NCEA Maths: help and resources for students

Do you need help with NCEA Level 1 Maths, Level 2 Maths, or Level 3 Maths or Stats?

This article outlines strategies for improving your performance in NCEA Mathematics and Statistics. It includes worksheets, resources, tips, and techniques recommended by experienced teachers, for use by students in New Zealand schools.

In this article:

• NCEA maths overview
• Why get a good grade
• More about unit standards and achievement standards
• 17 top tips to achieve in NCEA maths

What is NCEA Maths?

NCEA Maths is a subject that can be taken by high school students in Year 11, Year 12, and Year 13. There are a number of different NCEA Maths classes and courses available, covering different parts of the curriculum.

NCEA stands for the National Certificate in Educational Achievement and is the main national qualification for secondary school students in New Zealand. NCEA is used as a selection tool by universities and polytechnics and is recognised by employers. You will choose your syllabus each year from a range of courses and subjects offered by your school. Each subject is made up of a number of standards, which you will be assessed against.

For example, an NCEA Level 1 mathematics standard is “Apply algebraic procedures in solving problems”.

An NCEA Level 3 statistics standard is “Conduct an experiment to investigate a situation using experimental design principles”.

Why is a good grade in mathematics important?

Maths is an important part of everyday life, from working out if you’ve got enough money to buy your latest must-have gadget, to calculating whether you’ve got enough time to walk to school versus running to catch that bus.

But success in maths is not just about knowing how to do the maths. Getting a good grade in your maths classes is important in its own right. Good grades are helpful for getting into the university or polytech course that you want, for landing that scholarship or apprenticeship, and for having independent proof of your skillset without needing to constantly demonstrate it.

But before we jump into the tips and techniques for tackling NCEA Maths, you might have some bigger questions about NCEA.

What is an NCEA unit standard or achievement standard?

There are two types of assessment standards in NCEA, unit standards and achievement standards. All but one of the NCEA Maths standards are achievement standards, which means they can receive a grade of Not Achieved, Achieved, Merit, or Excellence, and follow the New Zealand curriculum.

Some standards are assessed internally by the student’s class teacher using a range of tests or projects. Other standards are assessed externally at the end of the year by NZQA (the New Zealand Qualifications Authority) either by exam or portfolio. Each standard is worth a number of credits, which the student earns by achieving the standard.

If the student performs particularly well, they will receive their credits at a Merit or Excellence level, and if they achieve consistently high marks across several standards, then they can receive a Merit or Excellent endorsement for the subject or course as a whole.

NCEA Scholarship Exams

If you do well in maths, you may wish to sit New Zealand Scholarship examinations at the end of your NCEA Level 3 year. The subject requirements are the same as Level 3, although assessed to a much higher standard, and successful candidates receive a financial award.

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So, with that out of the way, let’s discuss some strategies for fulfilling your potential!

Tip 1: Always write down your working

This is really important. An examiner or teacher can’t give marks for something they can’t see -so make sure they can see all the steps that you’re taking.

As you answer the question, explain your thinking using clear numeric reasoning, and correct notation and terminology. This will not only help your teacher or marker to follow your working, but it also helps keep you on track as well, especially if you decide to move onto a different question and then come back to answer this one later.

The clearer you can be in your working, the more likely you are to get full marks.

“Too many students did not show sufficient working to provide appropriate evidence for their answers. Candidates need to be aware that a correct answer only is likely to be awarded a grade u.”

2019 Examiner Assessment Report, NCEA Level 2 Mathematics and Statistics

Write out each step, even if you can do it in your head or use a calculator.

It may feel pointless or obvious, but it is not a waste of time to show your working.

An examiner will have a guide in front of them for how to allocate marks for the question, and some of those marks will be for correctly demonstrating your working.

Remember that the assessment standards for NCEA maths are almost all about applying particular skills in solving problems, which means you need to show how you are applying that skill to the problem in front of you – not just giving the answer.

