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Series Convergence/ Divergence

7 Mar

Super nerd alert. I have been working the last 3 hours on perfecting this flow chart that took me 3 years to figure out how best to go about proving series for convergence or divergence. If you have any idea what I’m talking about, I hope you find it useful. =)

(Please be aware that these are not perfect descriptions of the tests, only summaries. The second page is a fill in the blank copy. I think it would be fun to have a group competition to try to figure out every blank, then review/grade together as a class.)

screencapture

Length, Area, and Volume

17 May

Them-PosterDid you know… that if you DOUBLE up the size of a chair by doubling up its dimensions in every direction, you will need FOUR times as much paint to paint that chair and that it will weight EIGHT times as much?

Did you know… that if a gorilla was to really proportionally increase to be the size of King Kong, that his guts would just explode and gush out of him from the sheer weight of itself? (The same concept prevents an elephant from ever getting a c-section!)

Did you know… that if an ant was really the size of the ants in “Them!”, not only would they not be able to stand up on their legs, they would also suffocate immediately?

Did you know… that if you proportionally double up in size, you’ll only be able to jump the same height as you did when you were normal size? Likewise, if you got 100 times smaller, you will also be able to jump the same height as you did when you were normal size! (So how long should it have taken for the kids in “Honey, I Shrank the Kids” to get home?) 

tumblr_m78krxad8d1qiceiuo1_500Interesting, right?  All these facts  take into consideration that very first “Did you know…,” that changing a linear dimension changes the area and volume differently. Intuitively we all have some understanding of this. I mean, imagine making an itty-bitty model of a wee little ant out of Play-Doh. How easily will the ant stand on its tiny legs? Now can you imagine building a 2-foot model of the same ant? You’re going to have some difficulty making it stand on the same proportional legs.

So I found it all the more interesting when I went online to search for great lessons on this concept and found NONE. I had never been able to teach this topic to very much detail due to our crazy pacing plans and, frankly, due to my hazy knowledge of the science behind it. However, this year, I had a couple of advantages that allowed me to actually try to plan a lesson- time, of course, and an awesome student teacher who is ridiculously skilled in the sciences!

Anyways, Day 1, I started with a lesson on the basic concept* and followed it up with a follow-up to an exit ticket from the area chapter. It was pretty interesting to see how many students didn’t make the connection to the worksheet that was previously given. I’d say it was about 50/50. Half who figured it out and half who simply multiplied the 52 square foot tiles by 12 inches per foot… (I posted this student’s work because she had asked me if she could simply multiply by 12 or if she had to convert all the numbers on the page first and re-work it. Like a good teacher should, I said, “Exactly! Test it!”)

tiles

The next day was the fun day though. I had almost an entire block period to now talk about the implications of same concept in reality.


 

Again, looking online, I found almost nothing fun, nothing very deep, and nothing very useful to me, except these notes… which I devoured. And from these, we came up with this lesson

__________________________________________________________________________________________

FACT: If you double up the dimensions in every direction, then the area multiplies by ____four____________ and the volume multiplies by ______eight________.

QUESTION: If you have a chandelier barely hanging from a rope and you doubled up the size, including the rope that it hangs on, will the chandelier still hang or will the rope snap?

Why or why not? (Make your argument here. You will not be graded based upon whether your answer is right or wrong, but on how well you make your case.)

 By this point, most students understand that although the dimensions are doubled, the weight of the chandelier has now become 8 times heavier. However, italicizing the part about the rope has them also consider the fact that the rope has also gotten 8 times heavier proportionally. A very true fact. I have them in groups and ask them to talk about it and then write out their argument, emphasizing that they will NOT be graded on correctness. 

I then take a vote. The vote for all 3 of my classes was close to 50/50. I say awesome, and before I give away anything else, I have one person from each camp make their case. I then take out a piece of rope to begin my “Consider…” lecture:

Consider… a really heavy object hanging from this piece of rope. I want to make it stronger and so I double up the length, like so, because I know that this will increase the VOLUME by twice as much. Will that make the rope stronger? [Class agrees, no way!] So then, how do I make the rope stronger?? … exactly! I need to double up the rope by making it thicker. So what does that mean? What does the strength of the rope depend on? The volume? (no) The thickness (yes!). And how do we measure thickness?? … the cross-sectional AREA!

So will the rope snap of the chandelier snap or not?! Discuss…

So yes, the rope definitely snaps. The cross-sectional AREA only increased four-fold (which is what the strength of the rope depends on) while the weight increased eight-fold. In this case, two chords of the thicker rope will hold the larger chandelier.

