"Real Life" Math

Some of the math teacher bloggers that I follow started a practice of sharing their real-life math experiences on a regular basis. We all know that we use math all the time - no one just hands you math worksheets to complete - it just flows from the normal course of human events!  Here is my contribution.

Yesterday morning, I presented a warm-up problem to my students - a "real-life" math problem no less from my own life. My intention was that it be a WARM-UP not half of my class time. The understanding that I received from allowing this math experience to morph into something more was well worth while.

Here's the situation: I have those universal desks that defy group work. They just don't position well for sharing materials or being able to get into your seat without climbing. The legs of the desks and chairs have tennis balls to make them quieter and less damaging to the floor. These little bumpers prevent the legs from getting too close to each other. They get loose and become rolling objects. And they collect clumps of yucky stuff from the custodian's broom that I can't even describe....

I had the idea to begin removing the tennis balls and replace them with those little felt pads that you can by in any store. Here's where the math comes in. If the pads come in packages of 12, how many packages will I need to buy, and how much is this little project going to set me back?

I gave the problem to my students to solve in pairs, and as I walked around the room to see how they were approaching the problem, I noticed many of them were immediately drawing factor trees! HUH? Why did the students think finding the prime factorization, the GCF, or the LCM of the number of desks and chairs and pads in a package was the way to go?

I asked several of the students why they were doing the factor trees and the responses were generally to "find" the GCF or LCM.  How does that relate to this situation? They couldn't say how or why. My feeling on this was that it was the most recent mathematical concept we worked on last week. After some questioning, most of them abadoned this idea and started again.

Another common mistake was adding up the number of chairs and the number of desks (10 + 32) and then dividing by 12 (the number of pads in one package) to tell me I needed to buy 4 packages of 12 pads to complete my job.  Even some of my top-notch students made this mistake! So I asked them how many pads that was (48), and then had them count by 4's as I pointed to the desks in the vicinity. They quickly realized that this was not enough because they didn't multiply by the number of legs on the furniture.

The next thing I did was ask several groups to come to the document camera to show and explain their work. This gave them a chance to put into words a logical step-by-step explanation. Boy, this is a tough one for many of them.  To be able to go back and explain your thinking means you have to keep track of your thinking in some manner. Most of their papers looked like this:

We had an opportunity to talk about how to organize the work so that it could be easily followed by another person. This is how one student edited his work:

            

Today, we'll add the writing component to this. Perhaps, this is the most difficult aspect of the task. Writing (as I am doing right now...) takes time, thought, and effort to pull all the pieces together into a coherent form. To get the process (which happens quickly in the brain) down onto paper requires slowing that process down. Most of the students I work with have not developed that patience for the reader. They would rather that I be a telepath and cryptographer when it comes to their work!

Through all of this I didn't get discouraged because I realized that I provided a great learning opportunity for my students. I also see my skills with questioning and facilitating math tasks improving every day. It is a challenge to give up that "stage" of lecturing on a full-time basis. I am starting to see the glimmer of light at the end of the tunnel....it might be as small as a pinprick, but still there nonetheless.

Don't Forget the Math

I came across this little gem of a video by Phil Daro, one of the writers of the Common Core.  

 

 

This led to a look at the SERP (Strategic Education Research Partnership) web site of which he is a part. On this site are videos of Phil teaching a math lesson to teachers while modeling the mathematical practices. I watched the beginning clips which opened my eyes to some strategies to help my own students with solving word problems.

 

One strategy I tried this week was to give the stem of a word problem to the class and ask them to generate questions that could be answered with the information that I gave. Very interesting discussion!  I got some humorous questions, thought-provoking questions, and then the impossible-to-answer questions.

 

Isabelle is having a partyand is putting out the dessert on platters on the table. She has 15 pieces of cake and 6 pieces of pie.

 

Questions:

- How many total guests can she invite so that everyone gets one piece of dessert?

- How many guests can she invite if everyone gets one piece of each dessert?

- What is the greatest number of identical platters she can make with no leftover desserts?

- My favorite: Is she serving punch? LOL

 

I then revealed the question I wanted the class to investigate which was the third question in the list. Students worked in pairs to prepare a visual representation of their thinking as well as a written description. While the numbers in this problem are relatively easy to work with, my emphasis was to work with the mathematical practices of communication, collaboration, and reasoning. My students are still struggling with these but improving weekly.

 

Here are some examples of their work - one that is a correct representation and one that is not.

 

 

 

I had randomly called on partners that wanted to share their work at the document camera and explain how they found the answer.  The first pair had the idea and understood the mathematics involved. When the second group came up, inside I cringed AND rejoiced at the same time! It was a great opportunity to teach some peer evaluation and communicating with kindness.  After the second pair shared, I asked if anyone had feedback. The audience's hands sprang up. They called on one student, and she said that their work did not make sense. How could you distribute 21 pieces of dessert on 30 plates and call them identical?  TEACHER HEAVEN!

 

Not only was it the students  who spotted the mistake and then explained it to the others, but the students all were kind in their feedback and it was received with openness!  Turning over the talking to the students reaped great benefits on this experience. The girls who did the incorrect mathematics said, "Now we get you!" A positive growth experience for everyone.

 

 

"A" is for AMAZING and ARNE

Photo courtesy of the Dodge Magnet Middle School web site.



You know how you clean the house before company comes to visit? Especially when that company hasn't been to your home before or they are someone who is special in your eyes. That's what has been happening for the last month at our school. We found out that U.S. Secretary of Education Arne Duncan and U.S. Secretary of Transportation Anthony Foxx were coming to our school on September 11 and the housecleaning began! Anything that could be polished, washed, waxed, trimmed, raked, and made beautiful was tackled. Our campus sparkled thanks to all the hard work of so many.

