Math explained

Math help by course: every idea students search for, explained with one worked example, the wrong line, and the course it sits in

Math help by course: the unit circle, derivative rules, logs, asymptotes, vectors, matrices, standard deviation and more, each with one marked example.

The YSR Studies team · Published 7 November 2026 · 7 min read

Math help that works is not a formula sheet and not an answer. It is one example worked line by line, the line students write instead and why it is wrong, a sibling problem to try cold, and the course the idea sits in so you know what comes before and after. Every page below gives exactly that for one idea of Grade 12 functions, calculus, linear algebra or data management. Find your course, then your idea.

What "math help" should give you

Every explainer in this section has the same four parts, and each part is there for a reason the research has measured.

First, a worked example, marked line by line. Sweller and Cooper (1985) found that a novice who studies a worked example learns the method faster than one who is left to solve problems unaided. Reading a solution is not a waste of time when you are new to an idea. It is the fastest way in.

Second, the common wrong line, named. Booth and colleagues (2013) studied algebra students who were shown incorrect examples alongside correct ones, with the fault explained, and concluded that the incorrect examples, alone or paired with correct ones, may be especially useful for conceptual understanding. You learn where the mark goes by seeing where it went.

Third, a sibling problem to try cold, with its answer under a Check heading. Roediger and Karpicke (2006) showed that the attempt is the learning: students who were tested on material remembered more a week later than students who reread it. The sibling is your attempt.

Fourth, the place in the course. Every idea is named by the course it belongs to, with the idea before it and the idea after it, so the page reads as a chapter and not as a dictionary entry.

An answer with no marked line teaches nothing you can reuse. If you have been pasting questions into an answer app and copying what comes back, getting unstuck without losing the learning says what to do instead, and how to study math has the method on paper for a whole course.

One example, to show the shape

Here is the shape on one limit, so you can see what every page promises.

Find the limit of (x² − 4)/(x − 2) as x → 2.

Try substitution. Put x = 2 in: (4 − 4)/(2 − 2) = 0/0. That is not a number and not an answer. It is a signal to do more algebra.
Factor the numerator. x² − 4 = (x − 2)(x + 2), so the expression is (x − 2)(x + 2)/(x − 2).
Cancel. For every x except 2, this equals x + 2. A limit only asks about x near 2, never at 2, so the cancelling is allowed.
Evaluate. The limit is 2 + 2 = 4.

The common wrong line: "0/0 = 0, so the limit is 0", or "0/0 is undefined, so there is no limit". Both treat 0/0 as a value. It is not one. It tells you the question is not finished, and the factoring finishes it.

A sibling to try cold: the limit of (x² − 9)/(x − 3) as x → 3.

Check

The numerator factors as (x − 3)(x + 3). Cancel the (x − 3) and the expression is x + 3 for every x except 3. The limit is 3 + 3 = 6. If you wrote 0 or "undefined", the wrong line above is yours; the limits page below works a harder one by rationalising.

Advanced Functions (MHF4U) and Algebra 2 / Precalculus

Most of what a US student calls Algebra 2 and Precalculus is one Ontario course, MHF4U: polynomials, rational functions, exponentials and logarithms, trigonometry and transformations. The overlap is large but not total, so the pages are written by idea rather than by course name. The MHF4U course map lists all 26 ideas in teaching order, and the precalculus page in the courses section names the twelve that calculus assumes you own.

The MHF4U explainers, in the order the course teaches them:

Dividing polynomials takes one cubic through long division and synthetic division, the remainder and factor theorems, and the full factoring, with the sign in the box named as the fault.
Asymptotes and rational functions reads one function from the algebra: holes, vertical asymptotes, the horizontal asymptote by the degree rule, the oblique asymptote by division.
Logarithm rules derives the three laws from the exponent laws, then solves one exponential equation and one logarithmic equation, including the root you must reject.
The unit circle is built from two triangles and four signs instead of memorised, with radians to degrees and one angle worked.
Trig identities derives the eight basic identities and the double-angle formulas, proves one identity and solves one equation without losing a solution.
Domain and range, inverses and transformations reads one function three ways. These are the course's last strand, though most students need them from the first unit, and the page says so.

Calculus and Vectors (MCV4U) and AP Calculus

MCV4U is the Ontario Grade 12 calculus course, and the AP Calculus AB and BC exams test the same differential calculus with integration added. The MCV4U course map has the 24 ideas; AP Calculus by unit maps the exam's units to these pages.

The explainers, in the pack's order:

Limits covers tables, graphs and algebra, the limit laws, and evaluating by factoring and rationalising.
Derivative rules is the complete list, when each applies, and the two rules students confuse.
Product and quotient rules works one example of each and names the sign error in the quotient rule's numerator.
The chain rule is inside, outside, and the inner derivative students forget.
Implicit differentiation and related rates shows the one idea behind both, worked on a ladder sliding down a wall.
Vectors, the dot product and the cross product says which question wants which, with magnitude and direction first.

