Sweeney Math

f(u) Calculus Song!

2011-05-09T10:04:00.001-04:00

In case you haven't seen it, I made a song parody about some of the basic derivatives rules in Calculus to the tune of Ceelo's "Forget you." Enjoy!

updated with lyrics:

Chorus:
I see you derivin' in class during calc-u-lus
And you've got f(u)
Which you want to find the derivative of,
it's just f '(u) and a du too,
it's just the chain rule,
a great cal-cu-lus tool
and you use it when...(you use it when)
When there's an inside and an outside that you can write as,
f(u)
(ooo ooo ooo)

Another derivative tool,
is the product rule.
And I know you can get it done
The slope of the product
of functions 1 and 2
is 1d2 plus 2d1.
I pity the foo-ooo-oool
who forgets the product rule
(oh, that's how you get it done)
(it's just 1d2 plus 2d1)
OoooOooOoOh
There's just one more rule
Wait... what was the chain rule again?

-Chorus-

For quotient, quotient, quotient you don't have to, have to pull out your hair,
(your hair, your hair, your hair)
Cause it's just low di high minus hi di low all over low squared
(Low squared, low squared, low squared)
To find dy dx dy dx dy dx baaa-aaa-aaaby,
But what about, that dang chain rule?

-Chorus-

How I see exponent rules (and log rules)

2011-05-03T13:59:00.001-04:00

Exponent rules can be difficult to remember, and memory has never been one of my strong suits. When I was in high school myself learning exponent rules, I would get mixed up just trying to remember them individually, and had to come up with a different way of thinking to condense it into one idea. What I came up with deals with the levels of complexity of operations:

So, you've got your simple functions on the bottom, multiplication and division are a little more complex, and then exponents and roots are more complex. The actual chart above I created after the fact when trying to explain the idea to students later in life. So, basically when it came to exponent rules all I had to remember was to "go down a level" of complexity.

So multiplication becomes addition, division becomes subtraction, an exponent to an exponent becomes multiplication and a root with an exponent becomes division. The chart also helps for remembering when to distribute. Operations distribute on to the tier below them. (exponents distribute over multiplication and division for example)

My results with trying to get students to see the same thing I do has been mixed. I usually end up doing an exploratory learning exercise with exponents , then going through the rules individually and only quickly going through this chart idea on the side. While it doesn't really connect with every student, when a particular student gets the rules mixed up it can really help because it at least gives them a plan rather than just relying on straight memorization.

Later on, when I learned about logs it turned out that log rules (surprising to me at the time, not so much anymore) followed along the same lines, with exponents becoming multiplication, multiplication become addition, etc.

Anyone else ever think of it this way? Have some other strategy for helping students get exponent rules straight? Let me know!

Top 5 math blogs survey

2011-04-19T11:12:00.000-04:00

On and off I've been working on a "Welcome to the Math Teacher Online Community" website to help quickly orient people to all the great stuff we have going on between blogs and twitter. One thing I'd like to include is a not too overwhelming list of quality math teaching blogs to add to an RSS reader for someone new to blogs. You can help by filling out this form and writing in your favorite 5 math teaching blogs. Thanks!

Working With Students With Dyscalculia

2011-03-03T17:54:00.001-05:00

Today is International Dyscalculia Day, and it's high time I come clean about something: I teach at a school for students with learning disabilities. Now if you have ever met me in person you're thinking "Yeah, duh" but if you know me exclusively through shouting at each other between the internet tubes, then you're probably a little surprised. It's been my intention not to mention this fact before in my blogging career. In a way I've been trying to make a point, if only to myself, about teaching children with learning issues. The point is something like this: Great lessons and strategies for teaching students with LD are great lessons and strategies for all students. If you get up there and teach a differentiated lesson that uses multiple modalities and engages your class all of your kids will benefit, not just those with LD. Everyone struggles with something, and the students I teach are really just regular kids that happen to struggle with a few specific things. Education would be a lot easier if more people(especially the students themselves) realized that.

So, all that being said there are particular strategies that I've learned that can really help when you come across students with Dyscalculia. This post is going to detail some of the strategies I've found to be successful when working with students with those students. I may also share some ideas about students with dyslexia or ADD/ADHD on my blog in the future as well.

-Keep in mind, this is all from my admittedly limited interaction with children who have Dyscalculia and not the necessarily the result of years of studies and research.-

Dyscalculia is not as well known as it really should be, which can be seen in the fact that blogger's spell check doesn't recognize it as a word despite the fact that it is estimated around 1 in 20 people have it. If you aren't familiar with it try to imagine dyslexia, but only with math. Students with Dyscalculia will likely struggle with new material, but also have a lot of gaps in their background knowledge. Many things that are helpful for most kids in learning math such as mnemonics, hands on lessons, extra practice, talking out problems, diagrams and manipulatives become necessary for students with Dyscalculia. The term itself is broad, covering a wide variety of math issues kids can have. I'll try to address some commonalities, but understand that each student is different, and determining exactly what is giving the student trouble is the key to helping the student move forward.

I would argue that for high school students(and probably even middle school) students with Dyscalculia, their actual math disability isn't their biggest problem. You may be saying to yourself "Whaaatttt?!?", but stay with me here. Try to put yourself in the shoes of a high school student with dyscalculia. You've been forced to take a subject that you have struggled with every school day for 10+ years. You've probably been told by peers, parents and/or teachers that you're just plain bad at that subject and it's possible that no one expects you to get any better. Skills that you are supposed to have mastered, but didn't (multiplication, division, solving equations, etc) are now smaller parts of more and more complicated problems. Countless times that you thought you were right, you were told that you're wrong(possibly just due to an arithmetic mistake, but to you getting an answer 2% wrong feels the same as 100% wrong because the answer is just not correct). Now imagine trying to solve a somewhat complicated math problem given all that. Many times it ends up a juggling act between cursing math, feeling incredibly anxious and doubting oneself at every step. Actually breaking down the problem and going through the steps becomes nearly impossible with all those other nagging feelings. So at some point the original issue of struggling with math facts, skills and formulas becomes overshadowed by extreme lack of confidence and anxiety. This can often lead to times when students actually get the right answer, but then change it and make it wrong because they simply have no confidence in their math ability.

The successes I've had are a result of significant one on one work, which means the student must either be in a school with small class sizes, be assisted by a good tutor or at least having time in the day to meet with their teacher for one on one support. As an educator, the most important tools in your arsenal are patience and the real belief that the student will be able to improve. The more time you can spend actively believing in a student, the more he will start to believe in him or herself. A really important strategy is to recognize each improvement a student makes, and make them aware of it. They may get 10 similar questions wrong with help in between each one, but chances are they are getting closer and closer to the right answer each time if good explanations and help are being given. Students with math anxiety aren't going to recognize this gradual improvement, and will immensely benefit from you pointing them out. "Awesome, you're getting this part down, now let's look at that next step." "Oh nice, you just made a little mistake with that negative, but you're really starting to get this!" Word choice and tone of voice can make a big difference between a student with Dyscalculia getting up and trying again versus shutting down completely. Confidence will build as the student begins to show success, even if it is little by little. While this kind of positive affirmation may seem silly or cheesy, the results I've gotten from it have been tremendous.

I've also dealt with a few students with Dyscalculia who actually do pretty well with concepts in high school, but have such serious gaps from early math that it can be hard to recognize that they understand the newer material. Identifying and working with those earlier concepts can make a big difference in certain cases. My first year I worked with a student who was diagnosed with dyscalculia and would have complete breakdowns during any math exam. After working with her a few times, her Trig teacher saw me and said "I don't know what you're doing with her, but she's doing amazing now!" The truth was we would spend about 1-2 minutes on Trig, and then work on essentially arithmetic for the rest of a tutoring session. She had a really bad experience with her times tables when she was younger, and working her through that helped build her confidence and her number sense almost overnight. This added confidence led to a huge improvement in math class, and in her ability to get through her future math exams.

