Blog
Plain-language guides for Montreal students heading into AP Calculus and CEGEP math.
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When to start math tutoring: September, after the first report card, or May?
Starting in September, after the first report card or before June exams: what each can realistically fix, and the signs that say not to wait.
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What is a logarithm? The exponent you’re looking for
A logarithm is an exponent. Learn to switch between log and exponential form, and derive the product, quotient and power laws from the exponent laws.
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“I understood it in class but failed the test”: what’s going on
Why a teen can follow every example in math class and still fail the test, the signs to look for, and a 20-minute home routine to close the gap.
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Removable, jump or infinite: telling discontinuities apart
How to tell removable, jump and infinite discontinuities apart on a graph and from a formula, using one-sided limits. With a self-check and answers.
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Signs your teen is falling behind in CEGEP Calculus I
The early signs a parent can see when a teen is falling behind in CEGEP Calculus I, which ones need help now, and how to raise it without hovering.
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The squeeze theorem, and why sin x / x tends to 1
Why sin x / x tends to 1 as x tends to 0: the squeeze theorem on x² sin(1/x), the unit-circle proof, and the derivative of sin x that depends on it.
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Square root equations: solve, then catch the extraneous solution
Solve a square root equation by squaring both sides, then see why one answer can fail when you substitute it back, and how to spot it in advance.
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Seven signs your teen is falling behind in Sec 4 math
Seven signs your teen is struggling in Secondary 4 math that you can spot without knowing the math, and one thing to do about each this week.
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Should my teen take AP Calculus in high school? Questions to ask
Is AP Calculus worth it for your teen? What to ask the school first: the teacher, the May exam, the timetable, and the load beside Secondary 5 math.
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How to find the rule of an exponential function from two points and an asymptote
Read k off the asymptote, divide the two points to cancel a and get the base c, then check. Two worked examples and a self-check for Secondary 4.
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How to find the rule of a step function from its graph
Read the step length, the jump and the closed end of each step to get a, b, h and k, with one rising and one falling worked graph, and a self-check.
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Left, right, midpoint, trapezoid: which sums over- or underestimate
Increasing or decreasing settles left and right Riemann sums; concavity settles midpoint and trapezoidal sums. The four rules, why, and a self-check.
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Solving a second degree inequality with a quick sketch
Solve a second degree inequality in four steps: find the zeroes, sketch the parabola, read where it is above or below the axis, write the solution set.
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The product-sum method for factoring ax² + bx + c
Factor ax² + bx + c when a isn't 1: find two numbers with product ac and sum b, split the middle term, and group. Worked line by line, with a self-check.
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The Mean Value Theorem in one picture, and the two conditions it needs
The Mean Value Theorem as a secant and a parallel tangent, why both of its conditions matter, a counterexample for each, and a self-check with answers.
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The IVT: proving a root exists without finding it
Use the intermediate value theorem to prove cos x = x has a solution between 0 and 1 without solving it, and see why continuity cannot be skipped.
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Helping with AP Calculus when you don't remember calculus
You don't need to remember derivatives to help with AP Calculus: what to ask, how to tell real practice from rereading, and when to bring in a tutor.
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Logs, powers or exponentials: which grows fastest as x → ∞?
ln x, powers of x and e^x ranked as x grows: read limits like x³/e^x at a glance, prove the ranking with L'Hospital's rule, and test yourself.
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The greatest integer function, explained from scratch
What the greatest integer function does to 3.7, −3.7 and 4, why its graph is a staircase of half-open steps, and how it prices every started block.
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From the graph of f′ to the graph of f: what each feature means
Reading f from the graph of f′: where f′ is positive, crosses zero, touches the axis or peaks, and what each says about f. With a graph-based self-check.
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The A student who gets a C in Calculus I: what changed
Why strong Secondary 5 students can stall in CEGEP Calculus I, what the course asks for instead of speed and memory, and what fixes it week to week.
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General, standard or factored: which form of a quadratic function answers what
General, standard or factored form: which one gives the vertex, the zeroes, the initial value or the maximum, and how to get from one form to another.
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How to find the rule of a quadratic function
Find the rule of a parabola from its vertex and one point, or from its two zeroes and one point, then check it the way that catches a sign error.
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Why the derivative of ln x is 1/x, and why ln|x| has the same one
Why the derivative of ln x is 1/x, derived from e^x in three lines; why ln|x| has the same derivative on both sides of zero; and a self-check.
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Where the derivative of arctan x comes from
Where the derivative of arctan x comes from: implicit differentiation of tan y = x, one right triangle, the same steps for arcsin, and a self-check.
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The derivative of an integral with x in the limits
How to differentiate an integral with x in the limits: Part 1 of the Fundamental Theorem, the chain-rule version, and the rule when both limits move.
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CST, TS or SN: what each Sec 4 math sequence keeps open
The difference between CST, TS and SN math in Quebec: what each sequence teaches, who it suits, and which CEGEP programs require which.
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Continuous but not differentiable: corners, cusps and vertical tangents
Why a continuous function can lack a derivative: corners, cusps and vertical tangents, the one limit that tells them apart, and a self-check with answers.
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How to complete the square: from general form to standard form
Turn ax² + bx + c into standard form a(x − h)² + k in four moves, including the a ≠ 1 case, then read the maximum or minimum straight off it.