals. The course types are defined as follows:
- University preparation courses
These are designed to equip students with the knowledge and skills they need to meet the entrance requirements for university programs.
- University/college preparation courses
Theses are designed to equip students with the knowledge and skills they need to meet the entrance requirements for specific programs offered at universities and colleges.
- College preparation courses
These are designed to equip students with the knowledge and skills they need to meet the requirements for entrance to most college programs or for admission to specific apprenticeship or other training programs.
- Workplace preparation courses
These are designed to equip students with the knowledge and skills they need to meet the expectations of employers, if they plan to enter the workplace directly after graduation, or the requirements for admission to many apprenticeships or other training programs.
The expectations identified for each course describe the knowledge and skills that students are expected to acquire, demonstrate, and apply in their class work, on tests, and in various other activities on which their achievement is assessed and evaluated.
University Preparation: Functions
This course introduces the mathematical concept of the function by extending students’ experiences with linear and quadratic relations. Students will investigate properties of discrete and continuous functions, including trigonometric and exponential functions; represent functions numerically, algebraically, and graphically; solve problems involving applications of functions; investigate inverse functions; and develop facility in determining equivalent algebraic expressions. Students will reason mathematically and communicate their thinking as they solve multi-step problems.
The overall expectations are divided in four main categories each of which contains several subcategories. Every subcategory lists specific expectations for this course of grade 11 in more detail.
communicate their thinking as they solve multi-step problems.
The overall expectations are divided in three main categories each of which contains several subcategories. Every subcategory lists specific expectations for academic courses of grade 10 in more detail.
Characteristics of Functions
A.Representing functions:
Students will
1.explain the meaning of the term function, and distinguish a function from a relation that is not a function, through investigation of linear and quadratic relations using a variety of representations (i.e., tables of values, mapping diagrams, graphs, function machines, equations) and strategies (e.g., identifying a one-to-one or many-to-one mapping; using the vertical-line test).
2.represent linear and quadratic functions using function notation, given their equations, tables of values, or graphs, and substitute into and evaluate functions [e.g., evaluate given ].
3.explain the meanings of the terms domain and range, through investigation using numeric, graphical, and algebraic representations of the functions and ; describe the domain and range of a function appropriately (e.g., for , the domain is the set of real numbers, and the range is ); and explain any restrictions on the domain and range in contexts arising from real-world applications.
4.relate the process of determining the inverse of a function to their understanding of reverse processes (e.g., applying inverse operations).
5.determine the numeric or graphical representation of the inverse of a linear or quadratic function, given the numeric, graphical, or algebraic representation of the function, and make connections, through investigation using a variety of tools (e.g., graphing technology, Mira, tracing paper), between the graph of a function and the graph of its inverse (e.g., the graph of the inverse is the reflection of the graph of the function in the line ).
6.determine, through investigation, the relationship between the domain and range of a function and the domain and range of the inverse relation and determine whether or not the inverse relation is a function.
7.determine, through investigation using technology, the roles of the parameters a, k, d, and c in functions of the form , and describe these roles in terms of transformations on the graphs of , and (i.e., translations; reflections in the axes; vertical and horizontal stretches and compressions to and from the x– and y-axes).
8.Sketch graphs of by applying one or more of transformations to the graphs of , and and state the domain and range of the transformed functions.
B.Solving problems involving quadratic functions:
Students will
1.determine the number of zeros (i.e., x-intercepts) of a quadratic function, using a variety of strategies (e.g., inspecting graphs; factoring; calculating the discriminant).
2.determine the number of zeros (i.e., x-intercepts) of a quadratic function, using a variety of strategies (e.g., inspecting graphs; factoring; calculating the discriminant).
3.solve problems involving quadratic functions arising from real-world applications and represented using function
4.solve problems involving quadratic functions arising from real-world applications and represented using function
5.solve problems involving the intersection of a linear function and a quadratic function graphically and algebraically (e.g., determine the time when two identical cylindrical water tanks contain equal volumes of water, if one tank is being filled at a constant rate and the other is being emptied through a hole in the bottom).
C.Determining equivalent algebraic expressions:
Students will
1.Determining equivalent algebraic expressions: Students will
2.verify, through investigation with and without technology, that , and use this relationship to simplify radicals (e.g.,) and radical expressions obtained by adding, subtracting, and multiplying [e.g., ].
3.simplify rational expressions by adding, subtracting, multiplying, and dividing, and state the restrictions on the variable values.
4.determine if two given algebraic expressions are equivalent (i.e., by simplifying; by substituting values).
Exponential Functions
A.Representing exponential functions:
Students will
1.graph, with and without technology, an exponential relation, given its equation in the form define this relation as the function , and explain why it is a function.
2.determine, through investigation using a variety of tools (e.g., calculator, paper and pencil, graphing technology) and strategies (e.g., patterning; finding values from a graph; interpreting the exponent laws), the value of a power with rational exponent (i.e., , where and m, n are integers).
3.simplify algebraic expressions containing integer and rational exponents [e.g., ] and evaluate numeric expressions containing integer and rational exponents and rational bases [e.g., ].
4.determine, through investigation, and describe key properties relating to domain and range, intercepts, increasing/decreasing intervals, and asymptotes (e.g., the domain is the set of real numbers; the range is the set of positive real numbers; the function either increases or decreases throughout its domain) for exponential functions represented in a variety of ways [e.g., tables of values, mapping diagrams, graphs, equations of the form function machines].
B.Connecting graphs and expectations of exponential functions:
Students will
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).
2.determine, through investigation using technology, the roles of the parameters a, k, d, and c in functions of the form c, and describe these roles in terms of transformations on the graph of (i.e., translations; reflections in the axes; vertical and horizontal stretches and compressions to and from the x– and y-axes).
3.sketch graphs of by applying one or more transformations to the graph of , and state the domain and range of the transformed functions.
4.technology, that the equation of a given exponential function can be expressed using different bases [e.g., can be expressed as ] and explain the connections between the equivalent forms in a variety of ways (e.g., comparing graphs; using transformations; using the exponent laws).
