Trapezoid Problem (take 2)

Am I blundering fool? You decide!

It turns out the trapezoid construction I posted earlier today is trivial. Thanks to Alexander Bogomolny for pointing out my error. The construction is quite easy (and it does not require the height), and I quote Alexander:

No, you do not need the height.

Imagine a trapezoid. Draw a line parallel to a side (not a base) from a vertex not on that side. In principle, there are two such lines. One of these is inside the trapezoid. This line, the other side (the one adjacent to the line) and the difference of the bases form a triangle that could be constructed with straightedge and compass by SSS. Next, extend its base and draw through its apex another base. That’s it.

So I redid my Geogebra Applet and posted it here. It’s not really worth checking out, though, since it’s indistinguishable from my previous applet. (In truth, you can reveal the construction lines and see the slight differences.) But I did it for my own satisfaction, just to get the job done correctly . Anyway, three cheers for mathematical elegance, and for Alexander Bogomolny*.

*check out Alexander’s awesome blog & site, a true institution in the online math community!

Vi Hart’s Blog

It’s high time I gave a bit of press to Vi Hart’s Blog. If you haven’t checked it out, do so right away. It’s brilliant.  A number of people have pointed me to her blog, including one of my Calc students. Her little math videos are fresh, funny, and insightful. Denise, at Let’s Play Math, gave her some press too, which is what reminded me to finally make this post. Here’s the video Denise highlighted (the most recent of Vi’s creations):

This is particularly appropriate because there were a couple of us in our math department discussing this very question: In total, how many gifts are given during the 12 Days of Christmas song? It’s a nice problem, perfect for a Precalculus student. Or any student. Here’s a super nice explanation of how to calculate this total, posted at squareCircleZ. But before you go clicking that link, take out a piece of scrap paper and a pencil and figure it out yourself!

Here’s another nice video from Vi Hart:

You could spend a lot of time on her site. Here’s another awesome video. I’ll have to have my Precalculus class watch this one when we do our unit on sequences and series.

And you’ve got to love the regular polyhedra made with Smarties ,  right?

Plus, Vi Hart plays StarCraft, which is awesome too.  Back in the day, I really loved playing. I haven’t played in a while, and I certainly haven’t tried SC 2 yet, because then I’d never grade my students’ papers.

Bottom line is, you need to check out all the playful stuff Vi Hart is doing at her blog. Happy Wednesday everyone!

 

Answer to Wild About Math Giveaway Problem

A few weeks ago, I blogged about a calculator giveaway at Wild About Math. Since then, Sol has posted a submitted solution here (and here’s the direct link to the pdf solution by Nate Burchell).

Here’s the problem for those who didn’t see it:

One can create a triangle of consecutive positive integers as follows:

 1
 2  3
 4  5  6
 7  8  9 10
11 12 13 14 15
16 17 18 19 20 21
   . . .

Each row, R, has R numbers. Each column, C, has infinitely many numbers. Rows and columns begin at 1. We define a function F(R,C) for row R and column C such that F(R,C) gives us a value in the triangle. Thus, F(1,1) = 1, F(2,1) = 2, and F(2,2) = 3. Note that F(R,C) is only defined when 1 < = C <= R.

Part 1: Come up with a formula that computes F(R,C) in terms of R and C for any positive values of R and C when 1 < = C <= R. Show your work.

Part 2: Come up with a formula or algorithm that, given a positive integer n, determines R and C.

I also solved the problem and submitted a solution but I didn’t win. Here’s my own solution.

TI-Nspire giveaway @ Wild About Math

Everyone should go enter this giveaway @ the Wild About Math blog. In order to qualify, though, you have to submit a solution to the problem Sol created for us. I just submitted mine!

Home, Home on the Range

The first week of school I gave this problem and never came back to it:

Find the range of .

 

The answer is . Here’s why:

First consider the range of the quadratic in the exponent, .  It’s a parabola that opens up with its vertex at . So the range of is .

Now, consider the function .  If we let the domain of be the values coming from  the range of , we have the mapping . That is, we’re considering the composition . Since is monotonically increasing, for any , we know . So the range of in is . So the range of is .

Do you feel “at home on the range”?

Here are a few more for you to try. In each case, find the range of the function. These aren’t meant to be any harder than the original problem, just different. Though watch out for the third one .

Powerful Problem (hint)

A few weeks ago I posted this “powerful” problem:

Solve

Now, allow me to give you a major hint. Consider the simpler equation

What are the possible values of and ? Here are the possible combinations:

• and is anything
• and is any nonzero number

And here’s the tricky one that most people forget:

• and is even

You now have enough information to solve the original equation. I think you’ll be delighted with the solution!

Powerful Problem

I love this problem. I love it because it seems so complicated at first, just because we don’t teach students how to attack problems like this in Algebra class. There aren’t any “traditional” methods of attacking it, just a little mathematical reasoning/logic. Here it is:

Solve

And this is my new “super duper” problem which I post throughout the year on my board (I use a lot of the same problems each year). I first saw this problem at Messiah College where one of my professors shared it–either Dr. Phillippy or Dr. Brubaker, I can’t remember which.

So give it a try. It’s sure to delight you. My Precalculus class was sharp enough to solve it today in one period (albeit, while I was teaching about a completely different topic ).

Interesting Cube Problem

If the cube has a volume of 64, what is the area of the green parallelogram? (Assume points I and J are midpoints.)

Go ahead, work it out. Then, go here for a more in depth discussion, including a video explanation. Also, see my very simple solution in the comments on that page. (My Precalculus students should especially take note!)

And, welcome, SAT Math Blog, to the internet! Thanks for pointing us to this great problem and creating the nice diagram above.

Soda Mixing Problem (revisited)

I posted a problem back in December that I never got back to answering. Sorry about that. The problem statement was:

Two jars contain an equal volume of soda. One contains Sprite, the other Coca Cola. You take a small amount of Coca Cola from the Coca Cola jar and add it to the Sprite jar. After uniformly mixing this concoction, you take a small amount out and put it back in the Coca Cola jar, restoring both jars to their original volumes. After having done this, is there more Coca Cola in the Sprite jar or more Sprite in the Coca Cola jar? Or, are they equally contaminated?

I have had the worked out solution for a while, just haven’t posted it until now. I’m relatively new with , but I’ve typed up the solution here, if you want all the gory details .  And yes, Peekay, you got the right answer!

“Real” Cool Solution

This week I start complex numbers with my Precalculus class. This is a fun problem I found here (you can also go there for the solution when you’re ready).

Given , find .