RUNNING OUT OF GAS?

We're finishing up our proportional reasoning unit after several weeks of exploring different representations for our thinking.  We've converted decimals, fractions, and percents.  We've made tape diagrams, grids, circle graphs, and number lines.  But, are we ready for an assessment?

Yesterday I presented a problem-solving task to my students, hoping that they would jump in with excitement and tackle the problem with some struggle but with success.  I used a task from the Shell Centre for Mathematical Education called Sharing Gasoline Costs.  Students have to use proportional reasoning to calculate the part of the total cost of gasoline for each student in a carpool.  Sounds straightforward enough.

Cue *crickets chirping* and *deer-in-the-headlights* stares across the classroom.

 

My excitement was not enough to carry them yesterday.  Some students gave some effort to solving the problem but most just stared at the paper with no clue where to begin. It seemed like information overload to them.  Beginning a problem is a challenge.  Some students dove into the problem and were convinced they knew the answer rapidly but hadn't considered some key factors in the proportional parts of the problem. This is the "I want to be done" group. 

Teacher is still learning that worthy tasks take t...i...m...e. Slow down. Let the ideas simmer. Afterall, I have a Master's Degree and several (ahem...) years of math experience on them, and it took me a little bit of thinking before I got the problem going on my own.

Today we're going to work on the problem in small groups with some guiding questions.  The groups are going to pool their thinking and come up with one solution and explanation for their group.

I'm keeping my fingers crossed and feeling hopeful again today. Some days, I just want to throw my hands up in the air and grab some worksheets for some "drill and kill".  It just seems easier somehow.

I think I can....I think they can....I think I can....I think they can.....

"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.

 

 

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