Factorising is the process of writing an algebraic expression as a product of factors – similar to how the number 15 can be written as 5 x 3, where 5 and 3 are both factors of 15. Factorising algebraic expressions involves using one or more brackets (parentheses). The number of brackets required depends on the type of expression being factorised. Factorising is the reverse process of expanding (multiplying out) brackets. So, you can check if factorising has been done correctly by multiplying out your answer to see if it matches the original expression. Factorising is sometimes called factoring.

Algebraic expressions are factorised into one of three types – common factor, difference of two squares or trinomials. More generally trinomials are an example of a polynomial expression and these have their own techniques for factorising such as *synthetic division* or *algebraic long division* – but these are beyond the scope of what we’re looking at here. The first key to mastering factorising is knowing that there are three types and how to recognise them. Each of the three types of expression has features that will help you identify it. However, every algebraic expression looks a little different so it takes some experience to accurately identify the types.

Trinomials are probably the most easily identified. ‘Tri’ means three and ‘nomial’ refers to the power (exponent). So a ‘trinomial’ refers to an expression with three terms, each with different powers. A trinomial is a type of quadratic expression since the highest power (called the order) is 2. A trinomial has one term with a power of 2, one term with a power of 1, and one term with a power of 0 (which is the number term on the end). The following are all examples of trinomials. Although each looks a little different they share the same features which defines them as a trinomial.

A difference of two squares literally means ‘something squared minus something squared’ since ‘difference’ means substract in Math. So this type of expression has two squared terms with a minus between them. Often, differences of two squares have one algebraic term and one number term. The number term generally won’t have the power 2 on it – instead the number will actually be squared. For example instead of writing 4^2 (i.e. 4 squared) it would be wrtten as the result of 4 squared which is 16. Because a difference of two squares contains a squared term it is also an example of a quadratic expression like trinomials. The following are all examples of differences of two squares, sharing the same common features, but each is just a little different.

Algebraic expressions which contain common factors are less easily recognisable than trinomials or differences of two squares. This is because they can appear in a range of different ways. In fact, it’s possible for both trinomials and differences of two squares to also contain common factors, and they often do. Sometimes it’s best to identify a common factor type factorising by process of elimination i.e. if it’s not a trinomial or difference of two squares it must be a common factor as that’s the only other option if it can be factorised. An important point with this type is that sometimes the common factor is a number, sometimes a letter, and sometimes both. The following are all examples of expressions which can be factorised using a common factor. Note the wide range of formats.

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Of the three types trinomials are generally considered the most difficult to factorise. But the difficulty depends on the coefficient numbers in the expression and whether they are positive or negative. For example, small positive numbers are relatively straight-forward but a mix of large and small, positive and negative, is much more difficult to factorise. Check out our YouTube Video on Factorising for a complete explanation of factorising trinomials.

In many cases a difference of two squares is quite straight-forward to factorise since the expression is similar each time as is the format of the factors. A difference of two squares factorises into two brackets with the first term in each bracket being the same and the second terms also being the same. However one bracket contains a ‘plus’ and the other a ‘minus’. This format never changes for a difference of two squares. Check out our YouTube Video on Factorising for a complete explanation of factorising a difference of two squares.

Anytime you need to factorise you should always check for a common factor – even if the expression is also a trinomial or difference of two squares. Removing the common factor first makes factorising these expressions much easier (although they can be factorised without removing the common factor). An expression can sometimes contain more than one common factor, but can only ever have one ‘highest’ common factor. It’s the highest common factor that we look for and this goes outside (in front) of one bracket. Check out our YouTube Video on Factorising for a complete explanation of factorising using a common factor.

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]]>The Chain Rule is used to differentiate composite functions. A Composite Function is a ‘function inside a function’ but they can take many forms. The challenge with The Chain Rule is not just using the rule, but knowing when it can be used in the first place. The following are a selection of examples of composite functions – as you can see there are many different types.