Though one on one work seems to be essential at first, my experience has shown me that as a student's confidence improves he or she will become more and more independent as well. Students who I've had in class have needed to come in for extra help less and less and eventually not anymore. Students who I've tutored quickly begin to pick up more and more concepts and skills up from their classroom teacher and apply them without my help.

To sum it all up, any student with a severe lack of confidence in their math abilities is going to have trouble learning at every step of the way. Older students with Dyscalculia are almost certain to have anxiety issues with math and little to no confidence in their abilities, and helping to make improvements in these areas can have a profound effect on their performance when doing math at all levels. The go-to strategies such as hands on lessons, talking out problems, diagrams and manipulatives all can significantly help. However, the best chance for students with Dyscalculia, especially those with low confidence, is to have them work one on one with a patient teacher or tutor who can help them to see their progress and build their confidence bit by bit, showing them that while it may not come easily they can do math.

Angry Birds Geogebra

2011-02-25T09:48:00.000-05:00

Angry Birds is a pretty popular game with the kids nowadays. My students brought up the game when we started talking about parabolas and I've been working on a way to bring that connection into a class. So, I created a lesson using GeoGebra and some screenshots from Angry Birds mixed in with some inspiration from Dan's Will The Ball Hit The Can?

I created 4 GeoGebra files each with a piece of a different Angry birds shot like so:

Using GeoGebra, students worked in groups of 2 on their laptops to place points onto the bird's trail as accurately as possible to create a quadratic models in order to decide if the bird would score a direct hit on any of the pigs. If you had 4 points labeled A-D, for instance, the GeoGebra command would be FitPoly[A, B, C, D, 2]

Some commands that helped them place their points accurately:
CTRL= Zooms in
CTRL- Zooms out
CTRL CLICK DRAG Pans your view

Once students finished their files would look something similar to the file shown here:

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

We then discussed if they thought they scored a hit, what would happen when it hit, and then showed them the big reveal:

Overall, the students were engaged, worked hard to get their answers, and learned how to use GeoGebra to create quadratic models. If you'd like the files I used you can find them here: Angry Birds GeoGebra files

PS- I'm muddling through learning GeoGebra, so if you know of a better way to do things than I'm doing, let me know.

Online Math Teacher Community Survey

2010-11-19T15:24:00.000-05:00

Bet you were surprised to see a (2) next to my blog in your Google Reader!

After NCTM, Kate and I have been talking about working on a "Welcome to the internet, math teachers" page where we could help new teachers get involved with the online math teacher community more easily. To start the process we've created this survey for active members of the community.(commenters, bloggers, tweeters, lurkers are all welcome to fill it out) If you feel the online math teaching community has helped you out, please take the survey so we can help get others excited and involved!

Geogebra Derivatives with Limits Activity

2010-11-18T10:42:00.004-05:00

Yesterday morning I had an epiphany and started on a new idea for a lesson which was rolled out exactly 43 minutes later. I've been trying to use GeoGebra more in my classes and I thought of an idea that would not only let me do that, but also help out with the actual concept of limits which some of my students were struggling with. While they can calculate limits pretty well and can throw back some of the things I've thrown at them(phrases like "what the function is approaching" and "what the point should be"), I've had this underlying feeling that there's still something missing in their understanding. We're also moving into derivatives, and I wanted a nice visual way to show them what exactly we are going to do. So I whipped up a few of these:

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

I split the class into a few small groups, had each group use their laptops to manipulate the GeoGebra files and gave them this worksheet which has them find various slopes at points on the curves, then had them explain their process and discuss what exactly that has to do with limits. The graphs increased in difficulty so the first answer was 0, which is intuitive just by moving the point around. The second answer was 1, which is also pretty intuitive, but a little more challenging. For questions 3 and 4 they had to get a little deeper into how exactly how to calculate the slope.

It was important to me that I didn't tell them *how* exactly how to find the slope, so that they could figure it out on their own. I did mention that if they put the points directly on top of each other, they wouldn't have a line anymore. For most of the students, I felt this activity went over really well.(A few students did struggle with not being told exactly how to do it and how exactly to discuss it with their group to make forward progress) For the most part they were able to figure out how to get the different slopes. A couple had an issue with the question about how this dealt with limits until the following exchanges:

"Okay so what exactly did you do to find it"
-"We just moved the point as close as we could to... Ooooh!"

or

"Hey wait a minute, you were assuming that point was (4, 6) but it's actually not exactly"
-"So as I move the point it's getting closer and closer to (4, 6) so.. Ooooh!!!"

That "Ooooh" moment was awesome, and exactly what I was looking to see. Through this activity, a lot of my students ended up with a better understanding of what limits are, and the stage was set much better this year for coming up with the limit definition of a derivative. I'm hoping this will lead to my students to a much better understanding of the relationship between limits and derivatives this year, but we'll have to wait and see.

I didn't end up with a ton of time to plan this activity, and though I'm happy with what came of it I would welcome any comments you have for improving it.

NCTM thoughts aka We don't even exist!

2010-10-23T10:41:00.001-04:00

aka You won't care about any of this if you aren't a math teacher.

I went to NCTM last week, where I met up with Kate, Jessica, Nick and Jackie. I had a good time, and as I don't have a lot of real life high school math teacher friends, it was a lot of fun to have people that can completely relate to everything I do for my job everyday. Fun random story: In the first session I went to, this girl briskly walked out and said to the guy next to her "Pseu-do-con-text" I laughed because of what she said and because she tripped over a bookbag immediately after. Later when I met up with other online math teachers, it turned out that girl was none other than @smallesttwine! Anyway...

If I had to choose one thing that stood out that I learned from NCTM it's that a ridiculous number of math teachers are completely unaware of the online math teacher community. I naively had it in my mind that when I walked around the conference center with *the* Kate Nowak that math teacher groupies would be startstruck left and right. This was not the case. In fact, I was given the impression that the 5 of us that met to hang out were some of the only few people in attendance who were aware of the amazing online math teacher community that we have here between blogging, comments and twitter. I have learned quite a bit, found some awesome lessons, and gotten plenty of help from said community, yet there are so few math teachers out there seem to know about us. I mean when you think about it, with 4000 teachers in attendance, and 5 of us... that's 0.125% and I didn't forget to move the decimal point! To a statistician, WE DON'T EVEN EXIST! Granted, there could have been more in attendance that we were unaware of, but I doubt it could be that many. Out of curiosity I started asking people near me at a few of the workshops if they knew of Dan Meyer and none of them did. (a sample size of like 5 is good enough, right?)

At the conference, a workshop might go through one or two good problems or lessons in an hour, and it was only possible to go to a few workshops a day. There a tons and tons of great problems and lessons online right now that teachers could find if they knew where to look. So I think it would be awesome if we could come together and work towards building our little corner of the Internet. Some ideas I'm thinking about are working collaboratively on a "Welcome to the online math teacher community" introduction/roadmap/guide for newbs and working together to make a live presentation₁ that we could give in our own respective corners of the world. Watch this space, and let's get the word out and show off how awesome this community is.

_{1- We could use Google Wave! Oh wait...}

Pimp My Catapult

2010-09-10T12:22:00.002-04:00

*Fffwwwooooooooooooooooooosssshh* That is the sound of me blowing the dust off of my blog. Welcome back to my little corner of the Interwebs. Between getting married, buying a house and my math programming side project I've had little time to share anything on here, but I'm back in action and ready to start updating more regularly, so here we go...

I've been tweaking my catapult design a bit each year, and think I've finally found the best design to fire accurately(even in student hands). So if you tried the catapult project last year, but had trouble with student accuracy this post might help if you're willing to give it another shot. If you haven't tried the project yet but want to this year, I would highly suggest using the design and tips below. Here is the new design:

Highlights:

1) Mini clothespins give a more appropriate distance and height.