__________________________________________________________________________________________

FACT: The height that a body can jump vertically is directly proportional to the ratio of muscle mass to the total mass of the body.

jump height

, where C is a constant (enough).

QUESTION: If a human being who can jump 1.5 feet is doubled up in size (pretending like it’s actually possible), how high will the new larger human be able to jump? Explain.

 This was the section that my student teacher had helped me out the most in. He came back w/ an equation that we boiled down and simplified to the one above (apologies to any physicist who might get offended by this). I helped to explain what this equation meant and how it made sense. I made sure they all knew that working out actually made your muscles heavier, which makes total sense in this equation. I had them discuss and then take a vote. 

Their logic behind this one was so adorable. Most of them completely ignored the equation and went straight to guesses. “If twice as tall, muscles get bigger so they’re stronger… they can jump higher.” But the winner of all arguments…

He jumps shorter! Like Mario! When Mario grows, he can’t jump as high! 

The group cracked up and then voted, yes, he jumps less than 1.5 feet… like Mario. 

Final answer? This person will jump the same height of 1.5 feet. His/her muscle weight will definitely increase eight-fold, but so will the person’s total weight, canceling each other out. 

QUESTION: Now reverse the idea. If the same human being is shrunk to half the size, how high can the new small human jump?

Likewise, the smaller human can jump 1.5 feet.

(So how long would it have taken the kids in “Honey, I Shrunk the Kids” to go home?!)

It would’ve taken the same amount of jumps as when they were full size! This was an awesome discovery after reading that one professor’s notes.

__________________________________________________________________________________________

FACT: Ants don’t have lungs. Air is received through “blind tubes” that cover the surface of the ants’ bodies where air enters through.

QUESTION: If an ant was to be blown up as big as the ants in the movie, “Them!”
(pretending the physical structure is possible), what can you conclude about the
air intake of the giant ant?

 We didn’t have time in class to go over these, but these should be easier after all that we’ve learned. I should’ve given it to them for homework… 

 FACT: With larger animals came the development of blood circulation and lungs as a more efficient way of oxygen distribution. Lungs increase the surface area available for oxygen absorption by the blood.

QUESTION: If a human was blown up twice as big (again pretending like the physical structure is possible), how do the lungs have to increase along with the body in order to support the oxygen level needed to survive?

__________________________________________________________________________________________

Anyways, if there are any other lessons/facts out there that you think are good for this concept, let me know! I didn’t have this much fun lecturing in awhile!

*Some of the problems from that worksheet were either directly taken or modified from the NCTM Illuminations site.

And the winner of the Flashcards is…

12 May

www.QUIZLET.com!

After attempting my own templates at creating flashcards on Word and Excel, a student showed me this site and I cried, “HOORAH!!!!”

FINALLY! A website that gets it! At least for flashcard needs.

Screen Shot 2013-05-12 at 9.08.21 PMPROS:

  • It’s FREEEEEEE!!! At least until you need to upload your own images. Otherwise, it’s $15/ year. I happily paid it so that I can freely use screen captures and whatnot for geometry figures and algebra graphs.
  • They have an APP for it!!! Tap to flip, swipe for next card. Awesome.
  • Super easy and clean spaces for you to put in your words.
  • They have multiple languages you can type in, including MATH. =)
  • Your students can look you up and freely print your sets, or even create their own sets!
  • And my favorite?! They have 5 different print options!!! –>

CONS:

  • You can’t input images on both sides. Therefore, my cards are not all flipped in the same direction. I print, cut, and rearrange which way they should face, if and when it matters.

You can see the sets I have already created by searching my full name, jinnahwang (or by clicking here). I like to create class sets for them to practice with and have my own set to do my verbal vocab with. I’ve had students often ask me if they can get a copy and its always been clunky in printing out a PDF and then posting it on my website. I LOVE the print and fold flashcards as well so I can quickly print them on regular paper for students who are not-so-web-savvy and they can create them at home.

Love it!

Toblerone Challenge –> Can Top Challenge

28 Apr

Here was another fun one.

