'Tis the season to calculate the Surface Area of Mince Pies! Toby, one of our top Maths tutors, crunches the numbers.
As a Mathematics tutor, I’m continually astonished by the way examples drawn from life help struggling students to do calculations that they believe are beyond them. Money usually helps. For instance, I once had a student who needed to give an answer to 7 ÷ 2 as part of a GCSE question on a Non-Calculator practice test. She complained that it was impossible, as two doesn’t go into seven. I asked her to imagine that she and I had £7 between us, and we agreed to split it down the middle. How much would we each get? Without missing a beat, she told me £3.50.
A second example. Try asking the following question to a not particularly mathematical adult. “If x is directly proportional to y, and when y = 24, x= 3, what is y when x is 11?” When I asked my parents this, they both looked at me blankly, as if I’d just asked them to prove Fermat’s last theorem. However, if I’d said, “on Tuesday, Dan worked three hours, and got paid £24. How much would he get paid for working eleven hours at the same rate?” they’d pretty quickly tell me he’d get £88. What my former GCSE student and my benighted parents might not immediately realise is that I have asked them exactly the same question twice. All that has changed is the way that I have framed the problem.
Such imaginary scenarios provide the Maths tutor with a further advantage: they give him or her the riposte to that frequently asked question, “but when am I ever going to use this stuff in real life?” The answer, if you’re creative, is all the time. Ratios, for instance, can be taught with reference to cooking or, with adult students, mixing cocktails. Percentages and bank balances normally go hand in glove.
Recent experience offered up a perfect opportunity to use my Maths in real life, enabling me to fill my stomach more satisfactorily and to show up an old friend. It being the season, my friend Ryan and I were ordering novelty-sized mince pies. The pies came in two sizes: 7” and 10”. The smaller pies cost £12 and the bigger ones £20. Ryan, who is known for his appetite, said:
“I’m feeling hungry. Let’s get one of the small pies each.”
“That makes no sense.” I replied. “We should get one of the large ones and split it.”
“What are you talking about?” said Ryan, quite bewildered and a little cross. “Surely two seven-inch pies are bigger than one ten-inch pie.”
In fact, they are not, as I then went on to demonstrate. The measurements of the pies refer, of course, to their diameters. And though 2 x 7 is obviously more than 10, it’s not the diameters we’re interested in – you can’t very well eat a diameter. To compare the pies, we’d have to calculate their areas.
The formula for the area of a circle, as every good student knows, is πr². The key is the r² bit. It means that the area of the circle is proportional to the radius squared. In other words, if you double the radius, you increase the area by a factor of four, and if you treble the radius, the area increases by a factor of nine. To go back to our mince pies, the smaller pie has a radius of 3.5 (half of seven) and the bigger pie has radius 5 (half of ten.) Now let’s plug these numbers into our formula for area.
The area for the small pie is: π x 3.5² = 12.25 π square inches. So two of these pies together cost £24 (2 x 12) and have a total area of 24.5 π (2 x 12.25 π) square inches.
Whereas the area for the larger pie is: π x 5² = 25 π square inches. This is more than the combined area of two small pies, and at just £20 is cheaper too. What a bargain!
Having thoroughly convinced Ryan that he was wrong about the relative size of the pies, we bought a large one between us and a couple of glasses of port to wash it down.
Blog Post Crafted by Toby
Toby is one of our top English and Maths tutors, as well as our Chief Assessor, and he also works on our Admin Team. His proudest moment was when he tied Christopher Lee’s (AKA Saruman’s) bow-tie!
Toby is in charge of recruitment of new tutors. He conducts interviews with prospective tutors and assesses their lessons to get a feel for whether they have the teaching style we're looking for. As a member of our Admin Team, Toby advises applicants on the application process and books them in for interviews. He liaises with clients about their tuition enquiries and discusses potential jobs with tutors.
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