We’ve only been at it a week, but the new chapter we are studying – “Basic Statistics” – has been super fun. For now our talks have been limited to averages mostly, but the problems we’ve been doing have allowed him to pull a lot of different mathematical ideas together. Honestly, I never would have expected relatively straightforward basic statistics problems to bring together so many different ideas in the mind of a kid – I even skipped this chapter with my older son! But watching him work though these problems has been great.
The first problem from yesterday was this: The average of your first 7 test scores was 82. If you want to raise your average to 84 after the 8th test, what score do you need to get?
His first instinct is to assume that all of the first 7 scores were 82, but he gets a little stuck after that. He decides to see what happens if your 8th test score was 84. He sees that 84 doesn’t work, but makes a pretty cool observation – if the 8th test score is 2 higher than the previous 7, the average goes up by 1/4, so you’ll have to raise the score by 16 to raise the average by 2.
A nice observation that involves a little bit of thinking about fractions and a fairly good understanding of averages considering we’ve only been talking about them for a week. Nice solution!
Today’s problem was a problem from an old AMC 8 and involved thinking about the mean and the median of a set of 5 numbers. Here’s the problem: The mean of a set of 5 positive integers is 15 and the median is 18. What is the largest possible value of the largest integer in this set?
His first instinct is to make lots of numbers equal to 1, but that turns the median into 1 unfortunately. After noticing that only two of the numbers can be 1 and the 4th number has to be 18, he goes through a pretty clever calculation to find the 5th number.
His idea was to compare the list of numbers to the list 15, 15, 15, 15, 15, and look at the differences. If the mean is going to be 15, the sum of the differences has to be 0. If he first four numbers are 1, 1, 18, and 18, the differences are -14, -14, 3, and 3 which add to -22. That forces the largest number to be higher than 15 by 22. Again, great reasoning for a kid seeing averages for the first time!
So, never having going through this section before, I’m pleasantly surprised at the way the problems have gotten my son to think about many different ideas in math. Who knew that going through basic statistics could be so fun!!