The Word file here: Toblerone Challenge

Screen Shot 2013-04-28 at 11.20.36 PM

A few logistics:

  • They got to work either alone or in partners. Winner gets ONE Toblerone bar, so if partnering up, they must share.
  • 20 minutes was all that they had. After the Play-doh activity the previous day, they understood that the winner would be the BEST net for this Toblerone box in those 20 minutes. No extension of time.
  • Only one sheet of cardstock was actually passed out to the groups. All other materials were available to them for pickup, if they needed- rulers, scissors, compasses, protractors. They had to build everything until they needed the gluing, pretending like I was the one who will be gluing (there was no glue involved).
  • Collect all nets immediately and put them aside to determine winner when they are doing classwork.
  • At the end of class, when announcing the winner, I took apart the real Toblerone box under the doc cam to see how the actual engineers did it. I then wrapped the chocolate bar w/ the winning net and awarded the winners their chocolate.
  • Hw was to find the lateral area, base area, and total surface area of the box.

Immediately after they turned in their net and before any kind of clean-up, there was a 5-minute bonus challenge.

Screen Shot 2013-04-28 at 11.36.11 PM

I had the actual can in my hands w/ the actual label on it. The label was only taped down on one side so I could show how it wraps around the can and opens up to be a rectangle. I pretty much laid it all out save handing the can to them. I think I still only had about 2 groups per class understand how to find the perfect circle to go on top of the can. Most of them just eyeballed it. Ha!

Winners of that challenge got a mystery can that I had pulled the label off of last year for last year’s lesson. At least the expiration date was on there to prove it wasn’t expired yet! They just wouldn’t know what it contained until they actually opened it. Heh.

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 =/)…

Image

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.

Triangle Congruence Flashcards

1 Nov

I’m starting to wish every worksheet was available in flashcard form! They TALK about the problems, they ASK questions, they rotate/point to the problems as they explain, they debate, they make up games on their own… and they get to practice a million times over w/out wasting a million paper!

ImageSo with this set, in the BOX widget to the left, they simply have to identify if they triangles are congruent by which postulate/theorem or if it cannot be proven, say it cannot be proven. I gave them the cards right after teaching them SSS, SAS, ASA, and AAS. I had pulled out all the ones they did not know yet beforehand. After I taught them HL and the Isosceles Triangle Theorem, I added those cards in and gave it to them again.

TO PRINT:

Print pages 1 and 2 back to back, 3 and 4 back to back, and 5 and 6 back to back. Pages 5/6, however, are double sets so you only need to print half as many as the other two pages.

I printed a nice cardstock class set for myself (one per pair). Students use this in class and if they need to draw on it to explain to each other, I give them patty paper so they don’t mess up my cards. I also printed out a bunch of extra copies (not cut) on regular colored paper for students to take home if they need more practice.

ImageI threw this last card in today because after training them on how to identify AAS vs ASA, for example, they forgot what these were actually proving to start
!

Excel Flashcards template

12 Mar

Holy moly. Why didn’t I think of this sooner? I just created an excel template for 12 flashcards per page, where the inputs go into a table on the first worksheet and the flashcards are created on the second worksheet. This way, I don’t have to constantly try to align and rethink where the corresponding backs of each flashcard go.

It’s not perfect in that equations are a bit cumbersome. I would honestly just write in the exponents and such before I make the copies onto cardstock.

I’ve just been making class sets of useful flashcards for my Geo classes. This would be great for vocab! I created it originally for factoring perfect squares, as partner work. I’ll drop both into the Box widget, to be edited as you please. Please be careful of the “12flashcards ppg” file because it is a TEMPLATE file to be saved in your Templates folder…

1st worksheet where the data is input

2nd worksheet of the actual flashcards for print

Dear Basketball…

2 Mar

Kind of random, but I had this assignment come up in conversation and so I went back to re-read the letter before sending it out to a friend and I was reminded of just how good it was. Of just how good he was.

It’s a letter to basketball written by Michael Jordan after his second retirement. It was published in the LA Times on April 20 of 2003. Read it. You’ll enjoy it.

Dear Basketball,

It’s been almost 28 years since the first day we met. 28 years since I saw you in the back of our garage. 28 years since my parents introduced us.

If someone would have told me then what would become of us, I’m not sure I would have believed them. I barely remembered your name.

Then I started seeing you around the neighborhood and watching you on television. I used to see you with guys down at the playground. But when my older brother started paying more attention to you, I started to wonder. Maybe you were different.

We hung out a few times. The more I got to know you, the more I liked you. And as life would have it, when I finally got really interested in you, when I was finally ready to get serious, you left me off the varsity. You told me I wasn’t good enough.

I was crushed. I was hurt. I think I even cried.

Then I wanted you more than ever. So I practiced. I hustled. I worked on my game. Passing. Dribbling. Shooting. Thinking. I ran. I did sit-ups. I did push-ups. I did pull-ups. I lifted weights. I studied you. I began to fall in love and you noticed. At least that’s what Coach Smith said.

