Sep 022014
 

success

 

Today was mostly successful for my learners! Yay. I am behind in the book keeping / grading department so I won’t talk about that much. This is always my downfall, and it is so easy to have a conflict of interest between grading effectively and timely, and time spent producing interesting fun lessons that teach! It is worse this year for me because of grad school on top of it.  Enough about the Failure, on to the Success!

AP Statistics

I did a “I have, Who Has” exercise with vocab from Experimental Design and Surveys. It was tough going at first. I screwed up the instructions and suggested they trade cards in the first period. Don’t do that. I think better instructions are:

1. You all have a card with a word, the “I have experimental design” for example.

2. Underneath that word is a definition. It is NOT the definition of the word above, but the definition to a different word.

3. Find the person who has the definition of your word, and then stand next to them.

4. When you are done, you will have a giant circle of definitions.

5. AFTER the exercise is over, I let them take the 4 pages of cards with them. Yes, I made copies for every single learner. Now they did the exercise AND there is a set for them to take with them with which to study.

The biggest problem I had when doing this is the learners looking to me for validation. One class today dove right in and struggled with it and asked me almost no questions. The other class, oh boy. They wanted me to validate every answer. Every time I tried to get them to validate the answers themselves they were frustrated and really tried to get me involved. I finally had to tell the class, “No.” Stop and think about it as a group.

In the end, both classes today were successful, but I wanted greater fluidity. One problem is neither class really had a leader who took charge. It is a work in progress for sure.

 Algebra 2

This was a failure. I did @Cheesemonkey’s Speed Dating (and @mathymeg07 and I typeset files her files too) and was horrified by the lack of understanding of transformations of functions.

Sigh.

By the end of the period they were doing them okay. Not with any fluidity, not with any sense of understanding. I will do this exercise again. Absolutely.

They were frustrated and I was too. They wanted to do a table for every single function. Not good. By the end of class, I heard, “Aha, this is so much easier” and “Okay, I think I am getting it now.”

Next class we really need to get to, “Okay, Waddell, give me your best shot.” Eventually we will get there.

So, my lesson for tomorrow’s Alg2 class is already written. Do today over.

Time to grade more work again. Must get more things into gradebook!

Sep 012014
 

Sometimes the internet is your savior and sometimes your timesink, but when reading @Cheesemonkey’s material, it  is always golden delicious bits of luscious learning.

Take, for example, her post on Transformations in Precalc. It is a fabulous exercise, and I am going to do it this week. I was talking with my fellow teacher in Alg 2 Honors last week that the way to write translations is using function notation, and this exercise has everything wrapped up into a foil package of wonderfulness.

But…. I am really … particular… about how I write things, and I cannot just use her files. They are hand written and scanned, and that is not as nice as typed and printed.

So, I remade them. Here is the Word DocX and the PDF of her 36 transformations. I have included her instructions and link to the blog post in the file so that full attribution will always go to Elizabeth.

Thank you Elizabeth! I will let everyone know how it goes next week. EEK! Tomorrow!

——

edited same day

Due to some further delicious efforts by Meg Craig (@mathymeg07) we have Day 2 of the speed dating sheets up!

Speed-Dating-Cards Day 2 DocX format

Speed-Dating-Cards Day 2 PDF format

Aug 282014
 

Algebra 2:

mindblown

My learners blow my mind. I assigned my Honors Algebra 2 learners to write their name.

Seriously, all they had to do was write their first name.

In Desmos. With functions.

Heh, I am evil, right? It gets worse. They had just taken a quiz, so we only had about 10 minutes left in class. I showed them how to create an account on Desmos to save their work. I showed them how to type in a function. I showed them my name in Desmos. And then I showed them that if they scrolled down on the main Desmos page, they would see, well, they would see some amazing art created by learners like themselves.

That is it. That is all the prep, instruction, training, or anything else you want to call it, that I gave them. They had to take this bare bones instruction, write your name, and run with it. I never showed them circles, ellipses, or any other function. They learned it all themselves. Here are samples of what was shared with me:

skylar24 cloe10

Skylar = 24 functions, Chloe = 10 functions

 

janine35 gentry17

 

Janine = 35 functions and Gentry = 17 functions

Pretty representative. I did not say they had to use any special functions, just write your name. And I want to point out, I never showed them translations. They figured that out themselves by working with Desmos for ONE SINGLE DAY! Nice.

Then I started looking at the functions each person used. I noticed something very interesting that Chloe did. She only used 10 functions, but the “e” was especially interesting. You see, she did BOTH a domain restriction AND a range restriction on the same function.

cloeamaze

See what she did there? Mind Blown. I am still stunned by the creativity she used with Desmos. When I asked her why she did both her answer was, “I was just trying stuff until it worked and looked the way I wanted it.” Genius. Necessity is the mother of invention.

