Problem Set #4

1) The question I've picked is # 17. It's about a number line. Marked on this number line with , are integers that come one after the other, but you can't actually see the numbers. There are four bigger dots (I've marked them with 0 s), and two of them are multiples of 3, and the other two are multiples of 5. What we're trying to find is the point that is a multiple of 15 (A,B,C,D, or E).

_,_0_0_,_,_,_,_,_,_,_0_,_0_,_,

               A BCDE

2) When I first looked at this question, it seemed pretty tricky to figure out. The written explanation just seemed to make it more complicated, so I decided to look really hard at the diagram. What struck me was that the 4 bigger circles - the ones that are multiples of 3 or 5 - were really close together. I realized that this number line had to be pretty small numbers, because the larger the multiples of 3 and 5, the farther apart they would be on the number line. That got me thinking. 

These _,_0_0_,_,_,_,_,_,_,_0_,_0_,_, couldn't both be multiples of 3 or both multiples of 5, and neither

                         A BCDE

could these , because they have to be at least 5 or 3 apart. So, I was looking for a two smaller numbers right next to eachother that were multiples of 3 and 5. Since they had to be small, I started from the bottom. 3 and 5, obviously not. They're not consecutive. 5 and 6 works, but that makes the second two 14 and 16, which aren't multiples of 3 or 5. Not 6 and 10, either, but 9 and 10 works! That would make the second pair 18 and 20, which fit perfectly on the number line and are multiples of 3 (18/6=3) and 5 (20/4=5)!

So it would look like this:

_8_9_10_11_12_13_14_15_16_17_18_19_20_21_22

                       A   B   C    D   E

So, which number is a multiple of 15? 

15!!!

So, the answer is D, 15.

3) The reason I like this question is because it requires a bit of logic - the fact that the numbers had to be smaller, because otherwise the multiples of 3 and 5 would be farther apart. Without that, it would have been immensely frustrating, because there are unlimited multiples of 3 and 5 - just not the ones we want.

4) What I have learned about problem solving through this question is to use all the information you're given. You can't just look at the written explanation, and you can't just look at the diagram. You have to use both, and a bit of mathematical reasoning, in order to solve the question. So, even if the written explanation is super complicated and you don't understand at all, use the diagram to help you understand, and it will start to make sense.

Who Wants to be a Mathematician? Fieldtrip on Thursday, Oct. 21.

1) a squared + b squared doesn't always equal c squared - it only works on a flat plain.
2) There are an infinite number of Primitive Pythagorean triplets.
3)Chinese people are probably responsible for a lot of different theorems in math - like the Pythagorean. theorem.
4) There are a couple of million - dollar math questions out there - so you can be a mathematician AND a millionaire!
5) Chemistry and science changes, politics change, but math will always stay the same and will always be true.
6) Math can be very, very, difficult and complicated.
7) a cubed + b cubed = c cubed, but in this equation, c cubed CANNOT equal 1.
8) x squared + y squared = 1 is a circle on a graph.
9) One really important thing in math is not just being able to do math, but being able to explain math in a way that others can understand.
10) You shouldn't just take math equations for granted, like the Pythagorean theorem, but you should test things to find out what's really true.

From the talk, I learned not only about the millennium problems or the Pythagorean theorem, I learned to ask to see if things are true. How do we really know about Pythagoras, or sine and tangent, or even pi? I think that's what the talk was really all about, finding out if things are true, and finding out how we know that they are true, and being able to explain that. That's part of what math is about. At the end of the talk, however, I felt rather slow, because the math was way over my head. I did sort of understand x squared + y squared = 1 is a circle, but past that I was totally lost. So, I guess you could say that I learned that math is something that is very complicated and confusing.

The workshop helps you by preparing you to present what you know about math to everyone in a way that you can all understand. I know I had trouble doing this because Lauren started laughing and made me laugh too ... :) Also, the people at the workshop are examples of what you might look like if you consider a future in math. This entire workshop is an idea of what you might expect if you took math at UBC - if you include the bussing. Lastly, the workshop helped me with algebra, because there was a pretty tricky algebra question on it that I got help with.

Monday, October 18th HOMEWORK

4) If 5% of a number is 8, what is 25% of the same number?

If you have 5% of a number, and you want to get 25% of the SAME NUMBER, you need to figure out what number multiplied by 5 turns into 25. You multiply whatever 5% is by that same number, and it will give you 25%.

So, 5 x 5 = 25.
Therefore, 5% of a number (8) x 5 = 25 % of the same number.
8 x 5 = 40
40 = 25 % of the same number.

So the answer is (A) 40.

I really liked this problem because it seemed more complicated than it actually was. At first glance I was like, "Okay, some number, percentages, the number eight...." But then I realized that the question was actually more complicated than the answer, which was just 8x5.

I just think it's really satisfying when you look at a question, and realize after a second that you can definitely do it, that's it's easily within your abilities. I just hope in a year or two, I'll be looking at the stuff I find confusing now with the same sentiment.

What I learned about problem solving is that sometimes the answer is more simple than you think. You don't always have to over-think or over-complicate things to find the answer. Don't jump to conclusions about problems until you've actually looked at it.