How I Made Learning Logarithms EASY!

on Thursday, March 20, 2014
Teaching logs the past few years in my Algebra 2 class has always been stressful. I feel like the students never really understand the notation or how the pattern works... I have had other teachers tell me to teach them "little to the big", tried to connect it to the inverse of 10^x, tried to hammer translating from logarithmic to exponential equations, and it all ends in huffs and puffs of confusion and frustration... from me and my students.

Logs are dumb. Logs are weird. The notation doesn't make sense. I just don't get it.

I hate hearing these things! So I set out for a better way. A better way to teach and a better way to understand. And that's when I happened upon this gem of a video by Vi Hart.



I was inspired! Inspired to make a lesson that my students would not huff and puff at! So I took the big ideas from the video and made my own teaching script. The idea is to link students' understanding of regular number lines to a number line found by multiplying.

Start with reminding students that they know how a number line works. To get from one number to the next you take a step of size 1. If you want to do 1 + 3, you start at one and go three +1's. If you want to do 1 - 3, you start at one and go three -1's.


Not rocket science. 

Then you introduce the idea of a new kind of number line. Instead of a number line that works in an adding kind of way, we're going to look at a "times-y" or multiplying way.

So if we start at zero and label up to step 5, and have steps of size "x2", how does the ruler work? Well, start at 1, and times by 2 to get 2. Then times that by 2 to get 4, and that by 2 to get 8, and so on.


I asked them, how do the 5 and the 32 connect? And right away students see that 2^5 = 32. I asked, what does the 2 mean in our number line? They see it's the size of the step. What does the 5 stand for? It's how many steps you take! So in a "times - 2" number line, we can take 5 steps and get to 32!


This is a good point to have students try to explain why we always start with "1" above the "zero" step. (It's because any number to the zero power will equal 1!)

The next scenario to explore is this: Sometimes we know the size of steps we want to take and the number you want to get to, but not how many steps it will take us to get there. I draw the following on the board... If you are taking steps of "times 3" or x3, and you want to get to 81, how many steps should you take?
I tell students it's likely useful if we write this in a way that solves for x.... 3^x = 81 is what they usually come up with... but it still doesn't solve for x. So we introduce this new notation:
STC (System That Counts). So in a system that founds in a times 3 sort of way, you can get to 81 by... taking 4 steps! And it totally clicks! So I ask a few more, to make sure they understand -


And then come the awkward part of telling them you lied about this notation and that it's not really how it's written. But explaining how you only change a little bit of the notation to include the log doesn't seem to phase them.
After we go through the log to exponential equation translation ....
..... I take them through a bunch of examples.

You can explain the log(1/1000) by thinking of backwards steps or "divided by" steps. For log problems that have really odd fractional powers, I do not go into how this STC works, but Vi Hart does a great job of it and I might show my students that part of the video soon!

I know what you're thinking: "This really works? They get it?" YES! They do! After my first hour of teaching this, one of my struggling students said (no joke): "That's it? I think that's the easiest thing we learned all year!" The past couple of days this student has been helping other students understand logarithms!

I hope you can make learning logs easy for your students too! :)

Math Competitions

on Sunday, March 16, 2014
So some people may say that math competitions are really dorky - and having gone to one the past 3 years, I would have to disagree! My school doesn't really have a math club, but our local community college puts on a competition every year specifically for 9th and 10th graders. In putting together teams, I look for students who are reasonably good at math, have perseverance in problem solving, and are good at working with others. It's awesome to ask a kid to join who never thought they were good enough to be in a competition! It boots many spirits :)

Well, we went to a competition yesterday and my students earned 1st and 3rd place in their division! {And the 1st place team was 2nd overall!} Not too shabby :)

I would highly recommend finding a competition to bring students to. The first year I did this, I brought two teams. One of the teams earned 3rd and the other team didn't place. But they had a BLAST! I had a number of girls with me who talked to me about finding a competition they could do as juniors or seniors, or if there was a way to host a competition at our school! All of this after doing a day of math! :)

The Amazing Pi Race!

on Friday, March 14, 2014
Happy Pi Day!

This is my second year celebrating Pi Day with my students and I have had a BLAST every year! At first I didn't think that I could justify taking a day away from teaching, but I think it's important to remind kids that math can and should be fun, and we can do lots of really amazing things with it. Pi Day is a great day to celebrate that.

So, what do I do?

Well, about two weeks ahead of time I start talking it up to kids - letting them know we're going to celebrate it big time! Of course they ask if I will bring pie for them. I tell them: Are you kidding me??! There's 100 of you! But, you can bring in anything you want, and we'll eat it! 


A week ahead of Pi Day I created a sign up sheet for food. I suggest that they sign up to bring in circular or spherical food (for the fun of pi-day, of course) but if they bring in something that's not a circle, we'll still eat it. This year I did it using a google doc. I found out that no one will sign up for anything unless I have everyone take out their computers and look at the page and sign up.

On Pi Day people really go all out with food! Pie, cupcakes, circular tortilla chips, rings of pineapple, and so much more!

Also, being at a Christian school, we have pi-day devotionals and reach from 1 Kings 7:23 where there is an approximation of pi being 3.

While people are eating. we do the devotionals and watch a few random pi-day videos. These are good ones I found this year!
What Pi Sounds Like
Calculating Pi using Real Pies

Then after we have some time to sit, watch, and grab seconds we do the Amazing Pi Race. . . !

The past two years I have used this AMAZING activity created by Pam Burke called The Amazing Pi Race! She is wonderful and oh so generous to have posted her activity (and SOLUTIONS and HANDOUTS!) for all of us to benefit from! I hope you use it, because it seriously is a fantastic activity for students! I have used it in my Advanced Algebra 2 classes, and made an altered (easier) version for my Geometry class. I have a feeling the Advanced Geometry class could have handled it, but not my geometry kiddos.

I love that actually gets gets calculating things and doing math on a day that is considered a "fun day". It also has a fantastic element of competition and collaboration that gets people into the festivities!