Calculator Etiquette and Other Tips
By Evan Ma
(Math and Physics tutor at The Edge Learning Center)
Yes, this is the season of examinations. For those who are writing IB Mathematics Paper 2 (HL and SL) and Math Studies SL, where calculators are allowed and indispensable to getting the correct answers, knowing how to use your graphical display calculator(s) (GDC) is indeed essential to scoring high marks.
However, throughout my interaction with students, sometimes I watch with horror how careless mistakes are committed when using calculators, and the answer is neither evaluated for reasonableness nor steps checked for consistency. In short, sometimes students are not aware that even if the mathematics on paper is correct, the calculations are wrong because they fail to use their calculators correctly. There are a lot of GDCs in the market that offer the user the ability to enter a formula exactly as they write it on paper. Unless one is really skilled, it usually takes more time to enter calculations on such types of GDC than one will on a calculator that takes “linear” commands. For any type of GDCs that you may use, it is important to pay attention to the following rules of thumb.
1. Most calculators are built in with the rules of precedence that you have learnt since you started learning mathematics. For example, if you enter, the sequence of calculations are square root, multiplication, and then addition. If you intend to perform the addition first, you need to use brackets to change the precedence, which means .
2. Brackets are your friends. For example, many times when students entered the quadratic formula to solve for roots to a quadratic formula, brackets should be used appropriately. For example, when solving for the larger root of , one should enter
to calculate the positive root. Notice that there is a pair of brackets enclosing the calculation for the discriminant, without which the square-root will only be applied to the number 49, thus giving a wrong answer. Remember, the calculator does not know you want to apply the quadratic formula when given linear commands. It just does what you enter according to pre-set rules of precedence.
3. Notice in the previous example that the discriminant calculation was not entered as
as many students would have. The rule is therefore, whenever possible, to simplify the expression first in your mind. This will make your commands less prone to errors. I have seen a lot of students enter (-1)^4. This is 1 and should not require any input into a calculator.
4. A lot of students have also forgotten about the distinction between and , where > 0 and is an integer. The former is therefore always negative, whereas the latter, depending on whether is even or odd, could be positive or negative. Hence, if you want to perform -3.67 raised to the power of 4, please enter (-3.67)^4 , which will give the correct result of approximately 181.4. However, if the brackets are not there, the result will become a negative number, which is wrong.
5. When entering numbers in scientific notation, for example, , some students will enter it as 4.5 x 10^15. There is nothing wrong with it. However, it is better to use the capital “E” sign on the calculator. Why? For example, if I want to calculate , it is good practice to enter , which will give the correct answer of . However, if I were to enter 4.5 x 10^15^2, depending on how your calculator interprets it, it might give you an error message, or the answer , which is obviously wrong as it squares only and then multiplies the result to 4.5. Remember, your calculator does what you tell it to do. It won’t second guess your instruction.
6. When graphing a function, remember to size the window appropriately by setting suitable x and y limits.
7. Do not depend on GSolve too much for solving simple equations. For quadratic equations such as , I have seen students solving it by GSolve to yield only one of the roots, and most of the time it is . The other root is not displayed. For such simple equations, it wouldn’t be too hard to solve it by hand via simple factorisation.
The above is by no means an exhaustive list. There are also caveats when using statistical functions such as normal pdf, cdf, and inverse cdf. Know the distinction between them and especially pay attention to whether it is the right or left tail area that the calculator assumes you are entering when using the inverse cdf function. In short, know your calculator well.
Also, as a general rule, remember to write legibly when answering questions. Remember, the examiner may be very picky or trying very hard to award you as many marks as possible. Don’t make it hard for the examiner. I always tell students, if I were the examiner, the last thing you will want to do is to omit all the intermediate ‘=’ signs in steps, and inserting them where they don’t belong. To me, that shows a lack or disregard of logical flow. This will not stand well with the examiner who has to figure out what the student means.
Good luck to everyone who is writing the 2017 IB Examinations.