I've pulled together some math things I've thought about in recent months that could be used as problems for students at different levels.
world distance away
Which is further away, halfway around the world or all the way around the world?
This is one for grade school, before junior high.
I took me a while growing up to grasp that one. A bit of geometry. I'd heard it growing up and it was some years before I grasped the answer and why the saying is "halfway around".
temperature in Halifax and Fredericton
Halifax is a coastal city, while Fredericton is a bit inland. As a result of geography Fredericton summer is hotter than Halifax summer, and winter in Fredericton is colder than Halifax winter. Is there ever a time when Halifax and Fredericton have the same temperature? At any given moment there is exactly one official temperature in both Halifax and Fredericton.
This one could be used in junior high or in high school to demonstrate the concept and power of continuous functions. The junior high could just explain with words while high school could be a bit more formalized with equations.
An outline of a solution goes something like
Let F(t) be the temperature at time t in Fredericton
Let H(t) be the temperature at the same time t in Halifax
Let D(t) = F(t) - H(t), the difference in temperature between Fredericton and Halifax at time t
 |
| winter snow in Halifax |
increased voting share
In a two person election a candidate, Tim Tory, did not do well in one town. He only got 4 percent of the vote. The opponent, Larry Liberal, got 96 percent of the vote in that town.
In a later election Tim doubled his share of the vote in the town to 8 percent. Is Tim necessarily better off?
This one might be good in high or early high school. It's also useful as a practical math application, to avoid jumping to conclusion based on initial or incomplete information.
The answer is a bit subtle. On the face of it Tim seems better off. If the same number of people vote in both elections then Tim is better off. However with higher turnout Larry can overcome Tim's increased voter share and still win the town by a bigger margin.
Again the high school part might involve equations and expected value. It could be stated as (though it pretty much gives away the answer to the first question)
Suppose 1,000 people voted in the first election. How many people would Larry need to vote in the second election to overcome Tim's increase in voter share?
answer:
In the first election Tim had 40 votes and Larry got 960 votes. So the winning margin for Larry was 960 - 40 = 920 votes.
The question could then be restated as "At 92%, how many people does Larry need to vote to win by 920 votes" to equal his margin in the first election.
let x be the number of people who vote
0.92x - 0.08x = 920
x = 1,095.24, so need 1,096 voters
1,096 * .92 = 1,008
1,096 * .08 = 88
diff = 920
So Larry needs a 9.6% increase in the number of people voting to overcome the loss in voter share and maintain a 920 vote advantage in the town in the second election.