Notice that in each case there are two functions present – an ‘outer’ function and an ‘inner’ function. For each example these are the outer and inner functions:

The Chain Rule itself can be written in several different ways. Some of these are more useful than others, especially when it comes to applying The Chain Rule in different situations. But here is the key to mastering The Chain Rule . . . it’s just the derivative of the outer function multiplied to the derivative of the inner function. So a good way to remember The Chain Rule is:

(the derivative of the outer) x (the derivative of the inner)

Translating this intuitive notion into Mathematical form gives us the following. Let the outer function be f(x) and let the inner function be u – where u is a function of x. Then each composite function is in the form f(u) and the derivative, using The Chain Rule, is given by:

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Regardless of how well you understand and learn The Chain Rule, you still have to differentiate the outer and inner functions successfully. To do so, you have to learn and practice the rules for differentiating other types of functions. Common function types are Polynomials (which require The Power Rule), Trigonometric Functions, and Exponential & Logarithmic functions. Download our FREE Differentiation formula list HERE and see our Differentiation Q&A post HERE.

Have a go yourself at finding the derivative of the above example functions. Remember to just take the derivative of the outer and inner function and multiply them together. Then check the solutions below or watch our YouTube VIDEO on those questions.

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Differentiation FORMULA SHEET

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]]>Evaluate the infinite series

If you’re not familiar with evaluating infinite series this problem may seem odd. Essentially it’s saying to imagine that the terms of the series continue indefinitely (according to the rule for each term), then add them all together and see what you get. It may seem counterintuitive that if you add up infinitely many numbers the answer could be a finite, but that’s what sometimes happens with infinite series. Clearly you can’t manually just add them up because there are infinitely many – even if you added together the first thousand, or million, or billion terms you still have infinitely many to go. So we need a different approach.

We start the solution by using partial fractions to separate the expression into two fractions.

We can now rewrite the original series definition and start substituting values for n i.e. start writing out some of the terms of the series in the new partial fraction form.

Of course we want to evaluate the sum to infinity not just to the first ‘n’ terms. We can’t do this manually so we use a limit.

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To finish off our result we need to use a limit. We don’t have space here to explain the concept of a limit so if you’re unsure about this check for an online resource. Essentially by using a limit, we’re going to imagine what will happen to our series as the n value gets infinitely large. We can write

We can now write our final result which is

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]]>Find the sum to infinity of 2n/(n+1)!

This infinite series looks like this:

If you’re not familiar with infinite series this problem may seem strange. Essentially it’s saying to imagine that the terms of the series continue indefinitely (notice that each term follows a pattern), then add them all up and see what you get. It seems counterintuitive that if you add up infinitely many numbers the answer could be a finite number, but that is how infinite series sometimes go. Obviously you can’t manually just add them up because there are infinitely many – even if you added together the first thousand, or million, or billion terms you still have infinitely many to go. So we need another approach.

For a video explanation of the solution go HERE

The first thing to be aware of in this particular question is the presence of the factorial i.e. ! notation. Factorials work like this:

Notice that the series can be written using summation (Sigma) notation like this:

Then

So, we need to evaluate the sum:

And then multiply our answer by 2.

For a video explanation of this side working go HERE

We are going to use this result about factorials:

For completeness, let’s look at where that result comes from:

But, notice that

So we have

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We can now write

Writing out some terms gives:

Notice that up to the Nth term, the terms cancel (called a telescoping series) leaving just

Using a limit we can then write

Evaluating this limit gives:

So we can finally write:

We can now write

Drop us a message if you have a question about any of techniques on this page or about series in general.

