National 5 Mathematics SQA

This subject is broken down into 54 topics in 6 modules:

  1. Numerical Skills 7 topics
  2. Simultaneous Equations 11 topics
  3. Algebraic skills 15 topics
  4. Geometric skills 10 topics
  5. Trigonometric skills 7 topics
  6. Statistical skills 4 topics
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  • 6
    modules
  • 54
    topics
  • 18,897
    words of revision content
  • 2+
    hours of audio lessons

This page was last modified on 28 September 2024.

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Mathematics

Numerical Skills

Fractions

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Fractions

Fractions

Basics of Fractions

  • A fraction is a way of expressing a quantity as a part of a whole.
  • It is written as two numbers separated by a slash (/). The number above is the numerator and below is the denominator.
  • The numerator represents how many parts we have.
  • The denominator shows how many equal parts the whole is divided into.

Simplifying Fractions

  • Simplify a fraction by finding the largest number that divides both the numerator and denominator (the greatest common divisor).
  • Divide both the numerator and the denominator by this number to get the simplified fraction.

Adding and Subtracting Fractions

  • To add or subtract fractions, they must have the same denominator (be common fractions).
  • If they do not, find the least common multiple (LCM) of the denominators and convert the fractions.
  • The resultant fraction should be simplified, if possible.

Multiplying Fractions

  • To multiply fractions, multiply the numerators together for the new numerator.
  • Then, multiply the denominators together for the new denominator.
  • The resultant fraction should be simplified, if possible.

Dividing Fractions

  • To divide fractions, multiply the first fraction by the reciprocal of the second (flip the numerator and denominator of the second fraction).
  • That is, if you are dividing a/b by c/d, the result is (a/b) * (d/c).
  • The resultant fraction should be simplified, if possible.

Decimal to Fraction Conversion

  • To convert a decimal to a fraction, identify the place value of the last digit (for example, hundredths place for 0.02).
  • Use this to write the decimal as a fraction (0.02 = 2/100).
  • Simplify this fraction, if possible.

Fraction to Decimal Conversion

  • To convert a fraction to a decimal, divide the numerator by the denominator using long division method.

Course material for Mathematics, module Numerical Skills, topic Fractions

Mathematics

Algebraic skills

Completing the Square

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Completing the Square

Basics of Completing the Square

  • Completing the square is a method used to solve quadratic equations, convert quadratic functions to vertex form, and understand the graph of a quadratic function.

  • It involves rearranging a quadratic equation to create a perfect square trinomial from the quadratic and linear terms.

Steps to Complete the Square

  • Begin by ensuring your quadratic equation is in the form ax^2 + bx + c = 0. If 'a' is not 1, divide every term by 'a' to make it 1.

  • Move the constant term (c) to the other side of the equation.

  • Take half of the coefficient (number) of the 'x' term (b), square it, and add it to both sides of the equation.

  • The left side of the equation can now be written as a binomial squared: (x + h)^2, where h is half of the original 'b' value.

  • Solve for 'x' by taking the square root of both sides, remembering to consider both the positive and negative square roots.

Key Points to Remember

  • Completing the square only works directly on quadratics in the form x^2 + bx + c = 0. If the coefficient of x^2 is not 1, it must be made 1 by dividing every term by that coefficient.

  • After rearranging the equation, the left side becomes a perfect square: (x + h)^2.

  • Once the equation is in the form (x + h)^2 = k, solve for 'x' by taking the square root of both sides. Remember to consider both the positive and negative square roots.

  • This method is particularly useful for determining the turning point (vertex) of a quadratic function, and for sketching its graph. The vertex form of a quadratic is y = a(x - h)^2 + k where (h, k) are the coordinates of the vertex.

Course material for Mathematics, module Algebraic skills, topic Completing the Square

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