5.represent an exponential function with an equation, given its graph or its properties.
C.Solving problems involving exponential functions:
Students will
1.collect data that can be modelled as an exponential function, through investigation with and without technology, from primary sources, using a variety of tools (e.g., concrete materials such as number cubes, coins; measurement tools such as electronic probes), or from secondary sources (e.g., websites such as Statistics Canada, E-STAT), and graph the data.
2.identify exponential functions, including those that arise from real-world applications involving growth and decay (e.g., radioactive decay, population growth, cooling rates, pressure in a leaking tire), given various representations (i.e., tables of values, graphs, equations), and explain any restrictions that the context places on the domain and range (e.g., ambient temperature limits the range for a cooling curve).
3.solve problems using given graphs or equations of exponential functions arising from a variety of real-world applications (e.g., radioactive decay, population growth, height of a bouncing ball, compound interest) by interpreting the graphs or by substituting values for the exponent into the
Discrete Functions
A.Representing sequences:
Students will
1.make connections between sequences and discrete functions, represent sequences using function notation, and distinguish between a discrete function and a continuous function [e.g., , where the domain is the set of natural numbers, is a discrete linear function and its graph is a set of equally spaced points; , where the domain is the set of real numbers, is a continuous linear function and its graph is a straight line].
2.determine and describe (e.g., in words; using flow charts) a recursive procedure for generating a sequence, given the initial terms (e.g., 1, 3, 6, 10, 15, 21, …), and represent sequences as discrete functions in a variety of ways (e.g., tables of values, graphs).
3.connect the formula for the nth term of a sequence to the representation in function notation and write terms of a sequence given one of these representations or a recursion formula.
4.represent a sequence algebraically using a recursion formula, function notation, or the formula for the nth term [e.g., represent 2, 4, 8, 16, 32, 64, … as , as , or as , or represent as ; , as , or as , where n is a natural number], and describe the information that can be obtained by inspecting each representation (e.g., function notation or the formula for the nth term may show the type of function; a recursion formula shows the relationship between terms).
5.determine, through investigation, recursive patterns in the Fibonacci sequence, in related sequences, and in Pascal’s triangle, and represent the patterns in a variety of ways (e.g., tables of values, algebraic notation).
6. determine, through investigation, recursive patterns in the Fibonacci sequence, in related sequences, and in Pascal’s triangle, and represent the patterns in a variety of ways (e.g., tables of values, algebraic notation).
B.Investigating arithmetic and geometric sequences and series:
Students will
1.identify sequences as arithmetic, geometric, or neither, given a numeric or algebraic representation.
2.determine the formula for the general term of an arithmetic sequence [i.e., ] or geometric sequence (i.e., ), through investigation using a variety of tools (e.g., linking cubes, algebra tiles, diagrams, calculators) and strategies (e.g., patterning; connecting the steps in a numerical example to the steps in the algebraic development), and apply the formula to calculate any term in a
3.determine the formula for the sum of an arithmetic or geometric series, through investigation using a variety of tools (e.g., linking cubes, algebra tiles, diagrams, calculators) and strategies (e.g., patterning; connecting the steps in a numerical example to the steps in the algebraic development), and apply the formula to calculate the sum of a given number of consecutive
4.determine the radius of a circle with center (0, 0), given its equation; write the equation of a circle with centre (0, 0), given the radius; and sketch the circle, given the equation in the form .
5.solve problems involving arithmetic and geometric sequences and series, including those arising from real-world applications.
C.Solving problems involving financial applications:
Students will
1.make and describe connections between simple interest, arithmetic sequences, and linear growth, through investigation with technology (e.g., use a spreadsheet or graphing calculator to make simple interest calculations, determine first differences in the amounts over time, and graph amount versus time).
2.make and describe connections between compound interest, geometric sequences, and exponential growth, through investigation with technology (e.g., use a spreadsheet to make compound interest calculations, determine finite differences in the amounts over time, and graph amount versus time).
3.solve problems, using a scientific calculator, that involve the calculation of the amount, A (also referred to as future value, FV ), the principal, P (also referred to as present value, PV ), or the interest rate per compounding period, i, using the compound interest formula in the form [or ].
4.determine, through investigation using technology (e.g., scientific calculator, the TVM Solver on a graphing calculator, online tools), the number of compounding periods, n, using the compound interest formula in the form [or ]; describe strategies (e.g., guessing and checking; using the power of a power rule for exponents; using graphs) for calculating this number; and solve related problems.
5.determine, through investigation using technology (e.g., scientific calculator, the TVM Solver on a graphing calculator, online tools), the number of compounding periods, n, using the compound interest formula in the form [or ]; describe strategies (e.g., guessing and checking; using the power of a power rule for exponents; using graphs) for calculating this number; and solve related problems.
6.solve problems, using technology (e.g., scientific calculator, spreadsheet, graphing calculator), that involve the amount, the present value, and the regular payment of an ordinary simple annuity (e.g., calculate the total interest paid over the life of a loan, using a spreadsheet, and compare the total interest with the original principal of the loan).
Trigonometric Functions
A.Determining and applying trigonometric ratios:
Students will
1.determine the exact values of the sine, cosine, and tangent of the special angles: 0º, 30º, 45º, 60º, and 90º.
2.determine the values of the sine, cosine, and tangent of angles from 0º to 360º, through investigation using a variety of tools (e.g., dynamic geometry software, graphing tools) and strategies (e.g., applying the unit circle; examining angles related to special angles).
3.determine the measures of two angles from 0º to 360º for which the value of a given trigonometric ratio is the
4.define the secant, cosecant, and cotangent ratios for angles in a right triangle in terms of the sides of the triangle (e.g., ), and relate these ratios to the cosine, sine, and tangent ratios (e.g., ).
5.prove simple trigonometric identities, using the Pythagorean identity ; the quotient identity ; and the reciprocal identities , , and .
6.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 (including the ambiguous case).
