What do the Grade 12 Mathematics cards cover?

34 cards across 5 chapters of the national textbook: Sequences and Series, Introduction to Calculus, Statistics, Introduction to Linear Programming and Mathematical Applications in Business. You can take any chapter one card at a time on the page itself.

Do these help with the Grade 12 national exam?

Yes. Grade 12 sits a national exam, and these are the Mathematics rules and formulas it expects you to know. Past papers with a worked answer behind every question are on the same site, also free.

Where do these cards come from?

They are drafted from Grade 12 Mathematics, the Ministry of Education textbook for this grade. A second pass that cannot see the chapter then re-derives every formula, constant and table row, and anything it cannot confirm is held back instead of published.

Yes. Every card here is free to read and the printable sheet is free to download. Neither needs an account.

When was this last checked?

6 September 2026. Cards arrive chapter by chapter, and the line under each one says when that card was last read through.

Free short notes and revision cards

Grade 12 Mathematics

Grade 12 Mathematics on Temari has 34 revision cards, arranged by the chapters of the Ethiopian national curriculum. Every card says when the rule applies, what each symbol in it stands for, and the mistake students most often make with it. They are free to read and need no account.

34
Cards: 5
Chapters: 11
Formulas: 15
Reference tables: Grade 12 Mathematics
Textbook: Free
Price: 6 September 2026
Last checked

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Everything34Reference table15Formula11Common mistake5Method2Rule1

Across the whole subject

Statistics formulasReference tableMean · Median (odd count) · Median (even count) · +5

Rule	Formula
Mean	$\mu = \dfrac{\sum x_{i}}{n}$
Median (odd count)	the middle value of the ordered list
Median (even count)	$\dfrac{x_{n/2} + x_{n/2+1}}{2}$
Mode	the value that occurs most often
Range	$x_{\max} - x_{\min}$
Variance	$\sigma^{2} = \dfrac{\sum (x_{i} - \mu)^{2}}{n}$
Standard deviation	$\sigma = \sqrt{\dfrac{\sum (x_{i} - \mu)^{2}}{n}}$
Mean of grouped data	$\mu = \dfrac{\sum f_{i} x_{i}}{\sum f_{i}}$

When you use it

Use these whenever a question hands you a list of numbers and asks what is typical or how spread out they are. Work out the mean first, because variance and standard deviation are both built on it and neither can be found without it.

Watch out

Divide by $n$ for a whole population and by $n - 1$ for a sample. Exam questions say which one they mean in the wording, not in the numbers, so read the sentence before you reach for the formula.

Probability rulesReference tableProbability of an event · Range of a probability · Certain and impossible · +6

Rule	Formula
Probability of an event	$P(E) = \dfrac{n(E)}{n(S)}$
Range of a probability	$0 \leq P(E) \leq 1$
Certain and impossible	$P(S) = 1,\ P(\varnothing) = 0$
Complement	$P(E') = 1 - P(E)$
Addition rule	$P(A \cup B) = P(A) + P(B) - P(A \cap B)$
Mutually exclusive	$P(A \cup B) = P(A) + P(B)$
Conditional probability	$P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}$
Multiplication rule	$P(A \cap B) = P(B) \cdot P(A \mid B)$
Independent events	$P(A \cap B) = P(A) \cdot P(B)$

When you use it

Start every probability question by naming the sample space and counting it. Almost every rule here is a way of avoiding counting the same outcome twice, and you cannot see the double counting until you know what you are counting from.

Watch out

Mutually exclusive and independent are not the same thing and are the pair students swap most. Mutually exclusive events cannot happen together; independent events do not affect each other's chances. Two events with real probabilities can never be both.

Set counting formulasReference tableTwo sets · Complement · In A only · +4

Rule	Formula
Two sets	$n(A \cup B) = n(A) + n(B) - n(A \cap B)$
Complement	$n(A') = n(U) - n(A)$
In $A$ only	$n(A) - n(A \cap B)$
Three sets	$n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(A \cap C) - n(B \cap C) + n(A \cap B \cap C)$
Disjoint sets	$n(A \cup B) = n(A) + n(B)$
De Morgan	$(A \cup B)' = A' \cap B'$
De Morgan	$(A \cap B)' = A' \cup B'$
Subsets of a set	$2^{n}$ for a set of $n$ members

When you use it

Reach for these when a question counts people or things that belong to more than one group at once, which is what every Venn diagram question really is. Draw the diagram, fill the innermost region first, and work outwards.