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]]>Before we get into differentiation questions and answers, some background (yes, you may skip past this intro!). In the 17th century Mathematicians were working on a range of problems and attempting to solve unanswered questions – as Mathematicians do! One notable problem was working with tangents to curves; the so-called ‘tangent line problem’. This enquiry, building on the work of their predecessors, eventually led to the concept of a derivative and the process of differentiation. Differentiation would prove to be of huge importance in a range of fields and formed one half of what we now call CALCULUS – the other half being integration. The history of the Calculus is pretty interesting – if you like that sort of thing. If you do, you can read more about it HERE.

Back to the present. Differentiation is simply the process of finding the derivative of a function. When I say ‘simply’ what I really mean is ‘make me want to cry difficult!’ The definition of a derivative may be simple, but finding derivatives isn’t always so straight-forward. Firstly, you have to learn the rules (fortunately there aren’t very many but they’re easy to get muddled). Secondly, you have to know when to use those rules (i.e. not get them muddled). And finally, you’ll want to know what you can do with these rules (i.e. what can I use derivatives for?).

That’s right, differentiation does rule! In fact, arguably it does since so much of the modern world would be impossible without differentiation. In my experience of working with many many . . . many many differentiation students, I’ve found that most are competent with the rules themselves. It’s when they’re presented in weird ways or have to be combined . . . or sometimes one rule seems to masquerade as another . . . that’s when the trouble starts! We’ll look at some examples of this later. But let’s start with the rules themselves. You have to know these or the rest is irrelevant. See next section . . .

There are arguably a bunch of differentiation rules but for the vast majority of cases – and certainly when you first start with differentiation – you’ll just need to know The Power Rule, The Product Rule, The Quotient Rule, and The Chain Rule. There are also rules for Trigonometric functions, Logarithmic functions and Exponential functions but these are quite easily memorised (here’s a downloadable PDF FORMULA SHEET) and are, in large part, examples of The Chain Rule. Let’s look at each rule in turn.

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The Power Rule is without doubt the first rule you should learn to be successful in differentiation. The Power Rule is for differentiating polynomial type functions. Check out our post about THE POWER RULE or watch our VIDEO to start learning.

The Product Rule is for differentiating functions which are the product of two other functions and can be a real time-saver. And often it’s the only rule that can be used. Check out our post about THE PRODUCT RULE or watch our VIDEO to start learning.

The Quotient Rule is for differentiating functions which can be written as one function divided by another (i.e. a quotient). It’s usually easy to recognise when to use. Check out our post about THE QUOTIENT RULE or watch our VIDEO to start learning.

The Chain Rule is for differentiating composite functions. It’s a powerful rule and can be used in many different scenarios but it’s not always easy to recognise when to use it. Check out our post about THE CHAIN RULE or watch our VIDEO to start learning.

Since The Chain Rule can manifest is so many different ways it can be difficult for students to know when to use it. These typical examples should help you out. Check out our post about THE CHAIN RULE EXAMPLES or watch our VIDEO to start learning.

Differentiation is used across various disciplines such as science, engineering and economics. Differentiation, sometimes called ‘rate of change’ essentially measures how quickly one variable changes with respect to another variable (for example the rate of change of distance with respect to time is called velocity, which we commonly call speed). In the real world there are countless variables so you can imagine how applicable differentiation could be. But differentiation also helps us solve many Math problems and these are often called applications of derivatives. Once students learn the derivative rules they are then expected to apply them in various ways. Check out some common Math applications of derivatives below.

The grandaddy of differentiation applications! As mentioned in the introduction this is thee classic Calculus problem and a very common question for differentiation students to have to tackle on tests. But want to know a secret? It’s actually a really easy problem type . . . well, now that we have differentiation. Of course, for those guys in the 17th century is wasn’t so much fun! Check out our post about FINDING THE TANGENT LINE TO A CURVE or watch our VIDEO to start learning.

A function can be increasing (or decreasing) either at a single point or on a range of values. Put simply, a function is increasing if the *values* of the function (i.e. the Y or f(x) values) are increasing. This means that the graph of the function is sloping ‘up to the right’ at the point of interest or on the range of values of interest. Determining whether a function is increasing or decreasing is a key application of derivatives. Check out our post about INCREASING FUNCTIONS or watch our VIDEO to start learning.

