Thought Provoking Problem Lesson

Thought Provoking Problem Lesson

by Mr. Ellis

​Agenda for today

I want to help you think of projects and challenge your thought process with this week's lesson.

​

​Here is the way I will start this:

• Famous Mathematician...

• ​What are your thoughts?

• ​How many solutions (a.k.a. the Facebook problem)

• ​Famous Problems

Carl Friedrich Gauss

In the 1780s a provincial German schoolmaster gave his class the tedious assignment of summing the first 100 integers. The teacher's aim was to keep the kids quiet for half an hour, but one young pupil almost immediately produced an answer: 1 + 2 + 3 + ... + 98 + 99 + 100 = ????. The smart aleck was Carl Friedrich Gauss, who would go on to join the short list of candidates for greatest mathematician ever. Gauss was not a calculating prodigy who added up all those numbers in his head. He had a deeper insight, can you figure this out in less than 2 minutes (without a calculator)?

​Give it a try (without a calculator)!

​You have 5 minutes!

​The answer:

He had added the numbers in pairs - the first and the last, the second and the second to last and so on, observing that 1+100=101, 2+99=101, 3+98=101, ...so the total would be 50 lots of 101, which is 5050.

​

​The more general formula, for a list of consecutive numbers from 1 through n, is n(n + 1)/2.

​

​

​​Decimal to a Fraction?

​​It's easy to change a Fraction to a decimal, but how do we reverse the operation?

​

​Let's look at some examples on the board.

​

​At the end of the day, they are equal right?

​Now lets have fun...

​What if I told you that 0.9999999... was just really equal to 1?

​

​Do you believe me?

Poll

Is 0.99999... = 1?

• Yes

• No

• Maybe, who knows, Ellis is crazy and can prove ANYTHING!

I will show a proof and you let me know what you think!​

​The Facebook Problem

Spotted on The Daily Mail, the question was originally created by Go Tumble and shared on Wikr before taking off on Facebook and going viral.

​

Take a minute and try it!

Poll

What did you get?

• 96

• 40

• Neither

​Let's show how we arrived at our answer!

​Final Question

​Here is a real tough one...

​

​Give me the dimensions of a square and a circle that have the same EXACT area.

​

​By now we are probably running low on time, but give it a few tries...

There is no answer!​

Some question in mathematics have not been solved yet! ​

​

​Look to Schoology for your follow up assignment on this. I am basing the assignment off of our in class conversations!