Watch out

With three sets the signs alternate: subtract each pair, then add the triple back. Leaving that last term out is the single most common way a three-circle Venn question goes wrong, because the middle region ends up counted zero times instead of once.

Exponent and radical lawsReference tableSame base, multiply · Same base, divide · Power of a power · +8

Rule	Formula
Same base, multiply	$a^{m} \cdot a^{n} = a^{m+n}$
Same base, divide	$\dfrac{a^{m}}{a^{n}} = a^{m-n}$
Power of a power	$(a^{m})^{n} = a^{mn}$
Power of a product	$(ab)^{n} = a^{n} b^{n}$
Power of a quotient	$\left(\dfrac{a}{b}\right)^{n} = \dfrac{a^{n}}{b^{n}}$
Zero exponent	$a^{0} = 1$
Negative exponent	$a^{-n} = \dfrac{1}{a^{n}}$
Root as a power	$\sqrt[n]{a} = a^{1/n}$
Fractional exponent	$a^{m/n} = \sqrt[n]{a^{m}}$
Root of a product	$\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}$
Root of a quotient	$\sqrt[n]{\dfrac{a}{b}} = \dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}$

When you use it

Turn every root into a fractional power before you do anything else. Once the whole expression is written as powers, the three multiply, divide and power-of-a-power rules handle it, and there is nothing left to remember.

Watch out

Every rule with a division line in it needs the base under that line to be something other than zero. Beyond that: these rules only combine powers of the SAME base, and only across multiplication and division. There is no rule for $a^{m} + a^{n}$ , and $(a+b)^{n}$ is not $a^{n} + b^{n}$ for any $n$ but 1.

Area and perimeter of plane shapesReference tableSquare · Rectangle · Triangle · +5

$a$: Side or edge
$b$: Base, or breadth of a rectangle
$h$: Perpendicular height, never a slanting side
$l$: Length of a rectangle
$d_{1}, d_{2}$: The two diagonals of a rhombus
$s$: Half the perimeter, used only in Heron's formula
$r$: Radius, half the diameter

Shape	Perimeter	Area
Square	$4a$	$a^{2}$
Rectangle	$2(l + b)$	$l b$
Triangle	$a + b + c$	$\dfrac{1}{2} b h$
Triangle from its sides	$a + b + c$	$\sqrt{s(s-a)(s-b)(s-c)}$
Parallelogram	$2(a + b)$	$b h$
Rhombus	$4a$	$\dfrac{1}{2} d_{1} d_{2}$
Trapezium	$a + b + c + d$	$\dfrac{1}{2}(a + b) h$
Circle	$2 \pi r$	$\pi r^{2}$

When you use it

Perimeter is the distance once around the edge and area is the surface inside it, so the two answer different questions and never share a unit. Use Heron's formula when a triangle gives you three sides and no height.

Watch out

In a triangle, a parallelogram and a trapezium, $h$ is the perpendicular distance, not the slanting side drawn next to it. Reaching for the slanting side is the single most common way marks are lost here, and the answer it gives is always too large.

Arcs, sectors and segmentsReference tableArc length, angle in degrees · Arc length, angle in radians · Sector area, angle in degrees · +4

$r$: Radius, half the diameter
$\theta$: Angle at the centre
$\pi$: About 3.142, or 22 over 7 when a question says so

	Part of a circle	Formula
	Arc length, angle in degrees	$\dfrac{\theta}{360} \times 2 \pi r$
	Arc length, angle in radians	$r \theta$
	Sector area, angle in degrees	$\dfrac{\theta}{360} \times \pi r^{2}$
	Sector area, angle in radians	$\dfrac{1}{2} r^{2} \theta$
	Perimeter of a sector	$2r + \dfrac{\theta}{360} \times 2 \pi r$
	Chord length	$2r \sin(\theta / 2)$
	Segment area	$\dfrac{\theta}{360} \times \pi r^{2} - \dfrac{1}{2} r^{2} \sin \theta$

When you use it

Every formula here is the whole circle scaled down by how much of it the angle covers. Once you see that, there is only one thing to remember: what fraction of a full turn the angle is, and whether that turn is measured in degrees or radians.