7.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
B.Connecting graphs and equations of sinusoidal functions:
Students will
1.describe key properties (e.g., cycle, amplitude, period) of periodic functions arising from real-world applications (e.g., natural gas consumption in Ontario, tides in the Bay of Fundy), given a numeric or graphical.
2.predict, by extrapolating, the future behavior of a relationship modelled using a numeric or graphical representation of a periodic function (e.g., predicting hours of daylight on a particular date from previous measurements, predicting natural gas consumption in Ontario from previous consumption).
3.make connections between the sine ratio and the sine function and between the cosine ratio and the cosine function by graphing the relationship between angles from 0º to 360º and the corresponding sine ratios or cosine ratios, with or without technology (e.g., by generating a table of values using a calculator; by unwrapping the unit circle), defining this relationship as the function or , and explaining why the relation-ship is a function.
4.sketch the graphs of and for angle measures expressed in degrees, and determine and describe their key properties (i.e., cycle, domain, range, intercepts, amplitude, period, maximum and minimum values, increasing/decreasing intervals).
5.determine, through investigation using technology, the roles of the parameters a, k, d, and c in functions of the form , where or with angles expressed in degrees, and describe these roles in terms of transformations on the graphs of and (i.e., translations; reflections in the axes; vertical and horizontal stretches and compressions to and from the x– and y-axes).
6.determine the amplitude, period, phase shift, domain, and range of sinusoidal functions whose equations are given in the form .
7.sketch graphs of by applying one or more transformations to the graphs of and , and state the domain and range of the transformed functions.
8.represent a sinusoidal function with an equation, given its graph or its properties.
C.Solving problems involving sinusoidal functions:
Students will
1.collect data that can be modelled as a sinusoidal function (e.g., voltage in an AC circuit, sound waves), through investigation with and without technology, from primary sources, using a variety of tools (e.g., concrete materials, measurement tools such as motion sensors), or from secondary sources (e.g., websites such as Statistics Canada, E-STAT), and graph the data.
2.identify periodic and sinusoidal functions, including those that arise from real-world applications involving periodic phenomena, given various representations (i.e., tables of values, graphs, equations), and explain any restrictions that the context places on the domain and range.
3.determine, through investigation, how sinusoidal functions can be used to model periodic phenomena that do not involve angles.
4.predict the effects on a mathematical model (i.e., graph, equation) of an application involving periodic phenomena when the conditions in the application are varied (e.g., varying the conditions, such as speed and direction, when walking in a circle in front of a motion sensor).
5.pose problems based on applications involving a sinusoidal function and solve these and other such problems by using a given graph or a graph generated with technology from a table of values or from its equation.
University/College Preparation: Functions and Applications
This course introduces basic features of the function by extending students’ experiences with quadratic relations. It focuses on quadratic, trigonometric, and exponential functions and their use in modelling real-world situations. Students will represent functions numerically, graphically, and algebraically; simplify expressions; solve equations; and solve problems relating to applications. Students will reason mathematically and communicate their thinking as they solve multi-step problems.
The overall expectations are divided in four main categories each of which contains several subcategories. Every subcategory lists specific expectations for this course of grade 11 in more detail.
Quadratic Functions
A.Solving quadratic equations:
Students will
1.pose problems involving quadratic relations arising from real-world applications and represented by tables of values and graphs, and solve these and other such problems (e.g., “From the graph of the height of a ball versus time, can you tell me how high the ball was thrown and the time when it hit the ground?”).
2.represent situations (e.g., the area of a picture frame of variable width) using quadratic expressions in one variable and expand and simplify quadratic expression in one variable [e.g., .
3.factor quadratic expressions in one variable, including those for which a ≠ 1 (e.g., ), differences of squares (e.g., ), and perfect square trinomials (e.g., ), by selecting and applying an appropriate strategy.
4.solve quadratic equations by selecting and applying a factoring strategy.
5.determine, through investigation, and describe the connection between the factors used in solving a quadratic equation and the x-intercepts of the graph of the corresponding quadratic relation.
6.explore the algebraic development of the quadratic formula (e.g., given the algebraic development, connect the steps to a numeric example; follow a demonstration of the algebraic development, with technology, such as computer algebra systems, or without technology [student reproduction of the development of the general case is not required]), and apply the formula to solve quadratic equations, using technology.
7.relate the real roots of a quadratic equation to the x-intercepts of the corresponding graph and connect the number of real roots to the value of the discriminant (e.g., there are no real roots and no x-intercepts if ).
8.relate the real roots of a quadratic equation to the x-intercepts of the corresponding graph and connect the number of real roots to the value of the discriminant (e.g., there are no real roots and no x-intercepts if ).
9.determine the real roots of a variety of quadratic equations (e.g., 100x2 = 115x + 35) and describe the advantages and disadvantages of each strategy (i.e., graphing; factoring; using the quadratic formula).
B.Connecting graphs and equations of quadratic functions:
Students will
1.explain the meaning of the term function and distinguish a function from a relation that is not a function, through investigation of linear and quadratic relations using a variety of representations (i.e., tables of values, mapping diagrams, graphs, function machines, equations) and strategies (e.g., using the vertical-line test).
2.substitute into and evaluate linear and quadratic functions represented using function notation [e.g., evaluate , given ], including functions arising from real-world applications.
3.explain the meanings of the terms domain and range, through investigation using numeric, graphical, and algebraic representations of linear and quadratic functions, and describe the domain and range of a function appropriately (e.g., for , the domain is the set of real numbers, and the range is .
4.explain any restrictions on the domain and the range of a quadratic function in contexts arising from real-world applications.
5.determine, through investigation using technology, the roles of a, h, and k in quadratic functions of the form , and describe these roles in terms of transformations on the graph of (i.e., translations; reflections in the x-axis; vertical stretches and compressions to and from the x-axis)
6.sketch graphs of by applying one or more transformations to the graph of .
7.express the equation of a quadratic function in the standard form , given the vertex form , and verify, using graphing technology, that these forms are equivalent representations.
