Use the numbers 2 and 3 only once (but any combination of mathematical symbols you know), make the largest possible number you can. Make the smallest possible number you can. Make a number as close to zero as you can.
Bonus, use any numbers you like, and then look for patterns between the numbers. In general, which collection of operations will produce the largest number for any two given initial numbers?
Thanks to Michael Pershan for the idea.
Chris Hunter writes on his blog about a student explaining how they would express 0.500 using ten-frames:
One student expressed this as 500/1000 and 0.500. I assumed he was just extending the pattern(s). “Yeahbut where do you see the 500 and 1000?” I asked challenged. “I imagine that inside every one of these *points to a dot* there is one of these *holds up a full ten-frame*,” he explained. As his teacher and I listened to his ideas, our jaws hit the floor.
Showing 500/1000 or 0.500 using ten-frames
Have you seen examples where students come up with innovative ways of representing numbers?
This is a great video of some students sharing their thinking while solving a problem posed by their teacher.
See this blog post where Deirdre Bailey and Amy Park frame the problem posed for the students (as well to get a copy of the text of the problem itself).
This post was shared to me by Jennifer Silverman. What do you think of this student’s inventive use of a pair of calculators?
In a ninth grade Algebra 1 class, we were studying systems of equations. The problem for the day was about drag races, so we started with: http://youtu.be/LGEWCsP1bR4 & http://youtu.be/ggxsrVWEf9E.
One of my students, Ben, was tackling this problem with a partner, and they weren’t making a lot of progress. Suddenly, Ben jumps up and says, urgently, “Mrs. Silverman, do you have another calculator?” The “loaners” were gone so I gave him mine. Pretty soon, he looks up with a big grin and says, “The Mustang wins!” He had built two sequences, one in each calculator, and was carefully monitoring each of them as he simultaneously hit ENTER on each calculator. He knew that the Mustang won because that calculator got to 500 first. When I asked him how long the race took, he asked me to hang on a second. He repeated the process, this time counting the number of times he hit ENTER. “A little more than 11 seconds,” he said, “because at the eleventh hit, the first calculator was at 495.”
Eventually Ben was able to model and solve these problems algebraically, but on the day of the final exam, he had two calculators with him, in case he needed to go back to his own intuitive approach.
Pretty cool, huh?
I love this activity by Bruce Ferrington with his students.
What Is My Area?
Wait – before you say anything – I know your body is 3-dimensional and area is a measurement of two dimensions – so I probably should say, “What is the area of my silhouette?”
What is the area of my silhouette?
In the past week we have measured height and width, talked about Vitruvian Man, combined our heights, compared them with other classes.
Time to have a look at comparing areas.
My chief interest here is to look at how the kids are going to approach this question. I want to see if they can come up with any short-cuts that will speed up the calculations, so they don’t have to cover their entire body in 1cm grid paper and count out each square.
Read more about the activity at his blog.
My son and I played a game tonight. Each of us took turns giving patterns of numbers to the other who was to guess what the next number in the pattern was. Here are some of our number patterns we challenged each other with:
1, 2, 4, 8, …
1, 3, 5, …
2, 10, 18, …
1, 3, 9, …
150, 200, 250, 300, 350, 400, 450, 500, 550, 600, 650, 700, 750, 800, 850, 900, 950, 1000, 1500, ….
3, 5, 4, 4, 4, 4, … (this one is sneaky, so I’ll give away the answer. One has three letters, three has five letters, five has four letters, four has four letters, and so on. My son came up with this one. An aside: does this sequence always end up with a repeating four, no matter which number you begin with?)
I finally stumped my son with the following:
1, 1, 2, 3, 5, 8, 13, …
This is a riddle forwarded to me from the BCAMT Math list-serve (originally by George Pólya, according to someone on the list-serve):
A Life Insurance agent comes up to a ladies door to sell her a policy. He asks her how many children does she have and what are their ages. She replies, I have three children, but I will not even listen to your sales pitch if you cannot guess their ages. (Assume all ages are whole numbers)
Here is clue number 1: The product of their ages is equal to 36.
Salesman jots down some figures and replies, “I do not have enough information to answer that yet”.
The lady replies, “Okay, here is clue number 2: If you go the house next door (points to it), the sum of their ages is the same as the house number”.
Salesman goes next door, looks at the number, jots down a few more things and then returns to say,” I still don’t have enough information”.
The lady then replies, “Okay, your third and final clue: My eldest is learning to play the piano!!”
The salesman, replied with the answer, it was correct, he gave his sales pitch, she bought a policy.
What were their ages?
The puzzle itself is interesting, but what I found more interesting is what happened next. Another mathematics teacher from the list-serve gave the question to some of their students, and got this response back:
“Couldn’t sleep, wrote a script that not only solved your problem, but solves it with any number of children, and any product of their ages, if possible. It also will generate numbers that will give you valid variations of this question, given the number of answers that you want.”
This is a nice bit of mathematical reasoning by this student. They took a problem, solved it, and then found a way to generalize the problem, and solved the more general problem. You can’t ask for much more than that!
“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.
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?