2) I now make the basket out of a thin strip of paper, complete the circle with a small piece of scotch tape and use a bit of glue to hold the basket on. Having a basket that is just barely bigger than an m&m laying flat helps accuracy. Make sure students know to load the m&m laying flat each time.

3) Cut off the rounded edge of the popsicle stick behind the basket. The flat edge helps fire more consistently.

4) This new firing mechanism allows student to essentially just pressing a button to fire. There is still room for some error here, so make sure they know to hit the button SLOWLY each time. The back clothespin is positioned so that it just barely overlaps the back of the catapult when it is all the way down. Here's a close up with the catapult loaded:

5) The front of the catapult is tilted forward so that it goes further and not as high.

6) Okay, so there's no 6 on the picture, but thinking about it now it would make more sense to move the guide rails on the main catapult much closer to the back.

So there you have it. I hope this helps. Let me know if you have any other ideas for improvements in the comments.

Previous catapult posts:
http://sweeneymath.blogspot.com/2009/08/m-catapult-project-pt-1-catapult-plans.html
http://sweeneymath.blogspot.com/2009/09/m-catapult-project-pt-2-project.html

BONUS: I let one of my classes build their own catapults last year, and after finishing the project one group decided to come in to my classroom during some free periods and make this ridiculously awesome trebuchet.

Yearbook Signatures

2010-05-14T11:47:00.000-04:00

Posts have been lacking in the last 2 weeks because I've started developing a math note/workbook program for students to more easily take notes or do work on problems, which has been taking up all of my free time.

Pretty much every time I try to write words in a yearbook for a student it comes out completely and utterly cliche. (You're a great student! Have a good summer! Good luck!) So, I've decided to stop writing words altogether and usually draw a picture or do some sort of relevant-to-them thing. Here's a solid work of genius I came up with today:

The answer is pretty worth it, IMHO, so give it a try if you've got some free time! You can do the harder parts with wolframalpha if you want to cheat. (The integral from ___ of ___, the derivative of ___ at ___, the lim as x approaches ___ of ___) Sam and Dave, this is a no calculator question for you!

PS- I triple checked the answer would come out right, but I still have a sinking feeling I messed something up.

Exam Preparation Reflection

2010-04-30T15:13:00.001-04:00

Last year I tried something new with my seniors. A lot of my students(and I'm sure yours too) tend to not realize that their preparation for a test has a lot to do with the outcome. So many seem to think that you are just born with math ability and that directly determines how well you end up doing on tests. I wanted them to be able to reflect on how they prepared for an exam, and how that led to their grade. I started crafting a project for my students to help with this. I had the following goals:

1) Students who give 100% in every class and don't have to study much shouldn't be punished
2) Students should have the option of not doing anything at all. The fact is some kids don't study, and I want them to be able to face it without punishing them further.
3) Students who work hard should be rewarded.
4) Students would be forced to think about how they prepared while seeing their exam grade.

So, I decided that for the project they would hand in all of the materials they used to study and write up a one paragraph explanation detailing exactly what they did. I made it clear that if they wanted to they could hand in a paper that said "I didn't do anything to prepare" and it wouldn't be detrimental to their grade so long as they did well on their exam. The worst grade they could get on their project was their exam grade but they could do better on their project if they prepared well. I gave them the previous year's exam as practice and gave them some general suggestions on what they could do to best prepare.

I thought this would be a good reflection for my students and that they would learn a lot and it was, but it turned out that I learned from it too. I had been under the assumption that some students studied well, but struggled on my tests and that other students didn't prepare at all, but ended up doing well. To my surprise, students' grades lined up almost exactly with the way they prepared.

We spent the whole class on the Monday after exams reflecting on exam prep. I opened by having each student write down their answers to the questions below.

In preparing for your exam...

What did you do that worked?

What did you do that didn't work?

What could you have done better?

I handed back their graded exams and we went through the following stats one at a time(red results are from this year):

Class Average: 83% (83.9%)
Highest grade : 93% (99%)
Average of Students who didn't turn anything in: 72%
Average of Students who "looked over their notes": 74%
Average of students who didn't do any of the practice exam: 75%
Average of students who did the easy problems for them on the practice exam: 79% (76.8%)
Average of the students who did all of the hard parts of the practice exam: 87.4% (88.2%)
Average of students who identified and targeted what was difficult for them: 90% (97.25%)

Then we discussed the results and talked about effective studying. We talked about how "looking over notes" didn't do much of anything, and how even doing problems that are easy for them didn't really do much either. We talked about how the largest jump came from the people who made sure to figure out how to do the problems that were harder for them and practiced those problems. We went over some highlights of the Sweeney Study Method and their answers to the questions from the beginning of class, then used the remaining time to start on test corrections.

Overall, I was really happy with the results of this project. It forced my students to see how their study habits directly influence their exam grades, and was an interesting learning experience for me as well. I'd really be interested in seeing results from other classes, so if you try something like this, make sure to let me know how it works out!

The Banana Rule

2010-04-23T18:19:00.000-04:00

Another concrete rule I use to help my students is the Banana Rule. I noticed that my students often struggle with simplifying things like this(when we aren't directly learning about it):

Sometimes they don't realize that you can add them, but often times they realize they are like terms but end up messing with the exponents. To me, it's really just a matter of how they are looking at it. My best attempt at getting my students to see this problem the way that I see it is by telling them that if everything in a term is the same except for the coefficient(1) is the same you can think of it as a banana. Therefore, the problem above is simply:

Now, they see the answer is 4, but don't totally get it right away, so the banana explanation is quickly followed by:

"So, when you add a banana and 3 bananas, does the fact that you're adding bananas change?"

"No!"

"So the answer is 4 bananas, and what did we say was a banana?", etc

During that exchange the lightbulb goes off and they get it.

Just like the rainbow rule, I'll go through this with a class the first time time it comes up naturally in some problem, and then refresh it as it comes up throughout the year. This method seems to reinforce that it's a tool, and not just "this is how you do this kind of problem". If I have time, I might go off on a tangent about how it works with anything even if they haven't seen it before and go through some quick examples with trig, logs, integrals or crazy fractions and roots. Also like rainbow rule, it helps kids put a name to something they struggle with and attaches an intuitive process to it.

(1)Okay, I usually say "number in front" instead of coefficient at this juncture, but we're trying to make it simple right? Please let me keep my math teacher card.

The Rainbow Rule

2010-04-19T14:59:00.000-04:00

A couple weeks ago Sam urged people to talk about little tips and tricks done in their classrooms. Since I found THE CLAW so helpful, I thought it would be only fair if I shared a trick of mine.

The Rainbow rule, a Sweeney original, is my favorite as it's useful for all of my classes from Algebra 1 to Honors Calculus.

Whether my algebra students come across this...

or my calculus students come across this...

...they either don't seem to know how to do it, or are confident about doing it the wrong way.

Enter the rainbow rule:

Since we don't all have the luxury of excess time and image editing software when working out problems, the actual version looks and works like this:

This rule has been very helpful for my students because it gives a name to this situation which is both easy to remember and helps to avoid the common mistake of crossing the two lines (rainbows don't do that, after all). While not every students gets it totally right, it still has really improved my students' ability to deal with fractions, especially those who I've had for more than a year.

Student centered learning using WolframAlpha

2010-03-12T14:31:00.002-05:00

Recently I had a really successful lesson on exponent rules. Every year I go through exponent rules with my algebra 1 class, and though it's likely they've seen much of them before, their retention is such that it feels like they are seeing everything for the first time. This year I tried to change it up a bit in order to do two things that I've been trying to get right this year: Increasing my student centered learning and using wolfram alpha as an effective learning tool in class.

At the beginning of class, students were given what was last year's quiz I gave after teaching and practicing some exponent rules. The instructions:

1)       Answer every question with your best guess. We haven't learned this stuff yet, so I don't expect you to get many of the questions right, but I do want you to try to make some sort of guess for each.