I regularly see students overcomplicate and get confused by Stationary Points. If you’re one of those students then, don’t worry, this video will sort you out. In fact, the process of finding stationary points is, in itself, quite straight-forward. The challenge occurs when you’re asked to work with unusual functions which can easily throw you off. Also, determining the nature of a Stationary Point is often way overcomplicated! Check out our post about STATIONARY POINTS or watch our VIDEO to start learning.

Finding the maximum or minimum values of a function on an interval (also called finding local extrema) is a very common differentiation question, and one that students should expect to see on assessments. Although it sounds quite complicated the technique is actually straight-forward and really an application of Stationary Points – hopefully you already know about those?! If not, see our previous VIDEO. Check out our post about LOCAL EXTREMA or watch our VIDEO to start learning.

Differentiation FORMULA SHEET

Calculus PLAYLIST on YouTube (includes integration)

Calculus 1 MINI COURSE on YouTube

Our comprehensive CALCULUS 1 course (FREE one week trial)

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]]>The National 5 Maths course covers a lot of ground and is often underestimated by students. It’s the minimum required standard for many job roles and further education courses. I’m often asked whether a student should take National 5 Maths. This post aims to give you key information about the National 5 Maths course to help you answer that, and other, common questions.

National 5 Mathematics has SQA course code C847 75. It’s generally considered the minimum required standard for entry to many further education courses and job roles. However, this is changing as NATIONAL 5 APPLICATIONS OF MATHS becomes more recognised. Although the National 5 Maths course is considered to be a course in general Mathematics, it is largely focused on algebra techniques. This is often a source of frustration for students who do not consider algebra relevant to their future plans. The focus on algebra is primarily for learners who will go on to study Higher Maths where good algebra skills are essential. There are also several stand-alone topics in the National 5 Maths course, such as statistics and percentages.

The description of the intended learner for the National 5 Maths course in the course SPECIFICATION is somewhat vague. Essentially, the course is for students who intend to study Maths further – for example HIGHER MATHS – or who need a Maths qualification at level 5 demonstrating a range of Mathematical skills. Students can take ‘full’ National 5 Maths or NATIONAL 5 APPLICATIONS OF MATHS. The latter is essentially a course in numeracy and practical Maths skills. Until recently, almost all further education courses or job roles required full National 5 Maths but this is changing. Students should discuss which course to take with their guidance teacher. Students should not take both courses, although that has become increasingly common. I wrote a post about that HERE. National 5 Maths is a demanding course which students should only take if they performed well at National 4 Maths.

National 5 Maths has traditionally been taught in three units – Expressions & Formulas, Relationships & Applications. However, some teachers now prefer to order the course in their own way. I strongly dislike this practice as it throws students into topics without enough preparatory work. Essentially the first unit ‘Expressions & Formulas’ develops key algebra techniques. The second unit, Relationships – by far the most demanding part of the course – uses those algebra skills in various settings. And the third unit, Applications, is a mix of topics with a focus on numeracy. If students get the algebra skills from unit 1 down early, it makes the rest of the course significantly easier. If not, then it usually becomes a real struggle. You can view the full course syllabus HERE and read more about the course content in the SQA National 5 Maths SPECIFICATION.

We have the solution with our National 5 Maths Online Course. Featuring 85 step by step instructional videos, more than 500 exam relevant practice questions with full solutions and comprehensive course notes. Ideal to support your classroom work, help with homework or tests, and prepare for final exams. Learn more HERE or start your FREE one week trial.

