blog reboot and getting outside

I'm trying a reboot of my blog. My goal is to keep posts short, update what is going on with what I'm doing as a teacher and to make up for the years of blog posts that I spewed disgusting technology and data focused ideas...

This year I've made it a goal to get students out of the classroom on a weekly basis to see the math that we are learning appearing in the world around us. 

We walked the perimeter of the school grounds which I learned (after being here for 14 school years) is actually about a mile. Our shoes got wet but we now have a good memory to connect with perimeter!

We went into the cafeteria to find the area and that looked like counting floor tiles and subtracting areas of the room that had closets or protruding sections of wall. 

We went outside to look for right triangles and determine if objects being measured represented right angles.

I'm finding the students are enjoying getting out to be active and the more often we do this I'm able to see the students' creativity in finding ideas present I wouldn't have thought of. 

#codemath: Student created code on simplifying square root expressions

Today as an extension activity for students that have already mastered simplifying square root expressions, I offered students a chance to complete a symbolic example using my #codemaths structure.

Here was the minimal prompt given to students in a unit google document:

Fill in the gaps in the worked out solution in this #codemaths project

I gave a short explanation of how the variable "a" is randomly generated and when they type [a] it will output that value instead of the letter "a". (I recently switched from <a> to [a] to allow for simple html formatting to be used in code output. <b></b> for bold, etc.)
Students only needed to "fill in the gap" of the potential worked out solution from their perspective on how they would solve it--filling in enough detail that would help a fictional struggling student viewing it as a potential worked out solution.

The existing code in the worked out section looks like this:

And will print out:

For the sake of not giving away the solution to the problem, I will just share screenshots of the student output from the #codemaths worked out solutions they wrote and emailed to me:

"Remember to give credit to the makers!!!"--Melissa and Katelyn

"Here mr. schwen this is my super awesome code"--Ken

Pretty fun application for the kids and little time is spent explaining writing code because of the simplified process and kids are thinking more deeply about the mathematics embedded in an activity like this at a pretty highly symbolic level.
One student, wide-eyed and excited, said to me after getting his to work that "This is really fun!" I like too that there is a trial and error aspect to this as they refine and revise what they think will output and what actually does show up when they test it out.
In all honesty, not all my students are doing this but for the kids that are ready for it, it turned out to be a great discussion and application of where we had been in our mathematics and a good preview as to how math can be used in an increasingly more coded world.

Open-ended discussion on "Rules of exponents"

So earlier this week I read this post by Andrew Stadel and was challenged to approach the rules of exponents in a more perplexing way. First of all, starting with the mistakes and what kids already know about exponents was a great launching point for this lesson. I made my own 4 question mistakes not really wanting to get into exponents of zero, negative values or fractions just yet.

The following statements are all INCORRECT.1. Identify the mistake(s).2. Correct.3. Justify (show) your reasoning.
1. 43 · 42 = 165
2. (34)2 = 36
3. x3 · x2 = x6
4. x2y3 = (xy)5

We spend a large majority of the time on question 1 in the last hour and somehow managed to get to fractional exponents. Different from earlier classes today, I started my last class by saying there are two levels of answering these questions:

1) You grab a calculator and find the value of each side and say "These numbers are different, so it's wrong".
2) You apply what you already know about exponents and come up with an explanation that someone who knows less than you about exponents can understand. 

For number 1 I had students say that the value should be 45, 210 and  322
The fun came in asking students to explain why 45 was the same as 210 in visual form. Most at first said that because you take half the base number you must double the exponent but we needed to know why to develop their understanding. 
One student explained by pairing up the two's to make 4's by multiplication and stating there were 5 groups of these pairs--the whole class applauded his explanation after gasping that they got it.

So then we went to why 45 was the same as 322 and a girl student did this (again applause from the class followed):

Realizing we were getting dangerously close to logarithms in my 6th grade Algebra class I pushed the envelope and asked what the exponent would be in 8^x = 45  and they jumped right in!! Some students naturally assumed it was 2.5 and stopped but others weren't satisfied. 

A girl student in class offered to come up and draw a picture of what she thought was happening and we got this to look at as a class:

Her picture was an excellent launching point for discussion in thinking about what to do with left-over factors when we try to group to get factors of 8. Her mistake actually brought out a big difference in fractions vs fractions as exponents when another student tried what she said in his calculator. The class came to consensus that for the exponent, since it would take 3-2's to make another factor of eight the exponent should actually be 3 1/3 instead of 3 1/4. 

Wow. Great class, fully engaged students and some great thinking going on driven by the students.