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## Graphing Sine and Cosine – Graffiti Wall

Graffiti Graph-iti Wall. That was an amendment made by kids.

Students had filled in their own unit circles (measuring the angles with protractors etc.), created charts to help them easily identify the sine, cosine, and tangent of all the angles, and worked out several unit circle problems.

Next came graphing.

We began our journey with this “guided notes”/discovery activity.

I won’t lie – the intent was students would move seamlessly from their unit circle into this x/y table and plug in and graph away. They would work individually or with a partner discovering and having a grand ol’ time.

Uh no.  With my two lower level classes I did need to provide more scaffolding. I started off with a simple x/y table of a line and reminded them how we chose the values for x and so on. Then I placed this table on my smart board and explained I had chosen my x  values as certain angles around the unit circle and we used the unit circle to find the y values. I also drew a “regular” x/y plane with 1,2,3 but then explained those are not my x values so it doesn’t make sense… and so on.

The students did a fantastic job after the appropriate scaffolding and continued the project for the rest of class and then for homework. The next day we practiced graphing with the Graph-iti Wall. All around the room I had placed different charts like the one below.

Students began at one station and were timed for 5 minutes. Everyone had a marker and everyone had to write. Everyone had to write, even though they were writing the same thing. They did as much as they could until the timer rang (or the music stopped, I couldn’t get my music working though!) and then faced the center of the room. The next task was to head to the next station and check the work completed before them. They had to actually check off  each answer and again everyone was checking off then they could move on from where the last group left off. They rotated around to about 4 stations and then were sent back to their original to see what happened.

There was a lot of writing – in fact you could barely make out the first group. Hence the graffiti idea. It’s an awesome activity for anything  multi-step and I look forward to doing it again!

Students got the amplitude and vertical stretch/shrink DOWN after this activity – which made teaching period (horizontal stretch/shrink) easier without the confusion of the “inner” number and “outer” number.

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## Surprised! (I suppose I should just start expecting the unexpected)

I focused on surface area this week with my juniors in my catch-all-test-prep-this-should-definitely-be-illegal class. (To put my two cents in on forcing students to take Algebra 2 as well as a PSSA prep class… well a post for another time I suppose).  The past two weeks we were covering area and perimeter and I was astounded at the lack of understanding. Many misconceptions were evident but the most alarming was the inability to use the test prep formula sheet. For example, many students were unable to identify the triangle on their paper and match it to the triangle on the formula sheet and thus find the correct formula. I thought it might be because they were using a right triangle and the height was measured different, but when the test came and I matched the shapes to their formula sheet I was still surprised at the number of students who could not match the shapes. Thus the test on area and perimeter became a test on something totally different.

Only about 5 students in each class were still struggling with identifying parts of a shape and the rest of the students were starting to get frustrated and bored. I decided to move forward and try to squash area misconceptions during surface area, since we would still continue to use the area formulas for the shapes again and again.

Why I believe this unit worked so effectively had to with the use of scaffolding in their warm-ups, manipulatives, and lots of classwork practice time with limited homework. For each day I focused on two shapes and had the warmup focus on finding the area of the 2-D shapes that would ultimately make up the 3-D solids. For example, for the rectangular solid their warm-up was to find the area of three rectangles that later in the lesson we would put together to build the 3-D solid.

I made sure that for each shape they saw the deconstruction of the solid into their familiar shapes and I had cut out my own nets and taped and colored them to pass around the classroom for further understanding. What captivated the students most was asking them what shapes they believed made up the 3-D solid. For example with the cylinder they all noticed the two circles for the bases but most did not guess that the “middle” was actually a rectangle. When we got to the cone it was pretty cool how excited they were to guess what the 2-D shape would be. I used that as a great moment to say, “This is a funky shape- which is why it has a funky formula.” So the students didn’t really moan and groan at the obnoxious cone formula because they saw (sort of) where it came from.

Finally I made sure to have a practice sheet for the last 15 minutes of class everyday. I wanted students to practice the formulas in front of me and immediately after learning about it. I also read this blog about Results Only Learning Environment and the role of homework. Since homework completion and cheating are two huge obstacles to overcome I decided to focus on classwork and if students needed more time they could complete the rest of the worksheet at home. This way I could monitor who is doing their own work, the struggles they are having as well as ensure they are getting the practice I desire.

Overall I was very pleased with this unit and hoping to follow some of these teaching themes that worked to help me in the Volume Unit I start on Wednesday.  All suggestions, resources, and what you’ve done in the past for surface area and volume would be greatly appreciated as well!