First-year calculus (Calculus I)

A first-year calculus course carries the MCV4U ideas again at a faster pace, then adds the integral. The limits, derivative rules, chain rule, and implicit differentiation pages above are shared; the Calculus I pack carries them at ideas 1 and 5 to 7, with related rates at 17. First-year calculus, what changes is the honest account of the jump, and the guide to whether calculus is hard is for the student who is asking that question before the course starts.

New to first year:

L'Hôpital's rule and the Mean Value Theorem: when each applies and the hypothesis students skip.
Riemann sums and the Fundamental Theorem of Calculus: why the area under a curve is an antiderivative.
u-substitution and integration by parts: choosing between them, and two examples.

Linear algebra

First-year linear algebra has no Grade 12 course behind it in Ontario except the vectors strand of MCV4U, so most students meet matrices for the first time in September of first year.

Matrices: matrix multiplication, the 3×3 determinant, and Gaussian elimination to reduced row echelon form, which are ideas 1, 6, 17 and 23 to 24 of the linear algebra pack.
Vectors, the dot product and the cross product (above) shares the dot product with the pack's idea 5, "Length, dot product and angle".

Data Management (MDM4U) and introductory statistics

MDM4U is Ontario's Grade 12 data management course, and an introductory statistics course in first year covers the same descriptive statistics and the normal distribution before it adds inference. The MDM4U course map has the 24 ideas.

Standard deviation by hand: mean, median, mode, variance and standard deviation, quartiles, box plots and histograms, each computed on one small data set.
The normal distribution and z-scores: the table, the sketch, and the three questions it answers.
Confidence intervals and p-values: what each sentence of a result means, with hypothesis tests, the t-test and the chi-square test. The AP Statistics page maps the exam to these.

How to use an explainer so it sticks

Read the example once with the page open. Close it. Redo the example cold from the question alone, then do the sibling, then write the wrong line in your error log beside the right one. That is the whole method, and it is the attempt, not the reading, that the research credits with the learning.

Mix two ideas in one sitting rather than ten problems of one kind: Rohrer and Taylor (2007) found that students who practised shuffled problem types did worse during practice and better on the test a week later, because the test asks you to choose the method and blocked practice never does. Why mixing problem types beats blocked practice has the shuffling method.

Math help by course, on one screen

Course	Explainers, in teaching order
MHF4U, Algebra 2, Precalculus	Dividing polynomials · Asymptotes and rational functions · Logarithm rules · The unit circle · Trig identities · Domain and range, inverses, transformations
MCV4U, AP Calculus	Limits · Derivative rules · Product and quotient rules · Chain rule · Implicit differentiation and related rates · Vectors, dot and cross product
Calculus I	The MCV4U calculus pages, then L'Hôpital and the Mean Value Theorem · Riemann sums and the Fundamental Theorem · u-substitution and integration by parts
Linear algebra	Matrices: multiplication, determinants, RREF · Vectors (the dot product)
MDM4U, introductory statistics	Standard deviation by hand · Normal distribution and z-scores · Confidence intervals and p-values
Every page	One marked example · the wrong line named · a sibling under Check · the course it sits in

Every page carries those four parts, and the one-page method for a maths course is chapter 4 of the free playbook, on finding the line where the mark went.

Questions students ask

Where can I get free math help for Grade 12?

Start with the explainer for the idea you are stuck on, not the whole course. Each page here works one example line by line, names the line students get wrong, and gives you a sibling problem with its answer to check. Then do the sibling cold. If you still cannot, the course page says what idea comes before it, which is usually the gap.

Is Algebra 2 the same as Advanced Functions?

Mostly. Algebra 2 and Precalculus together cover what MHF4U covers in one Ontario course: polynomials, rational functions, exponentials and logarithms, trigonometry and transformations. The explainers are written by idea, so they fit either name.

What is the best way to use a worked example?

Read it once, then cover it and redo it from the question alone. Research on worked examples shows they teach fastest when they are followed by your own attempt, not read in a row. The sibling problem on each page is the attempt.

Why does every page show a wrong answer?

Because the wrong line is where the mark goes. Seeing a correct example and an incorrect one side by side, with the fault named, is what the learning-from-errors research finds helps students stop repeating it.

Which calculus explainers do I need for AP Calculus?

Limits, the derivative rules, the chain rule, implicit differentiation and related rates, L'Hôpital's rule, the Fundamental Theorem, and the integration techniques. The AP Calculus page maps the exam's units to these ideas.

