What do the Grade 9 Mathematics cards cover?

34 cards across 9 chapters of the national textbook: Further on Sets, The Number System, Solving Equations, Solving Inequalities, Introduction to Trigonometry, Regular Polygons, Congruency and Similarity, Vectors in Two Dimensions and 1 more. You can take any chapter one card at a time on the page itself.

Is there a national exam in Grade 9?

No. Ethiopia sets national exams in Grade 6, Grade 8 and Grade 12 only. These cards are for your school's own exams, and for the national exam that comes a few years later.

Where do these cards come from?

They are drafted from Grade 9 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?

29 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 9 Mathematics

Grade 9 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: 9
Chapters: 18
Formulas: 8
Reference tables: Grade 9 Mathematics
Textbook: Free
Price: 29 September 2026
Last checked

Everything34Formula18Reference table8Common mistake5Rule2Method1

Further on Sets

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.

Cardinality of Union of Two SetsFormula

n(A \cup B) = n(A) + n(B) - n(A \cap B)

n(A \cup B) = n(A) + n(B) - n(A \cap B)

$n(A \cup B)$: number of elements in the union of A and B $1$
$n(A)$: number of elements in set A $1$
$n(B)$: number of elements in set B $1$
$n(A \cap B)$: number of elements in the intersection of A and B $1$

When you use it

Use when solving word problems or calculating the number of elements in overlapping sets.

Watch out

Remember to subtract the intersection n(A and B), otherwise shared elements will be counted twice.

Drafted from Grade 9 Mathematics, pages 1-86, then checked twice before it went up

Number of Subsets and Proper SubsetsReference tableAll Subsets · Proper Subsets

Set Type	Formula for n Elements	Example for n = 3
All Subsets	$2^n$	$2^3 = 8$
Proper Subsets	$2^n - 1$	$2^3 - 1 = 7$

When you use it

Use to quickly find the total number of subsets or proper subsets for any finite set.

Watch out

A set is never a proper subset of itself, which is why proper subsets have 1 subtracted from the total.

Drafted from Grade 9 Mathematics, pages 1-86, then checked twice before it went up

The Number System

Euclid Division LemmaFormula

a = bq + r \quad (0 \le r < b)

a = bq + r \quad (0 \le r < b)

$a$: dividend $1$
$b$: divisor $1$
$q$: quotient $1$
$r$: remainder $1$

When you use it

Use when finding the unique quotient and remainder of positive integers, or when expressing division relations.

Watch out

The remainder r must always satisfy 0 <= r < b. It can never be negative or greater than or equal to the divisor.

Drafted from Grade 9 Mathematics, pages 1-86, then checked twice before it went up

Relation Between GCF and LCMFormula

\text{GCF}(a, b) \times \text{LCM}(a, b) = a \times b

\text{GCF}(a, b) \times \text{LCM}(a, b) = a \times b

$a$: first natural number $1$
$b$: second natural number $1$
$\text{GCF}(a, b)$: Greatest Common Factor of a and b $1$
$\text{LCM}(a, b)$: Least Common Multiple of a and b $1$

When you use it

Use when finding the LCM or GCF of two numbers given their product and one of the values.

Watch out

This product formula holds only for two numbers. It is not true for three or more numbers.

Drafted from Grade 9 Mathematics, pages 1-86, then checked twice before it went up

Simplification of RadicalsFormula

\sqrt{a^2 b} = a\sqrt{b} \quad (a > 0, b > 0)

\sqrt{a^2 b} = a\sqrt{b} \quad (a > 0, b > 0)

$a$: rational factor taken out of radical $1$
$b$: remaining radicand $1$

When you use it

Use when simplifying square roots or combining like irrational terms in addition and subtraction.

Watch out

Square roots can only be combined by addition if they have identical radicands, so simplify each term first.

Drafted from Grade 9 Mathematics, pages 1-86, then checked twice before it went up

Solving Equations

Absolute Value Equation RuleRule

When you use it

Use when solving an equation where an algebraic expression is enclosed inside absolute value bars.

Watch out

If the isolated absolute value equals a negative number, the equation has no solution immediately.