At the time, I wasn’t sure exactly what was going on. But, now I know. Coach Smith was teaching me how to love you, how to listen to you, how to understand you, how to respect you and how to appreciate you. Then it happened. That night, at the Louisiana Superdome, in the final seconds of the championship game againstGeorgetown, you found me in the corner and we danced.

Since then, you’ve become so much more than just a ball to me. You’ve become more than just a court. More than just a hoop. More than just a pair of sneakers. More than just a game.

In some respects, you’ve become my life. My passion. My motivation. My inspiration.

You’re my biggest fan and my harshest critic. You’re my dearest friend and my strongest ally. You’re my toughest competition. You’re my passport around the world and my visa into the hearts of millions.

So much has changed since the first day we met, and to a large degree, I have you to thank. So if you haven’t heard me say it before, let me say it now for the world to hear. Thank you. Thank you Basketball. Thank you for everything.

Thank you for all the players that came before me. Thank you for all the players who went into battle with me. Thank you for the championships and the rings. Thank you for the All-Star games and the Playoffs. Thank you for the last shots, the buzzer-beaters, the hard fouls, the victories and the defeats. Thank you for making me earn my keep. Thank you for #23. Thank you forNorth CarolinaandChicago. Thank you for the air and the nickname. Thank you for the moves and the hang time. Thank you for the Slam-Dunk Contest. Thank you for the will and determination, the heart and the soul, the pride and the courage. Thank you for the competitive spirit and the competition to challenge it. Thank you for the failures and the setbacks, the blessings and the applause. Thank you for the triangle. Thank you for baseball and the Barons. Thank you for forgiving me. Thank you for the assistant coaches, the trainers, and the physical therapists. Thank you for the announcers, the refs, the writers, the reporters, the broadcasters and the radio stations. Thank you for the Pistons and the Lakers, the Cavs and the Knicks, the Sixers and the Celtics. Thank you forPhoenix,Portland,SeattleandUtah. Thank you for the Wizards. Thank you for the believers and the doubters. Thank you for the education and the experience. Thank you for teaching me the game, behind, beneath, within, above and around the game… the game. Thank you for every fan who has ever called my name, put their hands together for me and my teammates, slapped me five or patted me on the back. Thank you for everything you’ve given my family. Thank you for the moon and the stars, and last but not least, thank you for Bugs and Mars.

I know I’m not the only one who loves you. I know you have loved many before me and will love many after me. But, I also know that what we had was unique. It was special. So as our relationship changes yet again, as well as relationships do, one thing is for sure.

I love you Basketball. I love everything about you and I always will. My playing days in the NBA are definitely over but our relationship will never end.

Much Love and Respect,

Michael Jordan

Pretty awesome, no? I have some of my math classes read it and write a letter to Math at the beginning of the school year. (I got the idea from some speaker about 8 years ago I can’t remember… =( ) The results and insights from their letters are pretty telling actually and I get a rare sense of their writing skills that I don’t normally get throughout the year. Maybe someone will find it useful for the end of this school year or as a midyear check…

Properties of Quadrilaterals

13 Jan

FINALLY, a lesson for this section that I’m happy with. I’d say that for the past 7 years I’ve been trying different ways of having students “discover” the properties of the special quadrilaterals (rectangles, rhombuses, squares, trapezoids, etc.), but it seemed like every time I tried they were either terrible at the drawings so that they couldn’t discover the properties or they didn’t seem to have very good ideas of what properties even were.

That second hindrance makes sense to me now in hindsight, but it really did take me this long to figure out that maybe I can give them a checklist alongside the quadrilaterals that I have started to create for them.

And it worked beautifully! I had them work in partners and work through together. They discussed, they asked questions (all the right ones), and they made their “discoveries.”

Anyways, below is a quick snapshot of what it looked like.

  • I did half of parallelograms with them because we had already covered these sections. They did the second half . We then went over all of it and I made the connection to what we had already learned before break.
  • I have block periods. I gave them the whole packet to do in one period, which I shouldn’t have. I should have given them all the parallelograms first, then a bunch of practice problems. The next day should have been the   trapezoids and kites and a bunch of practice problems.
  • Materials: rulers and protractors. One quadrilaterals packet per person, but one copy of the checklist per pair. This helps in getting them to work together.

(My Word doc has a bunch of extra lines that need to be whited out. I’m not going to spend time doing it right now. If anyone does and want to send it back, that would be beautiful!)