On Friday, I started exploring Translations with my Math 8 class. The idea that I had set for the day was to get them to see how a translation affects the points when a shape is plotted on a coordinate plane. All students were familiar with translations from previous years, so I decided to start this by playing a game. I asked students to give me two translations, any direction and any number of units, and then asked them to give me any point. Almost immediately after they gave these things, I wrote down the new ordered pair for the translated point. For example: if they said move left 7 units and up 4 units and gave the point (4, -2), I then wrote down (-3, 2). I then had students perform the translation with patty paper and write the ordered pair for where they ended up. We checked if they matched. I then asked “HOW did I do that?”. I demanded the students look for a pattern between the point we started with, the movements that were asked for, and the ordered pair we ended up with. You could see the light bulbs start flash on. Many students began to notice that I simply subtracted 7 from the x-value and added 4 to the y-value. I then proceeded with questions like “How did I know to subtract from x?” “How did I know to add to y?”. Asking students to look for the patterns in the math that I did worked a lot better than teaching the rules and asking them to memorize. Even though I did the math, the students were put in charge of identifying the pattern and developing a rule.
Every year, I ask my students the same question at the end of a math problem. “What assumptions did you have to make in order to solve this problem?”. It has always been one of my favorite questions, because it gets students questioning a problem and wondering if the answer they get is limiting those situations. Today one of those problems came up.
We looked at the following problem, and worked through it. And got the answer dictated to us by the Teachers Manual. Almost immediately students began questioning these answers. Not whether they are right or wrong, but if those were the only ones. Some student responses…
“What about when the ball hits the ground?”
“What about negative distances?”
“What about wind?”
“What if it’s deflected by another player?”
“What if players within that range aren’t in the path of the ball?”
“What if the defender is behind the quarterback?”
All really valid questions, and all assumptions we had to make in order to solve this problem.
We finally settled on an answer of: between 0 ft and 3.09 ft and also between 102.17 ft and 111.85 ft (when it hits the ground), assuming that the players are in a position to be in the path of the ball.
Today our class explored finding volume of a cone. To do this I first showed them a cone and a cylinder (of same radius and height) and just had them list similarities and differences. The things they came up with were awesome: the cone and cylinder are both circular, they have the same size circle for a base, they are the same height, the cone tapers off to a point, etc. The next question I asked is “Which one has more volume?”. When some students responded that they have the same volume because the radius and height is the same for both I tried to remedy this by placing the cone inside of the cylinder so they could see it doesn’t fill up the same amount of space….still weren’t seeing it. (any ideas why? Because I thought for sure that would prove to them the cone has less volume). Anyway, I let them stick with their original conjectures about the volume of a cone as compared to a cylinder. I used rice and I filled up the cone to the top and I poured it into the cylinder and just asked the students if that filled the cylinder. I FINALLY got them to see that clearly the cone and cylinder are not same volumes. Then asked them to make a guess at how many scoops of rice from the cone it will take to fill the cylinder. Most students chose 2 because they picture the cone in the cylinder and thought with the left over space they could make one more cone. I’m thinking this is because they’re viewing it from a 2-D angle and not from a 3-D perspective. The students really gave some awesome responses. I continued this process and allowed students to change their estimates (which they call estimation180) until the students realized that it would take 3 scoops of the cone to fill the cylinder! The students were blown away that the extra space actually creates two more cones, not just 1 more. It was a really cool activity that allowed students to visually construct the formula for volume of a cone. Sorry no pictures (i’m terrible at remembering to do that). If you’re interested in the activity sheet I used, let me know and I’ll drop box it.
I haven’t blogged in a while, and frankly can never think of something interesting enough to blog. Or when I do, I completely forget and then cannot find time. Every once in a while though, I teach a lesson that I want to share with others in hopes I can get insight on how to make it better than it currently is.
This past week with my Algebra 2 students, I started teaching Applying Linear Transformations. In the past, I’ve always just flat out taught the rules to applying them. Mainly because I, myself, didn’t fully understand it all and the book did a terrible job at helping me. (as did many internet resources..granted I never looked too hard). Last year, I thought I had THE lesson for teaching it and explained to students that equations are not written in terms of x, so horizontal movements must take that into account. At this point I had students tediously re-write equations in terms of x, apply the horizontal transformation, and then rewrite for y and look for pattern to make this easier. It went well for some, but confused many others. The time it took to perform this did seem worth the outcome.
