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.

I

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.

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

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

*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! :)

## 2 comments:

I love this! I'm wondering if you have continued to use this method of introducing logarithms? I am also interested in this idea for rational exponents. Just curious of your experience. Thanks!

@Trix -- I still use this method! I love it! And I am still always shocked when students say how easy logarithms are after I do it this way.

I hadn't thought of using it for rational exponents -- It seems like it would be a great way to help get students used to this multiplying idea! Let me know if you end up using it and how it goes!

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