Advice on Calculators

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:

1. Determine what operation you want to key into the calculator

2. Estimate what you think the answer should be

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.

Is that your final answer?

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.

ABC’s

Your textbook is divided into sections and you normally cover one section per class (sometimes the teacher may decide to spend longer on sections which are more difficult or more important). The section usually has some explanation and examples at the front followed by a summary of the concepts and formulas, followed by the actual questions.

The questions are divided into Part A, Part B and Part C.
(not all textbooks actually use the letters, but there is usually a still a heading to separate the groups)

Part A questions are very straightforward. The answers usually only take a few lines and if you get stuck, there are usually examples earlier in that section and you can follow the steps exactly.

Part B questions are a bit more difficult. Usually you have to start the question by doing something you recently learned (from within the last few sections) after which, the question will start to look an awful lot like a Part A question. Most test questions tend to be similar to questions in this section.

Part C questions are extremely difficult. They require you to take the information from this section and apply it in creative ways not described in the examples, or tie it in with earlier course material that seems unrelated. You are usually not assigned many of these questions for homework. They are not particularly important because they rarely show up on tests and if they do, there is usually only one question.

Just a little tidbit to help you along studying.

learn to think like a math geek

Tutors are very often asked, “What use does this have in the real world?” And unless you’re specifically interested in doing something in engineering, math or the sciences, the answer is probably “not much.” However while the math itself may not have much use, a new book demonstrates – quite entertainingly – how learning to think like a math geek can benefit you.

If you want to know how you can turn thinking like a math geek into making millions of dollars, you’ll really enjoy The House Advantage by Jeffrey Ma. Ma is the card counter from MIT who was the inspiration for the book Bringing Down the House and later the movie 21 both dramatizations of how Ma used math and statistics to gain an advantage playing blackjack in major casinos and winning millions of dollars over a period of several years.

In the book, Ma explains how his training (thinking in this particular numbers-based way) helps him to make better decisions in life and also to advise others, in particular major sporting organizations in the NFL and NBA. It even has terrific chapter called “People who hate math and what to do with them.”

It’s incredibly easy to ready: there are less than 5 actual calculations in the entire book and there’s very little in the way of math jargon. It’s just full of some very easy to understand and genuinely helpful advice for life and business.

Maybe some day soon, you’ll be using math to make millions!

Warm Up

In our last article we argued that math is like a sport. (We’re on the internet, bullies can’t come after us for saying stuff like that here.) We want to take the analogy one step further and talk about something important for sports and math: warming up.

It’s important for any athlete to warm up before they go out on the field. It improves performance and reduces the risk of injury. The idea is that the human body needs to get into activity gradually. Trying to rush it isn’t a good idea. And your brain is the same way.

This is what a good warm-up looks like for doing math homework:

1. Grab your gear
   Take out everything you need to do your homework. This includes pencil, eraser, notebook, textbook, calculator and anything special you might need. Taking breaks in the middle to go hunting for extra paper or trying to find your calculator break your concentration.
2. Understand the assignment
   Make sure you know what’s expected of you. Are there special requirements on the questions like only do odd numbers, or skip certain questions because you have not been taught how to solve them yet.
3. Collect your notes
   On a separate piece of paper, you need a list of all the formulas or important facts you need for the questions you are about to do. (This can be a sheet from your notebook, but it should be separate so that everything can be laid out on the table at once without needing to flip pages.)
4. Start at the beginning
   Do the easier questions first. This gets you accustomed to the kind of problems from that section and gets you thinking along the right lines. Questions also tend to build on one another gradually.

 

Try it, it works!

Math is a Sport

As a subject, math has more in common with gym than it does with history or geography. First we’ll explain why, then how that knowledge can help you study more effectively.

So the things that all sports have in common are:

1. They take practice.
2. The more you do them- and the more often you do them – the better you get
3. You get rusty if you don’t play for a while
4. To perform your best, you need to be in good shape
5. Proper equipment helps
6. You perform better if you warm-up
7. It’s easier to learn with proper instruction

Now this leads to several tips for better performance:

1. It Takes Practice
   Don’t think that you can get by just by listening and not doing assignments
2. The more you do it the better you get
   If something isn’t easy, do a few extra problems on top of those suggested for homework
3. You get rusty if you don’t play for a while
   Keep good notes and don’t throw them out at the end of the year. You might need to look something up later.
4. To perform your best, you need to be in good shape
   No one learns well when they’re tired or haven’t eaten properly (and energy drinks are no substitute for eating and sleeping properly)
5. Proper equipment helps
   We’ve done this one to death already.
6. You perform better if you warm up
   Math books are designed with the easier problems at the beginning and the harder ones at the end. By doing easier questions first you get your brain accustomed to thinking in the right way and you’ll think of solutions you would have missed otherwise.
7. It’s easier with proper instruction
   Follow the advice of your teacher. When they say to do something a certain way, there is a reason. Sometimes the reason doesn’t become clear until the next section, chapter or even grade but it’s there.

It would be nice to end this section with a good sports metaphor, but unfortunately, we’re all math geeks and don’t know any. Sorry.

Pictures are your friend

The one single piece of advice we find ourselves giving to students more than any other is:

Draw a picture!

Our brains are designed to process images much faster than words or lines of symbols – that’s why they say “a picture is worth a thousand words.” Many, many problems become easier to solve by simply drawing a picture or two to illustrate the situation.

It doesn’t have to be perfectly drawn and labeled (although if the question says “sketch” then it needs to be) but it will give you an idea of what is going on and help you keep track of what all the different variables represent.

Pictures also help you translate the abstract concepts in a word problem into math concepts you are familiar with. For example if you are given an equation that describes the motion of an object (jumping off a building, kicking a football) and you need to answer a specific question (when it lands, what is its maximum height) then sketching the graph will help guide you to the solution and determine what you are looking for. Do you need to find a y-intercept? An x-intercept? The vertex? Or something else entirely?

If a problem is giving you trouble, remember: Pictures are your friend!

Speak the Language

In math class you have to (very general) goals. It’s only by working towards both of these goals that you can get good grades in math.

One goal is obvious:

You need to learn math. (duh)

Not many people are aware of the second one:

To prove to the teacher that you’ve learned math.

This isn’t always as easy as it looks. It has to do with communication, which is a skill in itself.

Handwriting – We’ve said before your marks will go up if you improve your handwriting. Studies have shown that if you give a teacher two identical answers and one is written neatly and the other is messy, the neat one tends to get a higher score.

Explain yourself – There’s a piece of advice given to actors which will help you. They are told that to make it easier to for the audience to understand them when they’re on stage for every action they should: 1. say what they are going to do, 2. do it, 3. say what you have done. It seems like overkill, but it works very well. Here is what it looks like for a math problem:

Step 1: Say what you are going to do – Add in a line where you announce what you intend to do. For example, if you are solving by elimination (Grade 10), you can announce “Subtract 2 from 1“. So the teacher knows where you’re going even before you’ve started.

Step 2: Do it – This is the usual answer where you show your work.

Step 3: Say what you have done – this can be a concluding sentence in a word problem, or in a simpler calculation may just mean underlining or putting a box around your final answer.

We know your first reaction is going to be “What do I need to do all of that extra work for?” And the answer is “because it makes the teacher’s life easier.” And things that make the teacher’s life easier are usually worthwhile for you. The other advantage it gives is it makes it easier for someone to help you find and fix your mistakes if you make any.