2)      Use your laptop to go to wolfram alpha to check each of your answers. Write the correct answer separate from your original answer and try to figure out what each exponent rule is.

3)     For each answer you got wrong originally, make up 3 similar problems that use the same rule(s), answer them and check them using Wolfram Alpha.

4)    Once you are confident that you could get every problem right without help, I will give you your quiz.

The result? Students did better on the quiz than in previous years where I taught the material. Granted, there are some other variables involved in that, but it was clear from moving around and talking to them that figuring out the rules themselves gave them a better understanding of what was going on. The whole thing took two 45 minute periods, part of which was getting them used to typing the expressions and a discussion about why they shouldn't use wolfram alpha to do all of their homework for them.

You may be saying to yourself "Hey wait a minute, you could've just given them an answer sheet and done the same thing." Well yes, and no. Answer sheets would take away the flexibility of being able to check answers to the problems that they make up. The making up step also helps them to understand what they are really looking for and is a good strategy they can use to help study for tests. Also, part of the point(which I discussed with them) was to get them used to using Wolfram alpha to check their work any time, like for homework or studying for instant feedback.

Now as pleased as I am with how this went I'm certainly not ready to give up explaining things at the board ever. Exponent rules lend themselves well to this sort of activity as they are reasonable to figure out with the answers and I'm definitely going to look for more topics that would work well with this kind of activity in the future. Can you think of any?

Failure is not an option

2010-01-26T16:02:00.002-05:00

I created this blog to provide practical examples of things that teachers can do inside of their classrooms, and wanted to try to avoid posts that were just my opinions on general theory of teaching. So much for that. I've been kicking this idea around in my head over the last couple weeks and I feel it's just too important not to share.

"We need to let kids fail so that they can learn important lessons about working hard and resilience. People are going to have failures in life and they need to prepare for that." I hear ideas like this a lot throughout blogs, twitter, in professional development sessions and within my school. Whenever I hear it, I immediately agree. It makes a lot of sense, and I feel like nowadays kids are given things too easily and protected from setbacks and failure too much.

However, in math especially it simply can't be an option to let kids fail. When I say "failing" I mean failing a test, or even just failing on an important concept within a test. I'm not by any means suggesting giving students grades they don't deserve, what I am suggesting is forcing students to deserve good grades in math. Now this is easier for me because I have fewer students than most teachers, but I think the concept in general still applies to math courses across the board and we can build our classes in a way that helps students to succeed without sweeping problems under the rug.

So why am I presenting such conflicting statements? I agree that kids should be able to fail, but I don't "let" them fail in my class. The reason is because of how math is different than other subjects. Math always builds. That's not to say there's no building in other courses, but it's more gradual, more encapsulated and there's more chance to catch up.

In math courses, students need much of what they learn to be able to succeed in future math(and science) classes. Let's say, for instance, a kid fails my test on solving equations in algebra 1 and it's totally his fault. He didn't pay attention in class, didn't do his homework, didn't study and didn't ask for help. Now if you ignore what the topic is and just look at what he did, does he deserve to fail? Sure. Here's where the problem sets in though. If I just let it go here, I'm not just letting him fail. I'm setting him up for failure. Unless this kid is explicitly taught how to solve equations somewhere else(and he very likely won't be), then I'm setting him up for failure in every following high school and college math class as well as Chemistry and Physics. Am I taking too much responsibility here? I don't think so. It's my job to teach him the concepts of Algebra 1, and his future teachers will expect that he knows it if he passes the course. If I don't intervene it's not as simple as letting him learn his lesson, it's dooming him to future failure as well.

Therefore, it is essential that in math courses students have option to (or better yet, they must) improve upon past grades for topics that they didn't master. Dan seems to have a pretty good system which I'll get around to implementing at some point, but for now I just have mandatory retests in some cases and optional retests for anything. I'll also find ways to revisit topics that students generally understood but didn't master (like warm up activities where they have to get every question perfect for some sort of motivator).

Are there topics in math that are less important that I'll let go if a student doesn't totally understand them? Sure. In general though, I get freedom over my curriculum so there aren't a whole ton of topics that aren't important later. There certainly aren't enough side topics for a student to be able to fail my class despite mastering the important stuff.

So how are my students going to be taught that important lesson on failure? I say, let their history teacher do it. Seriously. Not necessarily history, but anywhere where the loss of one unit isn't possibly the loss of that entire subject. Living in a world where people advertise if not brag about how bad they are at math when I tell them my profession just solidifies my feelings on this topic. Students that miss out on really learning how to solve an equation, or other core building topics will grow up to be those people; There are already too many of them in the world, and I refuse to take part in creating more.

What are your thoughts?

The Sweeney Study Method

2009-12-14T18:04:00.003-05:00

To be honest, I'm not entirely sure of the origin of my studying method. My best guess is that I came up with it my freshman year of college and developed it as the years went by. I probably used ideas that my teachers had been telling me all along in high school, but weren't necessary at the time since I found if I paid attention in high school, studying wasn't really necessary. Since I came up with it to be as efficient as possible, I can pitch it to my students as a plan that helps you study quickly and effectively.

I usually don't have time to share the entire method with my classes, but I pick and choose parts to share with them to help throughout. I do try to share as much as possible with my seniors as they will definitely need to figure out how to study in college if they've just been getting by without doing so in high school.

I wrote this as if talking to a student to make it easier to write. Keep in mind this strategy works well for any subject. It was actually more useful for my other classes as I clearly excelled in math. So, on with the show...

Preparation: Find somewhere quiet to work with no distractions. No TV, no instant messages, no texts. There are educational reasons for this, but the most important reason for me was that if I did that stuff, I would actually have to study for LONGER. I could get more done in 10 minutes of good studying than I could get done in an hour of studying with distractions. Why not just get the studying done quickly and then go have fun without having to worry about studying for so long?

Step 1: Gather your materials. In theory this should take almost no time to do. Grab your book, notes and any old tests or quizzes that might be relevant to the test you're going to take. In math especially, old tests and quizzes are incredibly important because not only do they provide you with a good prediction of what your teacher is going to put on the test/exam, they also tell you where you messed up the first time around. This is most likely where you'll mess up again!

Step 2: Start with whatever you think is most important. For instance, old tests would be most important when taking an exam. Your book might be most important if your teacher takes a lot of questions right from the book. Now, get a couple blank sheets of paper and a pencil.

Step 3: Here's the important part. You're not going to be studying or even reading everything. It's time to separate the things you know from the things you don't know. There's no point wasting time with the stuff that you're already sure of, you just want to focus now on the info that you don't remember already. It's also important to be very honest with yourself. Scan the book for bold words, information in boxes, diagrams and anything that looks important. Before you read the whole thing, ask yourself if you know it already. For instance, you see "A quadratic equation..." If you can finish the sentence in your own words "an equation with an x squared in it" then you know it and you don't need to write it down. It's also worth while to think about the way your teacher tests before writing things down. You might forget some obscure detail that appears in your book. Will your teacher test on it? Maybe not. If you're sure she won't, then skip it.

If you can't remember something, write it down in your own words. Never ever write down a definition directly from the book. Math definitions in your book are oftentimes overly complicated and take a lot of decoding. Instead of decoding the mystery of what the math book is saying every time you study it, translate it to yourself once and write that down. You might use a picture to help. Here's an example: "A parabola is... shoot I don't remember. The vertex of a quadratic function is... ugh i don't remember that either." What you write only has to make sense to you. Here's what I came up with:

So, let's say you read something and once you read it you suddenly "remember" it from class. It's important to still write this down. Here you are just familiar with the topic. You might remember it, but you shouldn't assume you're going to after reading it. After all, you did forget it once already didn't you? So, write down the information you're just familiar with as well as the stuff you don't know at all.

