Tuesday, November 25, 2014

Hidden Squares Activity - Equations with Variables on Both Sides

I did this activity last year with the snowflake posters if you want to check that out here.

To start class:

Tell the students there are a total of 32 small blue squares on this poster.  Some are hidden under the purple flaps.  Let them know that there are the same amount of square hidden under each flap.  How many blue squares are hidden under the flaps?

Most of my students noticed that there were 17 exposed blue squares and that meant that 15 were hidden.  If there are 5 flaps that means there must be 3 under each flap.  

Me:  When we don't know the quantity or value of something what do we do?
Student:  We use x.
Me:  Right a variable.  Can we represent this poster as an expression using constants (numbers) and a variable?

The students were able to come up with 5x + 17 = 32, and then noticed that they needed to take exactly the same steps they took when they solved the poster problem above.  

Me:  Do you see the "bubbles"?  What do you notice about the bubbles?
Student:  They have the same things inside them.  
Me:  Let's use this information to write an equation.

With a little prodding the students came up with 2(2x+5) + x + 7 = 32
As we wrote each line of the solution, we made the connection with the poster to see where the numbers were coming from.

Taking Things Up a Notch:

Show the students this poster (without the algebra at first) and ask if they can determine how many squares are hidden under each flap.
They need to know that both posters have the same amount of small blue squares on them and the same amount are hidden under each purple flap. 

All of my students used trial and error for this at first.  But then as a class we were able to solve with algebra.  I like that after we finish the algebra, we are able to lift a flap to see if we're right.  

Now It's the Students' Turn:

I put students into groups of 2 to 3 and had them create their own posters.  The requirements were:
1) There must be the same number of small squares on each poster.
2) There must be the same number of small squares hidden under each flap.
3) Each poster must have a different amount of flaps.
I also wanted them to have "bubbles" on at least one of their posters.

The last image is very interesting.  The students and I had a nice discussion about it.  At first they said the answer was 1.  I agreed.  But then I asked if it could be 2.  They agreed.  3? 4? 5?  Why?  How can we write what all the answers are?  What does this look like algebraically?

Next time:

I enjoy this activity.  It's such a concrete way to talk about equations, the distributive property, combining like terms, and even consistent and inconsistent systems.  Next time I will focus more on the "why" on the poster requirements.  

1) The posters must have the same amount of small squares.  Why?  What would happen if they didn't?

2) There must be the same number of squares under each flap.  Why?  What would happen if they didn't?

3) There must be different amount of flaps on each poster?  Why?  What would happen if they did?

Monday, November 24, 2014

Linear Patterns

Students previously 'learned' how to find the equation if given a linear sequence of numbers.  They learned the procedure only and it looks like this:

6, 13, 20, 27, ...


The difference between each number is +7.  This is called the first difference.

To find the equation:  (first diff)(n) + (first term - first diff)
For this example it would be (7)(n) + (6 - 7)   OR    7n - 1

Day 1

Me:  "Here is a sequence of numbers 5, 7, 9, 11, ... take a few moments on your own to write an equation to represent this pattern."

Many of the students look at me like I have two heads.  Mind you, quite of few of the students have taken the test on this AND passed.  I see a few students moving in the right direction, so I take a picture of their work with my iPad and display it on the Apple TV.  Then a few more students have some recognition in their eyes.

Next I ask the students if there's a way we can get to the equation without remembering this ridiculous formula.  **Insert crickets chirping**  We discuss about how the pattern is changing each time, that the change is constant, and that's why it's the number multiplied by the variable.  But what about the other number?  We decide that we could plug in a 1 to try to find the 1st term then add or subtract something so that we really do get the first term.

Finally I get to the objective of the day.  That I would like the students to be able to make connections between five different representation of a linear pattern; table, equation, graph, visual, and verbal.  Before I told them what the 5 representation were I asked if they could figure it out.  Would you believe that one of my classes did.  I was so happy I could cry.  As a class we came up with all representations for the pattern above.  As we moved from one representation to another I asked the student how it related to the previous.  The students were very insightful and you could see how confident they felt with having this opportunity to show their stuff.

I gave the entire class the equation y = 3x + 1 and asked them to work with their partner to create the other 4 representations.  Each pair got poster paper, markers, graph paper, and glue.  I was please with the results and I wasn't the only one.  As the students were walking out of the room at the end of class they told me how much they enjoyed class today.  They told me how smart they feel.  They told me we should do this more often.  Even the paraprofessional said she enjoyed class and loved how we make connections between all 5 representations.  

Here are a few posers the students made.

Day 2

I created index cards each with different information on each one.  The pairs of students are given a card randomly and asked to create the other 4 representation.  

This didn't go too bad.  The major issue for most students was finding the equation (the only thing I actually taught them).  But we reasoned our way through it rather than try to remember an algorithm.  

From here I just need to clean up a few things with the students.  Like to make sure the labeling on the axes is consistent:

Or just labeling the axes for scale.