Watch out

The degree forms divide by 360 and the radian forms do not. Putting a radian angle into a degree formula gives an answer roughly 57 times too small, and it will still look like a sensible number, so check the units on the angle before anything else.

Surface area and volume of solidsReference tableCube · Cuboid · Cylinder · +6

$r$: Radius, half the diameter
$h$: Perpendicular height, never a slanting side
$l$: Slant height, measured up the sloping surface
$a$: Edge of a cube
$B$: Area of the base
$P$: Perimeter of the base

Solid	Curved surface	Total surface	Volume
Cube	—	$6a^{2}$	$a^{3}$
Cuboid	—	$2(lb + bh + hl)$	$l b h$
Cylinder	$2 \pi r h$	$2 \pi r (h + r)$	$\pi r^{2} h$
Cone	$\pi r l$	$\pi r (l + r)$	$\dfrac{1}{3} \pi r^{2} h$
Sphere	—	$4 \pi r^{2}$	$\dfrac{4}{3} \pi r^{3}$
Hemisphere	$2 \pi r^{2}$	$3 \pi r^{2}$	$\dfrac{2}{3} \pi r^{3}$
Prism	$P h$	$2B + P h$	$B h$
Pyramid	$\dfrac{1}{2} P l$	$B + \dfrac{1}{2} P l$	$\dfrac{1}{3} B h$
Slant height of a cone	—	—	$l = \sqrt{r^{2} + h^{2}}$

When you use it

Read the question for the word total or the word curved before you pick a row, because most solids have both and the two differ by exactly the flat ends. With a cone, work out the slant height first: none of its surface formulas can be used without it.

Watch out

The slant height $l$ and the perpendicular height $h$ are different lengths, and a cone question usually gives you one while its formulas want the other. Surface area is measured in square units and volume in cubic ones, so an answer with the wrong unit is wrong even when the number is right.

Standard limits as x approaches zeroReference table(x)/x · x/(x) · 1 - xx^2 · +4

Limit	Value
$\dfrac{\sin x}{x}$	$1$
$\dfrac{x}{\sin x}$	$1$
$\dfrac{\tan x}{x}$	$1$
$\dfrac{x}{\tan x}$	$1$
$\dfrac{1 - \cos x}{x^{2}}$	$\dfrac{1}{2}$
$\dfrac{1 - \cos x}{x}$	$0$
$\dfrac{\sin ax}{x}$	$a$
$\dfrac{\sin ax}{\sin bx}$	$\dfrac{a}{b}$
$\dfrac{\tan ax}{\tan bx}$	$\dfrac{a}{b}$
$\dfrac{\sin ax}{\tan bx}$	$\dfrac{a}{b}$
$\dfrac{1 - \cos ax}{x^{2}}$	$\dfrac{a^{2}}{2}$

When you use it

Substitute first, always. If putting the value in gives a number, that number is the limit and there is nothing else to do. These identities exist only for the case where substituting gives zero over zero, and each one is a way of recognising that shape.

Watch out

Every limit here needs the angle in RADIANS. In degrees $\dfrac{\sin x}{x}$ tends to $\dfrac{\pi}{180}$ , which is about 0.017 rather than 1, and the working looks perfectly correct all the way to the wrong answer.

Exponential and logarithmic limitsReference tablex → 0 · x → 1 · x → ∞

As	Limit	Value
$x \to 0$	$\dfrac{e^{x} - 1}{x}$	$1$
$x \to 0$	$\dfrac{e^{ax} - 1}{x}$	$a$
$x \to 0$	$\dfrac{1 - e^{-x}}{x}$	$1$
$x \to 0$	$\dfrac{a^{x} - 1}{x}$	$\ln a$
$x \to 0$	$\dfrac{\ln(1 + x)}{x}$	$1$
$x \to 0$	$\dfrac{\ln(1 + ax)}{x}$	$a$
$x \to 1$	$\dfrac{\ln x}{x - 1}$	$1$
$x \to 0$	$(1 + x)^{1/x}$	$e$
$x \to \infty$	$\left(1 + \dfrac{1}{x}\right)^{x}$	$e$
$x \to \infty$	$\left(1 + \dfrac{a}{x}\right)^{x}$	$e^{a}$
$x \to \infty$	$\left(1 + \dfrac{a}{x}\right)^{bx}$	$e^{ab}$

