ESAT Maths 1: topics, timing and working without a calculator

What Mathematics 1 asks of you

Every ESAT candidate takes Mathematics 1, whatever their other modules are. Most people take two more on top, depending on their course. Maths 1 has 27 multiple-choice questions in 40 minutes. You can't use a calculator or a dictionary, and a wrong answer doesn't cost you a mark.

The maths itself comes from school. The difference is that nobody marks your working. Half a page of algebra that ends in the right answer scores the same as a two-line argument that gets there in a quarter of the time. The hard part is seeing a quick way in when the question doesn't tell you which chapter it's from.

Maths 1 also matters beyond its own score. Rearranging, ratios, graphs and units turn up all through the science modules, so a weak spot here tends to slow you down there as well.

Making your topic list

Start from the official ESAT specification and the ESAT Guide on the UAT-UK website. List the number, algebra, geometry and statistics content for Maths 1 as it's written there. Don't borrow a topic list from A level or from a different test.

Next to each topic, write what you can actually do. 'Fine rearranging straight-line formulas, slow when there are fractions inside fractions' is far more useful than a tick. So is 'I know graph transformations but keep moving f(x + 2) the wrong way'. A note like that tells you exactly what your next practice session should be.

Then check the notes with questions. A heading can look familiar long after you've stopped being quick at it. If you use older ENGAA or NSAA papers, read UAT-UK's notes on them first, because some of their questions cover work that isn't in the current ESAT.

Simplifying before you calculate

A lot of no-calculator trouble comes from arithmetic you never needed to do. Before you work anything out, ask what cancels, whether anything factorises, and whether the answer has to be positive or negative.

Say a question asks you to simplify (√12 + √27) ÷ √3. If you try to estimate each square root as a decimal, you'll get a messy number that matches nothing. Write √12 as 2√3 and √27 as 3√3 first. The top becomes 5√3, so the whole thing is 5. All the work is in spotting the square factors, and that takes seconds once you look for it.

Algebra works the same way. If an expression factorises, factorise it before you expand anything, and keep fractions as fractions until the last line. If the question only asks which range the answer lies in, a rough bound may be all you need.

Ratio questions

Ratio questions are easy to make long. Suppose a bag holds red and blue counters in the ratio 3 : 5, and after 6 more red counters go in, the ratio becomes 3 : 4. You could set up two unknowns and solve simultaneous equations. It's quicker to call the counts 3k and 5k and write (3k + 6) ÷ 5k = 3 ÷ 4. That gives 12k + 24 = 15k, so k = 8, and the bag started with 24 red and 40 blue.

Keep the ratio as a ratio for as long as you can. The usual errors are adding to both parts when only one changes, and choosing 8 because it's the number you found when the question asked for the number of counters. Read the last line of the question again before you pick an option.

Using the answer options

The options are part of the question. If they're algebraic expressions that are supposed to equal the one in the question for every value of x, put in x = 2 and see which ones match. If they're ranges, test a value near the edge of each. If they're whole numbers, try them in the original condition and see which one works.

Only use these shortcuts when they're valid. Substituting x = 2 proves something only if the expressions are meant to be equal for all x, and 2 has to be an allowed value: it can't make a denominator zero, for instance. If two options both pass your test, try a second value. When you go over the paper afterwards, write one line saying why the shortcut was safe, so you can trust it next time.

Estimating first

Make a rough estimate before any exact calculation, even when you plan to work it out properly. It catches answers that are out by a power of ten, ratios turned upside down and percentages that can't be right. Sometimes the estimate picks out the answer on its own. When the options are close together, it's still a quick check on your exact working.

When an estimate goes wrong, find out how. Did you round both numbers up and forget your answer was now too big? Did you forget that dividing by 0.8 makes a number bigger? Did you work out a change in pounds when the question asked for a percentage change? Each of those points to a small, specific thing to practise.

Reading graphs and diagrams

With a graph question, say what it's really about: the gradient, an intercept, a turning point, the area underneath, or a shift or stretch. Then check the axes. A graph might plot y against x², or start its vertical axis at 50, and either one changes what you read off it.

