In the world of industry, engineers are constantly balancing the level of accuracy of their calculations with the practicalities and realities of manufacturing and construction constraints. For example, an engineer is unlikely to want a length measurement of 43.2156 mm as it is simply not practical to work to this level of accuracy. We expect our students not to round part way through their calculations. The downside of this is that it sometimes encourages students to go to the opposite extreme of writing all the figures down from their calculator display. When they do this it is quite common to see figures copied incorrectly.
In calculations involving circles we usually expect our students to use the π key on their calculator or at least 3.142 in their work to get an answer that is going to be correct to three significant figures. In the world of industry calculations are often carried out on spread sheets so premature rounding is not such an issue. With all of this in mind it got me thinking about how interesting π is and what a good opportunity it would be to do a PMI (plus/minus/interesting) activity with my students.
PMI can help students see both sides of an argument and think more broadly about an issue. Traditionally people tend to focus on how to back up an opinion that has already been formed. PMI is designed to open up students’ thinking, encourage creativity and give an opportunity for reflection. It is also a useful tool in preparing students for a life in the world of work; differentiating a task by outcome (each PMI can vary considerably); giving students a chance to follow their own mathematical line of enquiry; giving the opportunity to work collaboratively and passing the responsibility of thinking to the students.
Here is an example of a PMI for the question: What if we use 3 to estimate π in circle calculations?
|Calculations involving π will be a lot easier to estimate quickly.||There will be a significant loss of accuracy in calculations.||What effect could inaccurate calculations have in industry?|
|A calculator will be less important.||Estimations of circumference or area will always be too low which is a problem when using them in practical situations e.g. working out the amount of fencing required to go around a circular field.||What is the percentage error in calculations depending on the accuracy you use for π and the size of the circle?|
|What accuracy for π is used in the real world and who uses it?|
If you want to use PMI in your classroom, you could give the students just the top line of this table, leaving the other rows blank, and ask them to work alone or in groups to fill in two or more rows or cells. Afterwards they could be given the homework of following up one of their lines of enquiry from the interesting column. I love doing these activities because I am always astounded by the level of creativity, the volume of work produced and the difficulty level at which the students work when they are genuinely interested in their homework as they have come up with it themselves.
Questions where you could use PMI include:
- What if we used the median to best represent a set of data? (E.g. plus – ease; minus – may not be representative; interesting – how statistics can paint a picture depending which you use.)
- What if we work out the area of a trapezium by splitting it into triangles and rectangles? (E.g. plus – good if you can’t remember the formula for finding the area of a trapezium; minus – slower method; interesting – compare the accuracy of the answers/average time taken to complete the question between students who used each method.)
Debbie Barton is a teacher, examiner and maths consultant with over 20 years’ experience. She’s written a number of books including Complete Mathematics for Cambridge Secondary 1. She also worked as a Gifted and Talented trainer and is passionate about ensuring able students are challenged with exciting mathematical stimulus.