8.express the equation of a quadratic function in the vertex form , given the standard form , by completing the square (e.g., using algebra tiles or diagrams; algebraically), including cases where is a simple rational number (e.g., , ) and verify, using graphing technology, that these forms are equivalent representations.
9.sketch graphs of quadratic functions in the factored form by using the x-intercepts to determine the vertex.
10.describe the information (e.g., maximum, intercepts) that can be obtained by inspecting the standard form , the vertex form , and the factored form of a quadratic function.
11.sketch the graph of a quadratic function whose equation is given in the standard form by using a suitable strategy (e.g., completing the square and finding the vertex; factoring, if possible, to locate the x-intercepts), and identify the key features of the graph (e.g., the vertex, the x– and y-intercepts, the equation of the axis of symmetry, the intervals where the function is positive or negative, the intervals where the function is increasing or decreasing).
C.Solving problems involving quadratic functions:
Students will
1.collect data that can be modelled as a quadratic function, through investigation with and without technology, from primary sources, using a variety of tools (e.g., concrete materials; measurement tools such as measuring tapes, electronic probes, motion sensors), or from secondary sources (e.g., websites such as Statistics Canada, E-STAT), and graph the data.
2.determine, through investigation using a variety of strategies (e.g., applying properties of quadratic functions such as the x-intercepts and the vertex; using transformations), the equation of the quadratic function that best models a suitable data set graphed on a scatter plot, and compare this equation to the equation of a curve of best fit generated with technology (e.g., graphing software, graphing calculator).
3.solve problems arising from real-world applications, given the algebraic representation of a quadratic function (e.g., given the equation of a quadratic function representing the height of a ball over elapsed time, answer questions that involve the maximum height of the ball, the length of time needed for the ball to touch the ground, and the time interval when the ball is higher than a given measurement).
Exponential Functions
A.Connecting graphs and equations of exponential functions:
Students will
1.determine, through investigation using a variety of tools (e.g., calculator, paper and pencil, graphing technology) and strategies (e.g., patterning; finding values from a graph; interpreting the exponent laws), the value of a power with a rational exponent (i.e., where x > 0 and m and n are integers).
2.evaluate, with and without technology, numerical expressions containing integer and rational exponents and rational bases [e.g., ].
3.graph, with and without technology, an exponential relation, given its equation in the form ), define this relation as the function , and explain why it is a function.
4.determine, through investigation, and describe key properties relating to domain and range, intercepts, increasing/decreasing intervals, and asymptotes (e.g., the domain is the set of real numbers; the range is the set of positive real numbers; the function either increases or decreases throughout its domain) for exponential functions represented in a variety of ways [e.g., tables of values, mapping diagrams, graphs, equations of the form ), function machines].
5.determine, through investigation (e.g., by patterning with and without a calculator), the exponent rules for multiplying and dividing numeric expressions involving exponents [e.g., ], and the exponent rule for simplifying numerical expressions involving a power of a power [e.g., ], and use the rules to simplify numerical expressions containing integer exponents (e.g., ).
6.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).
B.Solving problems involving exponential functions:
Students will
1.collect data that can be modelled as an exponential function, through investigation with and without technology, from primary sources, using a variety of tools (e.g., concrete materials such as number cubes, coins; measurement tools such as electronic probes), or from secondary sources (e.g., websites such as Statistics Canada, E-STAT), and graph the data.
2.identify exponential functions, including those that arise from real-world applications involving growth and decay (e.g., radioactive decay, population growth, cooling rates, pressure in a leaking tire), given various representations (i.e., tables of values, graphs, equations), and explain any restrictions that the context places on the domain and range (e.g., ambient temperature limits the range for a cooling curve)
3.solve problems using given graphs or equations of exponential functions arising from a variety of real-world applications (e.g., radioactive decay, population growth, height of a bouncing ball, compound interest) by interpreting the graphs or by substituting values for the exponent into the equations.
C.Solving financial problems involving exponential functions:
Students will
1.compare, using a table of values and graphs, the simple and compound interest earned for a given principal (i.e., investment) and a fixed interest rate over time.
2.solve problems, using a scientific calculator, that involve the calculation of the amount, A (also referred to as future value, ), and the principal, P (also referred to as present value, PV ), using the compound interest formula in the form ].
3.determine, through investigation (e.g., using spreadsheets and graphs), that compound interest is an example of exponential growth [e.g., the formulas for compound interest, and present value, , are exponential functions, where the number of compounding periods, n, varies].
4.solve problems, using a TVM Solver on a graphing calculator or on a website, that involve the calculation of the interest rate per compounding period, i, or the number of compounding periods, n, in the compound interest formula [or ].
5.explain the meaning of the term annuity, through investigation of numeric and graphical representations using technology.
6.determine, through investigation using technology (e.g., the TVM Solver on a graphing calculator, online tools), the effects of changing the conditions (i.e., the payments, the frequency of the payments, the interest rate, the compounding period) of ordinary simple annuities (i.e., annuities in which payments are made at the end of each period, and the compounding period and the payment period are the same) (e.g., long-term savings plans, loans).
7.solve problems, using technology (e.g., scientific calculator, spreadsheet, graphing calculator), that involve the amount, the present value, and the regular payment of an ordinary simple annuity (e.g., calculate the total interest paid over the life of a loan, using a spreadsheet, and compare the total interest with the original principal of the loan).
Trigonometric Functions
A.Applying the sine law and the cosine law in acute triangles:
Students will
1.solve problems, 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.
2.solve problems involving two right triangles in two dimensions.
3.verify, through investigation using technology (e.g., dynamic geometry software, spreadsheet), the sine law and the cosine law (e.g., compare, using dynamic geometry software, the ratios in triangle ABC while dragging one of the vertices).
4.describe conditions that guide when it is appropriate to use the sine law or the cosine law and use these laws to calculate sides and angles in acute triangles.
5.solve problems that require the use of the sine law or the cosine law in acute triangles, including problems arising from real-world applications (e.g., surveying, navigation, building construction).