When you use it

Reach for these when an exponential or a logarithm sits over a plain $x$ and substituting gives zero over zero. Each one is really the gradient of that function at a single point, written as a limit, which is why they all come out as tidy numbers.

Watch out

$(1 + x)^{1/x}$ tends to $e$ as $x$ approaches ZERO, while $\left(1 + \dfrac{1}{x}\right)^{x}$ tends to $e$ as $x$ approaches INFINITY. The two look almost identical and their directions are opposite, so read which way the variable is going before you write $e$ .

Limits as x approaches infinityReference tablea x^n + … b x^n + … · x^px^q, p < q · x^px^q, p > q · +7

Limit	Value
$\dfrac{a x^{n} + \ldots}{b x^{n} + \ldots}$	$\dfrac{a}{b}$
$\dfrac{x^{p}}{x^{q}},\; p < q$	$0$
$\dfrac{x^{p}}{x^{q}},\; p > q$	$\infty$
$\dfrac{1}{x^{p}},\; p > 0$	$0$
$x^{p},\; p > 0$	$\infty$
$c^{x},\; c > 1$	$\infty$
$c^{x},\; 0 < c < 1$	$0$
$\dfrac{\ln x}{x^{p}},\; p > 0$	$0$
$\dfrac{x^{p}}{e^{x}},\; p > 0$	$0$
$\sqrt{x^{2} + 1} - x$	$0$

When you use it

Divide every term by the highest power of $x$ in the DENOMINATOR. Each term that then has an $x$ underneath it goes to zero, and whatever is left standing is the answer. For a difference of two roots, multiply by the conjugate first so the leading terms cancel.

Watch out

Only the highest powers decide the answer, so a lower term never changes it. The ordering to remember is that an exponential beats every power of $x$ , and a logarithm loses to every one, which settles most of these with no working at all.

Sequences and Series

Arithmetic mean between two numbersFormula

m = \frac{a + b}{2}

m = \frac{a + b}{2}

$m$: arithmetic mean $none$
$a$: first number $none$
$b$: second number $none$

When you use it

Use this to find the single intermediate term between two terms that forms an arithmetic sequence.

Watch out

When inserting multiple arithmetic means, find the common difference first rather than averaging the two endpoints.

Drafted from Grade 12 Mathematics, pages 1-60, then checked twice before it went up

Arithmetic sequence testMethod

When you use it

Use this when checking whether a general formula defines an arithmetic sequence.

Watch out

Testing only the first few terms by substitution is not enough. You must subtract a_n from a_{n+1} and confirm that n cancels out completely.

Drafted from Grade 12 Mathematics, pages 1-60, then checked twice before it went up

Fibonacci and Mulatu sequencesReference tableFibonacci · Mulatu

Sequence	Initial terms	Recursive rule
Fibonacci	$F_0 = 1,\, F_1 = 1$	$F_n = F_{n-1} + F_{n-2}$
Mulatu	$M_0 = 4,\, M_1 = 1$	$M_n = M_{n-1} + M_{n-2}$

When you use it

Use this to recall the starting terms and recursion equations of Fibonacci and Mulatu sequences.

Watch out

Notice the zeroth terms. Fibonacci starts with 1 and 1, whereas Mulatu sequence starts with 4 and 1.

Drafted from Grade 12 Mathematics, pages 1-60, then checked twice before it went up

nth term of an arithmetic sequenceFormula

A_n = A_1 + (n - 1)d

A_n = A_1 + (n - 1)d

$A_n$: nth term $none$
$A_1$: first term $none$
$n$: term number $none$
$d$: common difference $none$

When you use it

Use this to calculate any term of an arithmetic sequence when the first term and common difference are known.

Watch out

Do not multiply d by n. The common difference is added n - 1 times to the first term.