Have a look at the names of these assessment standards:

• AS91026 – Apply numeric reasoning in solving problems
• AS91029 – Apply linear algebra in solving problems
• AS91030 – Apply measurement in solving problems
• AS91256 – Apply co-ordinate geometry methods in solving problems
• AS91257 – Apply graphical methods in solving problems
• AS91258 – Apply sequences and series in solving problems
• AS91573 – Apply the geometry of conic sections in solving problems
• AS91574 – Apply linear programming methods in solving problems
• AS91575 – Apply trigonometric methods in solving problems

Well, you get the idea. In order to get the full marks for each question, you need to not just solve the problem, you need to show how you are applying the mathematical methods to do so.

Which means writing out each step of the equation.

Marks can be given for working, even if the final answer is wrong

“Candidates should show intermediate steps in a logical manner and clearly communicate what is being calculated. By giving only the answer, candidates may lose the opportunity to provide evidence for grades or to have minor errors ignored and are unlikely to provide evidence towards a grade higher than Achievement.”

2019 Examiner Assessment Report, NCEA Level 1 Mathematics and Statistics

Another advantage of writing out each step is that even if you make a mistake, you can still get marks for the question.

This is because the question is marked not just on whether you solved the problem, but on how you applied the mathematical reasoning to the question.

Your teacher or examiner will be looking to see which equations you are using, and the numerical and logical steps you take to solve the problem.

Even if you make an error in your calculation at one step, by writing down all your working, you can show where you’ve followed the correct processes, and that you understood the question and how to arrive at a solution.

Make sure you’re eligible for the maximum marks for each question by showing your working. Remember, you can’t get marks for things the examiner can’t see.

Even if you’re not sure of how to solve the whole question, you can still get marks for your initial working

Showing your working is a great idea even if you aren’t sure how to answer a question.

Remember that an exam marker will have a marking guide in front of them showing how to allocate marks for each part of the answer. If you can make a start on your working and show how you might use a maths skill like geometry, or measurement, or graphs (depending on the achievement standard), then you may find that you’ve given them enough information to gain some marks – even if you aren’t able to follow the problem all the way to a solution.

Step one is almost always to state the problem clearly using numbers, and then choosing a maths strategy (like a graph, or an algebraic equation) that will help solve that problem.

And you might even find that once you’ve made a start by clearly writing out the problem, that the next step becomes clear as well.

Use the space given in the answer paper as a guide.

If you’re not sure how much working is expected for each question, the space given on the answer paper can give you a rough indication.

If the answer booklet gives you only a few lines, then you can be fairly sure that the person who set the exam is not expecting a great deal of working. On the other hand, if you are given a page or more, then you should be thinking that the answer will involve several steps, or perhaps need something like a graph or a diagram which will take up plenty of space.

Be aware that this is only an indication. You are not required to use the whole space given, and there is always the (very slight) chance that a mistake has been made, and that the space left for a question might be all wrong.

Tip 2: Understand your errors and how questions are marked

Make good use of the tests and examples given to you by your teacher throughout the year. When you get a question wrong, make sure you take the time to really understand where you made your mistake.

When your test and exam papers are returned, go through them and work out why you lost marks. Remember that marks are given for showing your working, and choosing the correct strategy, as well as for the final answer.

NCEA Maths exemplars and study guides can be a great help if you aren’t sure what is needed to achieve a certain grade. A good study guide will give worked examples and show you the type of marking schedule that will be used during the assessment.

Your teacher may give you access to NCEA Maths exemplars, but you can also do an internet search using terms such as ‘NCEA Level 2 maths calculus exemplars’, or ‘NCEA Level 1 maths external exemplars’.

Tip 3: Seek help from your teacher & engage in the classroom

It might seem obvious, but it’s amazing how many students don’t seem to know this tip: your maths teacher is an excellent resource for achieving high grades in NCEA maths.

In most New Zealand schools, the educational approach is one of interaction, discussion, and being responsive to learner’s questions. This means that to get the most out of your time in the classroom you need to be asking questions, listening to the questions that other students are asking, and engaging in classroom discussions. Remember, though, to balance this out with listening to the teacher when they are explaining something and ensuring that the questions you do ask are relevant to the subject.

By asking questions you do two things. The first and simplest thing is that you get your question answered, and have learned something new. The second thing is that you show the teacher that you are a student who is interested in improving and understanding maths.