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## Discovering Pi and Circumference Formula Project

It seems like forever ago but I wanted to finally blog about the success of the Discovering Pi/Discovering Circumference formula project I did with my students.

I used this as a warm-up to squash any questions students may have had about vocabulary etc.

That took about 5-10 minutes for us to answer as a class and reinforce the correct answers and why. I also made a point to let students know that they should pay attention or take notes because they would need to know all these things to successfully complete the project. I also explained that this project would count as a quiz grade and an opportunity to boost their averages. These were all (bribes?) incentives for students to engage with the project and do their best.

I handed out the worksheet and read the directions out loud. I also read the rubric out loud for students with a strong emphasis on the FOCUS ON TASK points.  I passed out supplies or in some classes I let students help me pass out supplies. We don’t have enough scissors or glue so students had to share those but each student got 2 different sized pre-cut circles and two different colored pieces of yarn.

I would probably not use yarn if I were to do this again, it stretches and makes a lot more opportunity for measurement error. I used yarn for a prettier poster option but in hindsight something without stretch would be better.

I also put a giant grid on the white board for students to put their name and their two decimal answers up on the board when they completed their calculations and before they could get construction paper. In the last 10 minutes of class we looked at the results on the board and at least one student in each class recognized we were awfully close to pi.

I then wrote C/D = pi and then solved for C = pi*D, and then we discussed radius and diameter and substituted and got C = 2*pi*r. A lot of students were shouting out “pi-r-squared” but when we got the end they were able to see that the circumference formula is not “pi-r-squared,” and why. They did identifying radius, diameter, and find the circumference procedures for homework and do-now’s the next day.

Worksheet:

Students got so creative and were so engaged and here are some of the results:

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## Introducing scientific notation using text messages

I don’t know what I would do without you guys. As I explained a couple posts ago my goal is engagement, engagement, engagement. I am teaching a lot of dangerously boring procedures in these test prep classes with students with short attention spans and not a lot of respect for authority and mathematics. Basically a recipe for a student teacher meltdown.

I searched and searched and searched and finally SUCCESS. I stumbled upon this set of postings (with worksheets) by  Mr. K and then I hit the jackpot by finding an amazing introduction by @ddmeyer here. He introduces the idea of scientific notation by asking students to write down names of states which ultimately leads to a discussion of common abbreviations of states. He also gives students a bunch of insanely long numbers and has them kind of invent scientific notation on their own. That’s amazing but I only had 38 minutes of awesome to provide to kids who barely want to listen to me (and no tech to display said long numbers) so I had to modify slightly. Luckily this woman Amanda commented on dy/dan’s post and offered the suggestion of using text messages abbreviations and VOILA I knew I had a winner.

I placed this on the white board.

Students got really excited, “I can write whatever I want?” Yes, absolutely, as long as it’s appropriate knock your socks off. So as I went around and checked homework students wrote their new text message, abbreviating whatever words they wanted. I gave the following lead off example, “I know you all probably say ‘hey’ instead of hello.” Then I asked them to raise their hand and tell me a word they changed. This was the result on the board (summary from all 3 periods):

Then I asked them if everyone would know what “wsp”  would mean (or whatever the odd man out was up there), most said maybe (it means what’s up) which transitioned beautifully into a conversation about abbreviations, usefulness, standardizing for understanding, etc. One student even said she added extra vowels for emphasis, her example, “I don’t knoooooooow.” This was awesome because that led right into abbreviating large words/text messages or making small words larger. Then I talked about how in mathematics we do the same thing – take the distance to the sun .. and so on.

It went great! The kids were talking and completely engaged. I kept emphasizing raising their hand in order to contribute to the board so it was reasonably controlled chaos. They even paid attention to the somewhat lame procedural lecture that followed when we discussed the previous nights pattern homework worksheet from the Math Stories blog. (*They could use a calculator until they found the pattern. It was assigned as a pattern hunting worksheet rather than a scientific notation one).

Love when plans go well, I know it’s rare so I’ll savor this moment.

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## Relax! It’s okay to be wrong

I took on 3 classes yesterday and… I survived!!!!! Now if you follow me on twitter you do know I made the small error of writing in sharpie on the white board but beside that things went really well. I used this exponent lesson from misscalcul8 (which was adapted from this by Sean Sweeney). It went great except for a few hiccups.

Students really resisted the “guess your answer” part of the worksheet. I knew students would probably not remember the properties but I assumed that after guessing and working with the numbers they might recall a few things  from their Algebra 1 days (last year!).  However they felt uncomfortable using any prior knowledge, number sense, or general guessing to come up with an answer. Many times as I was walking around the room I heard “I’m stuck” on that part – which was bananas because kids could literally put BANANAS and not be wrong. It makes me really sad that kids are SO scared of math that they won’t even put pen to paper in fear of failure (or maybe I’m reading too much into it).  This really opened my eyes to some general goals I want to have with these students.

