A.M. Ran with Tyler. First we did 5 miles with Benjamin in 39:24. During the run I gave Benjamin a math problem around mile 4 mark. Integrate x^7+x^5 from 0 to 1. It took him about a quarter mile to solve it. The answer was 7/24. Then I gave him another - find the minimum of x^2 - 6*x. He struggled some, but solved it after I reminded him to take the derivative and find the root. The answer to that one was -9 achieved when x = 3. A less violent method of solving it is to notice that x^2-6*x = x^2 - 6*x + 9 - 9 = (x -3)^2-9 . Since the smallest (x-3)^2 can be in the real domain is 0, the smallest x^2-6*x can get is -9 and it happens when x = 3. Then we took Jenny and Julia for 1.5. I was in the math mood, so I had them calculate 2^3. Jenny solved it right away, but to give Julia a chance to solve it showed me the answer (8) on her fingers using the traditional unary system (one finger for each count). The next problem was find the biggest number you can count to on your fingers. Jenny correctly observed that you can count up to 15 with only 5 fingers using the binary system, but struggled to name the number that can be reached with 10. With 10 fingers you can count up to 1023. The trick is that each finger can be in two states - up or down. With N fingers you can count up to 2^N - 1. That is if you agree that all fingers down is a zero. If you want to get fancy, you can agree that all fingers down is a 1, and shift everything else one up, then you get 2^N. One more problem for Jenny - 64^(1/2). She struggled some, then figured out it was 8. Then she did 121^(1/2) - the answer to that was 11, and 27^(1/3) which is 3. Then we ran 1.5 with Joseph and 0.5 with Jacob. Then 3 more on our own. We did a mile pickup in 5:16.2. Our splits were 79, 78, 81, 78. The 81 involved going under the bridge, which is a nasty rhythm breaker. So in essence it was a very steady run. This is incidentally a new mile PR for Tyler. I was pleasantly surprised by how good it felt. Last week I struggled to run 5:23 even though I did not feel like the race on Saturday had taken a whole lot out of me. This time I felt knocked out pretty bad by the heat on Saturday, and quite miserable for a while after the run. So I was expecting something similar to last Saturday, but it was not. I wonder if this was because I was stronger, or because I had Tyler with me. Of course, in this range of speed there is a lot of room for error and tricking yourself that you are feeling better or worse than you really do as it is still some distance away from all out race pace. The total distance was 11.5 miles. P.M. 2 miles with Sarah following me on a bike in 15:34.
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