In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.
I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).
This student submission comes from my former student Dale Montgomery. His topic, from Precalculus: mathematical induction.
Looking at Khanacademy’s video on mathematical induction, I feel like he has one of the better explanations of mathematical induction that I have heard. This lends itself well to starting class off with a video to engage, and then moving on to an explore where the students test what can or can’t be proved by induction. This quick explanation by Khan gives a good starting point, and the fact that his videos are interesting should be sufficient enough to engage the students. Another possibility is to have the students watch this at home, that way you have more time during class do work on learning how to use the principle of induction.
This problem, and proof (taken from Wikipedia) has flawed logic. In it, it uses the principle of mathematical induction. This would be a good engage because it has supposedly sound logic but it says something that is obviously not true. This will engage the students by showing them something that doesn’t make sense. This will cause a imbalance in their thinking, and make them want to make sense of the situation. I would probably present it as a bell ringer or similar problem, after induction has been introduced.
All horses are the same color
The argument is proof by induction. First we establish a base case for one horse (). We then prove that if horses have the same color, then horses must also have the same color.
Base case: One horse
The case with just one horse is trivial. If there is only one horse in the “group”, then clearly all horses in that group have the same color.
Assume that horses always are the same color. Let us consider a group consisting of horses.
First, exclude the last horse and look only at the first horses; all these are the same color since horses always are the same color. Likewise, exclude the first horse and look only at the last horses. These too, must also be of the same color. Therefore, the first horse in the group is of the same color as the horses in the middle, who in turn are of the same color as the last horse. Hence the first horse, middle horses, and last horse are all of the same color, and we have proven that:
- If horses have the same color, then horses will also have the same color.
We already saw in the base case that the rule (“all horses have the same color”) was valid for . The inductive step showed that since the rule is valid for , it must also be valid for , which in turn implies that the rule is valid for and so on.
Thus in any group of horses, all horses must be the same color.
The explanation relies on the fact that a set of a single element cannot have 2 different sets with the same element. Because this assumption cannot be made, the case of falls apart and tears the argument apart.
Dominoes have been talked about as a way to explain mathematical induction. The idea that if you can prove that the first one falls, and you can prove that in general if a domino falls, the one after it will fall, you can prove that the entire row of dominoes would fall. I think it would be fun to students to actually demonstrate this idea. It would even be fun to illustrate what would happen if you cannot prove that the first one falls by gluing the dominoes to whatever surface that you are using (not the table).
The idea would be to have it set up as the students walked in and ask them what would happen if you pushed over the first domino. After that test the hypothesis with one row (you should probably have multiple rows set up for this). Then introduce the concepts of base case and induction step using the dominos. Then you can ask well what if we cannot push the first domino over, does that mean we cannot show that all of the dominos will fall? After this you can start taking the concept of dominos and applying it to Mathematical induction.