Why is it important for your teacher to know that? Because if the teacher knows that you are interested in improving, then they will give you extra help. They may notice if there are questions you’re finding particularly difficult and offer you strategies. And if you need to talk to them outside of class about your coursework, they will be better placed to advise you if they already have some idea of how you think and what your maths strengths and weaknesses are.

Most teachers will have a system set up where students can contact them out of class to ask further questions, or really seek help with a problem area. Find out from your teacher, or your school office, what your school’s policy is about this and go ahead and ask those questions.

Your teacher is just as committed to you doing well at NCEA maths as you are.

Tip 4: Revise your weakest areas … or not?

There are two schools of thought on this. One the one hand, if you have some NCEA maths subjects that you know well and are confident in, then yes, you should spend some good amount of time revising the parts of the curriculum that you don’t know so well – after all, it’s pointless spending time on the subjects that you will already receive an Excellence in.

On the other hand, if NCEA maths is a subject that you struggle with as a whole, and there really aren’t any areas that you feel confident in, you should choose your strongest areas and really focus on improving them to a point where you know you will pass the achievement standard.

No matter whether the subject is your weakest or not, the process of revision is the same:

• Begin with the basic building blocks for the subject and really understand them. Don’t overwhelm yourself with the really complicated questions just yet. Take it right from the top and get the basics sorted in your mind first.
• If it’s starting to feel like NCEA maths is too hard, don’t give up. Persevere. You may have started with a question that is more complicated than it looks, you might need to take it even more slowly than you thought, or it may be that the notes from your teacher don’t explain it in a way that works for your brain. Try using a study guide, watching a tutorial video, asking a friend, or maybe even hiring a tutor.

If you’re not sure which achievement standard to start revising, the NZQA release notes each year on how students performed in NCEA the year before. For example, the 2019 NCEA Level 2 Maths Assessment Report notes that many students have difficulty understanding “rate of change” questions, so you may want to pay special attention to ‘rate of change’ for Level 2 subject revision.

Tip 5: Take clear notes during lessons

Take clear written notes during your classroom lessons. No matter how good your memory is, or how sure you are that you understand a subject and will never forget how to solve the problems, good notes are an essential revision tool.

Make sure you pay attention to any notes or formula sheets that the teacher gives you, read through them, and make clear notes alongside them to explain anything that isn’t immediately clear.

If your teacher writes notes or examples up on the board or projector at the front of the class, make sure to copy them down, taking care to be accurate.

If you are writing your notes up on a computer, make sure you label the file something useful and descriptive, such as ‘NCEA Level 3 Maths notes’ so that you can find your notes later in the year when you want to revise.

Tip 6: Understand how problems relate to real-world situations

Many problems in NCEA maths will relate to the real world. Whether it’s to do with measurement, statistics, graphs, or something else the problems are often put into a real-world context.

If you can understand how the maths is being used in these situations to describe an event or solve a problem, then that can help you solve the problem. It can also help you do a ‘reality check’ to see if your answer makes sense, given what you know about the world around us. If you calculate that a 20cm tall coffee cup can hold 300Litres, for instance, then you know you have made an error!

At higher levels, the questions may contain additional information, which is not actually needed to solve the problem. Again, this is just like real life where, as a problem-solver, you need to decide what information is relevant and what is not. Just because you are told that there are five coffee cups on a bench, and three more on a shelf, doesn’t mean that you will use that information to find out how much coffee will fit in one mug, for instance.

Understanding the real-world context of a question can help you solve it better.

“By necessity, the statistical information referred to in the problems needs to be a real-world context. However, this continues to be a stumbling block for some candidates.”

2019 Examiner Assessment Report, NCEA Level 1 Mathematics and Statistics

Tip 7: Become familiar with your graphing calculator

“Candidates who have a graphics calculator and good calculator skills are greatly advantaged in this standard.”

2019 Examiner Assessment Report, NCEA Level 2 Mathematics and Statistics

Get to know your calculator. Your calculator is there to make your life easier, not harder, so make sure you know how to use it well.

