Grade 11 Foundations for College Mathematics, College Preparation

Grade 11 Foundations for College Mathematics, College Preparation

A. Mathematical Models

A1. Make connections between the numeric, graphical, and algebraic representations of quadratic relations, and use the connections to solve problems

  • 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)
  • Determine and interpret meaningful values of the variables, given a graph of a quadratic relation arising from a real-world application
  • Determine, through investigation using technology, the roles of a, h, and k in quadratic relations of the form y = a(x – h)² + k, and describe these roles in terms of transformations on the graph of y = x² (i.e., translations; reflections in the x-axis; vertical stretches and compressions to and from the x-axis)
  • Sketch graphs of quadratic relations represented by the equation y = a(x – h)² + k (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 y = x²)
  • Expand and simplify quadratic expressions in one variable involving multiplying binomials [e.g., (½x + 1)(3x – 2)] or squaring a binomial [e.g., 5(3x – 1)²], using a variety of tools (e.g., paper and pencil, algebra tiles, computer algebra systems)
  • Express the equation of a quadratic relation in the standard form y = ax² + bx + c, given the vertex form y = a(x – h)² + k, and verify, using graphing technology, that these forms are equivalent representations
  • Factor trinomials of the form ax² + bx + c , where a = 1 or where a is the common factor, by various methods
  • 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
  • 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)

A2. Demonstrate an understanding of exponents, and make connections between the numeric, graphical, and algebraic representations of exponential relations

  • 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 exponent
  • Evaluate, with and without technology, numeric expressions containing integer exponents and rational bases (e.g., 2 to the -3rd power, 6³, 3456 to the 0 power, 1.03 to the 10th power)
  • 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., (½)³ x (½)²], and the exponent rule for simplifying numerical expressions involving a power of a power [e.g., (5³)²]
  • Graph simple exponential relations, using paper and pencil, given their equations [e.g., y = 2 to the x power, y = 10 to the x power, y = (½) to the x power]
  • Make and describe connections between representations of an exponential relation (i.e., numeric in a table of values; graphical; algebraic)
  • 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., simpl interest and compound interest, population growth)

A3. Describe and represent exponential relations, and solve problems involving exponential relations arising from realworld applications

  • 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, ESTAT), and graph the data
  • 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)
  • 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
  • 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

B. Personal Finance

B1. Compare simple and compound interest, relate compound interest to exponential growth, and solve problems involving compound interest

  • 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 time
  • Determine, through investigation (e.g., using spreadsheets and graphs), and describe the relationship between compound interest and exponential growth
  • 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 A = P((1 + i) to the n power) [or FV = PV((1 + i) to the n power)]
  • 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 I =^ A – P (or I = FV – PV)]
  • 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 A = P((1 + i) to the n power) [or FV = PV((1 + i) to the n power)]
  • 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

B2. Compare services available from financial institutions, and solve problems involving the cost of making purchases on credit

  • Gather, interpret, and compare information about the various savings alternatives commonly available from financial institutions (e.g., savings and chequing 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)
  • Gather and interpret information about investment alternatives (e.g., stocks, mutual funds, real estate, GICs, savings accounts), and compare the alternatives by considering the risk and the rate of return
  • 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
  • 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 balance
  • Solve problems involving applications of the compound interest formula to determine the cost of making a purchase on credit

B3. Interpret information about owning and operating a vehicle, and solve problems involving the associated costs

  • 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
  • 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
  • Solve problems, using technology (e.g., calculator, spreadsheet), that involve the fixed costs (e.g., licence fee, insurance) and variable costs (e.g., maintenance, fuel) of owning and operating a vehicle

C. Geometry and Trigonometry

C1. Represent, in a variety of ways, two-dimensional shapes and three-dimensional figures arising from real-world applications, and solve design problems

  • 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)
  • Represent three-dimensional objects, using concrete materials and design or drawing software, in a variety of ways (e.g., orthographic projections [i.e., front, side, and top views], perspective isometric drawings, scale models)
  • Create nets, plans, and patterns from physical models arising from a variety of real-world applications (e.g., fashion design, interior decorating, building construction), by applying the metric and imperial systems and using design or drawing software
  • 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

C2. Solve problems involving trigonometry in acute triangles using the sine law and the cosine law, including problems arising from real-world applications

  • 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
  • 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 a/sin A, b/sin B, and c/sin C in triangle ABC while dragging one of the vertices)
  • 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
  • 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

D. Data Management

D1. Solve problems involving one-variable data by collecting, organizing, analysing, and evaluating data

  • 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
  • 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)
  • 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)
  • 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
  • Identify different types of one-variable data (i.e., categorical, discrete, continuous), and represent the data, with and without technology, in appropriate graphical forms (e.g., histograms, bar graphs, circle graphs, pictographs)
  • Identify and describe properties associated with common distributions of data (e.g., normal, bimodal, skewed) 
  • Calculate, using formulas and/or technology (e.g., dynamic statistical software, spreadsheet, graphing calculator), and interpret measures of central tendency (i.e., mean, median, mode) and measures of spread (i.e., range, standard deviation)
  • Explain the appropriate use of measures of central tendency (i.e., mean, median, mode) and measures of spread (i.e., range, standard deviation)
  • Compare two or more sets of one-variable data, using measures of central tendency and measures of spread
  • Solve problems by interpreting and analysing one-variable data collected from secondary sources

D2. Determine and represent probability, and identify and interpret its applications

  • 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)
  • 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)
  • 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
  • Compare, through investigation, the theoretical probability of an event with the experimental probability, and explain why they might differ
  • 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")
  • 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)
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