Patty Paper - It’s Good For More Than Hamburgers

When I first heard about patty paper, I had no idea what it was (I obviously don’t make my own hamburgers), let alone what it had to do with mathematics. Tracing paper, transparencies, patty paper… they can all be used similarly to help students visualize key geometric ideas. Now that I am acquainted with the multifaceted nature of patty paper, I don’t know how I would teach certain concepts without it! Here are a some examples of how patty paper can be used in the classroom:

Transformations.

For “spatially challenged” people like me (yes, I get lost at the mall), patty paper is invaluable for visualizing transformations - particularly rotations and reflections.  Here is how one teacher uses patty paper to demonstrate a rotation. What are some connections the students would be able to make using this visual model? How would you do it differently?

Constructions

Constructions are typically learned with a compass and straightedge. This is important work which helps develop an understanding of constructions and the geometric reasoning that accompanies them. Patty paper can be a nice compliment to the work with a compass, giving students another avenue to “see” the relationships between various figures and thereby solidifying the understanding they started with the compass and straightedge.

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http://www.cpalms.org/Public/PreviewResourceLesson/Preview/51159

Now that we've seen some of the uses of patty paper, two questions commonly arise - Are we allowed to use patty paper? Does using patty paper lower our expectations for students?

First - Yes, we are allowed to use patty paper. In fact, several standards even call for it:

• 8.G Understand congruence and similarity using physical models, transparencies, or geometry software
• G.CO.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
• G.CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another
• G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).

Second - A tool by itself cannot raise or lower our expectations. How we use the tool defines the cognitive demand of the task. Standard for Mathematical Practice 5 says: “Use Appropriate Tools Strategically.” Patty paper is a tool that, when used strategically, can give many students access to understanding and connections that the students otherwise may have missed. Additionally, students may combine patty paper with other tools and strategies to check their work and to validate their own learning.

How have you used patty paper in your classroom? If it is new to you, are you willing to give it a try?

Gaining Momentum

The estimated reading time for this post is 1 minute.

Merriam-Webster dictionary defines momentum as “the strength or force that allows something to continue or to grow stronger or faster as time passes”.

It turns out, Momentum is also a google extension I added this week.  When I was working with a teacher, he opened a new tab in chrome and it looked something like this:

I admit, initially, I was drawn to the simple greeting, “Good afternoon, Sarah”.  Who wouldn’t like a nice picture and a lovely greeting instead of internet advertisements?

Upon further scrutiny, the question and the immediate blank felt like a slap in the face.  What was my main focus for today?  I wanted to fill the blank line with so many things.  But I soon found out that I could only put ONE thing, my MAIN focus.  What was my main focus?  What was driving me and giving me momentum for that day?

Today marks the 3rd week of classes in RUSD, the honeymoon might be over for some. Now comes the real work.  We are committed to these students for the school year.  As the year progresses, there will be many things to do, but what will give you the strength you need each day?  So I ask, what is your momentum, what is your main focus for today?  What is the strength or force that enables you to continue to grow stronger or faster as time passes?  

In a few weeks when we are knee deep in the semester and school year, we will need to remember our main focus.  Take a moment and find your momentum.

Math is Beautiful, Isn't It?

Disneyland just ended their Forever Fireworks show.  Beyond enjoying the fireworks and projected show, have you ever asked yourself, "What Math is involved in this?"

The first time that I watched the show I was amazed at all of the mathematics necessary for the show.  Stop for a second and think about all of the mathematical factors that go into a place like Disneyland.  When I think about the future "Emagineers" that we are educating today, I wonder what marvel they are going to create for our tomorrow.  How do the Mathematical Practices that we are developing in our students relate to places like Disneyland? Do you wonder like I did, if students would be able to connect their daily learning to the world around them?

Could you highlight and/or find the Mathematical Practices in this video?

• Make sense of problems and persevere in solving them
• Reason abstractly and quantitatively
• Construct viable arguments an critique the reasoning of others
• Model with Mathematics
• Use appropriate tools strategically
• Attend to precision
• Look for and make use of structure
• Look for and empress regularity in repeated reasoning 

During the spring of 2016 our sky in Riverside was given a rare display of mathematical brilliance by

the Air Force Thunderbirds.  While I looked in the sky I could not help to think about the hyperbolas, parabolas, circles, ellipse, planes, angles, etc that I was seeing.  The topic that my students were investigating during the time of the show was conics.  Would students find solving a system of conics more interesting if they understood the importance in these planes not intersecting.  As teachers could we provide more meaning to our Teaching Practices if we looked at them through the lens of the crowd drawing shows that we go to each year.  Where is your favorite place to go and what mathematics is necessary for you to enjoy it?

Mathematics Teaching Practices include:

• Establish mathematics goals to focus learning
• Implement tasks that promote reasoning and problem solving
• Use and connect mathematical representations
• Facilitate meaningful mathematical discourse
• Pose purposeful questions
• Build procedural fluency from conceptual understanding
• Support productive struggle in learning mathematics
• Elicit and use evidence of student thinking

In the comment sections, we would LOVE for you to share your thoughts:

Did your students give you insight to the mathematics that they saw in the Forever Fireworks Display or Thunderbirds show?  What mathematics did you uncover in watching these?  What Mathematics Practices or Mathematics Teaching Practices do you see as most necessary for our students to be successful in a future career?

What if you became the "Meaningful Monday Math Teacher"?  What if on Mondays' your warm up was to present a video for students to stop and be marveled by the mathematics that is in the world all around them?  What is you favorite math video?