Drafted from Grade 12 Mathematics, pages 1-60, then checked twice before it went up

Sign of common differenceCommon mistake

When you use it

Reach for this whenever finding the common difference of a decreasing arithmetic sequence.

Watch out

Always calculate succeeding term minus preceding term. If terms decrease, the common difference is negative.

Drafted from Grade 12 Mathematics, pages 1-60, then checked twice before it went up

Introduction to Calculus

Average Rate of ChangeFormula

\frac{\Delta y}{\Delta x} = \frac{f(b) - f(a)}{b - a}

\frac{\Delta y}{\Delta x} = \frac{f(b) - f(a)}{b - a}

$\Delta y$: change in output $1$
$\Delta x$: change in input $1$
$b$: final input value $1$
$a$: initial input value $1$
$f(b)$: function value at b $1$
$f(a)$: function value at a $1$

When you use it

Use to calculate the overall rate of change or the slope of the secant line between two endpoints on an interval [a, b].

Watch out

Do not divide f(b) - f(a) by a - b. The order of points in the numerator and denominator must match.

Drafted from Grade 12 Mathematics, pages 61-158, then checked twice before it went up

Confusing Secant and Tangent RatesCommon mistake

When you use it

Use when distinguishing between the rate of change across an interval and the rate of change at an exact point.

Watch out

Do not compute the average rate of change formula over an interval when the problem asks for the instantaneous rate at an exact point. An instantaneous rate requires taking the limit as h approaches zero.

Drafted from Grade 12 Mathematics, pages 61-158, then checked twice before it went up

Geometric Meaning of Rate of ChangeReference tablePositive · Negative · Zero · +1

Slope Value	Graph Direction	Function Behavior
Positive	Slopes upward from left to right	Increasing
Negative	Slopes downward from left to right	Decreasing
Zero	Horizontal line	Constant
Undefined	Vertical line	Undefined

When you use it

Use to interpret what the sign of a slope or derivative says about the graph and behavior of a function.

Drafted from Grade 12 Mathematics, pages 61-158, then checked twice before it went up

Instantaneous Rate of ChangeFormula

\lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h}

\lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h}

$x_0$: point of evaluation $1$
$h$: increment in input $1$
$f(x_0)$: function value at x_0 $1$
$f(x_0 + h)$: function value at x_0 + h $1$

When you use it

Use to find the exact rate of change or the slope of the tangent line of a function at a single specific point x_0.

Watch out

Do not substitute h = 0 before simplifying the expression, as this produces an undefined division by zero.

Drafted from Grade 12 Mathematics, pages 61-158, then checked twice before it went up

Statistics

Absolute vs Relative DispersionReference tableAbsolute dispersion · Relative dispersion

Measure Type	Unit of Measurement	Main Purpose	Examples
Absolute dispersion	Same unit as the data	Measures the spread of a single data set	Range, Inter-quartile range, Mean deviation
Relative dispersion	Unitless (dimensionless)	Compares variability between two or more data sets	Coefficient of range, Coefficient of variation

When you use it

Use this comparison to decide whether to report spread in original units or use a unitless ratio to compare different data sets.

Watch out

Never use absolute measures of dispersion to compare two data sets that have different measurement units.

Drafted from Grade 12 Mathematics, pages 159-236, then checked twice before it went up

Mean Deviation for Ungrouped DataFormula

MD = \frac{\sum |x_i - A|}{n}

MD = \frac{\sum |x_i - A|}{n}

$MD$: Mean deviation $unit of data$
$x_i$: Individual observation value $unit of data$
$A$: Central average used: mean, median, or mode $unit of data$
$n$: Total number of observations $dimensionless$

When you use it

Use this formula to find the average absolute distance of raw data points from the mean, median, or mode.

Watch out

Always take the absolute value of each difference before summing. Summing deviations without absolute values about the mean always yields zero.