Drafted from Grade 9 Mathematics, pages 87-129, then checked twice before it went up

Discriminant and Nature of RootsReference tableD = b^2 - 4ac > 0 · D = b^2 - 4ac = 0 · D = b^2 - 4ac < 0

Discriminant Value	Nature of Roots	Number of Real Roots
$D = b^2 - 4ac > 0$	Real and distinct	$2$
$D = b^2 - 4ac = 0$	Real and equal	$1$
$D = b^2 - 4ac < 0$	No real roots	$0$

When you use it

Use to determine the number and type of roots of a quadratic equation without solving it completely.

Watch out

A negative discriminant means there are no real roots. Do not take the square root of a negative number in real numbers.

Drafted from Grade 9 Mathematics, pages 87-129, then checked twice before it went up

Extraneous Roots in Radical EquationsCommon mistake

When you use it

Use when solving equations containing variables under square roots or fractional exponents.

Watch out

Squaring both sides can introduce extraneous solutions. You must check every candidate root in the original equation.

Drafted from Grade 9 Mathematics, pages 87-129, then checked twice before it went up

Quadratic FormulaFormula

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

$x$: roots of the quadratic equation $none$
$a$: leading coefficient where a is not zero $none$
$b$: coefficient of the middle term $none$
$c$: constant term $none$

When you use it

Use to find the solutions of any quadratic equation in the standard form ax^2 + bx + c = 0.

Watch out

The division by 2a applies to the entire numerator including -b, not just the square root part.

Drafted from Grade 9 Mathematics, pages 87-129, then checked twice before it went up

Solvability of Linear SystemsReference tablea_1/a_2 is not equal to b_1/b_2 · a_1/a_2 = b_1/b_2 = c_1/c_2 · a_1/a_2 = b_1/b_2 is not equal to c_1/c_2

Ratio Condition	System Classification	Number of Solutions	Geometric Lines
$\frac{a_1}{a_2}$ is not equal to $\frac{b_1}{b_2}$	Independent	One unique solution	Intersecting
$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$	Dependent	Infinitely many	Coincident
$\frac{a_1}{a_2} = \frac{b_1}{b_2}$ is not equal to $\frac{c_1}{c_2}$	Inconsistent	No solution	Parallel

When you use it

Use to classify a two-variable linear system as independent, dependent, or inconsistent using coefficient ratios.

Watch out

Make sure equations are aligned in the same standard form before comparing coefficients.

Drafted from Grade 9 Mathematics, pages 87-129, then checked twice before it went up

Solving Inequalities

Absolute Value InequalitiesFormula

|k| < c \iff -c < k < c \quad \text{and} \quad |k| > c \iff k > c \text{ or } k < -c

|k| < c \iff -c < k < c \quad \text{and} \quad |k| > c \iff k > c \text{ or } k < -c

$k$: algebraic expression $dimensionless$
$c$: positive real number constant $dimensionless$

When you use it

Use when solving linear inequalities where an expression is enclosed in absolute value bars.

Watch out

Treating a greater-than absolute value inequality as a single compound inequality instead of two separate disjoint conditions.

Drafted from Grade 9 Mathematics, pages 130-161, then checked twice before it went up

Graphing Two Variable Linear InequalitiesMethod

When you use it

Use when finding the solution region of a linear inequality or system of linear inequalities on the coordinate plane.

Watch out

Selecting a test point that lies directly on the boundary line fails to determine which side of the region to shade.

Drafted from Grade 9 Mathematics, pages 130-161, then checked twice before it went up

Interval Notation and Boundary TypesReference table(a, b) · [a, b] · (a, b] · +1

Interval	Set Inequality	Boundary Dots
$(a, b)$	$a < x < b$	Open dots at $a$ and $b$
$[a, b]$	$a \le x \le b$	Solid dots at $a$ and $b$
$(a, b]$	$a < x \le b$	Open dot at $a$ , solid dot at $b$
$[a, b)$	$a \le x < b$	Solid dot at $a$ , open dot at $b$

When you use it

Use when expressing the solution set of an inequality on a number line or in interval format.

Watch out

Using square brackets or solid dots for strict inequalities includes boundary points that are not part of the solution.

Drafted from Grade 9 Mathematics, pages 130-161, then checked twice before it went up

Multiplying or Dividing by a Negative NumberCommon mistake

When you use it

Use when simplifying or solving an algebraic inequality by multiplying or dividing both sides by a negative value.