Every guide in this section

Asymptotes and rational functions: vertical, horizontal and oblique asymptotes, holes, and how to read the whole graph from the algebra
Asymptotes from the algebra: factor, cancel for holes, read vertical asymptotes from what remains, compare degrees for the horizontal. One function worked.
The chain rule: inside, outside, and the inner derivative students forget, with ln x and eˣ worked
The chain rule as a habit: outside, inside, times the inside's derivative. One example worked, the forgotten factor named, with ln x and e^x.
Confidence intervals, p-values and hypothesis tests: one of each computed by hand, and what every sentence of the result does and does not mean
A 95 % confidence interval, a t-test and a chi-square test computed by hand, and the two sentences students get wrong about intervals and p-values, fixed.
Derivative rules: the complete list, when each one applies, one example worked line by line, and the two rules students confuse
Every derivative rule on one page: power, product, quotient, chain, exponential, log and trig, with when each applies and one example worked line by line.
Domain and range, inverse functions and transformations: how to read a function's shape from its equation, worked on one function
Domain and range read from the equation, the inverse found with its domain restricted, and the transformations that built the graph, all on one function.
Implicit differentiation and related rates: the one idea behind both, worked on a curve and on the ladder problem
Implicit differentiation and related rates as one idea: the chain rule with a hidden variable. A curve and the ladder problem worked, the skipped step.
L'Hôpital's rule and the Mean Value Theorem: when each applies, one example of each worked, and the hypothesis students skip
L’Hôpital’s rule and the Mean Value Theorem: the hypothesis to check first, one example of each worked line by line, and the case where each theorem fails.
Limits: what they are, how to find them from a table, a graph and the algebra, the limit laws, and what to do when you get 0/0
Limits explained three ways: a table of values, the graph and the algebra. The limit laws, what 0/0 means, factoring and rationalising, one example worked.
Logarithm rules: the three laws, where each comes from, and how to solve an exponential equation and a logarithmic equation without a false root
The three log rules derived from the exponent laws, the change of base, and two equations worked, one exponential and one logarithmic, with the false root.
Matrices: how to multiply them, the 3×3 determinant, and Gaussian elimination to reduced row echelon form, each worked line by line
Matrix multiplication entry by entry, the 3×3 determinant by cofactors with the sign students drop, and Gaussian elimination to reduced row echelon form.
The normal distribution and z-scores: the sketch, the table, and the three questions every normal-curve problem turns out to be
The normal distribution and z-scores on one mean and standard deviation: the area below, the area between, the value at a percentile, the table misread.
Dividing polynomials: long division, synthetic division, the remainder and factor theorems, and one cubic factored all the way
Synthetic division and long division on one cubic, the remainder and factor theorems, the sign in the box students get wrong, and the cubic factored.
Product and quotient rules: one example worked line by line for each, and the sign error in the quotient rule that costs the mark
The product rule and the quotient rule, each worked line by line, the subtraction order students reverse, and a sibling problem to try with its answer.
Riemann sums to the Fundamental Theorem of Calculus: why the area under a curve is an antiderivative, with one integral done both ways
Riemann sums and the Fundamental Theorem on one integral: left, right and midpoint sums, then F(b) − F(a), and why they agree. The evaluation fault named.
Describing data by hand: mean, median, mode, variance and standard deviation, quartiles, the box plot and the histogram, all from one set of eight numbers
Standard deviation by hand on eight numbers, with the mean, median, mode, variance, quartiles and box plot from the same data, and when to divide by n − 1.
Trig identities: the eight you derive, the rest you look up, the double-angle formulas, and how to solve a trig equation without losing a solution
Trig identities grouped by origin: Pythagorean, quotient, reciprocal and compound-angle, with the double angles derived; one proof and one equation worked.
u-substitution and integration by parts: the one question that chooses between them, the LIATE order for u, and two examples worked line by line
u-substitution or integration by parts: one question decides. Each worked line by line, the du scaling and the sign students drop, LIATE, and two siblings.
The unit circle: how to build it in two minutes instead of memorising it, and how radians become degrees
The unit circle built from two triangles and four signs, not memorised: exact sin, cos and tan of the special angles, radians to degrees, one angle worked.
Vectors: magnitude, direction, the dot product and the cross product, worked on one pair, and which question wants which
Vectors worked on one pair: magnitude, the dot product and the angle, the cross product and the area, the sign error in the middle component, which to use.

Put tonight's studying on rails

YSR Studies turns your notes, lectures and course into cards that come back before you forget, a tutor that makes you try first, and a map of what you know. Free, five hours a month, no card asked.

Open YSR Studies

Sources

Sweller, J., & Cooper, G. A. (1985). The use of worked examples as a substitute for problem solving in learning algebra. Cognition and Instruction, 2(1), 59–89. https://doi.org/10.1207/s1532690xci0201_3
Booth, J. L., Lange, K. E., Koedinger, K. R., & Newton, K. J. (2013). Using example problems to improve student learning in algebra: Differentiating between correct and incorrect examples. Learning and Instruction, 25, 24–34. https://doi.org/10.1016/j.learninstruc.2012.11.002
Roediger, H. L., & Karpicke, J. D. (2006). Test-enhanced learning: Taking memory tests improves long-term retention. Psychological Science, 17(3), 249–255. https://doi.org/10.1111/j.1467-9280.2006.01693.x
Rohrer, D., & Taylor, K. (2007). The shuffling of mathematics problems improves learning. Instructional Science, 35(6), 481–498. https://doi.org/10.1007/s11251-007-9015-8
OpenStax (2016). Calculus Volume 1, chapter 2, Limits. Houston: OpenStax. The standard open treatment of the limit; the hub's example is our own.
YSR Studies (2026). The MHF4U, MCV4U, Calculus I, linear algebra, MDM4U and statistics course maps: the ideas in teaching order, by our own titles.