Monday, November 17, 2014

There's Still Hope For Me

This year, I am teaching all of my classes using Flipped Mastery.  Basically, I have (or will have) all of my lessons and lectures on video for the students to view online.    The students work at their own pace during class, assign their own homework, ask questions when they are unsure or confused, and take tests when they feel they are ready for them.  I provide the students with a pacing guide to make sure they will finish the curriculum by the end of the school year.  You can purchase this book to read more about it.

I decided to give it a trial run last Spring to help me work out any kinks.  That was helpful.  It helped me to realize that students really don't understand how to learn.  They are so dependent on teachers to tell them exactly what to do that some are almost paralyzed with all of the freedom.  I needed to be more proactive with encouraging the students to come up with a plan.  

This school year, I started with tables around the room, where 4-5 students could work together.  I created a video and a worksheet for each lesson, and a review for each outcome.  Each outcome has at least two versions of the test and high performance questions/projects for those that prove proficiency.  I keep track of the students who play games on the computer every time I turn my back, the students who do work for other classes in my class, and the students who just sit there and talk.  

The students are required to understand or at the very least perform the procedures that I teach during the videos.  So, I have the procedure part of my job well underway.  The problem is that this is so time consuming that by the time I am finished grading all the tests for that day and planning the next lesson to record, my prep period is over.  I need to bring work home with me every day just to stay ahead of the game.  I'm not feeling great about this.  My students this year are learning procedures mostly, and once in a while I'm able to find the time to do some class activities or lessons to support their understanding of concepts and application.   

But I'm not feeling totally down on myself.  Next year I will have more time.  More time because I will only have to tweak my videos and worksheets and not recreate them.  My plan for next year is have an activity or project each week.  Maybe a Mathalicious or shell lesson one week and a 3-act the following week.  As far as this year goes, there is hope.  I will soon catch up to where I started last year and be able to be more creative.  

I'm still trying to find my balance.  The old rules for cell phones doesn't make sense to me anymore.  Some students are more productive when listening to music while others are not.  Do I allow some students to use their phones while others are not allowed?  Some students want to do work for other classes during my class.  Do I stop them or do I let them go?  What if they prefer to watch the math videos at home while they complete their Spanish worksheet in my class?  Is that really a problem?  I'm walking a tight light between allowing students to find what works for them and saving them.  They are mostly Freshman and still need that guidance, but do I let them fail, so they see that it doesn't work, or do I save them so they pass?  

A few of my colleagues have said some not so nice things to me and I wasn't really prepared for that.  They have told me that my class is a joke and a free-for-all.  I was expecting resistance from the students and some parents, but it never occurred to me that my equals would take my changes so personally.  They are not comfortable with what I'm doing and it defies the rules they have in their classrooms.  Like the bathroom rule.  According the school, students are not allowed to use the restroom during the first and last 10 minutes of class because they will miss the intro and closure to the lecture.  That doesn't apply to my classes most days.  Does this really make my class a free-for-all?  Allowing the students to get up whenever they want to get a tissue, get a pair of headphones, help another students, grab a large whiteboard, or ask me a question must make my classroom a joke.  

But it's not all bad...

This whole flipped mastery thing is changing my career.  Now that I'm not lecturing every period all period long, I have time to get to know my students.  There are some students with baggage that they just want someone to hear about.  I have had more conversations in the hallway with my students this year than all of the other years of my career combined.  One student declared the other day that I was her favorite teacher because I actually care about her. 

I usually lose my voice twice a year.  Once in the fall and once in the spring.  I didn't lose my voice this fall.  I guess not having the project my voice for 5-6 hours a day will prevent that.  

As the students are getting accustomed to this model, they are finding that their classmates are very knowledgeable and willing the help.  I have a student who beautifully explains the vertical line test.  Each time a student has a question about determining if a graph is a function, I refer them to her.  Many students will use a large white board and teach each other, or work on problems together.  I also like this because they are writing larger and I can see their work as I walk by. 

I would love to have a testing center, a place where students can go to take tests when they are ready. I just need to get a few more teachers on board with flipped mastery.

Will I ever go back?  Not likely.

Saturday, November 8, 2014

Systems of Inequalities and the Crazy Hot Matrix

One day when I saw one of my less-than-motivated-yet-bright students "teaching" his table mates with a large white board, I was intrigued.  Basically, he was repeating what he learned on youtube about the Crazy/Hot Matrix.  Unfortunately, this video is not appropriate for the classroom, but look at all the possible ideas you could get from it.

Here's how I see using something like this in the classroom:

The students are grouped in 3s and 4s and decide if they are going to create the male chart or the female chart.  Or even the parent chart, best friend chart, business partner chart, etc.  The students pick two characteristics for the axes.  They could use crazy (I like sanity) vs hot (I like looks).  Or they could come up with their own, such as work ethic, age, education, personality, whatever.
Next they would need to come up with zones.  I'd say at least 3 to 4 zones.  And create the inequalities that bound those zones.  Maybe even a few ordered pairs representing celebs or fictional characters??????