Choosing a method

Choosing a method is the skill of turning the information in a question into a useful first mathematical step. It is not guesswork. Good choices come from recognising structure, identifying the required output and testing whether the available facts fit the conditions of a method.

What you should know first

You should be familiar with the main techniques from pure mathematics, statistics and mechanics. In particular, you should be able to:

• rearrange equations and sketch familiar graphs;
• differentiate and integrate standard functions;
• use common probability distributions;
• draw force diagrams and apply constant acceleration formulae.

If the algebra needed to carry out a chosen method is unreliable, review algebraic fluency first.

A reliable selection routine

Before calculating, write down three things:

1. Target: What exactly must be found, shown or decided?
2. Evidence: What equations, values, graph features or modelling assumptions are given?
3. Bridge: Which theorem, definition or model connects the evidence to the target?

Then check the method’s conditions. For example, the binomial model needs a fixed number of independent trials with constant success probability. Integration by parts needs a product, but it is useful only when differentiating one factor makes the product simpler.

The following question stems often signal a mathematical action.

Wording or structure	Likely action	Essential check
”Show that”	Work towards the stated form	Do not assume the result you must prove
”Hence”	Use the preceding result	Identify what the earlier result replaces or simplifies
”Exact”	Keep fractions, surds, $\pi$, $e$ and logarithms	Do not use rounded decimals
”Maximum” or “minimum”	Form an objective function, then differentiate	Check endpoints and classify stationary points
”At least”, “at most”	Translate to an inequality or cumulative probability	Include the boundary value correctly
Product inside an integral	Substitution or integration by parts	Look for an inner derivative first
Repeated independent trials	Binomial distribution	Verify fixed $n$ and constant $p$
Forces and acceleration	Resolve forces, then use $F=ma$	Fix a positive direction first

These are clues, not automatic commands. The whole structure decides the method.

Simplify before using a major technique

A sophisticated method is not automatically a good method. First ask whether factorising, cancelling, using an identity or changing representation reveals something simpler.

Worked example 1: an equation disguised by fractions

Solve

$$\frac{x^2-9}{x-3}=8.$$

The denominator gives the restriction $x\ne3$. Factorise before multiplying out:

$$\frac{(x-3)(x+3)}{x-3}=8.$$

For allowed values of $x$, this becomes

$$x+3=8,$$

so $x=5$. The cancelled value $x=3$ was never in the domain. Cross multiplication would work, but simplification makes both the method and the restriction clearer.

Worked example 2: differentiate after rewriting

Differentiate

$$y=\frac{x^3+2x}{x}, \qquad x\ne0.$$

Rather than using the quotient rule,

$$y=x^2+2,$$

so

$$\frac{dy}{dx}=2x, \qquad x\ne0.$$

The derivative retains the original domain. See product, quotient and chain rules for deciding between differentiation rules.

Work backwards from the target

When a question contains too much information, the requested result tells you which information matters.

Worked example 3: finding a tangent

The curve has equation

$$y=x^3-3x+4.$$

Find the equation of the tangent where $x=2$.

The target is a straight line. A line needs a point and a gradient.

First find the point:

$$y(2)=8-6+4=6,$$

so the tangent passes through $(2,6)$.

Next find its gradient by differentiation:

$$\frac{dy}{dx}=3x^2-3,$$

hence at $x=2$,

$$m=3(2)^2-3=9.$$

Now use the point gradient form:

$$y-6=9(x-2),$$

or

$$\boxed{y=9x-12}.$$

The method chain is therefore

$$\text{tangent} \longleftarrow \text{point and gradient} \longleftarrow \text{substitution and differentiation}.$$

Review tangents and normals if this chain is unfamiliar.

Choose between plausible methods

Sometimes several methods are valid. Prefer the one that uses the visible structure, preserves exact values and creates the least avoidable algebra.

Worked example 4: substitution or integration by parts?

Evaluate

$$I=\int_0^1 2x e^{x^2}\,dx.$$

The integrand is a product, which might suggest integration by parts. However, $2x$ is the derivative of the exponent $x^2$. That is a stronger structural clue.