Watch out

Failing to reverse the inequality direction when multiplying or dividing by a negative number changes the solution set.

Drafted from Grade 9 Mathematics, pages 130-161, then checked twice before it went up

Product Properties for Quadratic InequalitiesRule

When you use it

Use when solving factored inequalities of the form m times n greater than 0 or m times n less than 0.

Watch out

Assuming both factors have the same sign when their product is negative. They must have opposite signs.

Drafted from Grade 9 Mathematics, pages 130-161, then checked twice before it went up

Introduction to Trigonometry

Confusing Opposite and Adjacent SidesCommon mistake

When you use it

Check this before setting up sine, cosine, or tangent ratios for a given acute angle.

Watch out

The hypotenuse is always opposite the 90 degree angle, but opposite and adjacent sides swap places when you switch from one acute angle to the other.

Drafted from Grade 9 Mathematics, pages 162-180, then checked twice before it went up

Pythagoras TheoremFormula

b^2 + p^2 = h^2

b^2 + p^2 = h^2

$h$: hypotenuse $m$
$b$: base $m$
$p$: perpendicular side $m$

When you use it

Use this to find an unknown side length of a right-angled triangle or to verify if a triangle is a right-angled triangle.

Watch out

The hypotenuse h must always be the longest side opposite the 90 degree angle. Never substitute a leg in place of h.

Drafted from Grade 9 Mathematics, pages 162-180, then checked twice before it went up

Trigonometric RatiosFormula

\sin A = \frac{\text{opposite}}{\text{hypotenuse}}, \quad \cos A = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan A = \frac{\text{opposite}}{\text{adjacent}}

\sin A = \frac{\text{opposite}}{\text{hypotenuse}}, \quad \cos A = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan A = \frac{\text{opposite}}{\text{adjacent}}

$A$: acute angle $^\circ$
$\text{opposite}$: side opposite to angle A $m$
$\text{adjacent}$: side adjacent to angle A $m$
$\text{hypotenuse}$: hypotenuse $m$

When you use it

Use these ratios to relate an acute angle with the side lengths in a right-angled triangle.

Watch out

The opposite and adjacent sides depend on which acute angle is chosen. Make sure they are identified with respect to angle A.

Drafted from Grade 9 Mathematics, pages 162-180, then checked twice before it went up

Trigonometric Values of Special AnglesReference tableA

Ratio	$0^\circ$	$30^\circ$	$45^\circ$	$60^\circ$	$90^\circ$
$\sin A$	$0$	$\frac{1}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{\sqrt{3}}{2}$	$1$
$\cos A$	$1$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{1}{2}$	$0$
$\tan A$	$0$	$\frac{1}{\sqrt{3}}$	$1$	$\sqrt{3}$	undefined

When you use it

Use these exact values when evaluating trigonometric expressions with standard acute angles without a table.

Watch out

The value of tan 90 degrees is undefined because division by zero occurs.

Drafted from Grade 9 Mathematics, pages 162-180, then checked twice before it went up

Regular Polygons

Area of a Regular PolygonFormula

A = \frac{1}{2} a P

A = \frac{1}{2} a P

$A$: area of the regular polygon $\text{m}^2$
$a$: apothem of the regular polygon $\text{m}$
$P$: perimeter of the regular polygon $\text{m}$

When you use it

Use this formula to calculate the area of any regular polygon when its apothem and perimeter are known.

Watch out

The apothem is the perpendicular distance from the center to a side, not the radius to a vertex.

Drafted from Grade 9 Mathematics, pages 181-208, then checked twice before it went up

Interior Angle of a Regular PolygonFormula

\theta = \frac{(n - 2) \times 180^\circ}{n}

\theta = \frac{(n - 2) \times 180^\circ}{n}

$\theta$: measure of each interior angle $^\circ$
$n$: number of sides of the regular polygon $1$

When you use it

Use this formula to find the measure of a single interior angle in a regular polygon where all angles are equal.

Watch out

This formula applies only to regular polygons. Irregular polygons do not have equal interior angles.