B.Connecting graphs and equations of sine functions:
Students will
1.describe key properties (e.g., cycle, amplitude, period) of periodic functions arising from real-world applications (e.g., natural gas consumption in Ontario, tides in the Bay of Fundy), given a numeric or graphical representation.
2.predict, by extrapolating, the future behaviour of a relationship modelled using a numeric or graphical representation of a periodic function (e.g., predicting hours of daylight on a particular date from previous measurements, predicting natural gas consumption in Ontario from previous consumption).
3.make connections between the sine ratio and the sine function by graphing the relationship between angles from to and the corresponding sine ratios, with or without technology (e.g., by generating a table of values using a calculator; by unwrapping the unit circle), defining this relationship as the function , and explaining why the relationship is a function.
4.sketch the graph of for angle measures expressed in degrees, and determine and describe its key properties (i.e., cycle, domain, range, intercepts, amplitude, period, maximum and minimum values, increasing/decreasing intervals).
5.make connections, through investigation with technology, between changes in a real-world situation that can be modelled using a periodic function and transformations of the corresponding graph (e.g., investigate the connection between variables for a swimmer swimming lengths of a pool and transformations of the graph of distance from the starting point versus time).
6.determine, through investigation using technology, the roles of the parameters a, c, and d in functions in the form , and describe these roles in terms of transformations on the graph of with angles expressed in degrees (i.e., translations; reflections in the x-axis; vertical stretches and compressions to and from the x-axis)
7.sketch graphs of by applying transformations to the graph of , and state the domain and range of the transformed functions.
C.Solving problems involving sine functions:
Students will
1.collect data that can be modelled as a sine function (e.g., voltage in an AC circuit, sound waves), through investigation with and without technology, from primary sources, using a variety of tools (e.g., concrete materials, measurement tools such as motion sensors), or from secondary sources (e.g., websites such as Statistics Canada, E-STAT), and graph the data.
2.identify periodic and sinusoidal functions, including those that arise from real-world applications involving periodic phenomena, given various representations (i.e., tables of values, graphs, equations), and explain any restrictions that the context places on the domain and range.
3.pose problems based on applications involving a sine function and solve these and other such problems by using a given graph or a graph generated with technology from a table of values or from its equation.
College Preparation: Foundations for College Mathematics
This course enables students to broaden their understanding of mathematics as a problem-solving tool in the real world. Students will extend their understanding of quadratic relations; investigate situations involving exponential growth; solve problems involving compound interest; solve financial problems connected with vehicle ownership; develop their ability to reason by collecting, analyzing, and evaluating data involving one variable; connect probability and statistics; and solve problems in geometry and trigonometry. Students will consolidate their mathematical skills as they solve problems and communicate their thinking.
The overall expectations are divided in three main categories each of which contains several subcategories. Every subcategory lists specific expectations for this course of grade 11 in more detail.
Mathematical Models
A.Connecting graphs and equations of quadratic relations:
Students will
1.construct tables of values and graph quadratic relations arising from real-world applications (e.g., dropping a ball from a given height; varying the edge length of a cube and observing the effect on the surface area of the cube).
2.determine and interpret meaningful values of the variables, given a graph of a quadratic relation arising from a real-world
3.determine, through investigation using technology, the roles of a, h, and k in quadratic relations of the form , and describe these roles in terms of transformations on the graph of (i.e., translations; reflections in the x-axis; vertical stretches and compressions to and from the x-axis).
4.sketch graphs of quadratic relations represented by the equation (e.g., using the vertex and at least one point on each side of the vertex; applying one or more transformations to the graph of ).
5.expand and simplify quadratic expressions in one variable involving multiplying binomials [e.g., ]or squaring a binomial [e.g., using a variety of tools (e.g., paper and pencil, algebra tiles, computer algebra systems).
6.express the equation of a quadratic relation in the standard form , given the vertex form , and verify, using graphing technology, that these forms are equivalent representations.
7.factor trinomials of the form , where or where a is the common factor, by various
8.determine, through investigation, and describe the connection between the factors of a quadratic expression and the x-intercepts of the graph of the corresponding quadratic relation.
9.solve problems, using an appropriate strategy (i.e., factoring, graphing), given equations of quadratic relations, including those that arise from real-world applications (e.g., break-even point).
B.Connecting graphs and equations of exponential relations:
Students will
1.determine, through investigation using a variety of tools and strategies (e.g., graphing with technology; looking for patterns in tables of values), and describe the meaning of negative exponents and of zero as an
2.evaluate, with and without technology, numeric expressions containing integer exponents and rational bases (e.g., ).
3.determine, through investigation (e.g., by patterning with and without a calculator), the exponent rules for multiplying and dividing numerical expressions involving exponents [e.g., ], and the and the exponent rule for simplifying numerical expressions involving a power of a power [e.g., ].
4.graph simple exponential relations, using paper and pencil, given their equations [e.g., ].
5.make and describe connections between representations of an exponential relation (i.e., numeric in a table of values; graphical; algebraic).
6.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).
C.Solving problems involving exponential relations:
Students will
1.collect data that can be modelled as an exponential relation, through investigation with and without technology, from primary sources, using a variety of tools (e.g., concrete materials such as number cubes, coins; measurement tools such as electronic probes), or from secondary sources (e.g., websites such as Statistics Canada, E-STAT), and graph the
2.describe some characteristics of exponential relations arising from real-world applications (e.g., bacterial growth, drug absorption) by using tables of values (e.g., to show a constant ratio, or multiplicative growth or decay) and graphs (e.g., to show, with technology, that there is no maximum or minimum value)
3.pose problems involving exponential relations arising from a variety of real-world applications (e.g., population growth, radioactive decay, compound interest), and solve these and other such problems by using a given graph or a graph generated with technology from a given table of values or a given equation
4.solve problems using given equations of exponential relations arising from a variety of real-world applications (e.g., radioactive decay, population growth, height of a bouncing ball, compound interest) by substituting values for the exponent into the equations.
