In support of #DiaryDays: (-A-Diary-a-Day-keeps-the-Pie-Fights-away-DiaryDays) — Trying to make mathematics relevant to students is a very hard task. Relating mathematics to day to day life and finding examples of why it is important to learn mathematical methods is even harder. Apart from a generic background sense that it is important to study, giving a very demonstrative example is arduous.
Young students, you just hope that they can view this as slightly better than just solving puzzles. Adults usually come with a pre-conceived notion that mathematics is hard. But the ones that I encounter are there because they have decided they need to learn, or they specifically want to do something regularly that requires rudimentary mathematics and occasionally people who have suspicion that their payslip does not match. Adults are much easier since there are numerous examples and most already know which elements of day to day life would be improved with some basic mathematical skills.
Perhaps I should have mentioned, I usually teach adults who need basic mathematical skills. For whatever reason they do not have it. Lets not dwell too much on why they do not, as it is not relevant to the story. Added to that small amount of time per week on young children.
One of the life application examples that we bumped into has been very fun for me, and surprisingly also fun for the adults in their class and young people in their class.
There are many pizzerias in the locality, some eat in places, some delivery places and some do both. There are always some leaflets with latest deals pushed through the door. Every two weeks new leaflet with new deals just for next two weeks. Somehow one of them found its way into my pile of papers I took with me to adults class. When people saw the leaflet in my hand, naturally conversation turned into speculation about my possible plans to order a pizza when I got home late, and some surprise at me having a pizza leaflet especially since I am known to have a heavy bias for spicy Nepali/Indian food, and perhaps I was planning a pizza for when my young children came for a visit at the weekend.
Suddenly, one of the students asked, if I worked out which of the two deal this week is the better value for money. I was surprised to see that there was some excitement and people were already saying Deal1 or Deal2 or small pizza deal, medium pizza deal.
Deal 1: Buy 1 small pizza, get another small pizza free. (I recall the prices for small pizza were cheaper than last month). Price 8-00.
Deal 2: Buy 1 medium pizza, get a pet bottle of drink free. Price 9-00.
I declared that we were studying mathematics so we should be able to work through it easily as a group. (In the adults group you get to be a dictator about what gets done next and there is never a discipline issues. There might be social issues but never a discipline issue.)
After various ways of working out which is a better deal and a few blind alleys, we settled on this sequence. Usually focus is on having a rough idea and polishing it further to be progressively more accurate. Things get calculated and someone mentions something we missed out, and that gets taken into account, and someone else mentions some other aspect and we factor that in as well. Some things get removed as unnecessary. But we record all these, as we are trying to make a habit of recording these, that were factored in and things that were discounted. Blind alleyways necessary to record so that we can spot same errors in other calculations in future.
And then drop the lot and change the whole thing from calculating individually to calculating ratios as we would get the answer faster without needing to do too much calculation. {Yes I hate unnecessary mathematics too}
And finally because at the end of the term/year they would get a formal certificate confirming their skills and equivalency to formal education, we had to write up a formal answer.
Generally however this is what we ended up with as a systematically calculated response. {My side comments in italics in this section}
We agreed that the face size of pizza (surface area) mattered most because of the cheese and toppings. More surface would mean more toppings. Followed by thickness of the base (amount of dough/bread).
Data available to us:
Small pizza was 7Inches diameter. Medium Pizza was 10.5 Inches Diameter.
2 Small pizzas costs 8-00. 1 Medium Pizza + pet bottle of drink costs 9-00. A pet bottle of drink costs 1-50 if buying separately.
Volume of pizza would be proportional to surface area so we could just ignore the thickness and assume pizzas would be same thickness irrespective of which one you bought. We only needed to calculate surface area to find relative value of the pizzas.
using formula, Area of Pizza (circle) = Pi.r.r {That thing probably everyone remembers, pi are square}
Let r be small pizza radius, let R be medium pizza radius. {for anyone who has momentarily forgotten: Diameter = 2 x Radius}
Calculating the ratios or radius: r/R= (7/2) / (10.5/2) = (7/10.5) = 2/3
3r=2R (Equation 1) {Oh yes, all these important equations have to be labelled so that we can reference them later if we use it elsewhere}
therefore, r=2R/3
and R=3r/2
Deal1_Surface_Area = 2.Pi.r.r {There are two small pizzas}
Deal2_Surface_Area = Pi.R.R = Pi.3.r.3.r/(2.2) = 9.Pi.r.r/4
Now we know Deal2 pizza has more surface than Deal1.
Ratio of surface areas, Deal1_Surface_Area/Deal2_Surface_Area = 2.Pi.r.r / (9.Pi.r.r/4) = 8/9
Deal1_Surface_Area/Deal2_Surface_Area=8/9 (Eq 2) {recording for reference}
Looking at costs.
Deal1+Drink = 800 + 150 = 950.
Deal2 = 900
Substituting from Eq 2,
9 x Deal1 / 8 +Drink = 900
Therefore Deal2 is better.
In retrospect I should not have been surprised when the same calculations were even a bigger hit with the 11/12 year olds the next week. There are probably now parents who will constantly dread every other Thursday when leaflets with new deals arrive.
I am however partly mortified that children will use mathematics to get fizzy soda and pizza.