Let

$$u=x^2, \qquad du=2x\,dx.$$

Change the limits:

$$x=0\Rightarrow u=0, \qquad x=1\Rightarrow u=1.$$

Therefore

$$\begin{aligned} I&=\int_0^1 e^u\,du\\ &=\left[e^u\right]_0^1\\ &=e-1. \end{aligned}$$

Integration by parts is possible but inefficient. For

$$\int x e^x\,dx,$$

there is no inner function whose derivative supplies the other factor, so integration by parts is the more natural choice. Compare integration by substitution and integration by parts.

Worked example 5: exact algebra or numerical solution?

Consider

$$x^3+x-1=0.$$

Testing the possible integer roots $\pm1$ gives neither root, and no simple factorisation appears. The derivative

$$3x^2+1>0$$

shows the function is strictly increasing, so it has at most one root. Also,

$$f(0)=-1, \qquad f(1)=1,$$

so a root lies in $(0,1)$. A numerical method is appropriate if the question asks for a decimal approximation.

Newton Raphson gives

$$x_{n+1}=x_n-\frac{x_n^3+x_n-1}{3x_n^2+1}.$$

Starting with $x_0=0.5$,

$$x_1=0.5-\frac{0.5^3+0.5-1}{3(0.5)^2+1} =0.714285\ldots$$

and further iterations converge to $0.6823\ldots$. Do not choose a numerical method if the question requests an exact answer. See change of sign and Newton Raphson.

Selecting a statistics model

In statistics, the assumptions are part of the method. Matching a familiar word such as “number” or “mean” is not enough.

Worked example 6: identify the distribution

A biased coin has probability $0.3$ of landing heads. It is tossed independently $20$ times. Find the probability of at least $8$ heads.

There is:

• a fixed number of trials, $n=20$;
• one of two outcomes on each trial;
• independent trials;
• a constant success probability, $p=0.3$.

Thus

$$X\sim\operatorname{B}(20,0.3).$$

“At least $8$” means $X\ge8$. Most calculators evaluate lower tail probabilities, so use the complement:

$$\begin{aligned} \operatorname{P}(X\ge8) &=1-\operatorname{P}(X\le7)\\ &\approx 0.2277. \end{aligned}$$

If the probability changed from toss to toss, or tosses influenced one another, the binomial model would not be justified. Use choosing a distribution to compare statistical models.

Worked example 7: choose the test from the claim

A manufacturer claims that $40\%$ of customers choose option A. In a random sample of $50$ customers, $27$ choose A. Test at the $5\%$ significance level whether the proportion has increased.

The data count successes in a fixed sample, so under the null hypothesis

$$X\sim\operatorname{B}(50,0.4).$$

The word “increased” determines a one tailed upper test:

$$H_0:p=0.4, \qquad H_1:p>0.4.$$

The observed result is $x=27$, so the relevant probability is

$$\operatorname{P}(X\ge27).$$

The direction comes from the alternative hypothesis, not from whichever tail produces a smaller number. Review binomial hypothesis tests and hypothesis testing language.

Selecting a mechanics model

Start mechanics questions with a diagram and a positive direction. Then separate the modelling stages:

$$\text{forces} \xrightarrow{F=ma} \text{acceleration} \xrightarrow{\text{kinematics}} \text{motion}.$$

Worked example 8: forces before kinematics

A particle of mass $4\text{ kg}$ is pulled horizontally by a force of $18\text{ N}$. A resistance of $6\text{ N}$ opposes the motion. It starts from rest. Find its speed after travelling $16\text{ m}$.

The kinematics formula needs acceleration, which is not given. Find it from the forces first. Taking the direction of motion as positive,

$$F_{\text{resultant}}=18-6=12\text{ N}.$$

Using $F=ma$,

$$12=4a \quad\Rightarrow\quad a=3\text{ m s}^{-2}.$$

Now $u=0$, $s=16$ and $a=3$. Time is absent from both the data and the target, so choose

$$v^2=u^2+2as.$$

Hence

$$v^2=0+2(3)(16)=96,$$

so

$$\boxed{v=4\sqrt6\text{ m s}^{-1}}.$$

The negative square root is rejected because $v$ here denotes speed. The positive direction and the physical meaning decide the sign.

