A teacher trying his best to make math enjoyable for students.

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Sunday, November 22, 2015

Constant Difference: Building Understanding

Fair warning: I typed this up quickly and posted, so there is a high probability of nonsensical statements, which you are free to comment on and ridicule me for; I can take it.

While teaching high school, I found that many students could calculate slope because they remembered the equation. Teaching juniors and seniors in Pre-Calculus and AP Calculus AB, I found that every time a student made an obvious mistake when calculating slope, it was rooted in their lack of understanding of what slope is (and as a Calculus teacher, this was a horrifying discovery). Now that I teach middle schoolers, I have the ability to try to develop that understanding so that Calculus teachers of the future do not have the same stomach sinking experience. This short story is all to set up the activity I recently used with my students to develop an understanding of slope (a word I had not yet presented them with) as the constant difference. Who says students aren't engaged by exploring patters rather than "real-world applications" (buzz word for non-math teachers or those who do not see the beauty of math for math's sake)? Below is the sheet I hand out at the start of the lesson.


The Google Doc for this activity can be viewed here and of course, feel free to use it.

After having kids read through it, I then give them baggies of toothpicks they can use to construct the design. Using my document camera (I have a fairly cheap USB camera that works great with my Mac once you download the free software, thought the PC version is not as good), I then build the first two designs and ask kids how many toothpicks are in each. Kids respond 4 and 7, then I have them finish counting designs three and four, once again taking their answers of 10 and 13. At this point, I give them 8 minutes of independent time to tackle the questions where no talking is allowed. I take this time to do some formative assessment, looking at student reasoning. Kids are allowed to build with toothpicks, count in their head, or draw out the designs, giving multiple access points, which is important for multiple learning modalities.

After the 8 minutes is up, kids are then allowed to talk with their neighbors (they sit at tables of three to four students). The beauty of this activity is they become so engaged in figuring out the patterns that almost no redirection is required and I love the student reasoning.

For question 1, kids simply count three more segments/toothpicks and come up with 16.

For question 2, a few strategies are common:
  • Some kids continue to count/build by threes until they come up with the solution of 31.
  • Others simply double their solution to question 1 and come up with 32 (an expected misunderstanding).
  • Finally, some kids pick up on the fact that each design increases by three each time, but the extra toothpick at the beginning has to be accounted for and they also come up with 31.
For question 3, similar strategies to question 2 were used:
  • First, "Mr. D, you can't expect me to count all the way up to the 100th one?!?!" To which I reply, "can you find a faster way then?"
  • Others multiply their answer to question 2 by 10 or question 1 by 20 to get 320 (once again, a predicted misunderstanding). Another common error is 31x10 = 310.
  • Finally, the kids who understand the pattern can be very clever. Some will simply do 3x100 + 1 = 301. Others will cleverly use their answer to question 2 by doing 31x10 = 310, then subtract by 9 because they know that multiplying by 10 means they have counted that initial toothpick too many times.
The beauty of talking to their neighbors is that it can lead to productive conflict where students are so sure they are right and then have to argue their case, often discovering that their assumptions in question 2 were incorrect. after 10 - 15 minutes of collaboration, I will call on certain students to bring up their work to the document camera and show their work for questions 1, 2, and 3, but being a sneaky teacher, I often call up students who fell into predicted traps, like thinking the answer to question 2 was 32. Or the answer to question 3 is 310 or 320. This is not to humiliate them, but to applaud them for their reasoning, which is correct in many ways. But then I will call on challengers, often a student who also made an error. And finally, I will call on a student with the correct logic such as 3x100 + 1 = 301 and have them explain their reasoning. I will then show them that 4+3+3+3+3+...=301 in the end.

At this time, I then give students an additional 3 minutes together to try to come up with a rule or equation, which could be as simple as you multiply the pattern number by 3 to get how many added toothpicks there are, then add 1 to include the extra toothpick at the beginning. Others will come up with something more complex such as 3 x P + 1 = N, where P = the pattern number and N = the number of toothpicks. Once again, when time is up, I have students present their solutions and we run a couple of samples to make sure they produce the correct values for design 8 for example, which is easily verifiable. This gives students a chance to use their reason to come up with something more abstract.

Finally, there is the challenge problem, which is much more accessible than it looks at first sight. Some of my students that struggle with mechanics were very clever and said no, 5000 could not be the number of toothpicks because they guess and checked with multiplication to find that 4999 and 5002 would work, which means 5000 wasn't possible. Students with a deeper understanding of mechanics simply did (5000 - 1)/3 and knew that because the answer was not a whole number, then it was not possible.

In the end I love this task because it builds up to abstraction after being grounded in very concrete representations. And by having students teach, while I facilitate and push them to go deeper with their explanations, they have ownership over the task and retention is increased. I follow this lesson up with a class of problem solving where students are presented with various patterns and have to work out similar problems, or I create simplistic models of diseases spreading and ask more advanced kids to make predictions about numbers of infections. In this way, the same reasoning is required of all students, but they can be pushed at appropriate levels.

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