Drafted from Grade 12 Mathematics, pages 159-236, then checked twice before it went up

Quartiles for Grouped DataFormula

Q_k = L + \left(\frac{\frac{k n}{4} - cf}{f}\right) w

Q_k = L + \left(\frac{\frac{k n}{4} - cf}{f}\right) w

$Q_k$: k-th quartile $unit of data$
$L$: Lower class boundary of the quartile class $unit of data$
$k$: Quartile order, where k is 1, 2, or 3 $dimensionless$
$n$: Total number of observations $dimensionless$
$cf$: Cumulative frequency of the class preceding the quartile class $dimensionless$
$f$: Frequency of the quartile class $dimensionless$
$w$: Class interval width of the quartile class $unit of data$

When you use it

Use this formula to calculate the first quartile, median, or third quartile in a grouped frequency table.

Watch out

The term cf is the cumulative frequency of the class before the quartile class, not the quartile class itself.

Drafted from Grade 12 Mathematics, pages 159-236, then checked twice before it went up

Range for Grouped DataFormula

R = B_U(H) - B_L(L)

R = B_U(H) - B_L(L)

$R$: Range $unit of data$
$B_U(H)$: Upper class boundary of the highest class $unit of data$
$B_L(L)$: Lower class boundary of the lowest class $unit of data$

When you use it

Use this formula to calculate the range of a continuous grouped frequency distribution.

Watch out

Do not subtract raw class limits. Always find the true class boundaries first.

Drafted from Grade 12 Mathematics, pages 159-236, then checked twice before it went up

Using Class Limits Instead of Class BoundariesCommon mistake

When you use it

Review this when finding range, class width, or quartile positions in grouped distributions.

Watch out

Discrete class limits leave gaps between classes. Calculate class boundaries by subtracting 0.5 from the lower limit and adding 0.5 to the upper limit for integer data before computing range or quartiles.

Drafted from Grade 12 Mathematics, pages 159-236, then checked twice before it went up

Median of grouped dataFormula

\text{Median} = L + \frac{h}{f}\left(\frac{n}{2} - C\right)

\text{Median} = L + \frac{h}{f}\left(\frac{n}{2} - C\right)

$L$: lower class BOUNDARY of the median class
$h$: class width, upper boundary minus lower boundary
$f$: frequency of the median class
$n$: total frequency, the sum of every f
$C$: cumulative frequency of the class BEFORE the median class

When you use it

Use when the data arrives as classes with frequencies and no individual values, so there is no middle number to read off. Build the cumulative frequency column, find the first class whose total reaches n over 2, and that is the median class the formula works inside.

Watch out

The width h comes from the BOUNDARIES, not the written limits. A class written 60 to 69 has boundaries 59.5 to 69.5, so h is 10 and not 9. Taking 9 shifts the answer by half a class, which is enough to move a grade, and it is the error most often copied off revision sheets.

Mode of grouped dataFormula

\text{Mode} = L + h\left(\frac{f_{m} - f_{1}}{(f_{m} - f_{1}) + (f_{m} - f_{2})}\right)

\text{Mode} = L + h\left(\frac{f_{m} - f_{1}}{(f_{m} - f_{1}) + (f_{m} - f_{2})}\right)

$L$: lower class BOUNDARY of the modal class
$h$: class width, taken from the boundaries
$f_{m}$: frequency of the modal class, the largest one
$f_{1}$: frequency of the class just before it
$f_{2}$: frequency of the class just after it

When you use it

Use when you need the most common value and the data is in classes, so no single value repeats. The modal class is simply the one with the highest frequency, and the formula places the mode inside it by leaning towards whichever neighbour is smaller.

Watch out

Two classes tied for the highest frequency means the data has two modes and this formula does not apply to either. Check the neighbours before writing anything down. Classes must also be equal in width for the result to mean anything, which is worth checking when a table's last class is left open.

Introduction to Linear Programming

Boundary Lines and Inequality TypesReference tableStrict · Slack

Inequality Type	Symbols	Boundary Line	Points on Line Included
Strict	$<, >$	Dashed or broken	No
Slack	$\le, \ge$	Solid	Yes

When you use it

Use this table to choose the correct line style and determine whether the boundary line is included in the solution region.

Watch out

Drawing a solid line for a strict inequality or a dashed line for a slack inequality is penalized in coordinate graphs.

Drafted from Grade 12 Mathematics, pages 237-302, then checked twice before it went up

Inequality Sign InversionRule

When you use it

Use this rule whenever multiplying or dividing both sides of an inequality by a negative number.

