Hello friends! It has been awhile since I last posted – over 2 months – whew! Life gets busy. I am starting a unit on Exponential, Power and Logarithmic Functions in my Advanced Algebra Class. Last year on a whim I created some visuals that I loved. I thought I might be alone, but at the end of the school year when I debriefed the year with the teacher I co-plan with daily – Morgan Fierst – I asked her what her favorite thing we did all year was and she named the pattern in this blog post. SO, as we start this unit again this year and work to visualize the mathematics in this unit even better than the year before I thought I would share with you our journey (so far).

Last year I started my blog shortly at the end of this unit (soon I’ll be celebrating my 1-year blog-o-versary – that’s a thing, right?). Two of my personal favorite posts from my blog – though not ones you all read according to the metrics on the back side of the bog – were titled ‘Quick Thoughts: Exponents‘ & ‘Quick Thoughts: Logarithms‘. You should read these 2 posts because through the work of Andrew Stadel, Michael Fenton, Kate Nowak and others I changed how I talked about exponents and logarithms. Note – this blog post – ‘Add-Em-Up: My favorite Review activity‘ – is also from this unit. This review is soooooo00 good too.

One part of the unit I did not blog ago came from the screen shots I took at this site that visualizes numbers starting at 1 and on up by breaking the numbers apart into their factors visualized by dots. **STOP** what you are doing and click **HERE to see the magic of STEPHEN VON WORLEY’s work**. Pretty cool – huh? I loved it. I started by making some poster’s on 8.5×11 inch paper to hang up in my room. You can find these posters here if you want them:

Word Docs: dot-posters & dot-posters-with-numbers & powers-of-3-pictures

I then started arranging them into sets to use in pattern talks with students.

**As I was arranging them, I arranged them into the following 5×5 grid. **

I showed my friend Morgan and told her that I thought we should do some noticing and wondering (thanks Annie & Max) with this visual.

- Word Doc for noticing & wondering: what-do-you-notice
- This is the one we printed in color for students notebooks: (word doc: lots-of-dots)

We did lots of noticing/wondering with this pattern and students started noticing some things in the patterns. All this noticing & wondering by both students and Morgan & I surprised us. We had never seen the connection between exponential and power functions like we did when we spent time with the visuals above. Slowly – over days – this visual morphed into one with numbers and finally equations for each row and each column.

**The dots became numbers….**

**The dots became numbers with exponents….**

**Then we explored patterns within columns….**

**& Rows….**

** The dots & numbers with exponents became rules….**

Pretty cool stuff. I don’t know about you – but I had not really seen the connection between these rules in this way before. I think why Morgan & I loved this pattern so much is the discussion that students had about this 5×5 pattern and how we kept coming back to it.

Here are some ways I pulled these images back into our work with exponents and logarithms.

- When we were solving equations involving exponents in different bases, the visuals were key. For example – here is a problem we gave in class…. and then we gave the same problem using the visuals we had used earlier the week prior in class…. Convincing students that to change 81 into 3 raised to the 4th power was a breeze with this visual.
- Here is what we are hoping will help a bit with logarithms this year. We have not used these visuals before. I will update you with what I learn.

We think there will be some cool conversations about bases & exponents and how that connects to logarithms. We’ll see.

Here is a Word Doc that includes the Logarithm Dot Pictures: visualizing-logarithms

That is all for now. I love hearing from you. If you have ideas for Morgan & I for doing a better job teaching Exponents, Power Functions or Logarithms to our students tweet us @saravdwerf or @MsFierst or comment below. More soon (I promise).

**Side Note:** I came across this link in tweets today researching a different article I was writing. More pretty cool stuff. From Mathhen comes these 3 GIFs:

I have a slightly less complicated array — but I put a line of “to the zero power” along the top, too (leaving the zero to zero out of the picture). I have amendments I want to make but so many times powers have come up and I’ve just been able to point at a giant visual 🙂

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