A1–A3 - Algebraic notation, substitution and vocabulary

Reading algebraic shorthand

Do $3a^2$ and $(3a)^2$ give the same answer when $a = 2$? They use the same symbols, but the brackets give different instructions. Keep that question in mind.

A variable is a letter that represents a number. Its value may change between calculations. The same letter has the same value wherever it appears within one calculation. Algebra leaves out some multiplication signs, but it never leaves out the multiplication itself.

Write the number before the letters: $5x$, rather than $x5$. A coefficient of $1$ is normally invisible: $1x = x$. Similarly, $-1x = -x$. Two different letters may happen to have the same value; $ab$ does not say that they must be different.

Use fractions for non-integer algebraic coefficients. Write half of $x$ as $\frac{1}{2}x$ or $\frac{x}{2}$, rather than $0.5x$. Three quarters of $y$ is $\frac{3y}{4}$. A fraction bar groups its whole numerator: $\frac{x + 4}{2}$ means $(x + 4) \div 2$, whereas $x + \frac{4}{2}$ divides only the $4$.

Give an algebraic answer in its simplest form even when the question does not say “simplify”. For example, reduce the coefficient in $\frac{2x}{4}$ to give $\frac{x}{2}$.

A power acts on the object immediately before it. In $3a^2$, only $a$ is squared. In $(3a)^2$, the whole product $3a$ is squared. These notation rules are not on the formulae sheet: learn them.

Reading the shorthand first prevents a wrong calculation later. This question checks the meaning of a product and a power together.

Expressions, statements, terms and factors

An expression is a calculation; an equation makes a claim that two expressions are equal. That distinction tells you whether a question asks for a value, a new expression or a statement about possible values.

In $6x^2 - 5x + 2$, the three terms are $6x^2$, $-5x$ and $2$. The number $2$ is a constant term, because it has no variable. The factors $6$, $x$ and $x$ multiply to make $6x^2$. In $3(x + 4)$, the whole bracket $x + 4$ is a factor. A factor can itself contain a sum.

A formula is an equation used as a rule relating quantities. These categories can overlap: $A = lw$ is both a formula and an equation. An inequality uses $<$, $>$, $\leqslant$ or $\geqslant$. An identity is stronger than an equation that works only for some values. The symbol $\equiv$ emphasises “equal for every allowed value”. These meanings are not on the formulae sheet: learn them.

For example, $x + 2 = 7$ works when $x = 5$, but fails when $x = 0$. It is not an identity. By contrast, adding $x$ to itself really does make twice $x$, whatever number $x$ represents. Testing one value can expose a false identity; it cannot establish that a statement works for every value.

The sign belongs to its term. Keeping that sign matters when you simplify expressions in the next lesson.

Substituting values into expressions

Substitution means replacing each letter by its given value, then carrying out the calculation in the usual order.

Start with $3x + 4$ when $x = 2$. Replacing $x$ gives $3 \times 2 + 4 = 10$. The answer is a number because the variable has been given a value. You are evaluating an expression, not finding an unknown.

Put negative values in brackets. This keeps the replacement together until the correct operation is applied. Powers still come before multiplication and addition. The substitution method is not on the formulae sheet: learn it.

The second example changes only the sign of the substituted value. Brackets now protect the negative sign.

Here is a neat limitation of squaring: $2x^2 + 5$ gives the same value for every pair of opposite inputs. Knowing the output is $23$ cannot tell you whether the input was $3$ or $-3$. Squaring has lost that sign information.

The next expression keeps the square and changes the final operation from adding to subtracting.

Substituting into scientific formulae

A scientific formula uses exactly the same substitution method, but its letters describe measured quantities and its answer needs a unit.

An unfamiliar formula will be given in the question. Read what each letter means rather than guessing from the letter. The formula below connects initial velocity $u$, final velocity $v$, acceleration $a$ and time $t$. You do not need to learn its physics to substitute correctly.

The product $at$ means acceleration multiplied by time. The initial velocity $u$ is then added once; it is not inside that product.

The next example changes only the time. Leave the other values and the formula in the same places.

A fraction in a formula needs the same care as a bracket. For example, if a question gives $D = \frac{m}{V}$, then $m = 18$ and $V = 3$ give $D = 18 \div 3 = 6$. The question must supply or establish the units and the meaning of the letters. This example formula is given in the question; it is not on the formulae sheet.

You can now answer the opening question too: with $a = 2$, $3a^2 = 3 \times 4 = 12$, but $(3a)^2 = 6^2 = 36$. The bracket changes what is squared.

Quick Recap

• Algebra leaves out multiplication signs: $ab = a \times b$, while $a^2 = a \times a$.
• Terms are added or subtracted; factors are multiplied. A coefficient is a numerical factor.
• Expressions describe calculations. Equations and inequalities make statements; formulae connect quantities. An identity works for every allowed value.
• Substitute the whole given value, put negatives in brackets, and follow the order of operations. Scientific formulae also need compatible units.

The opening expressions give $12$ and $36$, so $3a^2$ and $(3a)^2$ are different instructions.