The first thing to check is that your calculator is one of the approved ones for your course. Check the make and model against the NZQA Approved Calculator List. Other calculators may well do the same job, but only approved calculators will be allowed during an external exam assessment. You don’t want to arrive at the exam and have your calculator confiscated!

The second thing to do is really get to know the functions on your calculator. Read over the manual or find a tutorial video online showing how to use it to solve the types of problems in your course.

And of course, the third thing to remember when using a calculator is that you will still need to show your working and your understanding of the problem, by writing it all down in the exam paper. Remember, you can’t get marks for things that the examiner doesn’t see, and they sure can’t see your calculator screen!

“Candidates will be expected to demonstrate an understanding of the mathematical concepts, rather than directly transferring results from a graphing calculator.”

Assessment specifications for Level 1 Mathematics - NZQA

Tip 8: Draw diagrams to illustrate problems

Don’t be afraid to draw on your question or answer booklet in an external exam. If you find it easier to think about a problem by drawing out a table or diagram of the problem, then go ahead and do it. And if the question has a diagram in it, drawing extra lines or markings on that diagram to think the problem through can help you to solve it.

The exam paper is not a textbook, and the examiners won’t be reusing them, so if graphing the equation out helps you, then go ahead. Just be sure that your drawing doesn’t make it hard for the examiner to see your working and your answer. If the diagram is a part of the answer, then make sure it is clearly labelled so it can be understood as part of your working and be eligible for marks.

“Candidates need to be aware that it is permissible to draw extra lines onto the diagrams, which may help them solve the problem.”

2019 Examiner Assessment Report, NCEA Level 1 Mathematics and Statistics

Tip 9: Complete maths homework before school, or early in the afternoon, rather than late at night

The idea here is to use your brain when it is fresh and rest it when it is tired. Your brain uses energy and oxygen as it thinks and doing maths problems and equations is just easier if your brain is well-rested.

Experiment with different times of the day for doing your maths homework. It might be that getting up a bit earlier to do it in the morning works for you. Or maybe right after school before dinner is better. Or the early evening.

The key thing is not to leave your hard brainwork, like calculus and statistics, until the last thing at night. By late night your brain and body are either tired and ready for sleep, or they’re starting to get overtired and wired. Either way, it’s not a good time to be doing maths.

“I have tried many, many times to do Calculus or Physics late at night, after 12 or 1 am, but you are just doing yourself a disservice. I have stared at problems for hours because I just could not sleep until I knew how to solve it…then I finally fell asleep out of extreme fatigue…but when I woke up it just seemed so simple how to proceed with the problem.”

Jason Gibson, Top 10 Strategies to Improve Your Math Grades

Tip 10: Practice past NCEA maths exam papers

Completing official examination papers from previous years is one of the best ways to test how you are performing in a subject. NZQA publishes old exam papers on their website, which are available from the Mathematics and Statistics resources page.

By completing past papers you will become familiar with the style of questions that will appear in the external exams, and how the paper is organised. Then, when you arrive at your exam, you will have a good idea of what to expect when you open the exam paper in front of you. Exams can be stressful enough without adding unnecessary surprises to them. Take special note of instructions that say that you only need to answer SOME of the questions, and make sure you understand which questions they mean.

Another good reason to practice questions from exam papers ahead of time is to get a feel for the level of difficulty of the exam questions. This way you’ll know whether you are on track with your revision, or if you need to a bit more work before-hand. Some people find it stressful to look at old exam papers and are shocked at how hard the questions look, but if you are going to be shocked, it is far better to get that out of the way in the comfort of your own home, instead of waiting for the exam to start.

Try to spread out your use of past exam papers, as there are only a few available. Leave most of them for the week or two before the exam, giving yourself enough time to do more revision if needed, without using them all up before you’re really ready to answer the questions. It is a great idea to do at least one past exam paper within the time-frame of your exam. That way you can practice the questions and answers, and get a feel for how quickly you’ll need to work. Remember to try and leave some time in the exam to go back and revise, if you can.

Tip 11: Do the easy questions first in tests and exams

“Some candidates appeared to assume that they needed only to answer what they perceived as the Excellence level questions. Occasionally very good candidates were greatly disadvantaged by this strategy because of simple errors.”

