This post was motivated by something two of my A level students noticed quite by chance. They both typed the same calculation into their calculators but were presented with different results! I’ll share that particular example later in the post, along with a summary of the polemic it caused on Twitter. In each example below, I suggest you have a look at the calculator screenshots first and figure out what’s happening for yourself before reading what I have written underneath. Let’s get started with a warm-up.
Calculators do as they’re told: -4²=-16
This is a classic and one that I’m sure every teacher highlights to their students at least by the time they are substituting values into a quadratic, if not much earlier. As we all appreciate, the calculator is following the precedence order of operations (aka BODMAS, BIDMAS, PEMDAS, …) and it is thus squaring the 4 before applying the unary minus operation to make it negative.
Yes, -4 squared is 16 but that’s not what we asked the calculator to tell us.
Hopefully the size of the numerals in this one gives a big clue as to what is happening here, but it only occurred to me after seeing a student consistently make the same mistake. The disadvantage of these ‘natural’ displays is that you can type 8½ in two ways, with two different meanings – similar to the ambiguity that can arise in handwritten mathematics, and precisely the reason why I hate mixed fractions!
Typing 8½ using the mixed number button on the calculator creates the value 8.5 but typing an 8 next to ½ entered with the simple fraction button implicitly creates the product 8×½. When these two different values are added, the result is 25/2. (By which I mean (25)/2 and not 2+5/2 or 2×5/2…)
This is the one that I posted on Twitter and that caused such a heated debate – do have a look at some of the responses. This time there is no difference in the expression entered on the calculators, but the results are different!
For some reason, the two calculators are interpreting the input in different ways. The pink one has decided to infer brackets around the surd term whereas the silver hasn’t. Rather than arguing about mathematical correctness, the real issue here (and the most interesting point for a discussion with students) is ambiguity. What was the intention of the user of the calculator? It’s likely the pink calculator interpreted the expression in the way the student intended, but the silver one has strictly followed the order of operations.
Clearly, this ambiguity can be avoided by including explicit parentheses or making (careful) use of the fraction key etc.
Another curiosity brought to light by a student of one of my colleagues this week:
I never considered the idea of taking a negative root of a number but it does make mathematical sense. I am cautious though that teaching my students this too early will stop them aiming for a deep understanding of indices: I think I’d prefer them to analyse this from a position of sound knowledge about how indices operate. When I shared this one on Twitter, I was given another example by @ggerardk:
Next week I will share that with my students and just watch what happens.
Think, Pair, Share
- Do you know of any other unusual pitfalls when using calculators?
- At what point do you share these ideas with students? As an advance warning or as a post-topic discussion point?
- Have a look at the minutephysics video on Order of Operations that @comicalengineer shared in response to my first tweet.