What to do when you are stuck

If no method is obvious, do not search your memory randomly. Use this recovery sequence:

1. Rewrite the target using a definition. For a stationary point, write $f'(x)=0$. For independence, write $\operatorname{P}(A\cap B)=\operatorname{P}(A)\operatorname{P}(B)$.
2. Label every known quantity, including units, restrictions and parameters.
3. Draw a sketch, tree diagram or force diagram if it exposes relationships.
4. Change representation. Factorise, take logarithms, use vectors, or replace a repeated expression by a new variable.
5. Work backwards one step from the target and forwards one step from the data.
6. If a part begins “hence”, inspect the preceding result before starting again.

Worked example 9: change representation

Solve

$$3^{2x}-10(3^x)+9=0.$$

The repeated expression $3^x$ suggests the substitution

$$u=3^x, \qquad u>0.$$

Then $3^{2x}=(3^x)^2=u^2$, giving

$$u^2-10u+9=0.$$

Factorise:

$$(u-1)(u-9)=0.$$

Thus $u=1$ or $u=9$. Returning to $x$,

$$3^x=1\Rightarrow x=0,$$

or

$$3^x=9=3^2\Rightarrow x=2.$$

Therefore $\boxed{x=0\text{ or }x=2}$. The key step was not a special exponential rule. It was recognising a quadratic in disguise. See exponential and logarithmic equations.

Common misconceptions

”A keyword determines the method”

Words provide evidence, not certainty. “Rate” may lead to differentiation, a differential equation, a compound measure or a rate parameter in a probability model. Read the mathematical relationships around it.

”The longest method earns the most marks”

Marks reward valid progress. An exact identity or simplification may replace a page of algebra. Efficient mathematics is usually easier to verify.

”If my chosen method starts badly, I must continue”

Pause when expressions become unexpectedly complicated. Check whether you missed a factorisation, an inner derivative, a symmetry or a result supplied in an earlier part.

”A calculator can choose the model”

A calculator evaluates the model you enter. It cannot decide whether independence, constant acceleration or a distributional assumption is reasonable.

”Two valid methods must look alike”

Different correct routes can produce different intermediate forms. Compare them by simplifying, substituting a test value or checking against the original conditions.

Self check

Choose a method before attempting any detailed calculation.

1. Solve $2^{2x}-5(2^x)+4=0$.
2. Which integration method is most direct for $\displaystyle\int x\cos(x^2)\,dx$?
3. A particle moves with constant acceleration. You know $u$, $v$ and $t$ and need $s$. Which constant acceleration equation avoids an unnecessary unknown?
4. A random variable counts defective bulbs in $30$ bulbs sampled without replacement from a box of $40$, of which $8$ are defective. Is a binomial model exact?
5. To find the minimum of $f(x)$ on the closed interval $[a,b]$, is solving $f'(x)=0$ sufficient?

Answers

1. Set $u=2^x>0$. Then $u^2-5u+4=(u-1)(u-4)=0$, so $2^x=1$ or $2^x=4$. Hence $x=0$ or $x=2$.
2. Substitution. Let $u=x^2$, so $du=2x\,dx$. The factor $x$ supplies half of the inner derivative.
3. Use $\displaystyle s=\frac{u+v}{2}t$. It uses exactly the known quantities and the target.
4. No. Sampling without replacement makes the trials dependent and changes the success probability. A hypergeometric model is exact, although that distribution is outside standard A level Mathematics content. A binomial approximation would require explicit justification.
5. No. Evaluate $f$ at every stationary point inside the interval and at both endpoints. Also consider points where $f'$ does not exist if they lie in the domain.

A final exam checklist

Before committing to a method, ask:

• Does it connect the given information to the exact target?
• Are all its assumptions satisfied?
• Can I simplify or change representation first?
• Is there a supplied result I should use?
• Will the method preserve the required accuracy, exactness and units?
• Can I check the result by substitution, estimation, a graph or another method?

Next, learn how to make the resulting argument easy to award marks in communicating mathematical reasoning, then test the outcome systematically in checking answers.