With example problems, it's important to test yourself. Try each example in your head or on paper if necessary BEFORE looking at the answer in the book. This will help you decide what you don't know and need to write down, and help you focus on what you don't know. If there are problems you don't understand and can't figure out: ask your teacher, ask a friend or ask Google. In math you should really strive to understand every topic with help, and the test should just boil down to testing what you remember.

With old tests and quizzes focus on the problems that you got wrong, but take a look at some of the more difficult problems you got right as well just in case.

Writing down all the information you don't know won't take nearly as long as you think it might. With a focused goal and no distractions, you'll fly through the information. The best part is you're actually studying by putting things into your own words, testing yourself and writing things down. Once you've written everything down, you're mostly done.

If your teacher allows a sheet of paper on an exam and it only took one piece of paper to write everything down, you're done studying! Everything you don't know is on a sheet of paper and you can use that sheet for your final. Sweet!

Step 4: This step is key. Now that you've used all of your resources to compile a list of everything you didn't know, get rid of all the resources(tests, books, etc). Everything you need to know is on this sheet. Get a new blank sheet of paper and start from the top of your note paper. Now, repeat step 3 with your note sheet replacing your book/tests and a new sheet of blank paper.

As you work you're going to realize that you remember a lot of what you didn't know the first time around. In fact, on your second sheet you'll probably have far less that half of what you had on your first sheet. There will still be some things you just don't get, and that's okay, just write them down on your NEW note sheet.

You can repeat this part a third or even fourth time if you think it will be useful to you. If your teacher lets you use an index card, you can stop whenever you think you can fit the information onto an index card, put it on and then you're done!

At some point, there will just be a few lingering things that you can't seem to get in your head. If that's the case, move onto step 5.

Step 5: Now that you have a short list of things you just can't seem to get into your brain, it's time to intensely focus on those topics. Use flash cards, make up mnemonics (PEMDAS for example), make up a song, do a bunch of practice problems or just do whatever you can to help you remember those last couple details. I would usually just write the information over and over again because that seemed to work for me, but what's best for me not be what's best for you. Try to do something here that suits your learning style.

Once you feel confident with everything, go have some fun!

----

So that's it. I hope this is helpful to you or your students. Feel free to share this with your students or other teachers. If you're feeling extra ambitious you could have students read this post, do the study method for a test or quiz and have them hand in their notes sheet(s) for a project.

I'll have a more studying tips in the future as well as studying project I gave a class last year with some really interesting results. Stay tuned.

Questions, comments or suggestions? Let me know!

Sweeney Math has been nominated for an award!

2009-12-11T10:31:00.000-05:00

The annual Edublog nominations are out, and Sweeney Math has been nominated for the "Best New Blog" category! I'd like to give a big thanks to those who nominated me.

If you enjoy my blog, I'd be honored to have your vote.

If you haven't perused the Edublog Awards already, it's a great place to find some new and exciting blogs to follow. Check it out!

Videos galore!

2009-12-09T21:26:00.003-05:00

First things first I got engaged this weekend. It was awesome.

The studying strategies post is coming, but turning out to be a beast. In the meantime, I thought it was about time I make all of my videos available for download for teachers with filter issues or what have you. While I'm at it, I may as well include a previously unreleased Graphing Stories video. I felt I needed a bit of a softball to start off with to work out some of the kinks the first time around. I was happy with how it worked with my students.

Download AVI (9 MB, right click and select Save as...)

Everything below is old hat, but include direct download links. Relevant youtubes are above so you know what you're downloading.

Download Graphing Stories Bike (9.2 MB, Right click and Save as...)

Download Graphing Stories Figaro AVI (9 MB, Right click and select Save as...)

Download Graphing Stories Systems Running (8.9MB, Right click and Save as...)

Download Grocery Math AVI (15.5MB, Right click to Save as...)

Download Dance Instructional AVI (18.1 MB, Right click and Save as...)

Download Dance Steps to Solving an Equation (5.8 MB , Right click and Save as... )

Download Slope Rida (7.11MB, Right click and Save as...)

Let me know if you have any issues, I feel like I must have made a mistake somewhere!

Selling study strategies (Math studying strategies pt. 1)

2009-11-12T17:05:00.000-05:00

My school is very big on teaching kids strategies to help them succeed. This makes me happy because I feel that there are a lot of strategies for learning and test taking particularly in math that may have been obvious to me, but aren't obvious to many kids. With my 9th and 10th graders, I focus mostly on how to effectively learn in class from your teacher and review strategies for test taking. In 12th grade, I like to specifically take a decent chunk of time to model, explain and discuss how to learn and study from a math book. I believe this is an essential skill as it is inevitable that at some point in college students are going to need to learn directly from the book. This may be due to the fact that they didn’t understand the lesson, they misunderstood a portion of the lesson, or they missed a class entirely. Surprisingly, many kids don't understand how to read from a math book or even how to study for math at all. I feel certain that we have all had the experience of hearing a student say, "Well you can't really *study* for math..." or "I just look over my notes to study for a math test."

So, in my senior classes I take a chapter from one of our tests, and exclusively read through it together with the students. We focus on figuring out what the book is saying, and how to organize and study the concepts in preparation for college courses. I started this process today in class; and to be honest, we covered very little math material. Nonetheless, I really believe that today was infinitely more important for them than any single math topic could ever be.

I'll talk about my specific process and strategies in upcoming posts, but I want to intentionally stop after I make my next point because it's very important: I think many teachers go about teaching study skills in the wrong way. There are a lot of great teachers out there providing countless useful studying strategies, and I'm not saying those strategies aren't equal to or more awesome than mine. It's essentially the introduction that in my experience (as a student) needs improvement.

My selling of study strategies goes like this:

"Studying strategies are for people that don't want to study a lot! I promise you that at some point you're going to want to learn a lot about something in life whether you just have a strong interest in a topic, want to learn a specific skill, or just want to get a better grade in a college course. I never liked studying. I wanted to spend my time going out with friends, playing video games, enjoying a nice sunny day, but definitely not studying! However, I still wanted to do well in my classes. What I realized is that if I used effective study strategies, then I really didn't have to study nearly as much in order to understand the material and be successful. So again, good studying strategies are used because you don't want to spend that much of your time studying!"

At this point, I have their attention. To really drive the point home, I give an example from my own experience.

"My first year in college I was in the First Year Science and Engineering dorm. My floormates had mostly gotten 4.0's in high school, and there were a decent number of valedictorians. With my mere 3.5, I was feeling a little intimidated. Throughout the first semester I was always trying to get my friends to go out and do fun things, but was frequently met with "I've gotta study, you should try doing it sometime, Sweeney." This happened especially in the weeks leading up to exams. As the semester came to a close, a number of my friends were convinced I was going to fail. What happened? I was one of three people on my floor to make Deans list. (There was a dinner for us, so I knew). I ended up doing better than a lot of intelligent classmates who had studied much more than I did. Why? I'm not a genius, and (as all of my students know at this point) I have a terrible memory! The reason for my success was that I used effective study strategies and knew my strengths and weaknesses as a student when I did study."

I had pretty high hopes when I first decided to share this concept with my students a few years ago, and I wasn't disappointed. They can't argue with the logic that good studying = less studying. Many students just don't study. To me, it seems like if students feel they have to study a lot only to still maybe not do well, they're just going to choose to not study. When I suddenly present this third option of studying not that much, but doing it really well it's effective enough to give me their attention and eager discussion of strategies for a few periods. I can't be sure how much of the strategies we go over will stay with them, but at least I know for certain they're getting in there.

Sick of security camera problems...

2009-11-09T13:49:00.000-05:00

I cover a couple units in my Intro to Calc class that cover some topics from trig. Sick of security camera examples with sum and difference identities, I came up with this goofball problem to change it up a bit. I guess it's a little gruesome, but come on... We all know the evil professor will step out of the room and our hero will escape!

