A tangent line meets a curve at a single point, the point of tangency

A tangent line is a ‘line’ so we find it’s equation using y – b = m (x – a)

To use y – b = m (x – a) we need the gradient, m, and a point on the line, (a,b)

(a,b) will be the point of tangency

The gradient, m, we find by taking the derivative and evaluating at x = a

In other words, m = f ‘(a)

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Cancelling in a fraction means dividing the numerator and the denominator by the same value

To square a bracket write the bracket twice and multiply them together

Quadratic equations are solved by factorising and separating into two equations

Don’t confuse quadratic equations with linear equations which use a different technique

When you square a negative the answer should be positive

To square a negative use a bracket and put the squared outside the bracket

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]]>Students are being misinformed about the necessity to take the National 5 Applications of Mathematics course.

National 5 Mathematics (course code C847 75) and National 5 Applications of Mathematics (course code C844 75) are two distinct courses with different objectives and learning intentions. The University of the Highlands and Islands sums it up quite well when they state on their website, ‘if you require a Mathematics qualification at level 5 but are looking for an alternative to National 5 Mathematics, then National 5 Applications of Mathematics may be for you.’ (source: https://www.inverness.uhi.ac.uk/courses/national-5-national-5-applications-of-mathematics/).

National 5 Applications of Mathematics, or Apps as it is commonly referred to, is a rebranding of a legacy course called National 5 Lifeskills Mathematics. As the name suggests the laudable intention of that course was to develop learners practical Mathematical skills, mostly numerical skills, which could be applied to real-world situations such as finance, travel and everyday geometry. The Apps course has essentially the same content but is assessed by final exam where Lifeskills was assessed continuously at the end of each unit. Lifeskills was considered a ‘school-leavers’ qualification which gave students some practical skills to take into adult life and somewhat demonstrated to employers a candidate’s ability to use numerical and basic Mathematical concepts.

The Apps course retains this intention but extends the scope beyond practical skills to, as the name suggests, *apply* various Mathematical techniques to both everyday life __and__ other disciplines. For example, the skills developed in the Apps course could be applied to science courses, technology or various others. Recent studies suggest that current school-leavers will, on average, change job role 20 times in their life, so the ability to apply skills – not just Maths – to solve different problems will be key for this group. And this is one of the aims of the Curriculum for Excellence – to prepare learners for the world they’ll actually find themselves in. So, the Apps course is well intentioned and potentially meets the needs of a wide range of candidates. Furthermore, some higher education institutions are now starting to recognise Apps as evidence of numerical / Mathematical skill in courses such as primary teaching, nursing and some undergraduate programs. This is understandable since the full National 5 Mathematics course does not contain much numeracy or practical skills and is more focused on ‘pure’ Mathematical content such as algebra which lays the foundation for students going onto Higher Mathematics and associated disciplines. Most degree courses which require a high school Maths qualification still have National 5 Maths, not Apps, as the minimum level.

But it has become clear that schools are not pitching the Apps course to their students as an, albeit potentially useful, *alternative* to National 5 Mathematics. Rather they are slipping it into the National 5 Mathematics course – somewhat under the radar – as though the two are necessarily taught together. Increasingly students are being signed up for the Apps course not realising that it is a separate qualification requiring __additional__ class time and __additional__ preparation for the __additional__ final exams that it is assessed by. In my experience as a private tutor parents are unaware that this is a separate course and just accept that it’s part of ‘Maths’. At exam time students are already under pressure and depending on how close their exams are in the diary may be under significant time pressure to prepare for exams in a range of subjects. Adding two additional exam papers – Apps is assessed by two final exam papers, one calculator and one non-calculator – is just going to add further stress.

For some students Apps is absolutely the appropriate course choice since the skills taught in National 5 Maths may be less relevant to their future plans or the content may be beyond their ability – National 5 is significantly more challenging than Apps. However, for many students taking National 5 Mathematics, the Apps course is not particularly relevant or important and simply detracts from class time and adds more stress when preparing for the exams that are actually relevant to them. It should also be noted that there is significant cross-over between National 5 Mathematics and Apps, so students taking both will inevitably be answering questions on the same, or very similar, topics. This seems illogical but then the two courses were never designed to be taken together! The very existence of these two courses speaks to the age-old frustration of most students; ‘when am I ever going to use this?’ That is a question for another time but it does point to a need to radically update the content of Maths courses taught in high schools in Scotland. The National 5 and Higher courses are identical to when I took them 25 years ago (Nat 5 was called Standard Grade back then), yet look at how much the world has changed in that time.

Students and parents should decide whether National 5 Mathematics, National 5 Applications of Mathematics, or both is right for them. But they should realise it is their choice and this increasingly ‘standard’ practice of enrolling students in both is bizarre and underhand, and clearly weighted in the schools favour since many more students will produce high grades in the Apps course than in full National 5 Maths.

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The Quotient Rule is for differentiating a quotient of functions

A quotient of functions is simply one function being divided by another

Often you can use indices rules and then the power rule, rather than the quotient rule

The quotient rule can be written several ways – choose the one you can remember

Learn the rule and practice using it in different scenarios

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The Product Rule is for differentiating a product of functions

A product of functions is simply two functions multiplied together

Sometimes you can use indices rules and then the power rule, rather than the product rule

The product rule can be written several ways – choose the one you can remember

Essentially the rule says ‘the 1st x derivative of the 2nd + 2nd x derivative of the first’

Learn the rule and practice applying it in different scenarios

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Completing the square is a type of factorising for quadratic expressions

You must learn what the completed square form looks like

There are two methods of completing the square – one basic and one more complex

The basic form should be a one step process

With a number in front of the X squared term the working is trickier

Practice both

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The power rule is for differentiating polynomial style functions

If a function is not in the correct format you cannot use the power rule

it may be possible to manipulate it into the correct format using exponent rules

To master the power rule start by reviewing exponent rules

Try as many different variations of functions as possible to perfect the power rule

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Closed intervals are ranges of values which include the endpoints of the range

Open intervals, on the other hand, do not include the endpoints

Maximum and minimum values are the greatest and least ‘Y’ value in the closed interval

The maximum and minimum values correspond to the highest and lowest point on the graph

The maximum and minimum values always occur at a stationary point or at an endpoint

Find the stationary points and test their ‘Y’ values. Test the ‘Y’ values of the endpoints.

Then compare all of the values to find the greatest and the least, which are the max and min values

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Curve sketching refers to sketching the graph of a function

Curve sketching relies on finding key points on the graph of a function

Key points include the X & Y intercepts and the stationary points

The X intercept is found by letting y = 0 and solving the resulting equation

The Y intercept is found by letting x = 0 and solving the resulting equation

Stationary points are found by setting the derivative equal to zero and solving the equation

The ‘nature’, or type, of stationary point is determined using a nature table or 2nd derivative

The key points should fit together on the sketch to produce a coherent graph

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The discriminant tells us whether a quadratic equation will have 0, 1 or 2 real roots (solutions)

The terms ‘solutions’ and ‘roots’ are often used interchangeably

The discriminant rules work both ‘left to right’ and ‘right to left’

In finding undetermined coefficients we use the discriminant rules ‘right to left’

The terms ‘solution’, ‘solutions (plural)’ and ‘range of solutions’ indicate how your answer should look

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