Drafted from Grade 9 Mathematics, pages 181-208, then checked twice before it went up

Side Length of Inscribed Regular PolygonFormula

s = 2r \sin\left(\frac{180^\circ}{n}\right)

s = 2r \sin\left(\frac{180^\circ}{n}\right)

$s$: side length of the regular polygon $\text{m}$
$r$: radius of the circumscribed circle $\text{m}$
$n$: number of sides of the regular polygon $1$

When you use it

Use this formula to calculate the side length of an n-sided regular polygon inscribed in a circle of radius r.

Watch out

Remember to multiply by 2 because the perpendicular bisector splits the full side into two equal halves.

Drafted from Grade 9 Mathematics, pages 181-208, then checked twice before it went up

Congruency and Similarity

Linear Scale Factor vs Area ScalingCommon mistake

When you use it

Check this whenever a word problem asks how changing side lengths changes the area or perimeter.

Watch out

Multiplying side lengths by 2 doubles the perimeter, but multiplies the area by 2^2 = 4.

Drafted from Grade 9 Mathematics, pages 209-246, then checked twice before it went up

Ratio of Areas of Similar FiguresFormula

\frac{A_1}{A_2} = \left(\frac{s_1}{s_2}\right)^2 = k^2

\frac{A_1}{A_2} = \left(\frac{s_1}{s_2}\right)^2 = k^2

$A_1$: Area of first similar figure $\text{m}^2$
$A_2$: Area of second similar figure $\text{m}^2$
$s_1$: side length of first figure $\text{m}$
$s_2$: corresponding side length of second figure $\text{m}$
$k$: scale factor $1$

When you use it

Use when finding the area of a similar plane figure given side lengths or the scale factor.

Watch out

Remember to square the scale factor. The ratio of areas is k^2 and not k.

Drafted from Grade 9 Mathematics, pages 209-246, then checked twice before it went up

Ratio of Perimeters of Similar FiguresFormula

\frac{P_1}{P_2} = \frac{s_1}{s_2} = k

\frac{P_1}{P_2} = \frac{s_1}{s_2} = k

$P_1$: Perimeter of first similar figure $\text{m}$
$P_2$: Perimeter of second similar figure $\text{m}$
$s_1$: side length of first figure $\text{m}$
$s_2$: corresponding side length of second figure $\text{m}$
$k$: scale factor $1$

When you use it

Use to find an unknown perimeter or side length when two geometric figures are similar.

Watch out

Match strictly corresponding sides in order when setting up the ratio.

Drafted from Grade 9 Mathematics, pages 209-246, then checked twice before it went up

Triangle Similarity CriteriaReference tableAA · SSS · SAS

Criterion	Given Equality or Ratio	Conclusion
AA	$\angle A = \angle D \text{ and } \angle B = \angle E$	$\triangle ABC \sim \triangle DEF$
SSS	$\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}$	$\triangle ABC \sim \triangle DEF$
SAS	$\frac{AB}{DE} = \frac{AC}{DF} \text{ and } \angle A = \angle D$	$\triangle ABC \sim \triangle DEF$

When you use it

Use these tests to prove two triangles are similar using minimum required angle or side conditions.

Watch out

In SAS similarity, the congruent angle must be the included angle between the two proportional sides.

Drafted from Grade 9 Mathematics, pages 209-246, then checked twice before it went up

Vectors in Two Dimensions

Adding Magnitudes DirectlyCommon mistake

When you use it

When combining two non parallel vectors to find their resultant.

Watch out

Never add the magnitudes of two vectors directly using scalar addition unless they act in the exact same direction.

Drafted from Grade 9 Mathematics, pages 247-262, then checked twice before it went up

Magnitude of a Column VectorFormula

|\mathbf{a}| = \sqrt{x^2 + y^2}

|\mathbf{a}| = \sqrt{x^2 + y^2}

$|\mathbf{a}|$: magnitude of vector a $\text{units}$
$x$: horizontal component of vector a $\text{units}$
$y$: vertical component of vector a $\text{units}$

When you use it

Use to find the length or magnitude of a two dimensional vector given in column form.

Watch out

Always square negative components to get a positive value before adding under the square root.

