Raffle Tickets to Teach Perseverence

8 Apr

Again, back to this area chapter. Once the basics of finding area is taught, it’s a great chapter to get them to start thinking of different ways to solve a problem. But how do we get our little trained monkeys to do so?! Especially when they are so used to teachers teaching them exactly how to do a problem and then having them mimic our process? I don’t know about you, but I can’t stand hearing “but you didn’t teach us how to do this!” when there’s really nothing new to have taught!

Here’s one solution: Raffle tickets!

I used this for group work on one very difficult problem that required multiple steps. For example, for this chapter, ONE problem that looked similar to the following examples. (This came after finding the area of a regular polygon given the apothem and a side length.)

  1. Find the area of a regular octagon with side length 6 cm.
  2. Find the area of a regular octagon with radius 20 in

  3. Screen Shot 2013-04-08 at 9.26.35 PM

  4. Screen Shot 2013-04-08 at 9.28.48 PM

Every group had these hints written under their one problem:

Hints:

  • Did you try drawing a picture?
  • What do you need in order to answer the main question?
  • Can you draw any other parts that might be useful?
  • Did you try it multiple ways? (Try to re-draw, rotate the picture, split the picture, etc.)
  • Did you try using our extremely useful RIGHT triangles?
  • Did you double check your work? (Make sure you did not make any assumptions, that your answer makes sense, used units, maybe even try it a different method, etc.)
  • REMEMBER: You already have all the information you need in order to solve this problem!

Raffle Rules:

  1. Every team receives 5 raffle tickets to start.
  2. Up to 3 bonus raffle tickets will be awarded for the correct answer with proper and clear work.
  3. One bonus ticket can be earned for working well together.
  4. Each question that the teacher answers or hint that the teacher gives will cost 1 raffle ticket.
  5. Answer check is free only when the problem is finished. If correct, receive award. If incorrect, it will be made clear that there is a mistake. Each group can either try to find their own mistake without losing any tickets, OR teacher can point out where the mistake is at a cost of one  raffle ticket.

20 minutes were put up on the timer to solve just one of these problems on their own as a group of 3 or 4. In my classroom, each group also had their own large white board to work on.

And wow! did they work their butts off! It gave every student an incentive to fight through the problem. I had a couple of groups refuse any hints and tried to find the mistake themselves. I had a few groups not get the extra 3 tickets because they did not finish on time, but they still kept the first 5 so they still had a chance at the prizes at the end. It was awesome. At the end, they all wrote their group number on the back of their raffle tickets and we pulled one ticket from a shoe box and the whole group won a prize. Can’t wait to try this again soon.

My worksheets used here.

I hate EXIT tickets… but not these!

8 Apr

I know every observer loves to come in and, especially if they don’t know math pedagogy, say that I should try incorporating exit tickets.

I see what they’re saying. It’s probably better for the students than my frantic end of the period shout of, “Don’t forget to copy down your homework! Put your stuff back! BYE!” I think I just don’t believe that they help me as much as my observers seem to believe they do. Of course, there are the occasional surprises still, but not usually worth the time and paper that exit tickets take.

UNTIL!!

My new exit ticket strategy. I choose one solid problem and have them explain or justify one of their answers. Nothing new. The new part is that I only comment on these and help them improve their terrible explanations. The next day’s exit ticket is to actually rewrite the explanation but this time for an actual grade.

And they do get better!

Each problem is short enough so that it doesn’t take a ridiculously long time to grade the writing that we math teachers dread so much. And for the effort, this is actually worth it to me and I actually wish I can remember to do this more often than I can remember to…

Here is the format that I actually picked up from a CCSS speaker who was saying the same thing about the importance of rough drafts for math arguments (I’ll try to post his name soon =/)…

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I just made a bunch of copies and put them in my file so I can make up a problem on the fly if I have to.

Area of a REGULAR Polygon

8 Apr

The area chapter is so much fun! This is the chapter where I can easily say, it’s not about the formula, but coming up with the formulas. It’s not about the answer, but explaining how you found your answer. Making connections is easy to incorporate and there’s just so many activities that can be done!

To start my posts for this chapter, an activity that I love. This worksheet is to discover the area formula for a regular polygon. I really did not want to introduce yet another formula when this one is so easy to just split into triangles. However, I also wanted them to know the formula the book uses because I remember last year a bunch of students thinking the ‘a’ in (1/2)ap representing area (of what? I have no idea and neither did they.)

Sequence: After covering the area of parallelograms, triangles, rhombuses, kites, trapezoids. I had been mainly focusing on coming up w/ the formulas for a rhombus, kite, and trapezoid using the area formulas for parallelograms, triangles, and rectangles only.

Time of activity: 30 minutes?