2019 Examiner Assessment Report, NCEA Level 2 Mathematics and Statistics

When you open your exam paper, the first thing to do is have a look through the whole thing to see what questions you are facing and decide which questions to start with. Take note of any instructions that say you can choose between different questions, and make sure you are clear in your mind about which questions they mean.

Then, start with the easy questions first.

By completing the easier questions first, you gain all the marks you can from them in the least amount of time, leaving yourself the remainder of the time to really concentrate on the harder questions.

If you do the hardest questions first, you run the risk of spending so much time on them that you run out of time altogether, and never circle back to get those easy marks.

Once you’ve completed the easier questions, then begin to answer the more complex ones. Think about how much time you have left in the exam and portion it out for each question.

For example, imagine you have an hour left in the exam and three more questions to complete. If you just start working on a question without thinking about the time, you could spend the whole hour on that one problem, miss the marks from the other two, and also run out of time to revise your earlier work. A better plan is to decide to spend 12-15 minutes on a problem, and then pause to see how you’re going. If you’re nearly complete, then finish up that problem and do another one. But if you still have no idea whether you’re on the right track, leave that problem and try a different one for 12 to 15 minutes, and so on.

Do attempt every question and remember that you can gain marks for your working, even if you don’t finish answering the problem.

Tip 12: Spend spare time checking your answers in tests and exams (don’t finish early)

It is a good idea to plan your exam time so that you have some time set aside to revise and check your answers.

“When an impossible or unlikely solution is obtained, this should be seen as a message to recheck prior working and attempt to ascertain where this careless mistake may have occurred.”

2019 Examiner Assessment Report, NCEA Level 2 Mathematics and Statistics

The first thing to do is go back through the paper and make sure that you haven’t skipped a question. Double-check any instructions about being able to choose between questions and make sure that you have followed them. Carefully turn each page of your paper to check that you haven’t accidentally turned two pages at once and missed a whole section (it happens to someone every year – don’t let it be you).

Then go through each question and revise your answers. There are several different strategies for revision that you can use:

• Does the answer make sense, given the real-world context of the question? For instance, have you said that an eagle weighs three thousand kilograms instead of three thousand grams?
• Make sure your answer has got the relevant units with it, if appropriate. For example, 7cm, instead of just 7.
• Make sure any graphs, tables, and diagrams are appropriately titled and labelled.
• Redo the question to see if you get the same result.
• Work backwards from the answer and see if you get the right starting parameters.
• If someone asked you to prove that the answer was correct, could you do that mathematically?
• Do you get the same answer if you try a different strategy?

Tip 13: Print and complete maths worksheets

Your classroom teacher or tutor will have a supply of resource sheets that you can use for revision. Working through these will give you extra practice with the types of questions that you will meet in your maths assessments. Using maths worksheets means that you can save your past NCEA exams for later in your revision cycle, and have a way to really practice an area by doing a lot of problems over and over.

You can download some additional worksheets and booklets for free at the NZ Centre of Mathematics website MathsCentre. Take note that not all years and subjects are available.

Study Guides and Workbooks are available to buy from LearnWell. You can choose from individual subject guides that cover only one maths standard, or books that cover the whole year’s work.

• NCEA Level 1 Mathematics Workbooks from LearnWell
• NCEA Level 2 Mathematics Workbooks from LearnWell
• NCEA Level 3 Mathematics and Statistics Workbooks from LearnWell
• LearnWell Digital Maths resources

Tip 14: Use high-quality, up-to-date textbooks

A good textbook or study guide is an essential tool for studying NCEA Mathematics and Statistics. Your classroom teacher or school will likely have provided you with one at the start of your course, but sometimes it can be helpful to look at a different textbook to see if the style of the notes and explanations suits you better.

When looking to buy or borrow an NCEA maths textbook, try to get one that is up to date. The current standard classroom texts are Gamma Mathematics by David Barton for NCEA Level 1, Theta Mathematics by David Barton for NCEA Level 2, and Delta Mathematics by David Barton and Anna Cox, although your class teacher may have a good alternative. Older textbooks may not cover all the material that you need to know and may have incorrect information about the level requirements or exam structures.