Watch out

Multiplying or dividing by a negative number reverses the inequality sign, but adding or subtracting any number leaves the sign unchanged.

Drafted from Grade 12 Mathematics, pages 237-302, then checked twice before it went up

Test Point Selection for Half PlanesMethod

When you use it

Use this technique to determine which side of the boundary line to shade for a linear inequality in two variables.

Watch out

Never select a test point that lies on the boundary line. If the line passes through the origin, test a point like (0, 1) or (1, 0) instead of (0, 0).

Drafted from Grade 12 Mathematics, pages 237-302, then checked twice before it went up

Translating At Most and At LeastCommon mistake

When you use it

Use when translating word problems and everyday constraints into linear inequalities.

Watch out

At most translates to less than or equal to, not greater than. At least translates to greater than or equal to, not less than.

Drafted from Grade 12 Mathematics, pages 237-302, then checked twice before it went up

Mathematical Applications in Business

Cross Product of ProportionFormula

a \cdot d = b \cdot c

a \cdot d = b \cdot c

$a$: First extreme term $1$
$b$: First mean term $1$
$c$: Second mean term $1$
$d$: Second extreme term $1$

When you use it

Use this relation to solve for an unknown value when two ratios are set equal, where b and d are non-zero.

Watch out

Multiply the extremes together and the means together. Multiplying terms belonging to the same ratio produces a false result.

Drafted from Grade 12 Mathematics, pages 303-413, then checked twice before it went up

Direct and Inverse VariationReference tableDirect variation · Inverse variation

Variation Type	Proportionality	Constant Form	Constant Property
Direct variation	$y \propto x$	$y = k x$	Ratio remains constant
Inverse variation	$y \propto \frac{1}{x}$	$x y = k$	Product remains constant

When you use it

Use this reference to set up the correct variation formula based on whether two quantities vary directly or inversely.

Watch out

For inverse variation, the product of the two variables remains constant, not their quotient.

Drafted from Grade 12 Mathematics, pages 303-413, then checked twice before it went up

Rate of ChangeFormula

r = \frac{A_f - A_i}{A_i}

r = \frac{A_f - A_i}{A_i}

$r$: Rate of change $1$
$A_f$: Final amount $1$
$A_i$: Original amount $1$

When you use it

Use this formula to calculate the relative increase or decrease of a quantity compared to its starting level.

Watch out

Always divide by the original amount in the denominator, never the final amount. A negative result represents a rate of decrease.

Drafted from Grade 12 Mathematics, pages 303-413, then checked twice before it went up

Ratios with Unmatched UnitsCommon mistake

When you use it

Review this whenever forming a ratio or a rate from physical quantities given in different measurement units.

Watch out

Never divide numbers before converting both quantities into identical units. For instance, comparing 6 Birr to 50 cents requires converting 6 Birr to 600 cents first, giving 600 to 50.

Drafted from Grade 12 Mathematics, pages 303-413, then checked twice before it went up

The papers these formulas are for

Real national exam questions from past years, with a worked answer for each one.

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The same subject in other years

An exam paper keeps asking for what the year below taught. Those cards are here too.

Your other Grade 12 subjects

Everything else in this year that has cards so far.

Questions students ask

What do the Grade 12 Mathematics cards cover?: 34 cards across 5 chapters of the national textbook: Sequences and Series, Introduction to Calculus, Statistics, Introduction to Linear Programming and Mathematical Applications in Business. You can take any chapter one card at a time on the page itself.
Do these help with the Grade 12 national exam?: Yes. Grade 12 sits a national exam, and these are the Mathematics rules and formulas it expects you to know. Past papers with a worked answer behind every question are on the same site, also free.
Where do these cards come from?: They are drafted from Grade 12 Mathematics, the Ministry of Education textbook for this grade. A second pass that cannot see the chapter then re-derives every formula, constant and table row, and anything it cannot confirm is held back instead of published.
Is this free?: Yes. Every card here is free to read and the printable sheet is free to download. Neither needs an account.
When was this last checked?: 6 September 2026. Cards arrive chapter by chapter, and the line under each one says when that card was last read through.

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