Students often underestimate the National 5 Maths course which is a significant step up from National 4 Maths. Much of the content of National 4 Maths is based on logic rather than Mathematical technique. National 5 Maths, however, is more formal and requires students to demonstrate formal solutions to problems. Many students get caught out by this – they understand the topics but struggle to write sufficiently precise solutions. The Relationships unit contains some new and demanding topics and almost every student finds them difficult at first. However, the key to these is to have developed solid algebra skills in the first part of the course. A lot of students only ‘bring it all together’ at the last minute. This isn’t necessarily because they’ve been lazy. Rather, it’s because the techniques require experience as much as technical skill which takes time to develop.

The topics in unit 3, Applications, tend to be done quite well. In my experience this is largely because many of the unit 3 techniques are procedural in nature i.e. students can replicate the same technique every time. Also, there are some straight-forward numeracy topics in unit 3 such as Fractions & Mixed Numbers, Statistics, and Percentages. The Volume topic, from unit 1, is also quite straight-forward. Note that none of these ‘easier’ topics involve algebra. Students who have done the work should expect to make most, if not all, of the marks in these topics on the final exam.

Having tutored many hundreds of National 5 Maths students I can say confidently that unit 2 gives, by far, the most trouble. In fact, almost every student struggles with at least one, but often more, of the unit 2 topics. There are two main reasons for this. Firstly, some of the topics use completely new types of Mathematics, such as Functions. Secondly, the topics are largely built upon algebra skills from unit 1. If students didn’t master those skills, they’re playing catch up while learning new material. The most difficult topics are Quadratic Functions, Graphs of Quadratic Functions, Straight Lines and Trigonometry. Fortunately there are common themes in these topics. So, if a student masters one or two key ideas, it can make all of those topics more manageable.

National 5 Maths has a diverse curriculum but most of the marks come from algebra techniques. On the other hand, National 5 Applications of Maths is essentially a course in numeracy, with very little algebra. You can learn more about National 5 Applications of Maths HERE. These two courses serve very different purposes and students should be careful to make sure they take the correct one. Schools often put students in the wrong course – you can read more about that HERE. Students who wish to take Higher Maths should definitely take ‘full’ National 5 Maths. For some students, National 5 Applications of Maths is absolutely the correct course. However, many careers and further education courses require full National 5 Maths. Students should check what they need carefully, although at that age it isn’t necessarily clear.

We have the solution with our National 5 Maths Online Course. Featuring 85 step by step instructional videos, more than 500 exam relevant practice questions with full solutions and comprehensive course notes. Ideal to support your classroom work, help with homework or tests, and prepare for final exams. Learn more HERE or start your FREE one week trial.

National 5 Maths is assessed by two final exam papers – one non-calculator paper and one calculator paper. For the 2022 final exams paper 1, the non-calculator paper, lasted 60 minutes and paper 2, the calculator paper, lasted 90 minutes. Students should be aware that not all of the questions in the calculator paper require the calculator. Also, students should practice using the calculator as some advanced functions are required. I often see students making computation errors with the calculator which is really a big waste of marks. Students sometimes perform poorly in paper 2 because they start to get tired. This is made worse since there are often demanding questions towards the end of paper 2. View the past exam papers and marking guides HERE.

Students should check the entry requirements for their chosen course as they are subject to change. Until recently National 5 Maths was the minimum required qualification for many job roles and further education courses. However, that has changed somewhat as more and more recognition is being given to the National 5 Applications of Maths course. Students following a technical career or university course are likely to need Higher Maths. Higher Maths is a challenging course and students really need to make an A or strong B grade at National 5 to take Higher. Two examples of university courses which now accept National 5 Applications of Maths are nursing and primary teaching. Again, check carefully as each course will be different.

These links may help you find additional information about the National 5 Maths course.

National 5 Maths Exam Papers & Solutions

National 5 Maths Exam Questions by Topic

National 5 Maths Exam Study Materials

National 5 Maths FREE Worksheets

National 5 Maths Video Solutions

National 5 Maths Key Skills Playlist

National 5 Maths Exam Tips & Tricks Playlist

National 5 Maths Online Course

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]]>I’ve taught many hundreds of Higher Maths students and have seen many successes and failures along the way. Higher Maths is a significant step up from National 5 Maths. I’m often asked by parents and students if they should take Higher Maths. This post aims to give you key information about the Higher Maths course to help you answer that, and other, common questions.

