This very common type of question appears so straightforward. So why do variations of it recur so often on SATs and other standardized tests and math contests?
Why not test it out with your students and ask them to explain their reasoning. I am still surprised by the creativity of our students when given the opportunity to display it!
Again, my boring disclaimer...
This is not a conundrum for the math problem-solvers out there. It is intended as a discussion point for helping students develop some important ideas in mathematics.
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Wednesday, February 20, 2013
So is 75 the avg of the pos integers from 50 to 100 inclusive?
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How about thinking about what is the average of all the positive REAL numbers from 50 to 100 inclusive...
How about thinking about what is the average of all the real numbers from 50 to 100 inclusive... I think the answer must still be 75 but both the sum and the cardinality are infinite in that case.
What about the mean of all the REAL numbers from 50 to 100 inclusive? I wonder what middle or high school students would think of such a question that can't really be formulated in terms that make sense to them, and yet still has a kind of "obvoius" answer...
(sorry if I posted twice -- browser woes)
Great to hear from you mathmom!
That was quite a leap from the integers to the reals!
One could make a case for the average value of a continuous function f(x) = x over the interval [50,100]. To avoid calculus, motivate by asking middle or high schoolers why the area of the resulting trapezoid is the same as the area of the rectangle with base 50 and height 75. This geometric visualization replaces the function with the constant function y=75.
This leads to one of my favorite arguments for finding the average of the terms of an arithmetic sequence:
To simplify, let's find the average of 73,74,75,76,77.
Rewrite these 5 numbers as
75-2,75-1,75,75+1,75+2. so the sum of these 5 numbers is the same as replacing them with 75+75+75+75+75 analogous to replacing the linear function y=x with the constant function y=75.
Students should see that this argument will work the same for 50,51,...99,100 which can be rewritten as 75-25,75-24,...75+24,75+25.
The sum would then be 75+75+75+...+75, n times, so the mean is (75n)/n. Even if the student mistakenly counts 50 numbers instead of 51, the average remains 75!!
This leads naturally to asking for the mean of the EVEN integers 50,52,...,100 and on to more general arithmetic sequences.
Even more importantly it helps students see the BIG IDEA that the mean equals the median of an arithmetic sequence!
You know my craziness about making connections...
I do your 73, 74, 75, 76, 77 example by telling them to imagine them as piles of jelly beans, and if we have to add them all up to find the mean, we can move some of them around first to make it easier.
I have talked about the mean of an arithmetic sequence a lot with my kids over the years. I think it is starting to be intuitive for many of them :)
Mom,
I really like your idea. For K-2 children, having them start with 3 piles of, say 1, 2 and 3 counters, and asking them to make equal groups by moving just one chip seems like a good way to lay the foundation. Revisiting often with other arithm seq and even some non-arithm like 2,2 and 5 sends a powerful message to their brain in preparation for the big idea that the mean "represents" the data, long before these words have meaning for them. 8-10 years later they will be solving moment of inertia problems by replacing separate point masses by their center of mass and will remember what Mathmom taught them in 1st grade! Hey, we can dream!
Even more,children will see moving counters as a puzzle as they are creating their own neural connections --- oh wait, that's called learning!
As for the real numbers, I'm not sure how I'd convince them that the area of the trapezoid or rectangle was related to the problem in the first place...
One of my big things is to differentiate mathematical thinking from arithmetic. Most elementary and middle school kids naturally think math is just arithmetic. But arithmetic, while a necessary foundation, is the most tedious and uncreative part of mathematics, and can ultimately be done for you by a one-dollar discount store calculator (although I rarely let them touch them). The cool part comes when you can use mathematical thinking to reduce or eliminate the need for the arithmetic. When they see that the mean is 75 times n divided by n and realize that this means they don't have to do any arithmetic at all to get the solution, that's the aha moment I'm looking for. ;-)
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