Introduce: Vocab words, “apothem” and “radius” of a regular polygon. (Review “regular”).

Execution: Partner work.

A snippet of what the activity looked like:

Screen Shot 2013-04-08 at 8.21.37 PM

It goes on for a few more questions, but my favorite part that I added is this one:

Screen Shot 2013-04-08 at 8.22.04 PMThey did a pretty swell job of figuring out why, I just had to help them a bit in actually getting it into writing.

Full activity here, on my very baby-bud of a Geometry site.

Note (to self): ADD problem w/ connection of A=(1/2)ap to the area of a circle!

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Trained gratitude

10 Mar

20130310-135644.jpg

Oh, the things that can make a teacher’s day! -finding nice rulers, 4 for a $1! And cute little erasers for a $1.50.

And the best thing about it? I’ve trained my classes with janky wooden rulers that are so bumpy and dirty that they will actually appreciate these rulers as raffle gifts!

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2012-13 Class set-up

15 Feb

2012-13 Class set-up

I swear it every year. No matter where I go, they change my classroom every year to make me clean it up and re-decorate it. It’s a conspiracy.

Christmas lights were kept up to brighten up my window-less dungeon. White boards designs are now transferable for my next classroom. Newly installed Smartboard… is still a glorified white board/monitor. Students were in charge of doing one problem from their homework on solving systems by elimination and then checking their solution. They then walked around checking their answers for the rest.

Similar polygons- Using ratios

14 Jan

Good group work problems are really hard to come by. Especially in math when looking for problems with multiple pathways to solving them. Here is one that seems quite simple, but kids definitely will struggle nicely with.

I ran it for the first time last year and I wasn’t quite ready for the struggle that students would have. This year went a  LOT better (which is why I’m ready to share). I gave them this activity after doing an activity on The Golden Ratio. I talked a little more in depth about setting up proportions and what that means better than I did last year too.

Anyways, the problem starts like this:

Before digital cameras became popular, 35 mm films used to take pictures in a width-length ratio of 2:3. Most standard picture frames were also created back then to be 4” x 6”. Why does this make sense? Explain.

 Nowadays, however, digital cameras take pictures in a 3:4 ratio, creating a need to crop pictures before they are printed if wanting to fit them in a 4” x 6” frame.

Below is an example of a digital photo the way it prints when sent to a printer without cropping. What is the problem when trying to fit it in to a standard 4” x 6” frame?

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What will the new dimensions be if the digital photo were to be enlarged to fit the longer length of 6 inches? Then how much will you have to crop off? Explain with words and drawings.

Larger picture frames are typically 5” x 7” and 8” x 10”. In the same way, if a digital photo were to be enlarged so that there is no gap in the frame, how much of the picture would need to be cropped in order to fit each of the frames?

It’s a bit prettier and roomier on the actual worksheet, but if you work it out, it does a nice little twist by the time they have to fit the digital print into an 8″ x 10″ frame so that it tests their “monkey status,” as I like to call it (basically, how robotic they can get in solving a problem procedurally without understanding the concept of what they are doing). The last two problems in the worksheet are also nice and challenging.

A bit about the execution 

Misconceptions to understand in advance:

  • Students think that the proportions are additive, instead of multiplicative. (I wish I had an app/ applet to tackle that misconception! Anyone know of one?)
  • Students will try to double the 3:4 ratio so that the short side of the photo matches the long, 6″ side of the frame. Might want to have props ready to explain that one.
  • The picture I provide where the cloud picture does not fit the frame will throw them off. They have a hard time understanding that a 3:4 photo can have a short length of 4 inches to start. I’ve considered changing this picture, but instead turned it into a great way to start explaining the problem without giving too much away.

Other things noteworthy:

  • Order-wise, I had it mandatory that they finish pages 1 and 2.
  • Pg 4 came next as optional for those finished early. **They must read page 3 in order to understand page 4!
  • Pg 3 is a great example to do in getting them to use a variable to represent all 16:9 rectangles. I think it’s a great example for a lecture also because it reinforces the multiplicative idea. OR, it can be for your super SUPER kids.
  • Be ready to listen to what they are thinking and encouraging their chosen route! Be open to being surprised by their outcomes!For example: as I was explaining the problem itself, one student yelled out the right answer by simply enlarging and shrinking the photo in his head.
    3:4 became 6:8 or 9:12, for example, which became 4.5 : 6, which was the ratio we needed. He doubled the ratio, tried tripling the ratio, then he halved that tripled ratio… I hope that makes sense.

Anyways, you can find the full packet here.

Different Variable Usages

14 Jan

I remember there being a time when I had identified at least 3 different ways to use a variable, but for now, I can only think of two.