Personal Finance
A.Solving problems involving compound interest:
Students will
1.determine, through investigation using technology, the compound interest for a given investment, using repeated calculations of simple interest, and compare, using a table of values and graphs, the simple and compound interest earned for a given principal (i.e., investment) and a fixed interest rate over
2.determine, through investigation (e.g., using spreadsheets and graphs), and describe the relationship between compound interest and exponential
3.solve problems, using a scientific calculator, that involve the calculation of the amount, A (also referred to as future value, FV), and the principal, P (also referred to as present value, PV), using the compound interest formula in the form [or ].
4.calculate the total interest earned on an investment or paid on a loan by determining the difference between the amount and the principal [e.g., using )].
5.solve problems, using a TVM Solver on a graphing calculator or on a website, that involve the calculation of the interest rate per compounding period, i, or the number of com- pounding periods, n, in the compound interest formula [or ].
6.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.
B.Computing financial services:
Students will
1.gather, interpret, and compare information about the various savings alternatives commonly available from financial institutions (e.g., savings and checking accounts, term investments), the related costs (e.g., cost of cheques, monthly statement fees, early withdrawal penalties), and possible ways of reducing the costs (e.g., maintaining a minimum balance in a savings account, paying a monthly flat fee for a package of services).
2.gather and interpret information about investment alternatives (e.g., stocks, mutual funds, real estate, GICs, savings accounts), and com- pare the alternatives by considering the risk and the rate of return.
3.gather, interpret, and compare information about the costs (e.g., user fees, annual fees, service charges, interest charges on overdue balances) and incentives (e.g., loyalty rewards; philanthropic incentives, such as support for Olympic athletes or a Red Cross disaster relief fund) associated with various credit cards and debit Cards
4.gather, interpret, and compare information about current credit card interest rates and regulations, and determine, through investigation using technology, the effects of delayed payments on a credit card Blance.
5.solve problems involving applications of the compound interest formula to determine the cost of making a purchase on credit
C. Owning and operating vehicle:
Students will
1.gather and interpret information about the procedures and costs involved in insuring a vehicle (e.g., car, motorcycle, snowmobile) and the factors affecting insurance rates (e.g., gender, age, driving record, model of vehicle, use of vehicle), and compare the insurance costs for different categories of drivers and for different vehicles.
2.gather, interpret, and compare information about the procedures and costs (e.g., monthly payments, insurance, depreciation, maintenance, miscellaneous expenses) involved in buying or leasing a new vehicle or buying a used vehicle
3.solve problems, using technology (e.g., calculator, spreadsheet), that involve the fixed costs (e.g., license fee, insurance) and variable costs (e.g., maintenance, fuel) of owning and operating a vehicle
Geometry and Trigonometry
A.Representing two-dimensional shapes and three-dimensional figures:
Students will
1.recognize and describe real-world applications of geometric shapes and figures, through investigation (e.g., by importing digital photos into dynamic geometry software), in a variety of contexts (e.g., product design, architecture, fashion), and explain these applications (e.g., one reason that sewer covers are round is to prevent them from falling into the sewer during removal and replacement).
2.represent three-dimensional objects, using concrete materials and design or drawing software, in a variety of ways (e.g., ortho- graphic projections [i.e., front, side, and top views], perspective isometric drawings, scale models).
3.create nets, plans, and patterns from physical models arising from a variety of real-world applications (e.g., fashion design, interior dec- orating, building construction), by applying the metric and imperial systems and using design or drawing software
4.solve design problems that satisfy given constraints (e.g., design a rectangular berm that would contain all the oil that could leak from a cylindrical storage tank of a given height and radius), using physical models (e.g., built from popsicle sticks, cardboard, duct tape) or drawings (e.g., made using design or drawing software), and state any assumptions made.
B.Applying the sine law and the cosine law in acute triangles:
Students will
1.solve problems, 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
2.verify, through investigation using technology (e.g., dynamic geometry software, spread- sheet), the sine law and the cosine law (e.g., compare, using dynamic geometry software, the ratios in triangles ABC while dragging one of the vertices).
3.describe conditions that guide when it is appropriate to use the sine law or the cosine law and use these laws to calculate sides and angles in acute triangles.
4.gather, interpret, and compare information about current credit card interest rates and regulations, and determine, through investigation using technology, the effects of delayed payments on a credit card Blance.
5.solve problems that arise from real-world applications involving metric and imperial measurements and that require the use of the sine law or the cosine law in acute triangles.
Data Management
A.Working with one-variable data: Students will
1.identify situations involving one-variable data (i.e., data about the frequency of a given occurrence), and design questionnaires (e.g., for a store to determine which CDs to stock, for a radio station to choose which music to play) or experiments (e.g., counting, taking measurements) for gathering one-variable data, giving consideration to ethics, privacy, the need for honest responses, and possible sources of bias.
2.collect one-variable data from secondary sources (e.g., Internet databases), and organize and store the data using a variety of tools (e.g., spreadsheets, dynamic statistical software).
3.explain the distinction between the terms population and sample, describe the characteristics of a good sample, and explain why sampling is necessary (e.g., time, cost, or physical constraints)
4.describe and compare sampling techniques (e.g., random, stratified, clustered, convenience, voluntary); collect one-variable data from primary sources, using appropriate sampling techniques in a variety of real-world situations; and organize and store the data.
5.identify different types of one-variable data (i.e., categorical, discrete, continuous), and represent the data, with and without techno- logy, in appropriate graphical forms (e.g., histograms, bar graphs, circle graphs, pictographs).
6.identify and describe properties associated with common distributions of data (e.g., normal, bimodal, skewed).
7.calculate, using formulas and/or technology (e.g., dynamic statistical software, spread- sheet, graphing calculator), and interpret measures of central tendency (i.e., mean, median, mode) and measures of spread (i.e., range, standard deviation).
8.explain the appropriate use of measures of central tendency (i.e., mean, median, mode) and measures of spread (i.e., range, standard deviation).
9.explain the appropriate use of measures of central tendency (i.e., mean, median, mode) and measures of spread (i.e., range, standard deviation).