Anyway, we finished up by rocking translations. They already had the main part of translations down because they played with Desmos. That was awesome. They are still trying to make things complicated, but I am almost finished breaking them of the assumption things have to be difficult.

 

AP Stats

I am short 40 books so far this year, so I need to do things to get my learners doing problems without the book. One way to do that is Relay Cards. This is how I play the game. I hand out problem 1. Everyone gets the same problem, so they can discuss it, but they have to write their own.  I use a magnet to hang the answer I previously made on the board. The learners can come up and read the key after they have tried it. Their answer must be the same as my key in meaning, not in words (usually. Sometimes it has to be exact, as in the probability section.)

Once card one is done, they come to me for card 2, and so on.

Having the key on the board keeps me free to answer questions and help while I hand out the new problems and double check the accuracy of the previous.

I just finished a set for Experimental Design.  I have other sets (created by Shelli Temple (@druinok) almost completely).

03 Relay Cards for Conditional Marginal Probabilities

Ch 6 – Relay Cards – normal models

These are a great way to get the learners talking about the stats, writing and working with stats, and the teacher does nothing but help, coach, and assist learners learn.

I like this activity greatly.

Aug 142014
 

2014-08-10 15.43.10

My 3 rules for Algebra. Meg Craig made them pretty, I framed them, and went over them in Alg 2 Honors class last period.  Today we moved into Literal Equations. That is how I teach the rules and reinforce the rules, with literals.

Typically, these are one of the hardest things learners to grasp and wrap their head around. I am taking my physics book from college to the board and writing down formulas from it and from chemistry. Go.

And it was tough. Learners who can tell you that 25-25 is zero in a heartbeat will balk at v – v is zero. So there is a lot of coaching an patience as they work through the ideas of “The Rules” and the differences in how addition and multiplication act with respect to distribution.

It was difficult, fun, and frustrating but they are getting the concept of solving equations, all equations, no matter what they look like.  I will spend one more day, so that means 2 and a partial period on this. I think it is time well spent.

 

AP Statistics

The 6 W’s a H:

Who, What, When, Where, Why, by Whom and How. This is the introduction to thinking and reading scientifically. ScienceDaily.com is my friend during these couple of days, and the biggest stumbling blocks were finding the variables (the What) and the population of interest (the Who). I have added a 6th W, “by Whom”. I have found that helps with the learners wanting to put the author’s name in the “Who” spot. If they have a place for them up front, they can’t be part of the “who”.

Yesterday’s assignment was to grab one article from ScienceDaily and detail the W’s and H.I don’t think one learner did it fully correctly, though. I know that every single learner I helped one on one did not (and that was most!).

To help, in class we did a couple of problems from the textbook, which are … okay. Nothing earth shattering; they are short, and the learners can use keywords to figure out the answers quickly. Then we did 1 article from ScienceDaily on the board. I projected it from the web and we went through it, finding all the information. That helped too.

Finally, I handed each group of 2 an article and had them discuss the W’s and H together. I circulated and answered questions. They then swapped articles and did it again.

At this point, they have read 4 to 5 different articles from ScienceDaily. That is 4 to 5 more than they have read in the past. Reading these articles is a different skill than reading the fiction that English teachers have them read, and I think this is useful to getting the learners to be in a Stats frame of mind. I hope so at least.

The assignment for next class is to redo last night’s homework. Cross it out, start over, and do it right.

Aug 132014
 

My last post brought up the fact I am on a block schedule. It is a good schedule and I like the way I have created interaction with learners because of it.

Our schedule goes like this.

First class every day is P.A.S.S. – Panther Academic Student Support. It is 45 minutes of work time where each teacher is either teaching a “for grade” class or a “S/U” class. The “for grade” classes are just like any other class, but meet 5 times a week. The “S/U” classes are flexible and can be anything the teacher wants them to be. I have 26 of my AP Stats learners in my class. They are all motivated, great learners, so I help them be successful in all their classes.

The rest of our schedule is more normal (if you can call what I describe now as normal).

We have 72 minutes classes every day but Wednesday. On Wednesdays we are released 45 min early for PD time, so we have 60 minute classes.

In the morning, we rotate through a schedule of 1-2-3, but see only two of them per day. So Mon we see 1, 2, Tuesday we see 3 and 1 again. Wednesday (the short day) will see 2, 3, then Thursday will be 1, 2, Friday will be 3, 1, and Monday will be 2, 3, etc. It is a rolling 1,2,3 in the morning.

The afternoon is the same except for 4,5,6.