• Become a critical thinker – learn how to be creative, take advantage of things you know, write down what you’re given, make the problem simpler, find an alternative way to solve something, construct the solution before knowing the formula/rule and general strategies to solving more challenging problems
• Have more fun with math! Most of the students seemed so hesitant about guessing; usually when I won’t guess it’s because I’m scared to be wrong. I really want to show them that great success can come out of failure.
• Provide meaningful work. So many handouts, homework assignments and lectures are based on busy work. I want to make math relevant and meaningful not just in the classroom but at home. I want these kids to find value in math and notice it around them.

I asked my co-op about the inability to guess/link prior knowledge and he explained that, “The students here only do well if  they completely understand what they are doing.” Let me dissect this a little bit for you all; “Do well” means work quietly and efficiently on what’s assigned and “completely understand” translates to replicate the example problems left on the board. I’m sorry – but that’s not thinking, hell that’s not even doing math. When I was having a conversation with the department head about the scary seniors (a post for another time) he responded the exact same way. The only way to “control” them is to give them something they understand. They don’t like to struggle, they don’t want to be wrong, they only regurgitate information which is why they don’t retain it. And that is exactly why we have spent 6 days covering chapter 1 sec 1 and 2.

It’s going to be important for me to walk the fine line between struggle and frustration with these kids and I know that’s going to take a lot of trial and error as a new teacher. I read these posts  by @delta_dc on gradual release of responsibility and I think this is the type of model I may want to begin with to get them more comfortable with individual discovery. He wrote about students constantly seeking help from the teacher and backing off and noticing the frustration on the kids face. Let’s just say this – I’m known for being the bitch tutor who won’t give you the answer when you ask the first time, I will always say TRY SOMETHING. But, I need to be careful not to frustrate them so much that they turn off, shut down and put their heads on their desks. I also don’t want to remove all the struggle (hey, that’s using the brain!) so like I said, trial and error. I’m going to aim for engagement, strong worksheets/supplementary material, and reading up on GRR and other methods some more!

Anyway, I literally went to bed at 7pm last night but it’s worth it – I’M FINALLY TEACHING!

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## If I just engage them enough they’ll probably stop chewing gum

Man oh man, I am stoked to start teaching Monday. I am dying to provide these students with an opportunity to play with and create (stolen from @mrschirles) math. I am a little apprehensive to take over since over the past week there has been very little structure in terms of expectations, learning and discipline. I already knew that classroom management would be a challenge but it is obvious there will be plenty of opportunities to perfect my skills. Is it too idealistic to think that if I just motivate and engage these students enough they won’t sleep, eat, or listen to their ipods? Probably.

I am also torn because my co-op uses a lot of direct instruction. Suddenly I find my thoughts gearing towards “How are you supposed to teach PSSA prep without direct instruction followed by individual practice tests in booklet?” This scares me. How do I expect kids to be motivated if I can’t even get creative and motivated for this stuff?  This morning I scoured blogs for discussions on standardized test prep, teaching exponents, scientific notation and engagement. No luck – will keep looking.

So far my plan will go as follows:

“Engage”: I was considering putting a doubling question up which will most likely have them struggle  for the first 1-3 minutes… unfortunately I do not think this is engaging unless that doubling question has to do with extremely relevant topics, this would only serve to get them working and thinking quickly. However I also know these kids are not used to struggling and getting them into the habit of using their prior knowledge, thinking and linking ideas on their own before I teach will probably come with great resistance.

Properties of exponents:  In the past I have taught negative exponents and zero power using inductive reasoning – based off of this video I found a while ago. For some students it works great, for others not so much and I usually show another mini-proof by using the properties of division to reenforce as well.

This summer when I finished teaching exponents I discovered a great worksheet on a blog but now I can’t find it again. Let’s just consider it some mathematical mirage – one of those resources I wish I would’ve tagged but am not even sure if it really exists. This summer I created one based on of Kate Nowak’s log laws worksheet. I found my worksheet took too long and was not as “guided discovery” as I had hoped, still needs tweaking.

Scientific Notation:  My main idea is to ensure I am not teaching memorization – go left/go right! I really want to show students that it multiplication and use the properties of exponents to help guide this. Individually they will complete practice test problems from the book individually but check answers/strategies with their partner when finished. Ask each other first before they ask me/co-op.

All feedback appreciated!!

Exponent Worksheet

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