“Daddy, I’m full. I had 1 and a half…no, one and a quarter slices of pizza which is the same as five quarters of pizza,” said my son at dinner tonight.
“Okay,” I said, “you can go play if you like.”
“You know you can do that right, have five quarters of pizza, or even 10 quarters of pizza,” he continued.
“How many slices of pizza is 10 quarters of pizza?” I asked.
“Hrmmm…,” My son thought for about 10 seconds, and then said, “Two and a half slices of pizza,” and then he went off to play.
My son can do this kind of manipulation with wholes, halves, and quarters of pizza because the idea of dividing food into smaller fractions is familiar to him, and because he has seen halves and quarters used in many different contexts quite a bit during his life. Simple fractions are familiar to him, and so he can manipulate them as needed. Note that I have never once taught my son how to convert between an improper fraction to a mixed fraction, or even what those words mean. I’ve just slowly and deliberately introduced him to the ideas of fractions as they naturally fit our daily lives.
I was playing an online game today with my son (who had just woken up) watching over my shoulder.
“What does ‘split 28’ mean, Daddy?” he asked me.
“Well, I just looted 28 coins and I’m sharing them with my friend,” I responded.
“Oh…hrmmm…….so you’ll get 18 each then!” he responded.
“How did you get that?” I asked back.*
“Well, I took 20 and split it into 10 and 10, and then I took the 8…oh…I forgot to split the 8 too. You and your friend will get 14 coins each.”
* I always ask that question, whether he is right or not.
In this video James Grime examines the “challenging” math problem given in the movie Good Will Hunting and points out that it is not actually all that challenging. Unfortunately he is pressed by the person interviewing him to give all of the solutions to the dots and lines problem given.
This problem could easily be extended to be more open-ended simply by leaving the number of dots open. Are there any patterns when you generate diagrams with 2 dots, 3 dots, 4 dots, 5 dots, and so on? What kinds of diagrams are essentially the same (homeomorphic)? What kinds of diagrams cannot be made more simple without changing the character of the diagram (irreducible)?
The Collaborative Mathematics project, created by Jason Ermer, looks like another excellent source of rich mathematical tasks for students. I recommend following the Problem a Day blog. Jason encourages the problems to be done collaboratively, hence the name of the project.
Here is a sample:
Notice how Jason takes a closed form question (What are the four digit numbers that can be flipped when multiplied by four?) and converts it into a much more open-ended investigation simply by making the restraints less restrictive. This is a useful general strategy you can use to make closed-form problems more open-ended.
Thanks to the Math Munch for sharing this project.
A colleague of mine at work shared this excellent resource with me for interesting and perplexing mathematics problems. The Galileo project looks like it has about 100 interesting mathematics problems for students to do for a variety of different age levels.
Here is an example:
How many parents do you have?
How many grand-parents do you have?
How many great grand-parents do you have?
How many great-great-grand-parents do you have?
How many great-great-great-grand-parents do you have?
Wait a minute! Do you see a problem with this?
Here is another interesting discussion between Malke and her daughter, this time about infinity. What kinds of conversations have you had with your child about the concept of infinity? What analogies have they come up with?
Malke Rosenfeld shares another example of her daughter’s mathematical thinking, this time when her daughter finds a protractor and decides to use it to make a map of some angles. Read more about her daughter’s mathematical thinking on her blog.