Going through a Kate Nowak exercise

Tonight on twitter Kate Nowak published an exercise she was planning to use in her geometry class tomorrow. I may need to, um, borrow this exercise for my son tomorrow since I may need to run off to work early. I decided to go through it carefully just in case I get any questions at work. That short exercise helped me see the problems much better than just reading the list. Here’s are my thoughts, but first the twitter post:

Okay #MTBoS how is this congruent triangles intro going to backfire? Also who has a quick pre/post assess? https://t.co/AhA9RRGzBX

— Kate Nowak (@k8nowak) January 5, 2015

(a)

For this one I used the ruler to draw the 3 inch side and the compass to draw the range of points 5 inches away from one of the endpoints of that side. Lots of possible triangles and hopefully a good start to the lesson.

(b)

Here I started with the 3 inch side at the bottom of the page. That was lucky because 5 inches and 6 inches are longer on the paper that I expected. If you put the 3 inch side in the middle of the paper, you’ll not have room (on 8 x 11 paper anyway!). I used my compass to draw a 5 inch circle centered at one of the endpoints of the 3 inch side and then used it again to draw a 6 inch radius circle centered at the other endpoint. One of the intersection points for these two circles was on my paper and allowed me to draw the triangle.

I used the compass again only for problems (d) and (i).

(c)

Had to use the protractor for the first time on this one. Lots of possible choices of triangles here.

(d)

I like this one. After drawing the 120 degree angle it took me a bit to figure out where to put the 5 inch side. Then I remembered the compass and picked a random point on one of the sides and drew a 5 inch radius circle. Just reading this one I missed that it would be more challenging than some of the other problems.

(e)

Having measured the 120 degree angles in the last problems in the right hand side of the protractor, I did the same thing here with the 30 degree angle at the start and accidentally began by nearly drawing a 150 degree angle. Oops! I’ll be interested to see if the students notice the similar triangles from the different groups on this one.

(f)

Now that I was a pro at drawing acute angles, this one went pretty smoothly 🙂

(g)

This one also went pretty smoothly.

(h)

I found it easier on this one to draw the 100 degree angle and then the 30 degree angle on the other side of the 6 inch side. I sort of feel like I cheated, but I didn’t see where the intersection point with the 50 degree angle would be right away so I solved a different problem.

(i)

Hey, two possible solutions!! I’d like to hear what the kids have to say about this one, too!

So, fun little exercise. If I do have to run off to work early tomorrow, I’ll definitely give the worksheet to my son and have him attempt to create all of these various triangles.

An interesting math example coming from the world of finance

Saw this article tonight on Bloomberg:

The Best and Worst Investments of 2014

What caught my eye was the description of the best performing bond fund (about 2/3 of the way down the page):

“Vanguard’s index fund invests in U.S. government bonds that don’t mature for 20 to 30 years. They did well in 2014, reflecting expectations that inflation will remain low for quite a while. If you’d invested $10,000 in VEDTX on Jan. 1, it would be worth $14,506 today.”

Here’s the math that students interested finance (and, of course, anyone else) might find interesting:

According to Bloomberg data, the 30 year US Government bond had a yield of 3.969% on December 31, 2013, and a yield of 2.752% on December 31, 2014. So, throughout 2014 the yields of long-dated US government bonds – the exact type of bonds that the Vanguard fund invests in – were quite low. In that sort of yield environment how could a fund that invests in US government bonds have had a return of 45%?

Answering this question will help you gain a little bit of knowledge about how bond prices move with interest rates.

Let’s start with a simple example. Say that on December 31, 2013 someone had offered to pay you a 3.969% per year return on your initial investment which would result in you receiving $1,000 in 30 years. Just to be clear, you’ll receive no interest payments on this investment – the only time cash changes hands is on day 1 when you hand over the initial investment, and at year 30 when you are handed back $1,000. How much would you have to give that person today to make this investment (ignoring taxes and all sorts of other potential complications – just to keep things easy)?

One way to get to the answer to this question is to take the final $1,000 payment and divide it by the 1.03969% return that you are promised each year for 30 years. That math tells you the initial investment would be $1,000 / (1.03969)^30 = $311.09.

Now, one year later these long-dated yields have gone down to 2.752%. How much is the $1,000 you receive in 29 years worth at that rate now? We can do the same math: $1,000 / (1.02752)^29 = 455.07. Wow – what a difference!

In just one year the so-called “present value” of the future payment has gone from $311 to $455 – a gain of 46.3%! That gain is due to combination of interest rates falling by 1.25% and the $1,000 payment being far off in the future so that the change in interest rates compounds for many years.

I’m cheating a little in the above calculation by using the wrong rates. For a single payment in the future I should be using so-called “zero coupon” rates, but I’m just trying to illustrate where these high returns that the Vanguard fund achieved can come from.

If we look at the 30 year coupon bond we can see a similar, though not exactly the same, move in prices. Again according to Bloomberg, the price of the official 30 year US government bone on December 31, 2013 was about $96 and that bond paid an annual coupon of $3.75. Roughly speaking that means you could have bought this bond for $96 on 12/31/2013, then received $3.75 every year for the next 30 years and then received $100 at the end of 30 years.

One year later the price of the exact same bond was about $120. So, at the end of the first year you could have sold the bond for $120 and kept all of the $3.75 in interest that you received during the year. You would have paid $96 and received $123.75 in interest and sales proceeds. That’s a return of about 29% on your $96 investment! Again, that return comes from the combination of falling interest rates and the fact that the maturity of the bond is so far in the future.

So, understanding the math on bonds helps us see how it is at least plausible that a US government bond fund could have 46% returns in a year when interest rates were quite low by historical standards. It may seem surprising that a seemingly small movement in interest rates – that is rates falling from 4% to 2.75% – could produce a 46% return, but that more or less exactly how the math works out. It was a lucky year to be taking bets on the values of long-dated bonds!