Simply Mathematics: Introduction to GCSE Maths

Advice of the video:

When we asked you to think of a number that when multiplied by itself equals 4, there were two possible solutions. 2 and -2. BUT, whenever you see the root symbol, remember that it is only asking for the positive answer. That is quite a big take away from this so we hope you're reading this.

We explain the need for BODMAS. It allows us to understand the problem much better and is considered the gold standard when it comes to Grammar.

The second highest ranked in the BODMAS hierarchy, Orders are simply numbers involving powers or roots.

Summary: Because division and multiplication share the same ranking, we calculate which is ever operator is on the left first. The same goes for Addition and Multiplication.

An introduction to why we use Algebra. Mathematics is a language after all!

We introduce the idea of f(x) as a machine.

Think of this as more of a review. We are using previously understood concepts to introduce the technique of simplifying. 

The idea is in the word "common". We're looking for things in common. Lowest common multiple is the smallest multiple that the specified numbers share. Highest common factor is the largest factor that the specified numbers share.

We finally talk about simplifying fractions.

This, along with simplifying fractions are the toughest parts to understand in this course. Why? Because, it requires a layered form of thinking. It comes back to that ladder analogy from before. In order for us to add fractions we need to know:

1. What a multiple is.

2. How to find the lowest common multiple.

3. What a factor is.

4. How to find the lowest common factor.

5. How to simplify a fraction.

If we didn't "climb" this ladder, there's no way we would know how to add Fractions. Hope this helps!

Subtracting fractions makes use of the same technique as adding them. This is the toughest part over. Multiplying and subtracting is a cakewalk compared to this.

Our Advice: Always remember to simplify when working with fractions to gain that additional mark in an exam.

Logarithms will be discussed in further detail on our full GCSE Mathematics Course but for now we introduce it.

Mixed numbers have both an integer component and a fraction component, whereas an improper fraction is improper because the numerator is larger than the denominator.

Fractions, decimals and percentages are three different ways to represent a proportion. You should feel confident in seeing a proportion being represented in these three different ways.

These were the first of the conversions. A key idea to note is that 075 = 075.0 = 75.0 = 75

This was a rather lengthy video but we do urge you to pause/replay sections as it can be difficult to understand. We will go through some questions after the next video.

There is a very high chance you will be required to convert one proportion into another, so practice as many questions as you can on these.

If a question does not specify how many decimal places or significant figures you are to round the numbers to, you will have to use your own judgment. For example angles (degrees) are normally rounded to 1 decimal place whereas currency is normally rounded to 2 decimal places. Although it is important to note that if you are required to round 2 different currencies, you should not round one value to 2 decimal places and the other to 3 decimal places. Both the numbers should be rounded to the same decimal place(s).

As you saw in the episode above, an increase in the number of significant figures used to describe a measurement, also increases the precision with which that measurement is made. Notice we use the word precision and not accuracy. It is not necessarily true that rounding a number to a higher significant value will always yield a more accurate result.

Lets look at some of the definitions used in this episode.

Lower bound

The smallest value that rounds to up to the estimated value.

Upper Bound

The smallest value that rounds up to the "next" estimated value.

Our Example

In our example the estimated value (rounded value) was 5.5. The lower bound was 5.45. The upper bound was 5.55 and the "next" estimated value was 5.6.

Other Definitions

<

This symbol reads as "less than".

>

This symbol reads as "greater than".

≤

This symbol reads as "less than or equal to".

≥

This symbol reads as "greater than or equal to".

Interval

The numbers lying between two specified numbers. In our case all the numbers between 5.45 and 5.55 (including 5.45 but not including 5.55).

Just to be clear, converting a number into standard form is not the same as rounding. You are not estimating anymore. You are re-writing a number into what we call the standard form of a number. In certain fields, you sometimes have to work with ridiculously large or small numbers. Converting these numbers to standard form makes it a lot easier to analyze. Examples could range from working with the speed of light to measuring how much weight the average weaver ant can lift (for those interested, 50 times its body weight which is still only 0.2834952 grams).

This concludes our segment on numbers and you know what that means. A quiz to follow on Standard Form! Now that we are familiar with some key definitions, it's time to finally introduce the behemoth of a section, that is Algebra.

Algebra has always been a part of our culture. Without it, advancements in ANY industry would be impossible. Algebra is in our opinion, the most important topic of our course and in your case, the gatekeeper to higher-level mathematics. Each subsequent chapter will include at the very least, an element of algebra.

We discuss some fundamental rules in Algebra.

When students see the letter x, they often feel comfortable solving the problem. But when a question uses a different letter, say v, some feel a bit overwhelmed, and rightfully so. Why can't we just use x for everything? I mean it is the go-to symbol for an unknown quantity. The reason why we do not just use x is that in real-world maths problems, there are often too many variables to deal with. Sometimes it would make more sense to use the letter t, if the variable in question is time itself.

Try not to be put off by different letters. Different letters will be used in the exam so understanding the following simple idea is of paramount importance: Any letter can mean any variable!

A single mathematical expression is what we refer to as a term. It can take the form of a single number, a single variable (a letter), or a combination of numbers and variables multiplied but never added or subtracted. The bullet points below give examples of the number of terms:

• 7a is a single term.

• 7abcdefghijklmnopqrstuvwxyz is a single term

• 7a + b are two terms

• 7a + 7b are also two terms

• 777 is a single term

• 7 + 10x are two terms

• 7a + 7b + 7c +7d + 7e + 7f +7g +7h +7i +7j +7k +7l +7m +7n +7o +7p +7q +7r + 7s + 7t +7u + 7v + 7w + 7x + 7y + 7z are 26 terms.

This should hopefully put things into perspective.

Terms that share the same combination of variable(s) or letters can be combined together.

There are 10 rules for powers and roots. Sadly, you will have to memorise each rule. The good news is that practicing questions with powers and roots makes it difficult for these rules to be forgotten. With that in mind, you will be happy to know that a quiz is being prepared.

One of the first definitions we came across in Chapter 1 was BODMAS. We prioritised the expansion of brackets and now we are learning how to expand brackets with unknown variables within them. The rule is simple. Each term inside a bracket must be multiplied by each term outside the bracket.

Factorising is a different way of seeing an equation. Setting an equation equal to zero and factoring is an invaluable technique.

For whole numbers and polynomials (which we will come across), it is often much more difficult to factorise than to expand. This asymmetry of difficulty is the basis of all cryptographic systems we use.

A square number is the result of a number being multiplied by itself. The first 10 square numbers are: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100

If even one of the numbers is not a square number, then we have to look for common factors to take out.