Friday, September 27, 2013

Pennies, Paper Clips, Buttons, and the Distributive Property

First, you must go over to I Speak Math and watch her video.  I am completely stealing and adapting from her.

Before we started the activity we discussed the value of each of the things below.  In hindsight, I should not have used pennies because we know the value of a penny.  I asked students to pretend that they didn't.  We realized that we would talk until we were blue in the face, we still wouldn't know the exact value of each object.  That's when we decided to use variables for each item.

The value of a penny = P

The value of a button = B

The value of a paper clip = C

One by one the students came to the front of the room to either place items in the box (I used orange paper in the photos so it would show up better) or take items out.  I limited them to only one type of item.

The first student placed three pennies in the box.  I asked the students what the total value of all the items in the box were, they told me 3P, so we wrote 3P on our papers.

The next student came forward and put four paper clips in the box, so we wrote + 4C.
Now we have 3P + 4C.  I asked if we should make that 7PC.  We agreed that wouldn't make sense (get it?  sense/cents??  Never mind).

The third student took 2 pennies out of the box.  Now we have 3P + 4C - 2P.

The last student for this round of the activity added 5 buttons to the box.  3P + 4C - 2P + 5B

Keep in mind that the students can't see what's in the box.  I asked students to simplify the expression and they easily came up with 1P + 4C + 5B.  I asked if it would be a different set of items in the box if we wrote 4C + 1P + 5B.  They responded that the order didn't matter.  Yay!  The Commutative Property that we just learned last week!

Now, on to the distributive property.
After we did a few round with the single items, I brought out some Ziploc bags with items prepackaged.

I had 5 bags with 3 buttons and 2 paper clips,...

...5 bags with 4 pennies and 1 button, ...

... and 5 bags with 2 pennies and 1 paper clip.

The students can either put in or take out single items or Ziploc bags.  The first student put in 3 buttons.  3B.

The next student placed in two bags that have 3 buttons and 2 paper clips.
3B + 2(3B + 2C)

The third student took out 1 bag with the 3 buttons and 2 paper clips.
3B + 2(3B + 2C) - 1(3B + 2C)

The students simplified the expression while I emptied the bags into the box to get a total count.
When students came up with answers that did not match the content in the box, we discussed how the mistake was made and why it didn't work.  Like the student who distributed a 1 rather than a -1.

We also discussed that since the 2nd students added 2 bags and the 3rd student took one back out, it was as if only 1 bag was added.  Looking at that algebraically:  3B + 2(3B + 2C) - 1(3B + 2C).  Combine like terms with parenthesis to get 3B + 1(3B + 2C), then distribute.  Same correct answer, interesting.  Does this work all the time or just this problem?

Extension questions:

Write an expression, ask the students to explain what someone would have had to put in or take out of the box to make that expression, then finally simplify.

I like the questions that I Speak Math asked at the end of her video (I always forget to emphasize terminology).  What are the coefficients?  What are the terms?  What is the whole thing called?  What are the constants?  Etc.

Tuesday, September 24, 2013

I'm Trying to Get Inside Their Heads

In the latest Mathematics Teacher Journal from NCTM and Jaime Marts wrote in to Reader Reflections.  You can read what she wrote on page 84.  Anyway, her reflection is about making students thinking visible by placing questions around the room and having students respond to those questions.  And some of you are doing this too, so I've been seeing a lot of this lately.

So here are the posters I placed on tables around the room before we starting our outcome on solving equations:

We just finished learning about the properties, so I thought asking about the Inverse Property was appropriate.  I don't feel like the activity went over very well.  Most of the time the student stood around the poster talking about anything but math, until I walked up to them and asked what they were thinking.

I had the students go to each poster twice so that they could see how their classmates responded to them.  Take a look at this.

I don't know why this is sideways, it's right-side-up when I go to upload it, sorry.  Anyway, see how little writing there is?  *Sigh*.  I know it's hard to see, but someone wrote "I don't know." and another person wrote "I agree."

For this class, there's more writing, but more irrelevant writing.  Someone wrote poems, then another person crossed them out, then the original person wrote back.

I know it's because it's the beginning of the year, but this is frustrating.  Either my questions aren't doing the trick, or the students need more practice with something like this, or some combination of both.  But I feel like this activity was a failure.

Have you tried this?  How did it go for you?

Sunday, September 22, 2013

I want the first 5 minutes of class to run like clockwork.   I want to come into my classroom from hall duty and see every student ready for the day's activities.  In order for this to happen, I usually have to write notes to each class on the board letting them know what materials they will need, and to check their folders EVERYDAY.  But I need my board space for, you know, math stuff.

This year I bought a cheap 8 x 10 plastic picture frame and wrote down for each class what they needed.  I ran into a few snags with that.  I kept forgetting to update after each class for the next class, and my handwriting was too big, that it didn't always fit on the frame.  Sigh.

Then I had the idea to create the following chart.  As the students walk into class they look at the column for their class and get the necessary materials.  The only thing I would change is to add a blank row to the bottom, so that I could add something if I had to.

I have this sitting on the first table as they walk into the room, on bright yellow paper that they can't miss.  It's been working like a charm.

Sunday, September 15, 2013

GCF and LCM Introduction

This week, my Algebra 1 classes started the outcome that includes GCF and LCM.  Before I said anything about the next topic, I told my classes that I had this problem that I needed help solving.

The principal asked me to take the other teachers through an activity during our next in-service day.  He wants the other teachers to work in groups to create lesson plans for our new anti-bullying campaign.  For this training I will have access to 48 iPads and 60 copies of the anti-bullying book (I am not allowed to make more copies because of copyright laws).  I've decided to have the teachers work in teams and compete against each other.  In order to have the teachers create the most lesson plans, I want to have as many groups as possible.  Also, since they are competing against each other I want each groups to have the same amount of resources (iPads and books).  Lastly, since I have these resources I want them to all be used.  No iPad (or book) left behind.

Next I put the students into groups of 3 or 4 and gave them 48 pennies to represent the iPads and 60 paperclips to represent the books.  Below are some photos of the students' work.

If a group was stuck I would ask them if I could have two groups.  Yes, because each group would have 24 iPads and 30 books.  Next, I would ask if I could get more groups, since that is one of my objectives.  Could I have 5 groups?  No, you can't split 48 into 5 groups evenly.  Great!  Now work from there.

Every time a group claimed to have an answer, my response was "Can I get more groups?"

Eventually each group discovered that I could have 12 groups of teachers working with 4 iPads and 5 books.

We had a class discussion on this that eventually lead to GCF.

Next I displayed this website on shipping routes.  I would put in what seemed to be random numbers for each boat and asked the class when the two boats would be at dock again at the same time.

I started with 2 and 4 minutes, then 4 and 5, then 4 and 6, and finally, 3.2 and 2.4 minutes.  You can have so much fun with this one.

I found shipping routes originally on 101.qs.

Thursday, September 5, 2013

GCF and LCM Foldable

I am all for students understanding how and why procedures work, but sometime you just need a foldable to get through to them.

Many of my students come to my class listing multiples of a particular set of numbers in order to find the least common multiple.  This is all well and fine until I ask them to find the LCM of monomials with variables.  Now they don't have a procedure to help them.  So below is a picture story of my foldable.  I forgot to put the stapler in the the first photo with the supplies, but I guess you could get away with not using it.