10.solve problems by interpreting and analyzing one-variable data collected from secondary sources.
B.Applying probability
1.identify examples of the use of probability in the media and various ways in which probability is represented (e.g., as a fraction, as a percent, as a decimal in the range 0 to 1).
2.determine the theoretical probability of an event (i.e., the ratio of the number of favorable 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).
3.perform a probability experiment (e.g., tossing a coin several times), represent the results using a frequency distribution, and use the distribution to determine the experimental probability of an event
4.compare, through investigation, the theoretical probability of an event with the experimental probability and explain why they might differ.
5.determine, through investigation using class- generated data and technology-based simulation models (e.g., using a random-number generator on a spreadsheet or on a graphing calculator), the tendency of experimental probability to approach theoretical probability as the number of trials in an experiment increases (e.g., “If I simulate tossing a coin 1000 times using technology, the experimental probability that I calculate for tossing tails is likely to be closer to the theoretical probability than if I simulate tossing the coin only 10 times”).
6.interpret information involving the use of probability and statistics in the media, and make connections between probability and statistics (e.g., statistics can be used to generate probabilities).
Workplace Preparation: Mathematics for Work and Everyday Life
This course enables students to broaden their understanding of mathematics as it is applied in the workplace and daily life. Students will solve problems associated with earning money, paying taxes, and making purchases; apply calculations of simple and compound interest in saving, investing, and borrowing; and calculate the costs of transportation and travel in a variety of situations. Students will consolidate their mathematical skills as they solve problems and communicate their thinking.
The overall expectations are divided in three main categories each of which contains several subcategories. Every subcategory lists specific expectations for this course of grade 11 in more detail.
Earning and Purchasing
A.Earning: Students will
1.gather, interpret, and compare information about the components of total earnings (e.g., salary, benefits, vacation pay, profit-sharing) in different occupations
2.gather, interpret, and describe information about different remuneration methods (e.g., hourly rate, overtime rate, job or project rate, commission, salary, gratuities) and remuneration schedules (e.g., weekly, biweekly, semi-monthly, monthly).
3.describe the effects of different remuneration methods and schedules on decisions related to personal spending habits (e.g., the timing of a major purchase, the scheduling of mortgage payments and other bill payments).
4.solve problems, using technology (e.g., calculator, spreadsheet), and make decisions involving different remuneration methods and schedules.
B.Describing purchasing power:
Students will
1.gather, interpret, and describe information about government payroll deductions (i.e., CPP, EI, income tax) and other payroll deductions (e.g., contributions to pension plans other than CPP; union dues; charitable donations; benefit-plan contributions).
2.estimate and compare, using current secondary data (e.g., federal tax tables), the percent of total earnings deducted through government payroll deductions for various benchmarks (e.g., $15 000, $20 000, $25 000).
3.describe the relationship between gross pay, net pay, and payroll deductions (i.e., net pay is gross pay less government payroll deductions and any other payroll deductions), and estimate net pay in various situations.
4.describe and compare the purchasing power and living standards associated with relevant occupations of interest
C.Purchasing: Students will
1.identify and describe various incentives in making purchasing decisions (e.g., 20% off; off; buy 3 get 1 free; loyalty rewards; 3 coupons; 0% financing).
2.estimate the sale price before taxes when making a purchase (e.g., estimate 25% off $38.99 as 25% or off of $40, giving a discount of about $10 and a sale price of approximately $30; alternatively, estimate the same sale price as about of $40).
3.describe and compare a variety of strategies for estimating sales tax (e.g., estimate the sales tax on most purchases in Ontario by estimating 10% of the purchase price and adding about a third of this estimate, rather than estimating the PST and GST separately), and use a chosen strategy to estimate the after-tax cost of common items .
4.calculate discounts, sale prices, and after-tax costs, using technology
5.identify forms of taxation built into the cost of an item or service (e.g., gasoline tax, tire tax).
6.estimate the change from an amount offered to pay a charge.
7.make the correct change from an amount offered to pay a charge, using currency manipulatives.
8.compare the unit prices of related items to help determine the best buy.
9.describe and compare, for different types of transactions, the extra costs that may be associated with making purchases (e.g., interest costs, exchange rates, shipping and handling costs, customs duty, insurance).
10.make and justify a decision regarding the purchase of an item, using various criteria (e.g., extra costs, such as shipping costs and transaction fees; quality and quantity of the item; shelf life of the item; method of purchase, such as online versus local) under various circumstances (e.g., not having access to a vehicle; living in a remote community; having limited storage space).
Saving, Investing, and Borrowing
A.Comparing financial services:
Students will
1.gather, interpret, and compare information about the various savings alternatives commonly available from financial institutions (e.g., savings and checking accounts, term investments), the related costs (e.g., cost of cheques, monthly statement fees, early withdrawal penalties), and possible ways of reducing the costs (e.g., maintaining a minimum balance in a savings account, paying a monthly flat fee for a package of services).
2.gather, interpret, and compare information about the costs (e.g., user fees, annual fees, service charges, interest charges on overdue balances) and incentives (e.g., loyalty rewards; philanthropic incentives, such as support for Olympic athletes or a Red Cross disaster relief fund) associated with various credit cards and debit cards.
3.read and interpret transaction codes and entries from various financial statements (e.g., bank statement, credit card statement, passbook, automated banking machine printout, online banking statement, account activity report), and explain ways of using the information to manage personal finances.
B.Saving and investing:
Students will
1.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.
2.determine, through investigation using technology, the compound interest for a given investment, using repeated calculations of simple interest for no more than 6 compounding periods.
3.describe the relationship between simple interest and compound interest in various ways (i.e., orally, in writing, using tables and graphs).
4.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 of changing the total length of time, the interest rate, or the compounding period.
5.solve problems, using technology, that involve applications of compound interest to saving and investing
C.Borrowing:
Students will
1gather, interpret, and compare information about the effects of carrying an outstanding balance on a credit card at current interest rates.