Slope, slope, slope, slope, slope, slope, slope, slope

2009-11-02T19:03:00.002-05:00

First, I have to say I can't take credit for this myself. The mysterious "Andrew" left a comment on this post

and it was too awesome to ignore. It didn't seem like he had a blog, so I figured I'd post this because it worked really well.

Second, I swear that my classes aren't all songs and dances. I just wanted to post this now because I know teachers will have just done slope or are starting it soon, and it went really well.

Andrew's suggestion was to use the tune to Flo Rida's "Low" but with the following words:

The difference of the y and the difference of the x

Also known as rise over run

Divide the two

And then reduce

Then you got slope, slope, slope, slope

I added the following verse in between chorus' to add a little excitement:

"When I'm sittin' in math and I'm tryin' to find

How to get the the slope, the slope of a line

I think about the rise, and the run all the time

Then I think of this song, and I'm gonna be fine

1/2 slope come on

1 slope come on

2 slope come on

now that's three slopes

You think I'm a dope?

I'd gotta say nope

I am gonna find that slope!"

So, I easily found an instrumental version of the song by searching google and played it in the background. I had a student from another class help out the first time to introduce it. This is how it went(the 3rd time through):

I cracked myself up today

2009-10-22T12:52:00.004-04:00

I was looking through my lesson on slope from last year, and there were a bunch of frames with this skateboarding guy to help show how to think about slope.

So I'm flipping through the frames and I come to one where I was trying to explain how the slope of a vertical line is undefined, which I completely had forgotten about...

I laughed pretty hard. Chalk up a positive for having a terrible memory.

(Now with 100% less titlefail!)

Being Less Helpful

2009-10-15T15:13:00.002-04:00

A few weeks ago, Dan Meyer (http://blog.mrmeyer.com/ like you didn't know...) gave an online talk about being less helpful and his WCYDWT. (http://www.oreillynet.com/pub/e/1450) I decided to immediately try to incorporate some of those ideas in an algebra class the very next day where we were going over word problems with formulas.

What I came up with was two questions, one given explicitly and one implied in a picture. The first one was "How fast does Mr. Sweeney drive to work?" Granted there was no piece of media attached to this, but it was an accessible question that every kid had an opinion on, and every kid wanted to know the answer to.
"Mr. Sweeney is young, I bet he drives really fast!"
"Is there traffic?"
"He must take the highway, I bet it's like 65!"
"The highway doesn't right from his house to school, so it must be lower" etc.

I wrote down each student's individual guess and told them there would be a piece of candy for the closest (I'm not above bribes). My goal was to "Be less helpful" so I was determined to only tell them where I live. They figured out they needed to use D=RT and know how long it takes and how far it is. They kept asking questions trying to get me to do work for them, and I resisted answering. Eventually, they realized I wasn't going to budge and someone suggested to go to google maps.(Which gave both time and distance) We came to the answer, and then I told them how long it actually takes me(there's *always* traffic on the expressway!) and we solved again and I was happy.

Next, I showed them these pictures:

All I told them (When they asked) was that I am exactly 6 feet tall. The question we decided on was "How far across the wall can I paint?" ("How many cans of paint would it take?" was the first question, but I insisted there was only one can in the picture so we saved that for afterwards) To answer the question, they:
Enlarged the picture on the smartboard so they could see how much the can holds.
Figured out they'd need the formula for area of a rectangle solved for width.
Copied and pasted the picture putting my feet on my head to figure out the height of the wall.
Went to Behr.com to find out the range of areas you can paint a gallon of interior-semi gloss paint.
Converted from gallons to quarts.
Solved for the minimum and maximum distance across the wall that the paint would cover.
Measured the back wall and found how many cans of paint they would need.

They also came up with a lot of other ideas that we could've done if we had other tools or information ("Let's measure the height of the wall" "Alright, who has measuring tape?") It was a lot of fun, and they really got into the lesson. The best part was that a number of kids who don't usually participate at all were brimming with excitement and ideas. It was so successful that I decided to call an audible on my second class and do the same activity with them too.

What kind of things like this have you done in your classroom?

The Dance Steps to Solving and Equation- The Lesson

2009-10-03T19:19:00.002-04:00

I generally do this lesson after I've taught solving equations entirely. At that point there are at least a few students that get really overwhelmed by the process, and I've found that this helps them to break it down into steps (and to actually remember what those steps are) and it's just a heck of a lot of fun for everyone.

The day before the lesson I tell students that their homework is to remember to NOT bring their bookbags to class the next day. (Otherwise we wouldn't have room). At the beginning of the period, I race to get all the desks stacked on the sides of my room to clear a nice dance area for everyone. Then, I give them this speech:

"Today is a math fun day. I *absolutely* guarantee that if you don't act "too cool" for this lesson that you'll have fun. In fact, this will most likely be the most fun you ever have in math class!" Cutting the too cool kids off at the pass right up front has always worked for me, and was my biggest fear before ever doing this lesson. (I also tell them participation is mandatory) I then stick my arms out and swing them back and forth and tell them that they need to be able to do that without hitting anyone so they have space. I start the beat, and sing the intro, then put the lyrics on the board with the smart screenshade hiding the moves we haven't done yet.

From there, the lesson goes pretty much like this:

After they are able to do the dance with some proficiency, I speed it up 10% and keep speeding it up each successive time until it all ultimately falls apart. Then, I have them grab a desk and go get their backpacks and work on a sheet of difficult equations to solve, telling them to think about the song as the go along. When they ask questions, I pretty much just prompt them with song lyrics.

Here are some important tips:

You should definitely try this lesson if at all reasonable. I'm fairly certain you could throw this lesson inside 179 other days of teaching like Ben Stein in Ferris Bueller's Day off and kids would still think your class was fun.
The video is only 3:49, but the dance part of the lesson actually takes me 15 minutes (and an extra few moving desks out and back in). So, if you are to use the video above to teach the lesson I would highly suggest pausing and going back a lot to give kids time to really learn it.
I'd suggest writing the lyrics on a side board even though they are in the video so that it's easier for the kids to follow.
If you use the instructional video, I would still highly suggest doing the dance along with them. If you are too cool to dance, they probably will be too.
Call them out as a group if some kids aren't singing along, I think sometimes they honestly get so wrapped up in the dance they think they are singing along when they aren't.
Have some sweet moves prepared for the check portion.(I like the lawnmower or the shopping cart)

Video files (right click to download):
Dance Steps instructional video (High Quality, 110 megs)
Dance Steps instructional video (Low Quality, 30 megs)

Audio Files (right click to download):
The beat (play it in a continuous loop)
Audio of the full song
Faster full song
Even faster full song
Fastest full song

The Dance Steps to Solving an Equation - The Story

2009-09-26T19:07:00.002-04:00

This lesson is by far my most well known lesson at my school. I'll post how the actual lesson goes soon, but I wanted to share the story of its creation because it was integral in forming the teacher that I am today. Feel free to skip to the bottom for video and lyrics.

A few years ago, I was looking for something to help the kids understand that even if an equation seems really long and difficult, there are solid steps that can be done to get through it. I'd taught multistep equations, distributing, combining, how to deal with variables on both sides and printed out colored sheets explaining each step. There were, however, a few students that got totally lost when we tried to put it all together. I was racking my brain on the ride to work thinking of some way I could get the steps to stick, and hopefully make it a little more interesting for them after we'd been working on solving for so long.

"The steps to solving an equation... The steps to solving an equation... The... DANCE STEPS to solving an equation!!!!"

As with all of my ideas, I knew I had to act on it right away or risk never doing it. I got through the school day and started to work. By 10 that night I was finished and ready to do the lesson the next day.