  1. Using variables to represent an unknown.
  2. Using variables as a pattern summary.

Well, w/ Common Core right around the corner, here are a couple of activities I loved, but rarely found the time to do well in the past. However, with my double Algebra this year, I have twice as much time w/ them to go over the same material so I got to work on these much more thoroughly… and I LOVE ’em even more!! I truly believe the lack of understanding variables in the different ways is what makes Algebra hard and not something that should make solving problems easier.

1. Variables to represent an unknown.

Usually used in solving simple equations. x + 5 = 9. “Some number plus 5 will give me 9”. I did a LOT more mental math to teach this part and I really do believe it went significantly better than ever before. Even up to equations like -2x + 6 = 0.

Another big way of using variables as an unknown that we neglect often, is in word problems. So this year, I have used a couple of worksheets that my former professor created which contains just word problems. After going over them pretty thoroughly for the first time this year, I fell in love with it even more because my students came up w/ different ways of writing the equations! Here we come, Common Core!

One example:
Lunch Money. Daisy’s mom gives her lunch money every weekday. On Monday and Tuesday she gave her an extra dollar. The rest of the week, Daisy received the regular amount. For one week, she received a total of $42 for lunch. How much money does Daisy get for lunch each day normally?
Here are the 3 different answers my students came up with:
Once I got them to start drawing pictures to represent the problems prior to this, they did this one on their own. =)
Execution Tips: Only pass out the first two pgs of Amazing Stories first (no answers). We did the problems, one by one, up to “Pocket Change.” Starting “Lunch Money” (the posted example), they were supposed to do it by themselves.
Once they started working by themselves, after a few minutes, I projected all the possible answers so that they had to find their equation (or the equivalent) up on the board. They got a copy of the answers in order to finish the worksheet at home for homework.
The next day, I gave them the follow-up worksheet of “Life of a High School Student” for them to get a few more practice problems on their own, without having a bank of answers.
I saved a few problems from “Life of a HS Student” to use as an exit ticket and on future quizzes.
                 2. Variables as a pattern summary
This is the type of variable usage where all those darn rules come from. For example, if 2^3 * 2^5 = 2^8
4^9 * 4^7 = 4^16
what does x^m * x^n = ?
Anyways, here is the activity on practicing this type. Again, I love it all the more due to the multiple solutions that are possible. I copied down my entry below.
Materials: Cubes. Preferably not too small. And the worksheets.
Students are then asked to build certain objects that build on top of each other and asked how many cubes it would take to build a structure of 10 layers… then 100 layers.
For example:
First question after this would include: “How many cubes will it take for 10 cows? Explain.
Next they would be asked, “How many cubes will it take for 100 cows? Explain.
You can then summarize the patterns by writing an equation, with the input (x) representing how many cows, in this case, and the output (y), representing the number of cubes.
TimingWe started Pattern 1 and 2 in the last half hour of one class, continued the next day for about an hour, and then they were doing Pattern 5 by themselves by the next day’s 50 minute class. Amazing how many students who do not understand Algebra actually are really good at figuring these puzzles out.
Anyways, to get a better feel, you have to take a look at the multiple solutions that these problems have to offer. I have written out what my students had come up with this year. Yep, these were my intensive students coming up with these different solutions. We also took time to go over how to write a good explanation also. I went around giving them feedback and having them re-write a lot of what they had written…
Again, here are the links to the two types of variable practice.

Bad Math!

17 Dec

Bad Math

Bad math! Bad math! Bad math! 

I really hate this workbook! I tutor middle school kids once a week and this is what they came in with. They pace themselves so this is what was supposed to be teaching them? This shouldn’t even get onto paper!!

Bad math! Bad math! Bad math!

And you wonder why this kid was confused!

For all you non-mathy folks, let me tell you what to watch out for when your kid ends up in middle school…

  1. First and foremost, and what trips up the students when they get to Algebra: wrong usage of that dangnabit EQUAL SIGN! (<– Oh, Xanga days! How I miss thee). 0.05 x 100 does NOT equal to 0.05. This is why they have SUCH  a hard time remembering that you NEVER add 6 to one side twice…
    Example:
    PLUS_5
    Shoot me now.
  2. A rule w/out any kind of explanation will not stick.  “To multiply by 100, move the decimal two places to the right” is no explanation at all.
    Bah. Humbug.

Publishing companies are the devil.  To math. I’m sure of it. Draw a happy pencil and math does not magically get easier.

12/12/12

12 Dec

Must. Date. As many things. As possible. TODAY.

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Snarky

7 Dec

Snarky

One of my Algebra kids changed my “current mood” to this today.

I yelled at them to hurry and leave when the bell rang. And, “thank goodness I have two days w/out you guys!”

I smiled in love. They’re the most annoying adorable kids ever.