1.Understand the application of algebra relevant to engineering problems. 

• 1.1 Demonstrate application of algebra i.e.

  • binomial expansion

  • factorisation 

  • using the principle of the lowest common multiple (LCM)

• 1.2 Simplify and solve algebraic equations.

• 1.3 Demonstrate how to solve linear simultaneous equations with two unknowns using graphical interpretation and algebraic method: elimination method, substitution method.

• 1.4 Demonstrate how to solve quadratic equations i.e. 

  • sketching of quadratic graphs using the formula x=(-b ±√b^2 – 4ac)/2a

Be able to use geometry and graphs in the context of engineering problems.

• 2.1 Demonstrate how to use coordinate geometry, including straight-line equations and curve sketching.

• 2.2 Demonstrate graphical transformation.

3. Understand exponentials, logarithms and trigonometry related to engineering problems.

• 3.1 Demonstrate problem-solving using exponentials and logarithms.

• 3.2 Demonstrate problem-solving with arcs, circles and sectors.

• 3.3 Demonstrate problem-solving involving right-angled triangles.

4. Understand calculus relevant to engineering problems

• 4.1 Demonstrate problem-solving involving differentiation.

• 4.2 Differentiate functions of the form:

  • y = xⁿ

  • y = sin ax

  • y = cos ax

  • y = tan ax

Understand the application of algebra relevant to engineering problems.

Course Coverage

• Learners should understand the rules of algebra to simplify and solve mathematical problems for example:

  • algebraic division

  • the remainder and factor theorems 

  • (x+3)(x+2)=x² +5x+6

  • (a+b) ⁴ =a⁴ +4a³b+6a²b² +4ab³ +b⁴

  • bx+by=b(x+y)

  • (x+2)/5 +(x+4)/3 = (8x+26)/15

Using an LCM of 15

• Learners should be taught to simplify and solve equations for example: 5(x−3)−7(6−x)=12−3(8−x) leading to a solution that x=5

• Engineering problems are often described using simultaneous equations. Learners should be taught to solve simultaneous equations graphically and by calculation for example:

  • electrical engineering problems using Kirchhoff’s laws forces in a mechanical system using 0.7F1 + 0.5F2 = 9 and 0.3F1 +0.4F2 =5, state that when two equations contain two unknowns

  •  such as 3x+7y=10 and x+4y=6, such that only one value of x and y exist that will satisfy both equations, are called simultaneous equations.

• Engineering problems can often be described using quadratic equations. Learners should be taught to solve quadratic equations for example:

  • bending moment (M) of beams M=0.4x²+0.47x−3.2

  • fabrication of steel boxes when the volume of the box is, 2(x − 4)(x − 4) where “x” is a required dimension

  • equations of motion

    • v =u+at

    • v² =u² +2as

Be able to use geometry and graphs in the context of engineering problems.

Course Coverage

• Straight-line equations i.e.

  • equation of a line through two points

  • gradient of parallel lines

  • gradient of perpendicular lines

  • mid-point of a line

  • distance between two points

• curve sketching i.e.

  • graphs of y = kxⁿ 

  • graphical solution of cubic functions

• The behaviour of engineering systems can be described using straight line equations. Learners should be taught how to solve problems using straight line equations for example:

  • force vs displacement for a linear spring or spring buffer

  • electrical problems using Ohm’s law 

  • Learners should be taught to sketch mathematical functions in order to visualise (and sometimes to solve) problems for example:

    • y =-3x²

    • f ( x ) = x ( x − 1) ( 2 x + 1)

    • m(x) = (2 − x) ³

*Learners can be taught to use spreadsheets to plot and solve cubic functions using trend lines.

• Graphical transformations i.e.

  • translation by addition

  • transformation by multiplication

• Learners should be taught graphical transformations, for example:

  •  translation in the y direction by adding a whole number to the whole function.

  • translation in the x direction by adding a whole number to x.

  • multiplying the whole function by a whole number.

  • multiplying x by a whole number

Understand exponentials, logarithms and trigonometry related to engineering problems.

Course Coverage

• Many engineering systems and devices can be characterised, and problems solved using exponentials and logarithms for example:

• Voltage and current growth in capacitor circuits (RC circuits)

• Voltage and current decay in capacitor circuits (RC circuits)

• Stress-strain curves for certain engineering materials

• Learners should be taught how to solve problems involving exponential growth and decay including use of the exponential and logarithmic functions and the log laws.

  •  y = e^ax

  • y = e^−ax

  • e^y = x

  • lnx=y

• Learners should be taught both how to produce and interpret sketch graphs showing exponential growth and decay.

• Problem-solving with arcs, circles and sectors i.e.

  • the formula for the length of an arc of a circle

  • the formula for the area of a sector of a circle

  • the co-ordinate equation of a circle ( x − a) ² + ( y − b) ² = r ² to determine: centre of the circle

  • radius of the circle

• Problem solving involving right-angled triangles i.e.

  • what is meant by the term “solution of a triangle”

  • Pythagoras’ Theorem

  • use of sine, cosine and tangent rule for right-angled triangles

  • the formulae for the area of a right-angled triangle 

Understand calculus relevant to engineering problems

Course Coverage

• Problem-solving involving differentiation i.e.

  • determine gradients of a simple curve using graphical methods

  • the rule to differentiate simple algebraic functions

  • determine the maximum and minimum turning points and the co-ordinates of the turning points by differentiating the equation twice

• Learners should be taught to solve problems involving differentiation, for example:

  • given that an alternating voltage v = 20sin50t where v is in volts and t in seconds, calculate the rate of change of voltage at a given time.

  • differentiate displacement to get velocity. 

  • differentiate velocity to get acceleration, where possible problems should be presented in an engineering context.