Suppose a question shows a curve through the origin that gets steeper as it goes, and asks which statement about y is true as x increases. Put it into words before you read the options: y goes up, and it goes up faster as x gets bigger. That rules out an option saying the gradient is constant, and one saying y grows more slowly. If your sentence turns out to be wrong when you check the solution, the problem is how you read the picture, and extra algebra practice won't fix it.

In geometry, write down what the question says is true and what only looks true. A diagram that isn't marked as drawn to scale can show two lengths that look equal and aren't.

Pacing the 40 minutes

40 minutes for 27 questions is just under 89 seconds each, though your actual times will vary a lot. The aim is to reach question 27 with some time left to go back.

Decide your move-on point before you start. About a minute with no idea how to begin is a sensible starting point: choose the likeliest option, flag the question and carry on. Change the number if your own mocks suggest a different one. Since a wrong answer costs nothing, make sure every question has an answer selected before the time is up.

With the time left at the end, go first to the flags where you'd got partway through a method or crossed out all but two options. Starting the hardest question again from scratch while three almost-finished ones are waiting is a poor use of those minutes. There's more on this in the guide to pacing a 40-minute module.

Going through your mistakes

After a paper, give each mistake a short, plain name: expanded when factorising was quicker, rounded too early, dropped a minus sign, ignored a restriction on x, assumed the diagram was to scale, got a percentage change the wrong way round, used the wrong quantity, or kept going with no plan. Add how long it took and how sure you were. A quick sign error, a slow but correct factorisation and a lucky guess each need different practice.

When you read the worked solution, look for the detail that should have told you what to do, such as 'both surds have a factor of √3' or 'only one part of the ratio changes'. Then try a few new questions of the same kind and finish with a short timed mixed set. Noetra's replay questions are kept separate from the mocks and test the same skills with different numbers and wording, so you're checking the method and can't lean on a remembered answer.

A four-week cycle

Here is one way to organise a month of Maths 1 practice. Repeat it with fresh papers for as long as you have.

Keep one short session a week on topics you're already good at, so they don't get slow. If the first paper shows a lot of content you haven't been taught, give week two more time and ask a teacher for help. If you know the maths but can't finish the paper, spend more of week three comparing methods and practising moving on.

• Week one: list the official topics and sit a full paper (27 questions, 40 minutes, no calculator).
• Week two: work on the two weakest areas that paper showed, whether that's a topic or a habit like expanding too soon, using short focused sets.
• Week three: mixed questions on a short timer, without a calculator, trying out your move-on point.
• Week four: sit another unseen paper, compare its mistakes with week one, and carry whatever's still going wrong into the next cycle.

What your practice score means

Official ESAT results are given for each module on a scale from 1.0 to 9.0, and UAT-UK says there's no pass or fail. The official sample tests don't give a scaled score, because the scaling for the real test depends on how everyone sitting it performs. A raw mark from practice is still worth tracking, especially between papers meant to be of similar difficulty, but it isn't an admissions result.

Noetra's 1.0–9.0 score is our own estimate, not an official result. Read it next to your raw mark, your topic breakdown and your time per question. Those tell you whether your next mark is most likely to come from algebra, graph reading, more careful arithmetic or moving on sooner.

Questions students ask

Is ESAT Maths 1 compulsory?

Yes. UAT-UK says every ESAT candidate takes Mathematics 1. Most people also take two more modules, depending on what their course asks for.

How many questions are in ESAT Maths 1?

Mathematics 1 has 27 multiple-choice questions in 40 minutes, and you can't use a calculator.

How can I get faster at ESAT Maths 1?

Time each question when you practise, look again at the ones you got right slowly, compare your method with a shorter one, and practise a rule for moving on so you keep a few minutes at the end.

Official sources

Current details were checked against these primary sources on the date shown above.

• UAT-UK: ESAT format
• UAT-UK: ESAT specification, guide and archives
• Noetra guide: ESAT 40-minute strategy