Ontario Grade 12 MAP4C Foundations for College Mathematics (Grade 12, College Preparation)
Chapters: 4
1. A. Mathematical Models
1 Solving Exponential Equations · 2 Modelling Graphically · 3 Modelling Algebraically
- Exponential Functions – An exponential function has the form y = a·bˣ, where a ≠ 0 is the starting value and b > 0, b ≠ 1 is the growth factor. If b > 1 it grows; if 0 < b < 1 it decays. Each step of 1 in x multiplies y by b (a constant ratio), unlike a linear function which adds a constant. The graph of y = bˣ passes through (0, 1), has domain all real numbers, range y > 0, and the x-axis as a horizontal asymptote. The general form y = a·b^(k(x − d)) + c stretches, reflects and shifts it.
- Mathematical Modelling: Using Maths to Describe the Real World – A mathematical model is an equation, graph or table that describes a real situation in a simple way. The modelling cycle: understand the real problem → choose variables and make assumptions → build a model (for example linear, quadratic or exponential) → solve and predict → check the answer against real data → improve the model or state its limits. No model is perfect; a good one is simple and close enough to be useful.
2. B. Personal Finance
1 Understanding Annuities · 2 Renting or Owning Accommodation · 3 Designing Budgets
- Annuities and Mortgages – An annuity is a series of equal payments made at equal time intervals. A mortgage is a big loan to buy property that you pay back with an annuity. Each payment is split into interest (the cost of borrowing) and principal (the part that reduces the loan). At first most of the payment is interest; later most is principal. An amortization table lists every payment, its interest part, its principal part and the balance left. The payment is PMT = PV × i ÷ (1 − (1 + i)^−n). A higher rate, a longer term or fewer payments per year all raise the total interest you pay.
- Consumer Mathematics: Earning, Buying and Travelling – Consumer mathematics is everyday money maths. Gross pay is all you earn: hourly wage (hours × rate), salary, commission (a % of sales), piecework, tips, bonus and overtime (often 1.5 × rate). Deductions such as income tax, pension and insurance are taken off, leaving net pay (take-home pay). When buying, take the discount first, then add sales tax (GST or VAT): price × (1 − d) × (1 + t). Unit price = price ÷ quantity, so you can compare packs. Count up to give correct change. A car costs fuel (distance × litres per 100 km ÷ 100 × price), insurance (higher for new drivers), licence, loan, repairs and parking; compare it with bus, train or plane for each trip.
- Smart Ways to Manage Your Finances – Inflation makes prices rise, so the same money buys less. Money kept in a bank earns interest: simple interest is paid only on the original amount, while compound interest also earns interest on earlier interest, so it grows faster over time. A budget plans income into needs, wants and savings. Savings are kept safe; investments can grow but carry risk, and higher possible returns mean higher risk. Insurance shares risk among many people. Income tax is paid on income above a limit, at rates that rise with income.
3. C. Geometry and Trigonometry
1 Solving Problems Involving Measurement and Geometry · 2 Investigating Optimal Dimensions · 3 Solving Problems Involving Trigonometry
- Surface Area and Volume of Cuboid, Cylinder, Cone, Pyramid and Sphere – Surface area is the total area of the outside skin of a solid, like the paper needed to wrap it. Volume is the space inside, like the water it can hold. A cuboid's volume is the number of 1 cm cubes that fit in it. A cone holds one third of a cylinder with the same base and height, and a pyramid holds one third of the matching prism.
- Law of Sines (Sine Rule) – In any triangle, each side divided by the sine of the angle facing it gives the same number: a/sinA = b/sinB = c/sinC = 2R, where R is the radius of the circle through the three corners. Use it when you know two angles and a side (AAS or ASA), or two sides and an angle that is not between them (SSA). SSA can give two triangles, one or none.
4. D. Data Management
1 Working With Two-Variable Data · 2 Applying Data Management
- Correlation: Scatter Diagram, Karl Pearson's Coefficient and Spearman's Rank Correlation – Correlation tells how two variables move together. It is positive when both rise together, negative when one rises as the other falls, and zero when there is no straight-line pattern. A scatter diagram shows it as a picture. Karl Pearson's coefficient r measures its direction and strength and always lies between −1 and +1. Spearman's rank correlation R uses ranks and works for qualities like beauty or honesty; tied ranks need a small correction.
- Data Analysis – Data analysis means turning raw data into answers. It follows a cycle: ask a question, collect data, clean it (remove errors, repeats and blanks), organise and transform it, analyse it with summaries such as mean, median, range and patterns, show it with a good chart, and draw a careful conclusion. Watch for outliers, small samples and bias, and remember that a correlation between two things does not prove that one causes the other. Data must also be stored safely and used with permission.