Morning came and I nervously told the head of the upper school before assembly that he should probably check out my algebra 1 class. After the words left my mouth, I started to panic. I started seriously thinking that I was about to do something wrong. After all, I was going way off the normal formula that was every math class I had ever known. Shouldn't I be lecturing? Is this a big waste of class time? Luckily, it was too late to do anything about it. I didn't have a back up plan, and I had already told my boss something interesting was going to happen.

I cleared the desks and chairs to the sides of the room. Class started, and I ensured my students it would be the most fun math class they ever had. Full of nerves, I started into my carefully planned dance lesson. The kids were all smiles. They loved it, and before I knew it there was a crowd forming at the door. Twenty minutes later the kids had mastered the dance, and knew the words by heart. We brought the desks back in, and started on a difficult equations solving worksheet. Students were stuck much less, and when they did ask questions they had a much stronger base to work with. "Well, what's the first thing you should look for? And then what and then what?"

Later that day, the head of our school came up to me and said he had already heard about my lesson, that he wished he would've known about it and that he most definitely wanted to be in attendance next year. (and he was) By the end of the week, my 10th and 11th graders were demanding that they get to do the dance ("Hey, we solve equations in algebra 2 too!"), and our 8th grade algebra teacher was asking me to guest teach it to her group.

I really grew as a teacher that day. I didn't fear taking risks in teaching anymore. When I've had legitimate reasons to do something a little crazy or different to shake things up and get kids learning I stopped questioning it so much. I learned that my school fosters a creative learning environment for not only the students but the teachers, and because of that I am able to thrive.

Okay, enough typing. Below you'll find the lyrics and video. The video of me doing it alone doesn't really do the lesson justice. The kids add electricity like you wouldn't believe. More to come soon with audio files and the flow of the lesson.

The Dance Steps to Solving an Equation

First you simplify
put your hands up in the sky

so you distribute
then you do a little scoot

Still need to simplify
Put your hands up in the sky

So you combine like terms
and do the squirm

Add and subtract, x terms alone on one side
So take a step back and do a big slide

Multiply and divide, the answer you will learn
when you jump to the left and do a full turn

Now check check check ch-check check check

f(a bag of skittles)

2009-09-24T11:52:00.001-04:00

Today, I realized mid-class my students desperately needed a review of working with functions to get through the limit definition of derivatives. The picture below is what spontaneously occurred. I wrote the left half and then had them tell me should be on the right. They seemed to really enjoy it, and understood what was going on much better when we tried to apply it to the definition of derivatives.

f(x) = 5x + 1

Systems of Equations Project - Sparking Interest

2009-09-21T17:13:00.002-04:00

I started this project last year, and I was amazed at the results. I modified a project I had done in previous years to allow students some room to use math to explore something that they were interested in. (Believe it or not, analyzing the amount of homework they got didn't do much to get them excited!) The vast majority of my students got really into it, especially ones that otherwise were not very motivated.

For this project, students find data online that they are interested in comparing. (Sales of video games v sales of movies, Wins of their favorite sports team v wins of their friend's favorite sports team, Women's race times v Men's race times, Success of movie with many sequels v another, Sales of Abercrombie v sales of American Eagle, etc) They graph and find best fit lines for each set of data, then answer some thought provoking questions about the results.

The most time consuming part of this project was having students find good data. Anything sports related is easy, finding movie sales is easy, but other things got pretty difficult to find. When things got difficult, students often wanted to take the easy way out and pick something they didn't really care much about, but could find easy data for. I discouraged that heavily because the key to this project is bringing in their specific interests and showing them how math is involved. When students worked hard, but couldn't come up with data, I did my best to point them in the right direction. (Try this search or website)

I used 3 40 minute classes for this, but that's because I only expected students to do work at home if they were getting behind. You could significantly cut the in class time down by giving most of it as homework. If you decide to do this or a similar project with your class, I would highly suggest making your students check their data with you before they continue onto the rest of the project.

See the project description here.

One of the coolest things about this project was that I stuck excellent projects on my back wall and a number of times saw students from other classes thumbing through and actually reading the results on their own!

Any suggestions for more conclusion questions? What kinds of things do you do to get your students working with what they are interested in? Let me know.

Graphing Stories Remix

2009-09-17T22:35:00.002-04:00

As you might have heard on Dan Meyer's blog I've started making some of my own Graphing Stories videos. Dan Greene (other Dan) on twitter suggested I make one that shows a system of equations. I filmed one of me in a no holds barred footrace versus myself. (Don't worry, I won) Here's how it came out:

Once I'm finished I'm going to make a bigger post about all of the videos for this, and by then I will hopefully have found a way to provide them in better quality than Youtube. In the meantime, I'm looking for some input...

I'm not sure how well it translates at this angle. What do you think? Do you think what's going on in this one will be discernible to students?

Also, I'm looking to make some more of these (it's really easy now that I have a template). Anyone have any more good ideas? Let me know in the comments.

Life skillz: Shopping

2009-09-14T16:15:00.001-04:00

(the z makes it cool)

For about 3 years between high school and college I worked as a cashier at a grocery store. If I had a dollar for every time someone came through my line and spent way more than they thought they were going to...(Wait, scratch that, I technically did make a dollar every time this happened) Anyway, from that experience I realized that many adults have trouble estimating the amount of money they spend when going shopping.

I decided my students could use some help with the matter. So, I went to the grocery store to buy some things for a cookout and video taped parts of the experience. It was really busy there, so the video didn't come out that great as I was trying to avoid getting anyone on camera. (Converting for youtube didn't do the video any favors either) This is the finished product:

Granted adding and estimating isn't complex math, but students have no trouble in seeing the activity's use in everyday life.

I had some extra time at the end of class on Friday in two of my classes, so I told them we would have a competition. Of course, the winner would receive candy. I didn't tell them what the contest was (though it became pretty obvious), I just told them to pay close attention and to follow these rules:

No talking
No writing anything down during the video

I stopped the video before it gave the answer, and told them to write their guess for the total on a small scrap of paper. Many of the answers were pretty off (up to about a $20 difference), and surprisingly my 9th graders guessed considerably closer than my 12th graders. (Which made me feel better about taking their class time for it) Then we did some discussion where we talked about estimating and general paying attention. Despite the lack of complexity, it was definitely a worthwhile activity. It certainly was an eyeopener for those whose guesses were way off, and all of the kids really got into it.

Anyone have any ideas on how to extend this to add some higher level math and make a full lesson out of it?

I was sure I would never sign up for twitter...

2009-09-11T20:28:00.000-04:00

But then I clicked on this: http://samjshah.com/2009/05/11/why-twitter/ Curse you Sam J Shah!

Follow me: http://twitter.com/SweenWSweens

Pick up - Algebra Game

2009-09-10T17:52:00.003-04:00

The first couple lessons I shared were from my Algebra 2 class, so I figured it was time to share something from Algebra 1. This lesson revolves around a game that I think a student taught me years ago, but was similar to a game that we played in my college game theory class. The kids have a lot of fun, because they get to compete and figure out strategy. I call the game Pick up.

I split students up into groups of twos, and give each group a few pieces of scrap paper. I tell each group to rip up the paper to get 21 scraps of similar size. On one of those scraps they write the words "Math Fun." Then I give them directions for the game, which goes like this:

The 21 pieces are placed down on a desk or the floor with the "Math Fun" piece showing and visible the whole game, like above.
Players take turns picking up at least 1, and up to 3 pieces at a time.
Whoever must pick up the Math Fun piece loses.

Then I let the students play. They get to practice for awhile to get a feel for some basic strategy, but soon we start a tournament. Students that don't win the first round play off for a wildcard spot or two later in the bracket. Excitement builds, someone wins candy and then we begin discussion.

I prompt them with questions like "Well, what worked?" The winner will definitely have figured out what to do at the end, but they won't need to step it all the way back to the start to win games, so they don't. Then we start discussion around the question "Well, in what situation are you sure to win?" We decide to not count the Math Fun piece because it's pretty much irrelevant and we go through each situation that occurs at the end of a player's turn assuming their opponent is playing perfectly. It's fun for them think their way through it, and my students have been able to figure out the situations below with minimal prompting.

