What learning math sometimes looks like: counting arrangements

A problem from an old MOEMS test gave my son a lot of trouble this morning. It was one of those times where you start looking at a problem one way and it is so hard to move away from that viewpoint.

It did make for a nice project this morning, though. Here’s the problem:

Sara places four books on a shelf. The blue book must be somewhere to the left of the green book. The red book must be somewhere to the left of the yellow book. In how many different orders can Sara place the books?

We started talking about the problem using snap cubes – one idea he has here is that the arrangements will come in pairs. You have to be careful with that idea, though, because you might already accidentally have a pair.

He also tries some counting ideas, but we also need to be a little careful to make sure the arrangements we are counting satisfy the conditions of the problem.

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Next we try to find a systematic way of counting the arrangements. The first thing we try is to find all of the arrangements with the yellow book in the first slot.

After a long conversation about that case, he had an easier time understanding the cases where the green cube is in the first slot.

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To wrap up the conversation this morning I tried having him look at other ways to divide the 6 cases into two groups of 3. He noticed that we could look at the blue and red cubes instead.

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So, a tough problem for my son. Hopefully this conversation helped him see a few ways that looking at patterns help you count. His original focus was on finding the number of different possibilities for each slot, but that’s a tough way to approach this problem. An alternate approach that we didn’t cover today involves picking two slots out of the four for the green and blue books – I’ll leave that approach until the next time we talk about this problem.

You have to be careful looking at pensions

I saw an interesting article on Zerohedge the other day about the ongoing pension problems in Illinois:

Illinois to Delay Pension Payments Amid Budget Woes

What caught my eye, in particular, were the charts at the end of the article showing the funding of the plans. It was surprising to see that the funding percentages had not improved since 2009 – the general stock market levels have basically doubled since then. Here’s the chart showing the funding levels of of the Teachers’ Retirement System in Illinois, for example:

To get a better understanding of what’s been going on with the teachers retirement system in Illinois, I dug into the financials on their website:

Illinois TRS website

Here is one surprising thing that I found in the numbers.

At the end of the 2004 financial reporting year, the actuarial accrued liability of the plan was reported as $50.95 billion. During the following 10 years, these were the payments that were made from the plan for (i) benefits, (ii) refunds, and (iii) administrative expenses:

2005: $2.61 billion
2006: $2.95 billion
2007: $3.19 billion
2008: $3.50 billion
2009: $3.72 billion
2010: $4.00 billion
2011: $4.32 billion
2012: $4.66 billion
2013: $5.00 billion
2014: $5.34 billion

For the 10 year period, total payments are $39.29 billion.

So, the actuarial accrued liability was roughly $51 billion at the end of 2014, and in the next 10 years a bit over $39 billion in payments were made. What would you estimate the reported actuarial accrued liability of the plan was at the end of the 2014 reporting period?