This year I tried something different in hopes the students would finally get it. I gave them a real context problem “Verizon Wireless offers a plan for $15 per month and an application fee of $50.” and gave them a situation to model. For example: “Verizon give you the first 2 months free” — students were able to identify this would be a horizontal translation right (due to a previous lesson). I then posed the question: “since this equation is in terms of cost, how will two free months affect cost?”. The answers I got were incredible. In the past where I’ve had to show them different ways to think about it, they were coming up with themselves. By the end of the lesson, I asked students to write their own rules for each transformation. Again, I got some really amazing responses. A few of my favorites were: “because it’s in the form y = , even if the transformation is related to x, we need to think of how that will change y.” and “to perform a horizontal stretch or compression we can just multiply x by factor a^-1.” It felt good because I literally taught nothing, just used questions to guide them to a conclusion they made on their own. I dropboxed (i made it a verb) the documents I used. I apologize now as my class responses are still on the doc, I’m just too lazy to reset it right now.
I’ve recently made myself a part of the #MTBoS. I skipped the week 1 blog post, because every time I sat down to write it, I couldn’t find the words to describe what made my room unique. The most unique thing I could think of is that I teach my boys how to shake a hand and tie a tie and help them turn into young men. I’ve had a really hard time blogging, as I keep trying to find the perfect words, or perfect scenario to blog about and it’s just not coming to me. However, this week I can definitely comment on as I just started using twitter professionally and it’s changed how I’ve been teaching.
First off, since using twitter, my eyes have been opened to all the programs, websites, and tech options that are out there for teachers. I discovered DESMOS for crying out loud! I’ve incorporated DESMOS in some way/shape/form into my lessons since I’ve found it. What an easy way to teach transformations of functions. I let the students play with the sliders and make conjectures of their own. They are in charge of their learning.
I’ve also discovered Geogebra which completely transformed how I teach the geometry unit (get it…transformed). The geometry unit was new to us this year and through a huge wrench in my system. Initially I was having the students graph and perform transformations to see the patterns…which became completely irrelevant if they performed it wrong. Once again, I used Geogebra to pose questions about what they think will happen, performed the transformation, and they make a conjecture (i hope i’m using that word correctly). They are in charge of their learning.
Probably one of the best sites i’ve discovered is Estimation180. My students come in every year with little to no number sense. This site has been a great way to briefly build in number sense every day. And best of all, my students LOVE it! Just the other day, we did an estimation on the number of pieces of paper in a ream. Just as I was about to reveal the answer, I told the students “I’ll show you tomorrow”. This was greeted with a unanimous NOOOOOO! It was great to see the students excited about something math related.
Finally, and probably the best thing that Twitter has offered, is a connection to other teachers. I’ve gotten a view into the classrooms of teachers from all over. I’ve been able to see some great things that are going on in classrooms. I’ve gotten tons of ideas on how to improve my teaching. Most importantly though, I’ve gotten to see and hear that I’m doing things correctly. That I’m going about teaching in the correct manner. Twitter has rejuvenated me, and made me excited to go in and teach and try new things. I’m excited to get the chance to share some of my ideas, and hopefully return the favor.
So i’m new at this blogging thing and just got myself involved in the MTBoS recently. This is my first post, and I know I was supposed to follow Shah’s prompt, but I started talking today about a question I had that led to a teachable moment and got excited enough to post about it.
First off, my name is Hunter, I teach 8th grade math, and I am by no means a professor. I used to have my students call me by a different name every week (depending on what I wanted to be — Jedi Patton and Post Master Inspector General Patton was my favorite) and professor stuck because the students liked the alliteration. I’m in my 6th year of teaching and to be honest, with the curriculum change, this year has made me question what I do.
In my Algebra 2 class we are working on parent functions so that students can place a set of data in the correct family, model its behavior, and use that to make accurate predictions. We were working on a problem about wave height and wind speed. — One of the last questions is “Why might comparing wave height and wind speed lead to inaccurate results?” – The places this question led us was awesome. We got in to correlation coefficients, underlying causation, does more than just wind speed create waves. I had students wondering does temperature cause larger waves since Hawaii has large waves. I got to talk about tsunami’s and energy transfer. Why we don’t see tsunami’s when they are further off in the ocean. Why do Earthquakes cause large waves. Does a butterfly’s wings in Japan cause a HUGE wave here in the States? Why the graph “bends down and starts to flatten”. Would a linear graph be an accurate predictor? How energy needed for a large wave is much greater than energy needed for a small wave. A co-teacher of mine happens to be an avid surfer and was able to answer the questions I couldn’t (like why Hawaii has larger waves). I had them excited about a problem for parent functions and eventually was able to tie it all back into what our objective was.
Anyway, this got me thinking about how teachable moments are missed EVERYDAY by teachers. Our conversation included very little about parent functions, but I had the students excited to learn and engaged in an intelligent, thoughtful conversation. And by the end of the lesson we covered everything needed for parent functions. I try to take every teachable moment I can and teach them about ANYTHING. Life, math, science, myself, etc. In the long run, I find teaching them about anything makes them more excited to learn anything. It kind of put teaching into perspective. Find a way to teach a topic that will force students to “create” a teachable moment for you.