Notation, vocabulary and manipulation

Welcome to the World of Algebra!

Welcome! If you’ve ever looked at a page of maths and wondered why there are suddenly letters mixed in with the numbers, don't worry—you’re not alone. Think of Algebra as a secret code or a shorthand way of writing instructions. Instead of saying "I have a number, and if I double it and add five, I get thirteen," we can just write \(2x + 5 = 13\).

In this chapter, we are going to learn the "grammar" of this new language. Once you know the rules, you’ll see that algebra is actually a powerful tool that makes solving puzzles much easier!

1. The "Secret Code": Algebraic Notation

In algebra, we use letters to represent numbers we don't know yet. To keep things neat, mathematicians use a specific shorthand:

• Multiplication: Instead of writing \(a \times b\), we just write \(ab\). If you see a number next to a letter, like \(3y\), it means \(3 \times y\).
• Addition: If you have \(y + y + y\), that is \(3y\). Think of it like one apple + one apple + one apple = 3 apples.
• Powers: \(a \times a\) is written as \(a^2\), and \(a \times a \times a\) is \(a^3\).
• Division: Instead of \(a \div b\), we write it like a fraction: \(\frac{a}{b}\).
• Coefficients: The number in front of a letter (like the 5 in \(5x\)) is called the coefficient. We usually write these as fractions (like \(\frac{1}{2}x\)) rather than decimals.

Quick Review:
- \(4 \times m = 4m\)
- \(p \times p = p^2\)
- \(10 \div x = \frac{10}{x}\)

2. Maths Grammar: Key Vocabulary

Just like English has nouns and verbs, algebra has specific terms you need to know:

• Term: A single part of an expression, like \(3x\) or \(5\).
• Expression: A group of terms (no equals sign!), e.g., \(2x + 3\).
• Equation: Two expressions that are equal to each other, e.g., \(2x + 3 = 11\).
• Formula: A rule that links different variables, like \(Area = length \times width\).
• Identity: Something that is always true, no matter what number you pick. We use a special symbol for this: \(\equiv\).

Analogy: An Expression is like a phrase ("the red car"), while an Equation is like a full sentence ("The red car is fast").

3. Substitution: "Plug and Play"

Substitution is simply replacing a letter with a specific number to find the answer.

Step-by-step example:
Find the value of \(3x + 5\) when \(x = 4\).
1. Replace the \(x\) with 4: \(3(4) + 5\)
2. Remember that \(3x\) means \(3 \times x\), so: \(3 \times 4 + 5\)
3. Calculate: \(12 + 5 = 17\).

Common Mistake: Be careful with negative numbers! If \(x = -2\), then \(x^2\) is \((-2) \times (-2)\), which is positive 4. Always put negative numbers in brackets when you substitute them into a calculator.

4. Manipulation: Cleaning up Algebra

Sometimes algebra looks messy. We "manipulate" it to make it simpler.

Collecting Like Terms

You can only add or subtract terms that are the same "kind."
Example: Simplify \(3a + 5b + 2a - b\).
1. Group the \(a\)s: \(3a + 2a = 5a\)
2. Group the \(b\)s: \(5b - b = 4b\)
3. Final answer: \(5a + 4b\).

Single Brackets

To multiply a single term over a bracket, imagine the term outside is a postman delivering a letter to everyone inside the house.
\(3(x + 4) = 3 \times x + 3 \times 4 = 3x + 12\).

Taking out Common Factors (Factorising)

This is the opposite of expanding brackets. You look for the biggest number or letter that fits into every term.
Example: Factorise \(6x + 9\).
1. What is the biggest number that goes into 6 and 9? 3.
2. Put it outside the bracket: \(3( \dots )\)
3. What do you multiply 3 by to get \(6x\)? \(2x\).
4. What do you multiply 3 by to get 9? 3.
5. Final answer: \(3(2x + 3)\).

Key Takeaway: Simplifying doesn't change the value of the expression; it just makes it easier to read!

5. Expanding Two Binomials (Additional/Higher Content)

When you have two brackets like \((x + 2)(x + 3)\), use the FOIL method to make sure you don't miss anything:

• First: \(x \times x = x^2\)
• Outside: \(x \times 3 = 3x\)
• Inside: \(2 \times x = 2x\)
• Last: \(2 \times 3 = 6\)
Now, add them together: \(x^2 + 3x + 2x + 6\).
Simplify the middle: \(x^2 + 5x + 6\).

Did you know? This type of expression is called a quadratic because of the \(x^2\) term!

6. Changing the Subject (Rearranging Formulae)

This is about moving the letters around so a different one is by itself. The goal is to get the letter you want on its own (like \(x = \dots\)).

The Golden Rule: Whatever you do to one side of the equals sign, you MUST do to the other. Use inverse (opposite) operations:

• Opposite of + is -
• Opposite of \(\times\) is \(\div\)
• Opposite of \(x^2\) is \(\sqrt{x}\)

Example: Make \(x\) the subject of \(y = 5x - 2\).
1. Add 2 to both sides: \(y + 2 = 5x\)
2. Divide both sides by 5: \(\frac{y + 2}{5} = x\).

7. Functions (Higher Tier Focus)

A function is like a machine. You put an input in, it does something to it, and gives you an output. We use the notation \(f(x)\).

• \(f(x) = 2x + 1\) means "take the input \(x\), double it, and add 1."
• Inverse functions (\(f^{-1}(x)\)) go backwards—they undo the machine.
• Composite functions (\(fg(x)\)) mean you put your number into function \(g\) first, and then put the result into function \(f\).

Memory Trick: For composite functions like \(fg(x)\), always work from right to left. Start with the function closest to the \(x\)!

Summary Checklist

• Can you identify the difference between an equation and an expression?
• Do you remember that \(ab\) means \(a \times b\)?
• Can you expand a single bracket by "delivering the mail"?
• Are you comfortable substituting negative numbers using brackets?
• (Higher) Can you use FOIL to expand double brackets?

Don't worry if this seems tricky at first—algebra is a skill that gets much easier with practice. Keep going!