Log Graphs

I was recently inspired by this tweet from Alex Overwijk:

I created my own set of logarithmic graphs - ten with base 10 (white paper) and ten with base 2, 3, 4 or 5 (green paper).

My students, in random groups on big whiteboards, had to determine the equation of the graph they chose. All groups started with base 10 graphs (many stayed with those for the whole activity) and checked their answers on Desmos (and with me). I had printed two sets so they had plenty of new graphs to choose from when they correctly identified their graph. I didn't give them a lot of hints as I circulated as I really wanted them to find strategies to help determine the equations. They were using what they could see about the graph (vertical asymptotes, for example) to help map points from the parent function to their graph. I was really impressed with their efforts. I did decide to write the base on the graphs that were not base 10 as I thought it was too much of a step up at that point.

Here is some of what I saw:

Here is the file. Thanks for the idea, Al!

My plan is to revisit these graphs when we solve logarithmic equations which will allow them to use algebraic skills to help find the equations without having to know/guess which point was mapped onto each new point.

Log Clothesline - My Post-Activity Post

Yesterday, I ran the log clothesline activity with two of my classes.

What I liked:

• The great math conversations. They had not worked with logs for a while now, so some remembered a lot and some, well, not so much. They were talking to each other about how to get going with tricky expressions at their desks initially and the conversations continued when they were at the clothesline trying to place the cards in the correct order.
• The collaboration. Those who couldn't even remember how to go from logarithmic form to exponential form got help in their groups. They asked each other questions, they checked each other's work and argued about who was right. 
• The struggles. Apart from the one card with a typo (oops!), all of the others worked out. However, I made this a no calculator activity (horror!) so they had to work smarter to reduce the amount of arithmetic required. Those log rules really became useful.

What I would change:

• I found this activity worked much more smoothly with my smaller afternoon class. I only had 13 students in that class yesterday and everyone was engaged and busy. My morning class, which had 24 students, had more "traffic jams" around the clothesline causing other students to step away and no longer be engaged in the activity. I would put up two clotheslines for a larger class next time. 
• I also liked having extra cards in my afternoon class for those who finished quickly, so I would make more cards next time or have two sets for a larger class.
• The change that would have the biggest impact, I believe, is adding more number markers to the line. I had a 0 marker, but the struggle to correctly arrange all the expressions on the line was greater than I had planned. I asked my afternoon class if they thought having more number markers would help, and they unanimously said yes! I loved that some were using the small whiteboards to work out the expressions from other students' cards that were already on the line, but it became overwhelming.

I definitely think this activity was worth doing and that all students got something out of it. Here is the final clothesline:

Log Clothesline

There has been quite the buzz around clothesline activities of late. There is even a clotheslinemath.com website! I first heard about it from Andrew Stadel when he wrote this post. More recently, Jon Orr made a cool one to practice finding slope. Here is a link to his blog post. I thought that it would be great to use a clothesline to practice evaluating logs, but was having trouble finding the time to actually create it. Then we got a snow day (school buses are cancelled, but teachers still have to report to work... all of the students at my school take the bus so we have a day with no students) but I had other things to do (like meetings) and we were allowed to go home early (we got 51.2 cm of snow by the end of the day!). And then, shockingly, they cancelled the buses again today! So here is the log clothesline for you. I plan on actually doing it with my two MCV4U classes on Friday and and will snap some pictures then. I did take a "fake" one of a colleague placing a card on the clothesline.

Here is the Word file, and the PDF version is here. And I'll even give you the answers. I numbered the pages so those correspond to the question numbers, which I have sorted in the Excel file so that I can quickly check for correctness. I printed two pages per sheet and cut them down the middle.

I will give out one card to each student and they will have to evaluate their expression and check at least one from someone else in their group. Once the whole group is confident in their answers, they will go to the clothesline to place their expression in the correct location. I will give them a 0 marker on the number (clothes)line. The trick is that I will tell them that they can't write on the cards so in order to figure out where their card goes, they will have to work out a number of other expressions. Those students who put their cards up first will be responsible for ensuring that all the cards that follow are in the right place.

I will try to write a "how it went" blog post after I run the activity.

Postscript

After hitting publish this morning I started to think more about this activity and started questioning whether it met the spirit of clothesline activities. I worried that as it doesn't really work students' number sense, it might not be an activity worthy of sharing. I sought advice and, though we agreed this would be better suited to an intro to logs activity (more on that in a minute), this practice, with built-in error analysis and collaboration, was worth sharing. 

Back to the intro to logs idea. This is what I envision: students get cards with powers of 2s from 1/64 to 32768 (or something like that, depending on the number of students in the class) and they have to attempt to place them on a clothesline that will have markers of 0 and 1 on it (not as shown below). It would look something like this:

Hopefully it will become clear that there are too many cards between 0 and 1 and that the larger numbers simply cannot fit on the clothesline. So what to do? I'm not sure how to introduce the idea of taking log base 2 of each of the numbers, but that will be the goal. The result will be a clothesline with a logarithmic scale which will allow all the numbers to be seen. I won't actually be able to do this until next fall, so please let me know if you try it out!

Log 5 X 5

If you have yet to read Sara VanDerWerf's post about her 5 X 5 game go here and read it now. It sounds awesome, doesn't it? I didn't read it until I was already on break, but I look forward to playing it with my students at some point down the road.

At a very early hour this morning, I thought that this game could be modified to help students practice working with logarithms. Instead of having the numbers 1-10 in their grid, they could have logs. They would still place numbers of equal value next to each other, but some would look like log₂8, others like log₃27 and others like 3. They would also still need to find the sum of squares that match. I would have them write the sum of each set of matching expressions as a single logarithm so that they would need to use their log rules.

Clearly, I need to actually work through this, but I wanted to get it down before I forgot. I think you could also do this with trig expressions (sin π/6, cos 5π/3, etc.), but have thought that through even less than the log idea.

If you have feedback, please let me know. Ideas to build on or telling me why this is a bad idea are both welcome :)