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Topics

Each topic shows what you learn, where it often goes wrong and how your teacher helps. Tick where you are stuck.

The MULO year is a guide. It differs per school and per book. Bring your book to the lesson.

Algebra and functions

  • Powers and roots

    MULO 1 and 2

    2⁵ = 32

    You learn
    Rules for powers, such as a² · a³ = a⁵ and (a²)³ = a⁶. Simplifying roots, such as √12 = 2√3.
    Where it often goes wrong
    The rules get mixed up. Then (a²)³ suddenly becomes a⁵.For example: (a²)³ · a
    How your teacher helps
    We first write the powers out in full. That way you see each rule appear by itself.
    Read the post: Powers and roots: the rules

  • Brackets and factorising

    MULO 2 and 3

    x²3x3x9

    (x + 3)² = x² + 6x + 9

    You learn
    Expanding brackets, including (a + b)² and (a + b)(a − b). Factorising, such as x² − 4x − 21 = (x + 3)(x − 7).
    Where it often goes wrong
    The best-known mistake is (x + 3)² = x² + 9. The 6x is missing.For example: Expand (x − 5)²
    How your teacher helps
    We draw the square as a square with four boxes. Then you see the missing piece.
    Read the post: Parabolas: vertex, roots and the quadratic formula

  • Solving equations

    MULO 1 and 2

    3 − 2(x + 2)7

    3 − 2(0 + 2) = −1 ≠ 7

    You learn
    Solving linear equations with the balance method, also with brackets and fractions. Checking your answer by substituting it.
    Where it often goes wrong
    Minus signs are where it goes wrong, especially a minus in front of brackets.For example: 3 − 2(x + 2) = 7
    How your teacher helps
    You write each step on a new line. That way you find a mistake quickly.
    Read the post: Solving equations: balance and graph

  • Systems of equations

    MULO 2 and 3

    x + y = 10x − y = 4

    P(3, 7): x − y = −4

    You learn
    Solving two equations with two unknowns. By adding them, or by putting one into the other.
    Where it often goes wrong
    When you subtract two equations, a minus sign often goes wrong.For example: x + y = 10 and x − y = 4
    How your teacher helps
    We put every answer back into both equations. If it does not fit, you see it at once.
    Read the post: Linear functions: the straight line y = ax + b

  • Intervals

    MULO 3 and 4

    −1 < x ≤ 3 · x ∈ (−1, 3]

    You learn
    Writing a piece of the number line as an interval, such as (−2, 3]. Solving inequalities and writing the solution as an interval.
    Where it often goes wrong
    With x < 3, the 3 is not included, so the bracket is open. When you divide by a negative number, the sign flips.For example: Write −2 < x ≤ 3 as an interval
    How your teacher helps
    You draw the solution on the number line first. Only then do you write the interval.
    Read the post: Intervals on the number line

  • Linear functions

    MULO 2

    12

    y = 2x + 1

    You learn
    The formula y = ax + b. The slope and the y-intercept. Parallel and perpendicular lines.
    Where it often goes wrong
    a and b often get swapped. The sign of a often goes wrong too.For example: The line through (1, 4) and (3, 10)
    How your teacher helps
    We draw every line from b: one step across, a steps up.
    Read the post: Linear functions: the straight line y = ax + b

  • Quadratic equations

    MULO 3

    x² − 4x − 21 = 0: x = −3 or x = 7

    You learn
    Solving by factorising, completing the square or with the quadratic formula. The discriminant tells you how many solutions there are.
    Where it often goes wrong
    x² = 8 has two solutions. The negative one is often forgotten.For example: x² − 4x − 21 = 0
    How your teacher helps
    For each question you first learn to pick the quickest method. Then we work it out step by step.
    Read the post: Solving equations: balance and graph

  • Quadratic functions and parabolas

    MULO 3

    y = x² − 6x + 1 · vertex (3, −8)