Higher Mathematics has SQA course code C847 76 and is suitable for learners who have performed well in the National 5 Maths course and either wish to study Maths further or are required to study Maths further for their future plans – for example, to meet university entry requirements. Higher Maths is often a necessary requirement for students following technical careers and further education courses. Otherwise, students only need general Maths skills and numeracy. These are fulfilled by either National 5 Maths or National 5 Applications of Maths. Very few university courses, or careers, require Advanced Higher Maths (some exceptions do exist). So, for most students, Higher Maths is the pinnacle of their high school Maths education. Higher Maths is where students start to ‘specialise’ in Maths to some extent, studying challenging topics beyond ‘general’ Mathematics.

The SQA state in the Higher Maths SPECIFICATION that the course is ‘particularly for students who have demonstrated an aptitude for National 5 Mathematics’ and who ‘are interested in developing Mathematical techniques to use in further study or in the workplace.’ From my experience I only recommend Higher Maths to students who need the qualification for their future, or who are really really up for it. Higher Maths is challenging and a significant step up from National 5 Maths. Students often take Higher Maths without a strong enough motivation and get caught out by the increase in course sophistication. Many technical further education courses require students to have completed Higher Maths, often at an A or B grade. Students don’t have to be ‘into’ Maths to do well at Higher, but all students have to be committed and practice the techniques.

Higher Maths has traditionally been taught in three units – Expressions & Functions, Relationships & Calculus, and Applications. However, some teachers now prefer to order the course in their own way. There are many sub-topics but primarily the course centers around functions, which is odd since the preparatory course, National 5 Maths, barely mentions functions. The big topics in the Higher Maths course are Polynomials, Trigonometry, Straight Lines, and Differential & Integral Calculus. In fact, students could ignore the other topics and still do very well, not that I’m recommending that! You can view the full course syllabus HERE and read more about the course content in the SQA Higher Maths SPECIFICATION.

We have the solution with our Higher Maths Online Course. Featuring 85 step by step instructional videos, more than 500 exam relevant practice questions with full solutions and comprehensive course notes. Ideal to support your classroom work, help with homework or tests, and prepare for final exams. Learn more HERE or start your FREE one week trial.

Higher Maths is a challenging course with a range of topics, techniques and skills. In my experience of working with Higher students of various abilities, commitment and diligence are the keys. If students make a strong start to the course and work diligently to practice the techniques throughout the year, they tend to do well. If they don’t, even if they have a lot of natural ability, they tend to struggle. Cramming Higher just before the final exam usually ends badly. Students should only take Higher Maths if they have an A or strong B grade from National 5 Maths. Even then I’ve seen many students get caught out and go from an A at National 5 Maths to a D at Higher. This is not because the course is beyond them, it’s just that they don’t adapt to the increased level and, therefore, aren’t as ‘on it’ as they need to be.

Easy is, of course, a subjective term but there are certainly topics that students tend to do well in or struggle with. Polynomials is one of the more manageable topics. So much so, that students who practice should expect to get most, if not all, of the marks. Straight Lines tend to be done quite well because students have experience of it from National 5 Maths. Recurrence Relations, which was removed from the 2022 final exams, is a small and straight-forward topic. The topics that students find easiest are those with procedural question types which can be anticipated and practiced.

