By popular demand (and because I need somewhere to organize this for myself), I’m going to put up a series of posts about what I learned at the NCTM conference. If you’re a math teacher, I hope this is as valuable for you as it is for me!
(If you’re not a math teacher, I promise I’ll include “NCTM Takeaway” in all the titles so you’re warned in advance. These are posts you definitely don’t have to be reading. I promise I’ll return to regular posts soon enough.)
Below are some great classroom problem-solving questions, collected by Stephen Krulik. I’ve included some notes and some good hints for kids from the session, but I’m not going to include answers. If you need them, you’ll have to email me for them ([email protected]).
1. During the recent census, a man told the census taker that he had three children. When asked their ages, the man replied, “The product of their ages is 72. The sum of their ages is the same as my house number.” The census taker ran to the door and looked at the house number. “I still can’t tell,” she complained. The man replied, “Oh that’s right. I forgot to tell you that the oldest one likes chocolate pudding.” The census taker promptly wrote down the ages of the three children. How old are they? (Consider only whole number ages) Hint: make an organized list is a good strategy. Start and see what happens.
2. The new school has exactly 1,000 lockers and exactly 1,000 students. On the first day of school the students meet outside the building and agree on the following plan: The first student will enter the school and open all of the lockers. The second student will then enter the school and close every locker with an even number. The third student will then “reverse” every third locker. That is, if it’s open, she will close it; if it’s closed, she will open it. The fourth students will then reverse every fourth locker beginning with locker number 4, and so on until all 1,000 students in turn have entered the building and switched the status of the proper lockers. Which lockers finally remain open? Try a simpler problem. What happens with 10 lockers? 20 lockers? There’s a pattern in the numbers that remain open. Extension: can you explain why that pattern occurs?
3. Laura has a pet white rabbit named Ghost. She has taught Ghost how to go up and down a flight of 10 steps. Ghost can go up one step at a time or two steps at a time, but he never goes back down. In how many different ways can Ghost go up the flight of steps? Try a smaller problem – how many ways if there is 1 step? 2 steps? 3 steps? 4 steps? (Spoiler alert) You’ll see a very famous sequence in the number of possible ways as you increase the number of steps by 1. Extension: can you explain why that pattern occurs?
4. Tennis balls come three in a can, stacked vertically and packed so that the balls just touch the sides, top and bottom of the can. Is the can taller than its distance around? You only need to know how to find circle circumference for this problem.
5. The factors of 360 add up to 1,170. What is the sum of the reciprocals of the factors of 360? Try it with a smaller number, like 12. What’s the sum of the factors? What’s the sum of the reciprocals?
6. A farmer sent his two children out into the field to count the number of pigs and chickens he had. They came back and the son reported he had seen 200 legs. The daughter said she had seen 70 heads. The farmer wrote down exactly how many pigs and how many chickens he had. How many pigs and how many chickens does the farmer have? Classic systems of equations problem, but can you do it another way? Try a guess-and-check table. Try simplifying to 7 heads and 20 legs. Try having all the animals stand on half their legs.
7. A bridge spanning a bay is exactly 1 mile long, and runs parallel to the water without arching. In the summer, the bridge expands a total of 2 feet, causing it to arch. How high is the center of the arch from the bridge’s winter height? Please think about what you’d predict before you start, because it’ll make the answer all the more interesting when you find it. Hint: he used Pythagorean Theorem to solve.
And here are 2 he didn’t cover in the session, but you can have them too:
8. Barbara has fewer than 100 rocks in her collection. She gave one half of them to the school museum. She kept 10 for herself and divided the rest equally among her four friends. How many rocks were in Barbara’s collection?
9. Joanne and her friends are sitting around a circular table. Her mother gives them a tray of cookies to pass around. Each person takes a cookie when the tray reaches him or her and passes it to the next person. This goes on until all of the cookies are gone. Joanne took the first and last cookie (and might have taken some in-between). How many friends are at the table?