Textbooks are not designed to be read cover-to-cover. When using a textbook, make use of the contents and index pages to find the right information so you don’t get bogged down in information that is not relevant to your study. The textbook is likely to have the subjects and problems in an order that is different from the order that your class teacher will be covering the curriculum. Follow your teacher’s guidance about what is important to know.

Other textbooks and study guides that are worth having a look at are those available from LearnWell, D&D Resources, and NuLake.

• LearnWell print resources were formerly published under ESA Publications — a long-established educational publisher in New Zealand. With Study Guides and Workbooks for primary and secondary schools covering the New Zealand Curriculum and NCEA, they are user-friendly. Find Maths resources here.

Tip 15: Make use of online resources

Many NCEA teachers and schools upload their NCEA notes to the internet. Some are just for internal use for their own students to access, but some make them available for anyone to find and use. If your teacher makes their notes and worked examples available online, do make good use of that resource. If they don’t, then you can search online and find what is out there.

Some tutoring and revision websites also publish free NCEA maths resources or tutorials for students. Some are NCEA tutorial videos, some are written texts, some are textbook ebooks, there is a wide variety of resources and help available. Some examples include:

• StudyTime: “StudyTime is an online platform dedicated to helping NZ kids make the most of high school.” The site contains checklists, tutorial videos, strategy guides, online tutoring options, and more.
• Maths Centre: “Welcome to the NZ Centre of Mathematics, the access point for quality mathematical videos, textbooks and maths games. Choose from many of the free and printable maths worksheets and homework sheets. Each topic can be found in the New Zealand Mathematics Curriculum and the work complements the New Zealand numeracy programme. All of the mathematics on the site is free of charge to download or watch.”
• NCEA on TKI: TKI is the online learning basket for the New Zealand Curriculum. The NCEA portion of TKI is mostly targeted at teachers, but students and whanau can also find some useful articles and information there.

Tip 16: Hire a maths tutor

A maths tutor is a great idea for students who like to learn by talking and discussion, or who need to see problems worked out for them a few times before giving it a go.

You don’t need to be struggling with maths to consider using a tutor. A tutor can also help students who are good at a subject extend themselves to achieve an Excellence, or for students who are wanting to sit the Scholarship exams.

There are different types of tutoring available:

Private in-person tutor

This is a person who lives in your area whom you meet on a regular basis to help you with your maths and stats work. Private tutors are often university students or trained teachers who prefer flexible work hours and may advertise their services through a local paper or noticeboard, or through your school. Your town may also have a local tutoring company that provides individual or small-group tutoring classes.

The upside of a private in-person tutor is that they can tailor the work and the description to your exact learning-style and needs. On the downside, they can be expensive, and many families may not be able to afford as many sessions as the student really needs.

Online tutor

Online tutors offer one-to-one tutoring through online chats such as Skype or Zoom. They can be just as personalised as an in-person tutor and offer many of the same advantages. A big plus for online tutors is that they are available for students everywhere in the country, so are especially great for families living in the regions, or in smaller towns. They cut down on travel time and mean you can choose from a wider range of tutors, rather than just the people who live near you. There are several providers of online tutoring, such as StudyTime, Inspired Education, and MyTuition.

Tip 17: Use a digital platform like LearnWell Digital

Digital platforms such as LearnWell Digital make learning mathematics more rewarding, allowing you to learn and revise NCEA maths online. It is an online learning management system (LMS) that provides an easy-to-use way for teachers and schools to create content pages, quizzes, assessments and more. Not only that, but the LearnWell team has a range of resources available, which teachers can use as they are or customise for their students’ needs.

Teachers can track their students’ progress through the program, gain insights into how they are responding to the topic and identify any areas where further explanation or in-class discussion might be needed. By having those quality NCEA resources right there at their fingertips means that teachers can get on with actually teaching the kids, instead of spending hours searching for good material.

By having a combined in-class and online programme, schools are better able to provide a flexible work environment. They can keep each child in the class supplied with suitable content at exactly their level, set work that can be done both in the class and from home, and provide an online discussion environment so students and teachers can discuss questions and answers together both in and out of the classroom.

If you would like to find out more about how LearnWell can help you, please contact us.