Monday Tuesday Wed Thursday Friday Monday Tuesday Wed
PASS PASS PASS PASS PASS PASS PASS PASS
1, 2 3, 1 2, 3 1, 2 3, 1 2, 3 1, 2 3, 1
Lunch Lunch Lunch Lunch Lunch Lunch Lunch Lunch
4, 5 6, 4 5, 6 4, 5 6, 4 5, 6 4, 5 6, 4

 

What I like about this schedule is that I see my learners two times consecutively, and then a day off. So I see period 1 on Monday and Tuesday, then not on Wednesday. That allows me to stage the assignments so there is a short brief, and a longer more thoughtful assignment.

I also do not see the same learners at the same time every day. At the end of the day, I see period 5, then period 4, then period 6, not the same period 6 every day. Seeing them at different times of the day allows me to see them when their energy levels are different (as well as mine) and at different attention levels.  That is very helpful in how I set up to teach them and give them what they need.

It also means that my planning goes in 3 day cycles. This makes it hard to do a 180 blog, but I will be doing my best at least to do a 120 blog.

 

AP Statistics

Today is my happiest day. I teach 3 AP Stats classes in a row. Because of the schedule above and my schedule (per 2, 3, 5 = AP Stats; per 4 = Alg 2; periods 1, 6 = prep and dept lead prep) on the “C” days [2,3,5,6] I get to have PASS, then AP Stats, AP Stats, AP Stats, Dept Prep.

There can not be a better schedule.

—-(time passes to afternoon)

Today we did the “W’s” and “H”. [My PPTX file w/ problems]

Whew. I always think this is one of the toughest lessons. The confusion about when numbers are categories and when numbers are quantities is confusing at first, and the notes are hard to do, and the reading is tough, and … and … Arg.

Plus, it was the short day, so we didn’t do the reading I wanted, just the notes. But that means tomorrow and Friday we can do lots of reading. Yay!

I added something to the book’s W’s and H, which is “By Whom”. I have found that adding the “By Whom” eliminates the learners trying to write “Gallup” for the “Who”.

The assignment was to go to ScienceDaily.com, find ONE article from there and do the whole breakdown.

Tomorrow we are reading lots of articles, as well as some problems from the textbook.

All in all, a great day.

Jun 292014
 

This post is really for a friend of mine who is very much math aware and capable, but does not teach math. He has twin sons who are absolutely amazeballs smart, and through talking with him one day I espoused my approach to algebra. He was intrigued and wanted more info. This is my attempt at more info.  I figured that others may find value in it, so I am publically posting it for all to see (and read, hopefully.) Please rip it to shreds if you feel I am in error or made a mistake. I want to do better, and I can’t if I am doing something wrong.

So let me begin with where this came from. At the NCTM Las Vegas regional conference in 2013 I was introduced by an elementary teacher to what she called 13 Rules that Expire.

13 rules that expire

There are some definite Algebra concepts on here like number 2) Use keywords to solve word problems and number 3) you can’t take a bigger number from a smaller number or what really kills me is number 8) multiply everything inside the parenthesis by the number outside the parenthesis.

Yea, right. Try that with f(x)=2x+5.

 

And from there we have Nix the Tricks. A more rigorous treatment of the stupid things we as math teachers do to mess up learners in the guise of teaching them to get an answer instead of understand the mathematics.

So I sat back and tried to come up with some rules that did not expire. Some essential rules that always work, that always build understanding and not destroy it. I ended up with 3.

1. When solving an equation, you can do absolutely anything you want, as long as you do it to all terms of the equation.

2. When working with expressions, you can only change it by adding or subtracting 0 (by using additive inverses), or when solving equations, you create zeros by adding or subtracting 0 (by using additive inverses).

3. When working with expressions, you can only change it by multiplying by 1 (in any form), or when solving equations, you create 1 by using the inverse functions/operations.

That’s it. When I am teaching math, I stress the idea of creating a zero or using a zero. When demanding written explanations, I demand they say that 5+-5=0 in their explanation. So, a nice short example.

Find the function that is the inverse of y = 1/3x – 4

Work                             Why did I do what I did?

x = 1/3y – 4                    Because the first step in finding the inverse is switching the x & y

+ 4        +4                     Add 4 to both sides because –4 + 4 = 0 (additive inverses = identity)

x + 4 = 1/3y                    result

3(x+4) = 3(1/3)y              Mult by 3 because 3(1/3) = 1, (multiplicative inverses = identity)

3(x+4) = y                       finished, but should check it by …….

So there is an example of the work I require. I want them to be using the language of inverses and identity. Why does the square and the square root cancel each other? Because the exponent of 1/2 (the square root) and the exponent of 2 (the square) when multiplied equals 1, the identity.

Along with this, there are some forbidden words and phrases in my classroom. One is above, “Cancel”. I do not allow my learners to use it. At all. Ever.

