But what are they? What would you call these problems? What are their essential features? Do you know of other problems like them?
Here are some points in the plane:
(4, 1), (17, 27), (1, -5), (8, 9), (13, 19), (-2, -11), (20, 33), (7,7), (-5, -17), (10, 13)
Choose any two of these points. Check with your neighbor to be sure that you didn’t both choose the same pair of points. Now find the rate of change between the first and the second point. Write it on the board. What do you notice?
Find the length of the third side of this triangle:
Choose three consecutive whole numbers and add them. Write your sum on the board. What do you notice?
Everybody pick a number.
Multiply it by four.
Divide by two.
Divide by two again.
Now subtract your original number.
On the count of three, everybody say the number you have.
On a graphing calculator with overhead display, graph f(x)=sqrt(x) and show the graph to your class, but with a viewing window that is very small, like 3.9 < x < 4.2 and 1.95 < y < 2.05 (without showing them what function is being graphed for them). Show the class the graph and ask them what function they think they see. They will say that it is a line. You can also trace along the curves and find two points on the graph, if you want them to find an equation for the line that they think they see. The line they get will be approximately the tangent line. Then zoom out and they can see that the function isn't linear. Credit
Problems 1, 2, and 5 are from Scott Farrand, a math professor at Cal State Sacramento, and his student Janelle Currey. Problem 3 & 4 are my own constructions.