There are two reasons why students tend to find particular topics challenging. Firstly, it’s technically demanding and they struggle to really understand it. Secondly, the question types may involve an element of problem-solving where memorising a procedure isn’t enough to get all, or sometimes any, of the marks. Trigonometry is challenging for many students because there are a range of techniques and some are very similar but answer different question types. Students often get them muddled. Differentiation and Integration, collectively the Calculus topics, are new types of Mathematics for students and can be technically challenging at first. Even when the techniques are mastered students will sometimes use them to answer the wrong question. Trigonometry and Calculus together account for at least 25% of the marks on the final paper. In some years, possibly way more.

National 5 Maths has a diverse curriculum with a focus on developing key algebra skills. However, Higher Maths assumes those algebra skills and focuses on more advanced skills such as Trigonometry, Functions and Calculus. You can view the National 5 Maths syllabus HERE. The two courses are very different in nature and Higher is not really the continuation from National 5 Maths that a lot of students expect. Rather, it’s a significant step-change in level with a focus on very different topics. That said, students who do not pick up key skills and techniques from National 5 Maths (in particular Algebra and Trigonometry) face the challenge of improving those skills while learning new topics at Higher. This is a very common scenario that I see a lot. It’s made worse since students tend to forget those skills over the summer holidays and it takes time to recall them again.

We have the solution with our Higher Maths Online Course. Featuring 85 step by step instructional videos, more than 500 exam relevant practice questions with full solutions and comprehensive course notes. Ideal to support your classroom work, help with homework or tests, and prepare for final exams. Learn more HERE or start your FREE one week trial.

Higher Maths, like National 5 Maths, is assessed by two final exam papers – one non-calculator paper and one calculator paper. For the 2022 final exams paper 1, the non-calculator paper, lasted 75 minutes and paper 2, the calculator paper, lasted 90 minutes. Maths students at all levels often assume calculator papers will be easier but that is generally not the case. In fact, they often contain longer, more strenuous and time-consuming questions. Furthermore, there are actually not many questions where a calculator helps. Students can get caught out by the use of the calculator which is, in itself, a skill to be developed and practiced. You can view the past exam papers and marking guides HERE.

Students should check the entry requirements for their chosen course as they are subject to change. In general, Higher Maths is only required for courses of a technical nature or which contains a Mathematical element. As you’d expect courses such as engineering, sciences, accountancy, architecture and others of a technical nature, will require Higher Maths. Some may require Advanced Higher Maths or, at least, it would be an advantage. If a student plans to take a higher education course which does not require Higher Maths then I would not recommend they take it as an elective. I’ve seen many students do that and run out of steam because it is a demanding course and, therefore, detracts from time and energy for their other courses. That said, if a student is particularly motivated to achieve Higher Maths, even if they don’t require it, it will look good on their record.

These links may help you find additional information about the Higher Maths course.

Higher Maths Exam Papers & Solutions

Higher Maths Exam Questions by Topic

Higher Maths Key Skills Playlist

Drop us a message if you have any questions about the Higher Maths course. We’ll get back to you shortly.

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]]>National 5 Applications of Mathematics is a relatively new qualification, although it’s essentially a rebranding of National 5 Maths LifeSkills, with similar content. Since it is new, and because it sits alongside the similarly named National 5 Maths, students and parents often have questions about it. This post answers the most common questions and links to useful material.

Let’s start by saying what National 5 Applications of Maths is not – it’s not part of the full National 5 Maths course or an additional requirement for students taking National 5 Maths. 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.

National 5 Applications of Mathematics, or Nat 5 Apps as it’s commonly referred to, is a rebranding of the legacy course National 5 LifeSkills Mathematics. The focus of that course was to develop practical Mathematical and numeracy 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, whereas LifeSkills was assessed continuously at the end of each unit. Achieving the National 5 Maths LifeSkills qualification demonstrated to employers a candidate’s ability to use numerical and basic Mathematical concepts. The National 5 Applications of Mathematics 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.