2.gather, interpret, and compare information describing the features (e.g., interest rates, flexibility) and conditions (e.g., eligibility, required collateral) of various personal loans (e.g., student loan, car loan, “no interest” deferred-payment loan, loan to consolidate debt, loan drawn on a line of credit, payday or bridging loan).
3.calculate, using technology (e.g., calculator, spreadsheet), the total interest paid over the life of a personal loan, given the principal, the length of the loan, and the periodic payments, and use the calculations to justify the choice of a personal loan.
4.determine, using a variety of tools (e.g., spreadsheet template, online amortization tables), the effect of the length of time taken to repay a loan on the principal and interest components of a personal loan repayment.
5.compare, using a variety of tools (e.g., spread- sheet template, online amortization tables), the effects of various payment periods (e.g., monthly, biweekly) on the length of time taken to repay a loan and on the total interest paid.
6.gather and interpret information about credit ratings, and describe the factors used to deter- mine credit ratings and the consequences of a good or bad rating.
7.make and justify a decision to borrow, using various criteria (e.g., income, cost of borrowing, availability of an item, need for an item) under various circumstances (e.g., having a large existing debt, wanting to pursue an education or training opportunity, needing transportation to a new job, wanting to set up a business).
Transportation and Travel
A.Owning and operating a vehicle:
Students will
1.gather and interpret information about the procedures (e.g., in the graduated licensing system) and costs (e.g., driver training; licensing fees) involved in obtaining an Ontario driver’s license, and the privileges and restrictions associated with having a driver’s license .
2.gather and describe information about the procedures involved in buying or leasing a new vehicle or buying a used vehicle.
3.gather and interpret information about the procedures and costs involved in insuring a vehicle (e.g., car, motorcycle, snowmobile) and the factors affecting insurance rates (e.g., gender, age, driving record, model of vehicle, use of vehicle), and compare the insurance costs for different categories of drivers and for different vehicles.
4.gather and interpret information about the costs (e.g., monthly payments, insurance, depreciation, maintenance, miscellaneous expenses) of purchasing or leasing a new vehicle or purchasing a used vehicle and describe the conditions that favor each alternative.
5.describe ways of failing to operate a vehicle responsibly (e.g., lack of maintenance, careless driving) and possible financial and non-financial consequences (e.g., legal costs, fines, higher insurance rates, demerit points, loss of driving privileges).
6.identify and describe costs (e.g., gas consumption, depreciation, insurance, maintenance) and benefits (e.g., convenience, increased profit) of owning and operating a vehicle for business.
7.solve problems, using technology (e.g., calculator, spreadsheet), that involve the fixed costs (e.g., license fee, insurance) and variable costs (e.g., maintenance, fuel) of owning and operating a vehicle.
B.Travelling by automobile: Students will
1.determine distances represented on maps (e.g., provincial road map, local street map, Web-based maps), using given scales.
2.plan and justify, orally or in writing, a route for a trip by automobile on the basis of a variety of factors (e.g., distances involved, the purpose of the trip, the time of year, the time of day, probable road conditions, personal priorities).
3.report, orally or in writing, on the estimated costs (e.g., gasoline, accommodation, food, entertainment, tolls, car rental) involved in a trip by automobile, using information from available sources (e.g., automobile association travel books, travel guides, the Internet).
4.solve problems involving the cost of travelling by automobile for personal or business purposes.
C.Comparing modes of transformation:
Students will
1gather, interpret, and describe information about the impact (e.g., monetary, health, environmental) of daily travel (e.g., to work and/or school), using available means (e.g., car, taxi, motorcycle, public transportation, bicycle, walking).
2.gather, interpret, and compare information about the costs (e.g., insurance, extra charges based on distance travelled) and conditions (e.g., one-way or return, drop-off time and location, age of the driver, required type of driver’s license) involved in renting a car, truck, or trailer, and use the information to justify a choice of rental vehicle.
3.gather, interpret, and describe information regarding routes, schedules, and fares for travel by airplane, train, or bus.
4.solve problems involving the comparison of information concerning transportation by air- plane, train, bus, and automobile in terms of various factors (e.g., cost, time, convenience).
List of Skills
More than 230 math skills are considered in the math curriculum for grade 11 some of which are common to grade 10. Please, use the detailed list of skills in the old LG for grade 11.
Evaluation
Objective evaluation is believed to be one of the most essential parts of teaching mathematics. In Genius Math, we use different tools and methods to evaluate the mathematical knowledge of students and their progress. Our evaluation process consists of three stages: before teaching sessions, during teaching sessions and after teaching sessions.
- Initial Assessment Test
Before starting our teaching sessions, we administrate an assessment test to obtain some insights on the strengths and weaknesses of students and their previous math knowledge. This key information helps us to come up with a special plan for every single student. - Standard Problems
During teaching sessions, we use a combination of different resources providing standard problems that are designed by famous mathematicians all over the world to improve the problem-solving skills of students. Among those resources are Math Kangaroo Contests, CEMC (University of Waterloo), AMC (American Mathematics Competitions), and even IMO (International Mathematics Olympiad), the latter might be considered for those who want to tackle more challenging problems or prepare for math olympiads. We use these problems to design homework, quizzes, and tests for our students based on their grades, needs and goals. As a matter of fact, such problems can be used to unveil the depth of students’ mathematical understanding. - Final Assessment Test
When teaching sessions are over, students are asked to take another assessment test aiming to show their real progress in mathematics.
Most Common Challenging Topics
The followings are among the most common challenges students face in grade 11:
- Quadratic functions
- Exponential functions
- Sequences
- Trigonometric functions
- Compound interest
- Data management and probability
What We Can Offer
Students have different goals and expectations according to their background, knowledge, or experience. This data along with the result of assessment session help us to design a unique plan for each student. There are different kinds of helps that we offer students in Genius Math:
- To review and practice their class notes and handouts
- To be helped with their homework, quizzes, and tests
- To improve their math skills in general
- To level up (e.g., moving from B- to B+)
- To get A+
- To learn topics beyond curriculum
- To prepare for math competitions