Math as Power, not Punishment! The need for Logs.
Was just at #MICTM16 this week and got to see Mr. Dan Meyer speak about creating mathematical headaches before giving students the mathematical aspirin. He even went as far to say that math should be like an infomercial because they do such a great job at selling. (Look up pootrap for a classic infomercial.) Dan pushed us to send him ideas on how we do this in our classroom and so here is an idea that I use in my classroom. Now, I definitely stole this idea a few years back from someone on the #MTBOS but I can't remember who it was. So if anyone knows who had this original idea please let me know so I can give credit to him/her.
I've used this both in Precalculus and in Intro to Statistics. Every time we have great conversations about how organizing, graphing, and understanding data is a big deal right now with big data. I start by asking students to give me a list of countries, and then students look up population numbers using WolframAlpha. I love using WolframAlpha for this because of all the other information that it gives and students really get into some of the other things that come up. The list of countries always varies but it usually contains either China or India, and a smaller country like Sweden or Denmark, or the person I stole this idea from put up Vatican City to really force the issue. What you need is a wide variety of data. Students then graph the data, by hand, and I stress that I need a SUPER accurate graph and the values on the y-axis must be consistent. Students ask if they can use the squiggle mark (what's that thing called?) on the y-axis, or ask for another piece of graph paper so they can fit in China or India. Both of which I refuse to give. Some students just jump head first into graphing, others just sit there and pause because they know they can't do it, but after awhile I grab some of the graphs and throw them up on the doc cam and have some rich discussions. Almost always the students are using a linear scale, but sometimes there's one student who uses an exponential scale. This is great because then we go back to how the y-axis must be consistent and students argue that it's not consistent even though we have usually just talked about exponential functions and even have laid heavy ground work for logarithms. Vsauce has a great video (1, 2, 3, 4, 5, ....., I'm sure you've seen this) that I use to help with this foundation and turning their thinking back to multiplicative that is more natural anyway. After the students come to terms with being able to count multiplicative-ly on the y-axis we then talk about how the numbers are too large to write down and that in math we can use a LOG! (Grumbles here since they all have bad experiences about logs from previous classes.) We then take our data of countries and their populations, and add another column by taking the common log of the populations. Then I let them sit and think about what these numbers mean. It doesn't take too long for students to figure out that the number is based on how many digits which leads to great decimal system talk, but then I pose the question, what does the decimal mean? This takes some thinking and more guiding (depending on the level of the class) to get the students to see that its how close the number is to the next power of ten, in a multiplicative sort of way. This again ties back to the VSauce video. So for instance Sweden has a population of 9 million, which log(9,000,000) = 6.954 and India has a population of 1.29 billion and log(1,290,000,000) = 9.11. So Sweden is closer to 10 mil than India is closer to 10 bil. Obvi. But usually we have the US on the list which is the perfect country for this because the population is 322 million and log(322,000,000) = 8.508. This means that since population growth is exponential the US is halfway to a billion people!! Then all chaos ensues!
I love this activity because it takes out the "mathemagic" when you take a number and hit the "log" button on your calculator. I also do a different activity for "sin" or "cos" on the calculator and getting another weird decimal as an answer. I really think that some students think that these functions on the calculator just randomly select these decimals when they really come from pretty simple ideas. I guess I'll have to write another post about this topic as well, but this is my first blog post in two years. I don't want to pull a muscle.