The SQA state in the course SPECIFICATION that National 5 Maths Apps is ‘a suitable course for learners who have achieved the fourth level of learning across Mathematics experiences and outcomes in the broad general education, or who have attained the National 4 Applications of Mathematics course, or who have equivalent qualifications or experience.’ They further state that ‘this course is particularly suitable for learners who wish to develop the Mathematical reasoning and numerical skills which are useful in other curriculum areas and workplaces.’ I’d say the Apps course is for students who want a Maths qualification at level 5 but for whom the full National 5 Maths course may not be relevant (it teaches more ‘pure’ Maths such as algebra, rather than practical Maths) or may be too challenging. Of course, students should check what qualification they need for their future plans. More about that later.

The high-level topics included in the Apps course are: Numeracy Skills, Financial Skills, Statistical Skills, Measurement Skills, Geometry Skills and Graphical Data & Probability Skills. Each of these topics contains various subtopics. To understand what’s included under each of these subheadings see the course SPECIFICATION.

A common question is how does National 5 Applications of Maths compare to full National 5 Maths. Full National 5 Maths contains a variety of topics but is primarily centered around algebra and algebraic techniques. It contains only a little numeracy and very few practical Mathematical skills which could be used in an everyday scenario. On the other hand, National 5 Maths Apps contains very little algebra and centers around numeracy, numerical techniques and practical Mathematical skills. These are applied to everyday scenarios such as managing money, finance topics, and basic geometry. The two courses are designed to fulfil completely different student needs. However, it has become an increasingly common practice for schools to ask students to take both, something which I wrote about in this POST.

Unlike National 5 Maths LifeSkills which was assessed continuously with no final exam, National 5 Applications of Maths is assessed by a final exam consisting to two papers – one calculator paper and one non-calculator. In the 2022 exams these lasted 50 minutes for paper 1 and 100 minutes for paper 2. These final exams are not easy and contain a significant amount of content and skills. This is why it’s inappropriate for schools to ask students to take both courses and have two additional final exams to prepare for. One of the greatest challenges for students taking the Apps final exams is the manual calculations required in the non-calculator paper. These can be time-consuming and potentially frustrating. Students taking the final exams should practice numeracy diligently. You can find past exam papers HERE.

Since National 5 Maths Apps is a new qualification it is only now becoming recognised and widely accepted by universities as evidence of a students Mathematical ability. For example, The University of Edinburgh state that ‘if National 5 Applications of Mathematics is accepted this will be stated in our entry requirements in the degree finder.’ (https://www.ed.ac.uk/studying/undergraduate/entry-requirements/scottish-qualifications/national-5) This implies that some courses accept National 5 Applications of Maths and others require National 5 Maths. At the time of writing this post, Aberdeen, Stirling, Glasgow and The University of the Highlands & Islands accepted National 5 Applications of Maths for entry to their primary teaching degree, and some other universities accept National 5 Applications of Maths as entry into their nursing degrees. Students should check carefully with prospective universities and colleges about which qualification they require.

These links may help you find additional information about the National 5 Applications of Maths course. Additional resources will be added if useful.

Drop us a message if you have any question about the National 5 Applications of Maths course. We’ll get back to you shortly.

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]]>One of the first applications of derivatives that students learn is how to find the equation of a tangent to a curve. There are many applications of derivatives but this one is particularly important. It was actually from trying to solve the tangent line problem that Mathematicians in the 1600’s developed Calculus techniques. History aside, all differentiation students should be prepared to demonstrate how to find the equation of a tangent line to a curve as it’s a very common exam question type. Any questions about this video drop us a message HERE.

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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]]>The first thing I check with all of my students is their algebra. No matter how well they understand and can apply theorems, formulas or rules, they must have solid algebra. Algebra is the grammar and puntuation of Math. In English you can’t write a great essay with bad grammar and punctuation, regardless of how good the topic is. In Math, you can’t solve problems without good use of algebra. Some algebra takes time and experience to develop. However, there are common algebra mistakes that I see many times every day. You can avoid making these now and never waste those marks again. Any questions about this video drop us a message HERE.

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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