1- You lose, opponent takes 1.
2- You lose, opponent takes 2.
3- You lose, opponent takes 3
4- You WIN, opponent has to leave you with 1, 2 or 3.
5- You lose, opponent takes 1, leaving you with 4.
6- lose
7- lose
8- win
9- lose
10- lose
11- lose
12- win

At #8, they might see the pattern, at 12 they are sure of it.
"So, could we.... write a linear equation that would tell us the winning numbers?"
"If we counted the Math Fun piece, how would our winning situations change? How would the equation change?"
blahblahblah Slope, blahblahblah y-intercept. Hooray for math!

Extensions:
     Which player has the advantage if both play perfectly?
     What if you could take up to 4 pieces? Only 2? 10?

Super Extension:
     What if you split it into 2 piles, with two special pieces, and could only take from one at a time?

First day of school

2009-09-04T18:10:00.001-04:00

School starts for me on Tuesday. There's always a lot going on during our first day of school, so the schedule for classes is shortened to about a half hour. Now, there's nothing particularly different or spectacular about the way I run my first day, but there are a few things I think are pretty important about what I do.

I start by handing out a kind of "Getting to know you" sheet that doubles as a record for book numbers. There's space for student name, what they did over the summer, and three things they like. (Vague, I know) After books and everything are handed out I go through and have all the kids share what they wrote, but first share about myself. I always try to pick things about me they'd never guess. (Like that I enjoy dancing, and I'm pretty awesome at it) If you have any trouble learning kids' names, as you're going through on the first day make sure you say the name a couple times out loud back to them ("And, Jamie, what three things do you like?") and listen carefully as they answer. This first day activity is probably pretty standard, but here's what I think is really important... As they give their answers, I do everything I can to make some connection with one of the things they've said. If they say they like video games, I tell them about the time I won a Halo tournament without owning an xbox. If they went to the shore, I tell them about when I went to the shore over the summer too and what happened while I was there. If I can't think of ANYTHING, I'll ask them some more questions until I can. It's pretty helpful to get you to remember each kid, but more importantly it helps make connections with them, making it easier for them to buy into what you're teaching and to respect you as their teacher. You can figure out some interesting things too. A few years ago, we figured out that when I was in high school I worked with the older sister of one of my students!

Anyway, I'm really excited about starting school this year. I hope your first day goes/has gone well!

M&M Catapult project pt. 2- The project

2009-09-01T02:31:00.011-04:00

Click here for the Candy Catapult Project Description

In all honesty, I have probably always liked this project a little more than the kids, but over the past few years I've improved the delivery so that they definitely get into it. My biggest mistake the first year with this was not giving the students an overview of what they were going to do and trying to let the packet speak for itself. That failed... miserably. Now I give an general overview complete with pictures and a discussion of why the lesson is important beforehand and they seem to enjoy and understand it a lot better.

I think one of the reasons I like this project so much is that it actually works. When the kids are consistent and do their calculations correctly, they will hit the center of the target with ease. It's one of the all too rare opportunities that students get to see that the math they did directly affected something in real life.

Some tips for the teacher:

I generally work the stopwatch for them. I've found myself wincing at how much the kids think the timing is "fine" when they do it on their own. I'll start a countdown out loud, and if I feel like my button presses weren't as good as humanly possible, I'll tell them not to count the trial. Letting them do it on their own (poorly) could I supposed be a "lesson learned" but I feel like after all the calculations they do, they won't realize what exactly went wrong or take such a lesson to heart. It would probably be worthwhile to let your kids try it on their own first, but keep a close eye on the timing aspect.

Strongly encourage the students to make sure they are shooting consistently before they do official trials, and to start over if their official trials aren't close together. You'd think the bold, caps, and underlining in the description would be enough, but my kids tended to be a bit overconfident and tried to rush through without frequent reminders.

This project takes me ~2 40 minute classes based on class size and skill level. I'm sure it could be done faster with more space and more student independence.

Having some backup Skittles might be a good idea in case of allergies or dislikes. (I mean, what fun would it be if they couldn't eat some leftovers?)

I realize that there is a bit of hand holding, some of which could be removed to get kids thinking more on their own (especially if you were to give this to an honors class), but it's mostly to get it to fit within time constraints. As with any lesson or project on here, I encourage you to use and edit this to meet your own needs. Let me know what you come up with or if anything is unclear.

M&M Catapult project pt. 1- The catapult plans

2009-08-31T18:49:00.013-04:00

This is a fun lesson I've been doing for a couple years now with my Algebra 2 class. This post will briefly explain what the project is about and show how the catapults were made. Part 2 will include the actual project packet, and go into more detail about how it all works.

Each group of students (2-3) gets a small catapult, and shoots M&Ms from it while it is on the ground. They measure how far each shot goes and how long it is in the air and use that to figure out how far the catapult will fire when they place it up on a desk. They place a target where they think their projectile will land, and get points based on how accurate they are. I check in with each group regularly so they aren't getting too far off course. It's probably my favorite project because it takes a large number of the things we've done in the chapter on parabolas and puts it all together, in a "real life" situation. (While firing M&Ms doesn't have much of a purpose, it's pretty easy to get them to understand the correlation to ballistics)

Unfortunately this project does require some setup (making the catapults), but it shouldn't take too long. I bought popsicle sticks, a few small pieces of wood, wood glue and some small clothespins at the local crafts store and used some scissors and masking tape to make this:

If you have Google Sketchup you can see my model of it here.

The plans are (I hope) pretty self explanatory from the pictures above. The base was a small piece of hard wood I also found at the craft store. The clothespin sits on a platform made of four popsicle sticks cut up with scissors. The basket to hold the candy is made out of masking tape. I added the guide rails on the sides to help with accuracy, in the picture directly above I removed one so that you could see the interior better. This year I was considering changing the plans to allow some sort of button to fire it to improve accuracy, but overall if the students were careful they came out with excellent results.

Stay tuned for part 2 which will include the actual project the kids do!

Quadratic Formula Rap

2009-08-26T17:01:00.009-04:00

If there's one thing my students learn by the end of the year and actually still remember in later years, it's the quadratic formula. The class where I introduce the formula goes down like this: First I tell them about the quadratic formula in a traditional way. I explain that now with the QF we can solve any quadratic equation, and do it much easier than we could with completing the square. I show them how to use it and they solve a couple quadratics themselves. I then tell them that for homework they have to memorize the quadratic formula overnight and there will be a quiz on it at the beginning of next class.(This is not my usual style) I always receive a chorus of groans. "But!" I interject "It will be much easier than you think. I've gotten someone to come in and help you all with this, let me go get him." I go into the hallway, put my tie around my head, half untuck my shirt and start a live performance of this. (my rapping name is SweenDawg, of course)

The live performance helps make it really fun for them, and I would highly suggest doing it if you decide to use a rap in your classroom. Any time I do a song in class (there are others) I typically do one "live" and then have a recording so I can play it for the kids multiple times and in later classes to help it stick in their memories. Now I realize this is not the most groundbreaking or new idea, but I want to stress its effectiveness and fun. The kids who have been generally uninterested throughout the year usually love this lesson the most and really get into it. Not only that, but I work in a small school and when I have students in later years they almost always remember how to solve quadratics without any prompting... or maybe just an "op-op-op" to get them started.

I also tend to plug the idea of making their own strategies when they have to memorize something, and how making a song is just one example of a memorization technique.

Shoutout to Mr. Mellor for helping lay down the track.

Have your own fun song that you like to do with your students? Tell me about it!

Welcome!

2009-08-24T15:36:00.000-04:00

Hello and welcome to my brand new blog! I hope to update my blog at least weekly with lessons that I teach and love, or that I just like and want to make better. I hope that you enjoy what you find.