Drafted from Grade 9 Mathematics, pages 247-262, then checked twice before it went up

Magnitude of a Vector from CoordinatesFormula

|\vec{PQ}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

|\vec{PQ}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

$|\vec{PQ}|$: magnitude of vector PQ $\text{units}$
$x_1$: x coordinate of initial point P $\text{units}$
$y_1$: y coordinate of initial point P $\text{units}$
$x_2$: x coordinate of terminal point Q $\text{units}$
$y_2$: y coordinate of terminal point Q $\text{units}$

When you use it

Use to calculate the magnitude of a vector between given initial and terminal points on a coordinate plane.

Watch out

Subtract the initial point coordinates from the terminal point coordinates, not the other way around, to preserve direction.

Drafted from Grade 9 Mathematics, pages 247-262, then checked twice before it went up

Triangle Law of Vector AdditionFormula

\vec{AB} + \vec{BC} = \vec{AC}

\vec{AB} + \vec{BC} = \vec{AC}

$\vec{AB}$: first vector from initial point A to terminal point B $\text{units}$
$\vec{BC}$: second vector from initial point B to terminal point C $\text{units}$
$\vec{AC}$: resultant vector connecting initial point A to terminal point C $\text{units}$

When you use it

Use to find the resultant vector when the initial point of the second vector connects to the terminal point of the first vector.

Watch out

Ensure the tail of the second vector is placed at the head of the first vector before connecting initial to terminal.

Drafted from Grade 9 Mathematics, pages 247-262, then checked twice before it went up

Statistics and Probability

Arithmetic Mean of Raw DataFormula

\text{Mean} = \frac{x_1 + x_2 + \dots + x_n}{n}

\text{Mean} = \frac{x_1 + x_2 + \dots + x_n}{n}

$x_i$: Individual data value $none$
$n$: Total number of observations $none$

When you use it

Use when finding the average of an unorganized list of numerical values.

Watch out

Divide by the total count of values, not by the maximum value in the set.

Drafted from Grade 9 Mathematics, page 263 onwards, then checked twice before it went up

Mean from Frequency DistributionFormula

\text{Mean} = \frac{f_1 x_1 + f_2 x_2 + \dots + f_k x_k}{f_1 + f_2 + \dots + f_k}

\text{Mean} = \frac{f_1 x_1 + f_2 x_2 + \dots + f_k x_k}{f_1 + f_2 + \dots + f_k}

$x_i$: Data value or class score $none$
$f_i$: Frequency of the corresponding data value $none$

When you use it

Use when calculating the mean of data given in a frequency table.

Watch out

Divide by the sum of frequencies, not by the number of rows in the table.

Drafted from Grade 9 Mathematics, page 263 onwards, then checked twice before it went up

Properties of Mean under Linear ChangesReference tableAdd p · Subtract p · Multiply by nonzero p · +1

Change to each data value	Resulting Mean
Add $p$	$\text{Mean} + p$
Subtract $p$	$\text{Mean} - p$
Multiply by nonzero $p$	$p \cdot \text{Mean}$
Divide by nonzero $p$	$\frac{\text{Mean}}{p}$

When you use it

Use to quickly determine the new mean when every data value is changed by the same constant operation.

Watch out

The sum of deviations of all original values from their mean is always zero.

Drafted from Grade 9 Mathematics, page 263 onwards, then checked twice before it went up

The Grade 9 Mathematics textbook

The student textbook for the new curriculum, free as a PDF. The cards on this page follow its chapters.

Ministry of Education edition, English
337 pages
Download the PDF

What the book covers

1.Further on Sets
2.The Number System
3.Solving Equations
4.Solving Inequalities
5.Introduction to Trigonometry
6.Regular Polygons
7.Congruency and Similarity
8.Vectors in Two Dimensions
9.Statistics and Probability

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 9 subjects

Everything else in this year that has cards so far.

Questions students ask

What do the Grade 9 Mathematics cards cover?: 34 cards across 9 chapters of the national textbook: Further on Sets, The Number System, Solving Equations, Solving Inequalities, Introduction to Trigonometry, Regular Polygons, Congruency and Similarity, Vectors in Two Dimensions and 1 more. You can take any chapter one card at a time on the page itself.
Is there a national exam in Grade 9?: No. Ethiopia sets national exams in Grade 6, Grade 8 and Grade 12 only. These cards are for your school's own exams, and for the national exam that comes a few years later.
Where do these cards come from?: They are drafted from Grade 9 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?: 29 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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