Safety Nets Part 3: Trig

Trigonometry can be very hard to wrap your brain around. Nobody (at least nobody in high school) can give you a really satisfying explanation of what’s going on inside of a function like sin, cos or tan, so we tend to just push some buttons on a calculator and hope that it all works out ok.

But there is a secret few people know is that you can check your answers, if not exactly.

The secret comes from the fact that it’s very difficult to make small mistakes doing trig. If you choose the wrong operation (multiply when you should divide or use sin when you should have used cos) it usually gives an answer that is way off.

The way to catch these mistakes is to make a quick sketch. It doesn’t have to be a perfectly labeled diagram, but you should use a ruler and make it as accurate as you can just by eye-balling it. Then compare the answer you get to your diagram and ask yourself, Does it seem reasonable? If your answer doesn’t fit the picture very well, chances are you’ve made a careless mistake somewhere.

While it doesn’t guarantee that you’ve got the right answer, this technique will catch about three quarters of careless trig mistakes before it’s too late.

Safety Nets Part 2: Equations

Equations (one equation or a system of 2 or more) have a built in safety net and you can always know in advance whether or not your answer is right. The steps are simple:

Take your answer and return to the original question. Substitute your answer for the unknown variable and work out both sides of the equation separately and see if what you get on each side is actually equal.

You may not have full marks (you may have lost some for form or skipping steps) but at least you’re certain you’ve got a correct answer and can hand in your test with that much more peace of mind.

Note: Many homework or test questions will actually say “Check your answer” and this is exactly what they mean. Don’t skip this step as you’re not only giving up your safety net, you’re passing up free marks that you already earned by figuring out the answer.

Safety Nets

Wouldn’t it marvelous, if you were writing a test, and you knew before you handed it in whether your answers were right or not?

While it’s not always possible to know, many questions have safety nets built into them that allow you to check your answer on the spot so you can move on to the next question with confidence (or go back and fix a mistake while there’s still time.)

Over the next few weeks, we’ll share some of our favourites ones so you can start using them to check your work.

Years ago one of us found an absolutely fabulous piece of advance in a math textbook about the proper way to use calculators. While we all agree the advice is fantastic and not to be underestimated, we’ve never been able to find the reference, so we apologize for not being able to give credit where it’s due.

Here then is their miracle formula for how to use a calculator:

3. Pick up the calculator and do the calculation

Step one is often overlooked. Before picking up the calculator you should know what you intent to key in. The order you punch in keys is very important and it takes only one slip up to ruin the entire answer.

Step two is your safety check. By knowing more or less what you should get, you’ll know right away if you get an answer that is obviously wrong (and therefore you made a mistake punching in the operation or somewhere earlier in the question.)

Step three is… well duh!

It’s important to remember that calculators are mindless machines and have no way of knowing whether you typed in the wrong numbers. By taking a breath to think about what you will do and what the answer will look like, you can avoid many careless mistakes.

We’ve watched many students loose easy marks on quizzes, tests and exams because they did not express their final answer properly. It’s particularly upsetting because most students know how to fix the problem but don’t as a result of laziness, rushing and bad habits.

How do I write my final answer?

The question itself tells you the format that your answer needs to take. If the questions are instructions (Solve the equation…, determine the height…) or are just numbers by themselves (3 x 7 = …) then all that is required is the answer.

If the question is in a full sentence (What is the height of the basketball after six seconds?) then your answer must also be in a full sentence (The height of the basketball is 4.2 m.)

Rounding and units

Forgetting to round or add units are the silliest reasons to loose marks, but it’s very easy to forget. Any question that involves either measurements (length, area, volume, distance, angles, etc) needs to have units.

Secondary Questions

Some questions require an extra step. For example you need to solve the question in order to answer a yes/no question (Will he arrive on time for his flight?) You’d be amazed how many students do all the math perfectly and forget the sentence at the end (No, he will arrive 10 minutes too late.)

Some times you are asked to make a decision (Which cell phone plan would Tyler prefer?) So you need to calculate the cost of each plan, but still need to look at the two answers and make a statement at the end: (Tyler should choose the Selus plan because it is cheaper.)

Occasionally the question will need an extra very simple step. (If Shane has a \$50 gift card, how much money will he have left over) which requires you to do all the work, and then at the very end throw in the extra baby step of subtracting your answer from \$50 to get the real final answer.

Look back

These mistakes seem stupid which means when they happen to you, they sting quite badly. But there is one sure-fire way to make sure they don’t happen to you.

At the end of each question, before going on to the next, quickly look back and re-read the question to make sure that you’ve answered it properly. That 3 seconds spent checking will save you a lot of frustration in the future.

Prove

Earlier this month, we talked about questions that begin with “Show….“. There is a more difficult variation and that is questions that begin with “Prove…”.

One of the simplest examples might be to “Prove that the sum of two odd numbers is an even number.” This seems like a simple question. It seems obvious that it must be true, and by taking some random odd numbers and adding them together, it seems to always work. But in order to actually prove it, you have to demonstrate that there are absolutely no exceptions. You can’t check all of the possible combinations of two odd numbers because there is an infinite number.

There are a few different strategies, and it usually involves trying to address all of the possible cases (e.g. 1 + 3, 27 + 5, etc) at the same time. The answer winds up being some mixture of math question and short essay question. There is no guaranteed formula to get them to work, and it takes a lot of practice (and we mean a lot) to get good at them. Most universities will dedicate a huge chunk of a first year course to learning how to properly prove things.

The good news is that they hardly ever turn up in high school. If you do see one, pester the teacher to go over it in class so you can see what a proper answer looks like. Failing that, ask a tutor to walk you through the though process.

Show

“Show” questions can be tricky because it can be difficult to figure out what they’re asking you to do.

For starters, a “show” question starts with the word “show”. A common example is “Show that x = 3 is a solution to 5x + 4 = 19.”

The wording throws you off because most problems are about finding the answer. This type of question says what the answer is right up front. This means you are being asked to demonstrate that their answer (in this case that x = 3) is correct.

The basic strategy is always the same. Begin by thinking about how you would answer the question if there were no answer given. In our example you would do some basic algebra, bringing the four over to the other side, followed by the five to wind up with x = 3. Here you are just answering the question normally, with the added safety feature that you know whether or not your answer is correct.

Another option is to look for a shortcut knowing what the answer is. In our example if we replace x with 3, we can work out the left hand side of the equation is 19 and since both sides are the same we know we have a correct answer.

This is a very simple example, but “show” questions appear regularly throughout high school and you should be on the lookout.

Note: There are questions which begin with “Prove” that start to show up in higher grades. While they are the same in principle, they tend to be more difficult and demand longer, more well thought out answers.