Teach 180: Log Properties by Discovery (Day 108)

When I was in high school, I was content with being told certain rules in math.  I could either see that they were true for myself or I willingly accepted them as true.  We rarely, if ever, discovered rules or were asked to make and test conjectures.  Discovery, reasoning, and sense-making is more common in my classroom now than when I first started teaching, but it can be easy to fall back into pulling notes out of a file cabinet and just working through examples of concepts with the students.  If discovery is better for learning, why not just create lessons based on noticing, wondering and discovery?  Two main reasons: time and collaboration. With inadequate time to plan and lacking collaboration with a colleague, the chances of moving closer to discovery are greatly reduced.

This year I decided to make the time to work with a colleague and change our lesson on logarithm properties.  I had reached out via twitter asking for ideas and got a response back from Ralph Pantozzi (a.k.a.@mathillustrated).  He gave my colleague and me something he had used with his students and that was where we started.

These were the objectives for the lesson:
1) Students will make observations about numbers from a table of values for x and f(x), where f(x) was log₂x.
2) Students will use the observations to create the product, quotient and power property for logarithms.
3) Students will practice using the properties to combine and expand logarithms.

Page 1

Page 2

Page one shows part of the full table of values we gave our students.  Page two was what we had planned to give to our students, if they had trouble coming up with some ideas.  But they had so many ideas!!! We didn't need to give them the page with the questions.  They just took off on their own and our full class discussion led to observations about each of the log properties. Photos of two of the boards from our discussion are shown below.



At one point, my students asked if what we were doing would work for all bases.  For some reason, I didn't anticipate that the students would ask this.  I didn't have a spreadsheet set up where we could change the base to be a different value.  Was there a way to test to see if what we were doing with the sum and product would work for all bases?

Desmos to the rescue!  I instructed students to type in two expressions in Desmos: log_bM + log_bN and log_bMN, where b was a base of their choosing and M and N where numbers of their choosing.  Students could see that the results of both calculations were the same!  I made expressions with sliders for M and N to show that changing values for M and N produced the same result. The screenshot here shows something similar with the quotient property.

Based on our results from the product property, we derived the power property. We also reviewed the values of log_bb and log_b1.  Next, I had the students type f(x) = b^log_bx into desmos. Students were instructed to choose a base for b and a value for x. Why did it make sense that the f(x) always equal to x?  Why was the input value the same as the output value?  Does this happen for all values of base b? I created a table of values and a slider and when the function b^log_bx was plotted, it resulted in a line.  (See the short screencast below.) This led to a discussion of the fact that b to a power and log base b were inverse functions.  So, it made sense for them to undo each other and for f(x) to equal x.

So, what lessons have I learned about moving closer to having students learn by discovery, by noticing, by wondering?  Two things: First, it's ok to not plan for everything.  I didn't think that my students would ask about other bases, but we were able to quickly examine a multitude of bases.  Second, don't be afraid to let the students take control of the lesson.  They have the ability to notice and wonder, if you let them.  Their insights might surprise you.

Discovery can come naturally.  Learning logarithm properties by discovery made the lesson more fun for me to teach and it made it more interesting for them to learn.