Everybody makes mistakes from time to time. It’s one of the ways we learn. It’s also something that distinguishes us from computers. When I was a very junior programmer, one of the team leaders in the office said to me, “The machine is doing exactly what you told it to do, lad. It’s just not doing what you thought you told it to do.” I repeat, everyone makes mistakes…
This week’s post is about ways of checking your work. Some of them give you an absolute guarantee that you are right. Some of them merely show that an error exists. They’re worth doing, either way.
Use these when you have to work it out by hand, or as checks against typos when using a calculator.
Last digit checks can highlight errors in multiplication. The most basic comes from the simple fact that you only get an odd answer when you multiply odd numbers. However big the multiplication, a single even number makes the answer even. One factor of two is all it takes… (sounds like an advertising slogan). So if you multiply a string of odd numbers and get an even answer (or vice versa), something has gone wrong.
A more sophisticated last digit multiplication check is the units make the units. For example: 27346255346277347 x 38482942904939 must have an answer ending in a 3. This is because 9 x 7 = 63 and although the 60 gets mixed in with the other tens, nothing else can affect the 3 in the units column. Try it with smaller numbers and you will see what I mean.
Ballpark estimates are always a good idea. See if your answer seems ‘about right’ by rounding to simple whole numbers. In particular, take π as 3, and gravity as 10 ms-2. In a recent lesson we looked at a question involving a stone being thrown vertically downwards from the top of a tower with initial speed 4ms-1, and taking 2 seconds to reach the ground. The problem was to find the height of the tower.
This is easy enough using s = ut + ½at2 = (4 x 2) + (½ x 9.8 x 22 ) = 27.6 m
But if you miss out the decimal point in gravity (and use 98) you get 276m as the answer. This is a tall tower, but not impossibly so.
The ballpark check 4×2 = 8, plus ½ x 10 x 4 = 20, giving 28 will highlight the error faster than you can write it down (or the stone takes to fall).
Counting minuses in multiplication (or division) is a simple way of making sure you haven’t got the sign wrong. If you have an even number of minuses, the answer is positive. If you have an odd number of minuses, the answer is negative. Easy!
Counting decimal places is worthwhile whenever you have to multiply decimals. The rule is that you always end up with the same number of d.p. that you start with. This may include trailing zeros. Here are some examples:
0.5 x 0.5 = 0.25 2 dp on either side
0.24 x 0.45 = 0.1080 4 dp on either side. (Calculators just show 0.108)
0.02 x 0.0004 = 0.000008 5 dp on either side
In the last example, counting the d.p. on the left hand side is a quick way of determining that we need five leading zeros to bring the total d.p. to 6.
Comparing answers. If you are working from a text book and checking your answers against the ones in the back, you sometimes get a situation where the forms of the answers differs.
For example you might have 0.805612004, and the book has 3e-5 + π/4.
A quick way of checking is to type in the book version, followed by divide and ANS (the button that recalls the answer you just had). If you are right, the new answer will be 1. You could make the same test by subtracting (answer = 0 if correct) but I always use division. No particular reason.
Simplifying algebra From time to time you may have to ‘simplify’ horrible pieces of algebra. For example:
To check this, pick a couple of easy values for x and y (Don’t choose anything that will lead you to divide by zero!). So with x = 1 and y = 2, we have:
It only takes a few seconds to type this into your calculator using the fractions button. You could even do it by comparing answers by entering:
This gives an answer of zero straight away.
Checking solutions to equations is easy with a calculator if you use the memory to hold variables. (I am assuming you have a typical school scientific calculator such as a Casio fx-83GT).
Step 1 Assign the value you want to check to the variable X by entering the value then pressing shift, RCL, ). For example:
7 Shift RCL ) sets X to 7.
You don’t need to press =. The display shows 7→X at the top. Any previous X value is overwritten.
Step 2 Enter your expression, using the Alpha button whenever you want X.
Alpha ) x2 – 2 Alpha ) + 3 gives you x2 + 2x + 3
Step 3 Now press =. The calculator works out 72 – (2×7) + 3 to give 38.
Step 4 Now press AC, then enter the next value you want to test. For example:
5 Shift RCL ) sets X to 5.
Step 5 Press the up arrow. (This scrolls you back through previous actions). The expression x2 + 2x + 3 will reappear. Now press = again, to get the answer with X = 5.
You can do this as many times as you want, and if you are dealing with awkward decimals or surds it is quicker than typing them into the expression directly. It’s probably overkill at GCSE but can be a great help at A level where you often have equations with multiple solutions. (Tip If you aren’t sure of the current value of X, press Alpha ) = to display it.)
These are just some useful checking tricks. Maybe you have your own favourites you would like to share.