There's an app on my phone that is a game of memory with 16 cards. Each cars has a point value. When you pair, the points go in your bank which can then be redeemed for free items. Pretty sweet deal!
You can miss four pairs. On the fifth miss, the game is over. Being a super nerd, I'm working on figuring out both the probability of clearing the board (matching all 8 pairs before having 5 misses) and the expected value of a play.
The 5 plays accumulate over time. Last night at dinner I introduced the game to a colleague, who started with 4 plays. I advised him to wait for the fifth play but grew played any way. He lost. No points. It got me thinking of the big difference between 4 and 5 plays, as well as the difference between 5 and 6 plays. At 4 plays, you are guaranteed nothing, at 5 plays you are guaranteed only a single pair and at 6 you are guaranteed to clear the board. Someone put some serious math into these probabilities! Well done app maker!
Wednesday, May 4, 2016
Tuesday, May 3, 2016
Standard Algorithm for Subtraction
Kay is working on money in her math class. She had homework that required her to compute two people's money based on the coins they have, decide who has more and by how much.
She incorrectly calculated the amount of the difference for the two people. I asked her how she decided who had more money, since there were some markings on the paper. Using the comparison strategy, she explained, she was able to show that one person had an extra dime and penny. But there was a mistake that she was able to catch. One of the coins in her comparison was a nickel, not a dime. She then totaled the two and came up with 36 and 42 cents. She came up with an answer of 16 initially.
Standard response, "How did you get that?" She thought a moment, then said she did it wrong and it should be 6.
"How did you get that?"
"I know that 6 plus 4 is 10, and another 2 makes 42." Content she settled down and moved on. 30 seconds later...
"I could have done that so much easier I had just remembered my doubles facts...6 plus 6 is 12!"
Love this kid! She's making sense of problem, developing and evaluating strategies, and coming up with alternate paths to solutions. She is not, however, using the standard algorithm.
When is the right time for this introduction? She had a good conceptual grasp. Looking back to my post, she can use models to explain her thinking. She uses strategies that make sense to her. I'm torn on deciding if standard algorithm well become just one more tool for her or become a confusing abstract tool. How do you know 5 right time for the algorithm?
She incorrectly calculated the amount of the difference for the two people. I asked her how she decided who had more money, since there were some markings on the paper. Using the comparison strategy, she explained, she was able to show that one person had an extra dime and penny. But there was a mistake that she was able to catch. One of the coins in her comparison was a nickel, not a dime. She then totaled the two and came up with 36 and 42 cents. She came up with an answer of 16 initially.
Standard response, "How did you get that?" She thought a moment, then said she did it wrong and it should be 6.
"How did you get that?"
"I know that 6 plus 4 is 10, and another 2 makes 42." Content she settled down and moved on. 30 seconds later...
"I could have done that so much easier I had just remembered my doubles facts...6 plus 6 is 12!"
Love this kid! She's making sense of problem, developing and evaluating strategies, and coming up with alternate paths to solutions. She is not, however, using the standard algorithm.
When is the right time for this introduction? She had a good conceptual grasp. Looking back to my post, she can use models to explain her thinking. She uses strategies that make sense to her. I'm torn on deciding if standard algorithm well become just one more tool for her or become a confusing abstract tool. How do you know 5 right time for the algorithm?
Sunday, May 1, 2016
Kay 35 + 7
So Kay (6) became intrigued about the calendar. She started reading dates as we completed our monthly ritual of changing the dates. She told me that being able to count by 7's would be a good skill for keeping track of the date. She then went over to the calendar, paused a minute and started reading off, "7, 14, 21, 28..." She paused as she calculated the next number. "35". She stopped. To egg her on I told her, "If you add another seven, you get my favorite number." (Hitchhikers' Guide anyone?) She asked what it was to which I replied, "I'm not just going to tell you".
She came up with the correct answer of 42, but instead of confirming, I asked her how she got that. Matter-of-factly she stated, "I used my Make 10 facts". I assumed she did what I did, which is break the 7 into 5 + 2 and add the 35 and 5 to make 40. Just to be sure, I asked her what fact she used. She told me "7 + 3 is 10". I was totally confused.
She wanted to show me on a calculator, but since one was not handy, she ran over to where we keep our base 10 blocks. "I can show you". It was hard to contain my excitement! What ensued, was a pretty amazing explanation, I think.
It was about more than just the addition. Kay showed me a completely new way to think about decomposition. I never would have thought to take from the larger number to make my 10. Here I am, a math teacher in my 11th year, with a Bachelor in Math and a Masters in Math Ed, learning about addition from a 6 year old. I was truly and genuinely interested in her method, especially since she's my kid, but she was oh so eager to share it with me. It took her a lot of processing time, and she made a few mistakes. Once the ball was rolling, my job was to listen. As always though, whenever she shares her method, I try to share a different one, just for comparisons sake.
So how can we replicate this in the classroom? How can we make sure that students have a place to voice their thoughts? How can we give students enough time to make sense and explore their sense making? How can we be genuinely curious about the voice of each of our students on a daily basis?
She came up with the correct answer of 42, but instead of confirming, I asked her how she got that. Matter-of-factly she stated, "I used my Make 10 facts". I assumed she did what I did, which is break the 7 into 5 + 2 and add the 35 and 5 to make 40. Just to be sure, I asked her what fact she used. She told me "7 + 3 is 10". I was totally confused.
She wanted to show me on a calculator, but since one was not handy, she ran over to where we keep our base 10 blocks. "I can show you". It was hard to contain my excitement! What ensued, was a pretty amazing explanation, I think.
It was about more than just the addition. Kay showed me a completely new way to think about decomposition. I never would have thought to take from the larger number to make my 10. Here I am, a math teacher in my 11th year, with a Bachelor in Math and a Masters in Math Ed, learning about addition from a 6 year old. I was truly and genuinely interested in her method, especially since she's my kid, but she was oh so eager to share it with me. It took her a lot of processing time, and she made a few mistakes. Once the ball was rolling, my job was to listen. As always though, whenever she shares her method, I try to share a different one, just for comparisons sake.
So how can we replicate this in the classroom? How can we make sure that students have a place to voice their thoughts? How can we give students enough time to make sense and explore their sense making? How can we be genuinely curious about the voice of each of our students on a daily basis?
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