Why? Because I have seen all of the following described by the word:  -5 + 5 cancels to make 0. 3/3 cancels to make 1. Log10^4 cancels to make 4, sqrt(5^2) cancels to make 5 and on and on and on.

If those all “cancel” then that word means nothing, and it does not mean anything at all. It is just a word used to hid the mathematical knowledge of inverses and identities.

Another phrase that I will not allow is the common answer to the question, “Why did we add 4 in the first step above?” They typical answer I get is, “Because we want the y by itself.”

Huh?  That is not why we added 4. We could do anything we want in the world. We could have added 6, or subtracted 3, or taken both left and right side and made them exponents with a base of 7.5. We added 4 because –4 + 4 = 0. That zero is important. I also write it down. I think most teachers do not. Or they draw a line through the –4 and 4. What does that line mean?  I tell them it means a 1, and did we make a one? No, we made a zero.

It takes some time to unlearn the bad habits, but this is a Nix the Tricks kind of endeavor. In the end, I think I am helping the learners understand math better and more deeply. At least, I hope I am. Only time will tell.

Mar 072014
 

My goals:

  1. Construct a consistent vocabulary of problems that can begin in Algebra 1 and extend through to Calculus, Statistics, and all courses in between.
  2. The problems must have the potential to be engaging to learners.
  3. The problems must hit at least 4 of the eight Mathematical Practices & high school math standards (CCSS).

My idea started with this idea for Algebra 2: Model the escape velocity of a rocket on the Moon and the Earth. ( PDF and Word DOCX) This ended up being a far more difficult task than I expected, mainly because the learners did not connect the idea of writing the equation of a line with the fact we had a function in front of us.

I Desmosed the project for a visual display, and we spent another day discussing it and achieved the goal. [Is it okay to use the name as a verb? I don’t care, I am doing it anyway.]  It turned out great in the end, but it made me start thinking hard about how to connect Algebra 1 through Calc and Stats and make the ideas more real, more understandable, and more connected.

From there came the idea of using an “off the shelf” structure in a new or different manner to extend the lessons. Enter http://graphingstories.com . Dan Meyer started the Graphing Stories with a long time ago, and they are awesome. But they also fit the idea of using the video / graph combination to write the equations of lines and finding area under the curves.

With that in mind, I offer the following Desmos files:

File 1:

  1. This uses the Graphing Story of water being poured into a graduated cylinder to create the graph. I took some points from the graph on screen, and wrote a function that goes through the point (0, 0) because we know it was empty at time 0.
  2. Notice that the line does not go through exactly all 4 points! That allows for discussion of variability and observation skills.
  3. I also used the (h, k) form to write the function f(x) because it is the easiest way to show the line.
  4. What does the slope MEAN?  A standard AP Statistics interpretation is: As the time increases by 1 second, the water increases by 40.67ml.
  5. Next, find the area under the curve. Move the slider for “b” to the right and you see the area highlighted.  Okay, standard triangle, ½ b*h, and you get 5205.33 ml*sec. ??? What does that even mean?
    1. It is called “absement” and it is the time-integral of displacement. Yes, we don’t need to discuss that for Algebra 1, but as teachers we should know it.
    2. The area is the sum of all the instantaneous moments of water before. With the Desmosed file, you can see and clearly communicate what it means. It means that you are adding up the area of the little triangle when b=1 with the larger triangle when b=1.5, and then with b=2, etc. Except the area is the sum of the instantaneous areas, not the discrete areas.

Notice that this one lesson required the learner to interpret a real life action, pouring water, into a graph, and then find the slope and write the equation of a line, and then interpret the slope, and then find the area under the curve.

These are all essential skills of the Calculus learner, done at the Algebra 1 level!

A second one.

  1. Now we are removing cups from a scale.  There are actually several questions that the video brought to my mind, like is this really a continuous line, or should it be more discrete? Time is continuous, but the weights really are stepped.  But, I left it as is though because I wanted to not change it from what the video shows. That is a larger conversation in class.
  2. We now have a negative slope to calculate, which does not really make a huge difference for interpreting the slope: As the time increases by 1 second, the weight of the cups decreases by 3 grams.
  3. The fact the line only hits 1 point absolutely creates some conversation about which point to pick, variability, ect.
  4. The area gets fun, however.
  5. Notice that the FULL area is still a triangle. However, if you move the “b” slider across, you notice the partial areas, the area at 5 seconds, 8 seconds, etc, are trapezoids! Now the learner can be challenged and pushed to incorporate some extra questions of find the area of trapezoids.
  6. We still are doing and absement calculation and not a displacement calculation.

Finally, the Desmosed Lunar Modeling I started with:

It is far more complex and involved, but that is why it is an Algebra 2 lesson and not an Algebra 1 lesson.