    You learn
    The graph of y = ax² + bx + c. The vertex, the axis of symmetry and the roots. What a, b and c do to the parabola.
    Where it often goes wrong
    The minus sign in x = −b ÷ 2a often goes wrong.For example: The vertex of y = x² − 6x + 1
    How your teacher helps
    We make a sketch first. Then you see whether your vertex can be right.
    Read the post: Parabolas: vertex, roots and the quadratic formula

Geometry and trigonometry

  • Pythagoras' theorem

    MULO 2

    512c

    c = √(5² + 12²) = √169 = 13

    You learn
    Finding a side in a right-angled triangle. Distances on a coordinate grid. Pythagoras in a 3D shape.
    Where it often goes wrong
    For one of the short sides, students often add instead of subtract.For example: Hypotenuse 13, one short side 5. How long is the other?
    How your teacher helps
    You point out the hypotenuse first. Then you know whether to add or subtract.
    Read the post: Pythagoras in the plane and in space

  • Area and perimeter

    MULO 1, 2, and 3

    d

    area = π · 3² = 9π ≈ 28.27circumference = π · 6 = 6π ≈ 18.85

    You learn
    Perimeter and area of a triangle, rhombus, trapezium and circle. Parts of a circle, such as a quarter circle. Leaving answers exact, with π.
    Where it often goes wrong
    Perimeter and area get mixed up. Composite shapes look hard.For example: Area of a circle with diameter 6
    How your teacher helps
    We first split every shape into pieces you know.

  • Transformations in the plane

    MULO 2 and 3

    y = x

    P(2, 5) → P′(5, 2)

    You learn
    Translating, reflecting and rotating. Working out the coordinates of the image. Two transformations one after the other.
    Where it often goes wrong
    The original and the image get swapped. When you reflect in y = x, x and y change places.For example: Reflect P(2, 5) in the line y = x
    How your teacher helps
    We draw on squared paper first. Then we work with coordinates.

  • Sine, cosine and tangent

    MULO 3 · maths B

    BC = 10 · sin 30° = 5AC = 10 · cos 30° ≈ 8.66

    You learn
    Finding sides and angles in a right-angled triangle with SOH CAH TOA. Using the calculator well, including sin⁻¹.
    Where it often goes wrong
    Which side is opposite and which is adjacent depends on the angle. That is where it often goes wrong.For example: Angle C is 90°, AB = 10 and angle A = 30°. Find BC.
    How your teacher helps
    We mark the angle first, then the sides. Your calculator is always on DEG.
    Read the post: Sine and cosine on the unit circle

  • Trigonometry on the unit circle

    MULO 3 · maths B

    α = 30°: sin α = 0.5 · cos α ≈ 0.87

    You learn
    Sine and cosine of any angle, also above 90°. sin²α + cos²α = 1. Equations such as sin α = 0.5.
    Where it often goes wrong
    Students learn tables by heart without seeing why they are true.For example: Solve sin α = 0.5 for α between 0° and 360°
    How your teacher helps
    On the unit circle you see why sin 140° equals sin 40°. Then there is less to remember.
    Read the post: Sine and cosine on the unit circle

  • Sine rule and cosine rule

    MULO 4 · maths B

    c² = 49 + 64 − 112 · cos 60° = 57c = √57 ≈ 7.55

    You learn
    Finding sides and angles in a triangle without a right angle.
    Where it often goes wrong
    Choosing the right rule is the hardest part.For example: a = 7, b = 8 and angle C = 60°. Find c.
    How your teacher helps
    You learn one fixed question: which three things do you know? The answer picks the rule.
    Read the post: Sine and cosine on the unit circle

  • Lines and circles

    MULO 4 · maths B

    M(2, −1): (x − 2)² + (y + 1)² = 13

    You learn
    The equation of a circle: (x − a)² + (y − b)² = r². Where a line meets a circle. The tangent to a circle.
    Where it often goes wrong
    The centre is read off with the wrong sign.For example: Centre and radius of (x − 2)² + (y + 1)² = 13
    How your teacher helps
    We draw first. The tangent is perpendicular to the radius, and you see that on paper.
    Read the post: Lines and circles: the equation of a circle

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