 

These visitors, of course, were not alone. They brought an entourage of Secret Service, staff, and the accompanying press corps that follows any big event in a community. We had local police, district security, and all our district dignitaries. The students knew we were having a special visit but they had no idea how special and wild this visit would be! This kind of activity, in their minds, was usually reserved for rock stars, multi-million dollar athletes, and movie stars.

 

The secretaries arrived on one of the school buses with some of our students. One of my sixth graders was thrilled that she sat with Secretary Duncan and shared about her science project that she was delivering to class that morning. I loved seeing that big smile of hers when she told me!   The press did their thing once the bus unloaded - then the real fun began. The kids picked up on the high energy and couldn't help but notice all the suits and ties.  We're a school with a tougher dress code than most of our schools so the lesson was not lost on them.

 

Mr. Duncan and Mr. Foxx were interviewed by one of our eighth grade Social Studies classes, played basketball on the playground with some of the PE kids, painted Kindness Coins with our Builders' Club service organization in the cafeteria, and stopped into two of the sixth grade classrooms for a closer look at how hard our students and teachers are working to implement the new Common Core standards. I have to say that as much as I wanted to just have a "normal" teaching day, I was thrilled and honored that the Secretary (and the press...) popped into my class while I was teaching the lesson on patterns. He dove right into the activity with the kids and asked about it. Right on cue, Josh told him we were looking at patterns and how to predict the number of tiles in another step. Yeah, Josh!

 

                                          Photo courtesy of Fernanda Echavarri, AZPM

 

As teachers, we all know how hard we work on a daily basis - often without words of encouragement or appreciation. When the Secretary of Education pats you on the shoulder and tells you to keep up the good work, it makes you feel like you can do most anything for the rest of the school year. Before he left, Secretary Duncan told our principal that he thought our school was "amazing." Those of us who work there know how fortunate we are to work with the best kids, parents, and staff we know. We are very proud of our "A" rating by the state and strive to be excellent each and every day.

 

I'm not a cynic by nature, but even I questioned the value of this experience and the purpose of the Secretary's bus tour. I could get all political here and talk about political aspirations, tax dollars, man hours, etc. but I won't. What I want to remember is that for a few hours, our school shone like a diamond in the eyes of the public. Our kids were in the spotlight for being so amazing. For just a little while, everyone else got to see what we see every day. Thanks to everyone who noticed there are great things happening at Dodge!

 

For more photos and information from the visit:

https://www.facebook.com/dodgems

https://www.azpm.org/p/home-featured/2013/9/11/26721-us-secretary-of-education-visits-tucson/

Least Common Multiple

We're working on finding the LCM of numbers now. After being introduced to these types of problems through the use of cycles, my students set about finding the LCM of numbers using the usual list-making strategy. Today we're going to extend our thinking into some algebraic reasoning with a little twist on multiples through pattern exploration. This activitiy is from Connected Math.

What do you notice?  What do you wonder?   What will the 20th step look like? When will the total number of tiles in a step equal 576 tiles? Explain how you found that answer. 

Then there is another pattern to explore:

 

Once again, what do you notice?  What do you wonder? What will the 20th step look like?  When will the total number of tiles equal 110?

 
Here's my graphic organizer and an exit ticket for Least Common Multiple.

Arrays, Number Puzzles, and Factor Trees

This week my students are tackling a task I came upon from the Georgia Performance Standards web site resources. It caught my eye because my students need a LOT of work with the mathematical practices especially collaboration, communication, persistence, and reasoning. This task fit the bill for all of those skills while they practice factors, multiples, and factorization. Students work in small groups (we're working in pairs). They have three sheets of paper to share with rectangular arrays, factor trees with missing numbers, and puzzle cards. The goal is to identify the number of each representation and match them.

There are several entry points for this task - some easier than others. Most students wanted to tackle the word puzzle cards first but then realized that the other two sheets were much easier to do first and then use as clues for matching up to the word puzzles. There is a second group of cards for those that finish quickly - although no one in my two advanced classes finished yesterday. My other three
classes will begin this activity today.

Today we will finish up this activity, make posters of the matches, and present their findings with some discussion by the students to justify their answers.

For our math notebook, we will make a number puzzle of a mystery number for others to solve.

Here is the link to the file:
https://ccgpsmathematicsk-5.wikispaces.com/file/view/4-6_Arrays_Number_Puzzles.pdf

 

 

Gosh, I Love Math!

The following is from an e-mail I received from one of my math colleagues at school this week.

 

"Ben B. just turned to me while doing a Word Problem Workshop and stated:

'Ya know Ms. M…a math class is the only place someone can buy 80 watermelons and no one questions their motives. Gosh, I love math.' "

 

I just had to share that with all my mathy friends because THIS is the fuel that keeps our fires burning in teaching.  When I hear something like this, I forget about all the paperwork, the long hours of planning and grading, the meetings, the paperwork, the frustration of not having enough time....did I mention the paperwork? I especially love knowing I had a teeny part of that young man's enthusiasm for math as he was my student two years ago and now I have his sister in my class. Ben, you make me want to be a better teacher!

 

This week some of my students told me that my class always goes by quickly. I take that as a great compliment from a sixth grader. The days that drag by for me are the ones where I am not engaged, not interested, and am bored.  The day is endless when I sit and watch them take standardized tests....*yawn*. I am enjoying their attitude towards learning these days. I know that the group work and discussion, the discovery of concepts through the messy mucking about is ALL worth it!

 

Gosh, I love math!