Equations & Problem Solving (CIE IGCSE Maths: Core)

Many questions involve having to form and solve equations from information given about things relating to shapes, like lengths or angles.

How do I form an equation involving the area or perimeter of a 2D shape?

• Read the question carefully to decide if it involves area, perimeter or angles
• If no diagram is given it is almost always a good idea to quickly sketch one
• Add any information given in the question to the diagram
  • This information will normally involve expressions in terms of one or two variables
• If the question involves perimeter, figure out which sides are equal length and add these together
  • Consider the properties of the given shape to decide which sides will have equal lengths
    • In a square or rhombus, all four sides are equal
    • In a rectangle or parallelogram, opposite sides are equal
    • If a triangle is given, are any of the sides equal length?
• If the question involves area, write down the necessary formula for the area of that shape
  • If it is an uncommon shape you may need to split it up into two or more common shapes that you can work out areas for
  • In the case you will have to split the length and width up accordingly
• Remember that a regular polygon means all the sides are equal length
  • For example, a regular pentagon with side length 2x – 1 has 5 equal sides so its perimeter is 5(2x – 1)
• If one of the shapes is a circle or part of a circle, use π throughout rather than multiplying by it and ending up with long decimals

How do I form an equation involving angles in a 2D shape?

• If no diagram is given it is almost always a good idea to quickly sketch one
• Add any information given in the question to the diagram
• This information will normally involve expressions in terms of one or two variables
• Consider the properties of angles within the given shape to decide which sides will have equal lengths
• If a triangle is given, how many of the angles are equal?
• An isosceles triangle has two equal angles
• An equilateral triangle has three equal angles
• Consider angles in parallel lines (alternative, corresponding, co-interior)
• In a parallelogram or rhombus, opposite angles are equal and all four sum to 360°
• A kite has one equal pair of opposite angles
• If the question involves angles, use the formula for the sum of the interior angles of a polygon
• For a polygon of n sides, the sum of the angles will be 180°×(n - 2)
• Remember that a regular polygon means all the angles are equal
• If a question involves an irregular polygon, assume all the angles are different unless told otherwise
• Look out for key information that can give more information about the angles
• For example, a trapezium "with a line of symmetry" will have two pairs of equal angles  

How do I form an equation involving the surface area or volume of a 3D shape?

• Read the question carefully to decide if it involves surface area or volume
• Mixing these up is a common mistake
• If no diagram is given it is almost always a good idea to quickly sketch one
• Add any information given in the question to the diagram
• This information will normally involve expressions in terms of one or two variables
• Consider the properties of the given shape to decide which sides will have equal lengths
• In a cube all sides are equal
• All prisms have the same shape (cross section) at the front and back
• If the question involves volume, write down the necessary formula for the area of that shape
• If it is an uncommon shape the exam question will give you the formula that you need
• Substitute the expressions for the side lengths into the formula
• Remember to include brackets around any expression that you substitute in
• It the question involves surface area,
  • You may need to add or subtract some expressions
  • Remember to consider any faces that may be hidden in the diagram
• STEP 1
  Write down the number of faces the shape has and if any are the same
• STEP 2
  Identify the 2D shape of each face and write down the formula for the area of each one
• STEP 3
  Substitute the given expressions into the formula for each one, being careful to identify the correct expression for the dimension
• STEP 4
  Add the expressions together, double checking that you have one for each of the faces

What is problem solving?

• Problem solving in mathematics involves using several stages, across a variety of topics, to answer a question
• In this set of notes, the problems will involve equations
  • These could be linear equations or simultaneous equations
  • Often, the problem will involve reading the question and setting up an equation, or a set of equations, that can be solved
• With problem solving questions, you never know exactly what’s coming, but you can follow a pattern to answer them
• In an ordinary mathematics question you would be given an equation to solve
• In a problem solving question you would have to generate the equation ...
  • ... using information from the question
  • ... using your knowledge of standard mathematical results

• A key feature of problem solving questions is to interpret the answer in context
• An answer on a calculator may be 1.2
  • If the question was about money then your final answer should be £1.20
• Do not necessarily expect whole number (integer) or “nice” solutions
• Especially where a calculator is allowed
• Rounding appropriately may be one of the skills being tested
• • eg Rounding a value in cm only needs to be to one decimal place;

    so it indicates mm

• Practice, experience and familiarity are the keys to solving problems successfully