What started with cubes ended with unfolding the cubes and adding an additional crease into the mix to allow us to pull off a stellated octahedron.
Algebra went through a couple different Shikaku puzzles which can be found here.
Some students were FAST at these so I pulled up a pretty difficult one, which took a group of three around 10 minutes to solve.
I am going to take a break for the blogging till school starts up next year! I enjoyed these activities as day-before-christmas-break-oh-gosh-what-will-we-do activities to keep students thinking and engaged. Let me know what you did in the comments below!
When I gave my students this, most of them quickly said 78. We talked about which problem they had just solved and how the song went. They came up with tables and other crazy ways of pairing numbers, it was fun.
There was moaning when I asked how many total gifts there would be if it was The 25 Days of Christmas. Some students were already working towards The X Days of Christmas, by pairing. Most classes ended after finding the 12 days….
But I was on a mission to find The X Days of Christmas.
I pitched this problem to a few colleagues, I threw together the following table:
Constant third difference indicates cubic…
One colleague put the ordered pairs of (Days, Total Gifts) into a calculator and fit a cubic regression to it. Another colleague came up with the following (through some magical hand waving). They both ended up simplifying to almost the same equation.
Long ago (like two years), I had a modeling class in college where we messed with these sort of things. I pulled out my book and found it… its called Newton’s Difference Formula, the powers at Wolfram Alpha say this about it.
I am not quite sure where this comes from… but the deltas are the differences, which is exactly what my colleague came up with. This problem turned out to be DEEP. I always try and rethink a problem if I find myself going into calculus, but this time I needed it to come up and understand The X Days of Christmas.
I dug around some and came across these fun resources relating to this as well.
Today, algebra reviewed finding the equation of a line when given two points for a concept test tomorrow. A few months ago, I came across this post over at Everybody is a Genius. The idea had been sitting in my bookmarks for way to long so I gave a try.
I used pretty much the exact format explained in the post. Random.org was my go to for generating the numbers and I created my own template for the board.
As a class we worked through a few of the difficult ones with fractions as y-intercepts as they came up and took some time to discuss ordered pairs with x-values of 0.It is true that by the end of the period students were begging for just one more question.
Today in geometry we started incenter/circumcenter, here is a problem one of my colleagues shared with me; I really liked it. So did my students.
I passed out a paper with the same text/image on it. Here it is. Almost all my students were about to locate an point to place the tower, however most of them did this by guess and check. When I asked them to explain why that point made sense they took their compass and drew a circle, indicating they were equidistant.
I lost ’em after this. This seems to happen quite a bit, I need to think through followup and extension questions. Students pushed back when I asked them to explain how they could use math to help make sense why the point they chose made sense.
Tomorrow we are going to follow up by creating the circumcenter by finding the intersection of perpendicular bisectors and look a little deeper into the congruent distances from the three cities and what sort of triangles are created.
In algebra we went over graphing lines, nothing really fancy here. I am looking for short activities relating to this concept that will get students into the swing of things but so far haven’t had in luck in finding those.
Geometry went over some SBAC openers, which I thought were fun. They did really well on them and we had some good discussions over multiple solutions. I threw in a SAT question at the end also… When I was in high school the first time I saw an SAT question was on the practice book, this one tied in well with the concepts we have covered this year.
It is inevitable that students are going to take the ACT or SAT in my school at some point so a little exposure to these questions is a good thing in my opinion. Variety is good.
Anyways all but one student in my class said the answer was D, which is the answer to a question, just not the one the SAT folks were asking. I really regret not firing back and asking the class what question D was the answer to. I will keep that in mind for next time!
My school had another incentive day today, plus it was a half day. I posted about the last one of these here.
I had a nice discussion with a colleague yesterday about interacting with students…
A lot of times when I ask students questions about there work it is because I see a mistake. The colleague pitched the idea of asking students to explain their correct work at times, instead of only having them go over it when there is a mistake.
This short conversation was really helpful for me.
I get so caught up in things at times that I don’t even realize some days every conversation I have with students revolves around picking and questioning their mistakes. I haven’t had them explain why their work is correct much. Ever.
I want to work on creating positive interactions with students more often.
I let them think about it for 5 minutes or so. After I asked them for smaller cases. We looked at 2^8 vs 8^2 since some were claiming whichever number had the larger base would have the larger value.
After a few more minutes of discussion they plugged them into their calculators and checked their claims as to which was larger. The issue was the answers were all in scientific notation, sure it says a lot about how much larger one is than other, but students were not able to see how larger the numbers actually were.
So I pulled up Wolfram Alpha.
What students found most interesting was the number name. I had no idea these existed… We tried to find where a google would be….