Basketball Math

It seems appropriate to focus a little bit on basketball since we are in the midst of “March Madness” and the final four looming this Saturday, April 2.  I myself 2016-03-29_14-33-42have a difficult time staying excited when my team, Virginia Tech, was not even in the tournament, though, our arch rival, Virginia, did make it to the Elite Eight. So – home-state pride and all!!  As a math teacher, events like the NCAA Basketball tournament, provide opportunities to connect mathematics to real-world happenings. Students tend to be excited to learn and use the mathematics in context because they are either watching the games or at the very least, aware of them so there is a connection. With these types of current events going on, it provides an opportunity to research and collect relevant data and explore numbers in a variety of ways.

Statistical measures and comparisons are the obvious mathematical focus here, though not the only one.  You could explore some physics, such as what’s the ideal arc and location to make a 3-pointer (or a foul shot), or some geometry, such as what’s the ideal volume of a basketball to create the perfect bounce (if there is such a thing).  Having your students come up with their own questions and then do some research and get their own data would be a fun exercise all by itself. Here are just a few sites with NCAA 2016 statistics you could use to support your students questions:

2016-03-29_15-26-59Obviously, you could use these statistics in several ways, but for me, the easiest, and most efficient way to get students asking questions and then use the data they find to help answer those questions is to a) provide them access to data; 2) provide them with technology tools to explore the data; 3) allow them to explore and make conjectures; 4) have them share out their findings and justify their conclusions.  (Very Common Core!) If you are like me, where access to technology in the classroom consisted of a projector screen and computer, and then calculators, getting the data becomes the biggest hurdle.Printing out data from sites (like those above) is one way around that, though a bit cumbersome and it does not allow for student-driven questions, since the data you print may not be what they are questioning. You could have students in groups, and give each group time to formulate some questions first, and then provide each group some time on the computer to search for data that will help support their questions. Some of you may be lucky enough to have a few tablets or computers in your classroom for students, or allow students to access their smartphones/internet to do searches.  In that case, each group can do their own searches on devices within their group, which will make the search process easier.  No matter how students search for and gather data, having calculators for each student would be important, since they can then enter the data and explore quickly and make conjectures.  Yes, yes, you are right – you could just have them do everything by hand. But – what is the point of all this research and data collection? Is it just to find measures of central tendency?  Is it just to plot a histogram? No – it’s to use the data to answer interesting questions, make comparisons, and explore. The use of the calculator (or other technology if you have it) opens the door for students to explore.  They can compare multiple teams or players or test out ideas, or plot points to come up with equations, show multiple graphs from different teams on one grid. Lots of things that ‘by hand’ become cumbersome and tedious and defeat the purpose of this type of mathematical activity. I personally loved using a graphing calculator mainly because every student had access to one. This allowed everyone to be involved and they could work on separate, but related questions or they could work together to verify their conclusions.

Hopefully this gives you some things to think about and maybe try with your students while the NCAA is still going on. Happy calculating and if you have entered one of those work-pools for predicting the winner, hope you are beating the odds and your brackets are paying off!



ASSM, NCSM & NCTM – CA Here we come!

It’s math conference time – San Francisco and Oakland in April.  Can’t wait! For me personally, it’s been two years since my stock-photo-59142328-new-san-francisco-oakland-bay-bridgelast math conference attendance. My hiatus was for personal reasons – little things like defending my dissertation among them.  I have to say, I have missed the conferences – the energy, the reconnecting with friends and colleagues, the learning of new things and exploring the host cities. There are teachers out there that say face-to-face conferences are losing their appeal and are not relevant, but I disagree. While these conferences may not be as intimate and personally focused as say Edcamps, which are one of the new professional learning non-conferences, I do think they still have value and purpose.

One thing national conferences do is bring together people from all over the USA as well as other countries, so there are opportunities to get national and global perspectives.  Another value is the ability to “pump up” and inspire teachers with new stock-photo-77760809-bay-bridge-and-san-francisco-skyline-at-sunsetideas or strategies or tools that allow them to go back to their classrooms rejuvenated and excited about teaching again. It’s also a great venue for asking questions and getting information about educational changes – for example, the new ESSA law, assessment changes, and standards.  If you can meet one new person, learn one new strategy, get excited about changing one thing in your classroom, then the conference will have value to you.

Are there things that NCSM/NCTM could do better? Sure – free wifi everywhere would be nice, as you find at an ISTE conference. This allows everyone to tweet and connect during the conference, making the whole experience more collaborative and informative. Perhaps that is happening this year – I don’t know. More informal “meet ups” would be nice too –  where people of like mind can get together informally in a designated area to discuss a topic.  Like a blogger meetup, or a Twitter Meetup area, or even an Edcamp meetup.  These things may be happening – it would be nice to see.

I am excited the conferences are in the San Francisco area and that I get to be there to interact with state leaders at ASSM, district leaders at NCSM, and then teachers at NCTM. It will be an interesting experience this year with NCSM being in Oakland and NCTM in San Francisco, but that’s what Bart and Uber are for, right? Having worked for Key Curriculum for KeyCurriculum_NCTM2012-0528seven years, I grew to love the Emeryville/San Francisco area, so coming back feels like coming home.  And most of my Key peeps are still there or are working for companies that are represented at NCTM, so it will be a reunion of sorts. Family time!

Going as a consultant with Casio will be a new experience for me, but I absolutely love being part of the whole exhibitor end of NCSM/NCTM because there are so many ways to connect with and share ideas with educational leaders and teachers. Casio is going to have a fun interactive booth this year – doing math and helping show the power of technology integration. We have lots of fun things in store – game shows, workshops, hands-on take-away activities, lots of prizes (Gshock watches, calculators, keyboard, projector).  All to support math teachers, the Common Core, real-world math applications, and teaching with technology. Check out some of what will be going on at Casio here.

Looking forward to some new experiences, new connections, meeting up with old friends and colleagues, and most importantly, learning something new to improve my own practice.  Collaboration, sharing, and learning – that’s what it’s all about.

Finding Roots of Quadratics – Casio Prizm vs. TI-84+ CE

Factoring quadratics to find the roots is a skill students are expected to have as they move through mathematics. Where the zero’s of a function are, or where the x-values are when y = 0, helps determine how fast the function grows.  In real-world terms, thinking about projectile motion, the roots help determine where a ball hits the ground when hit by a bat, or thrown, or hit by a golf club. Students, especially in Algebra courses, spend a lot of time working with quadratics, factoring them, determining the vertex, finding the roots, and using the quadratic formula. It is almost a given that on a standardized test students will be asked to find the roots of a given quadratic or be asked to write the factored form of the given quadratic equation, so an understanding of how the factored form, roots and graph of a quadratic equation are related is important.

As a teacher, obviously we want students to learn how to do this with and without technology.  Technology helps students see how the equation, both standard and factored form, and the graph are connected and where the roots are and how the factored form relates to the roots. I know as a teacher, I spent time helping students learn to factor and find roots both by hand and graphically. Using a calculator to find the roots of a quadratic helped students explore quadratics, allowing them to answer more in depth questions because they could get to the roots quickly with the calculator, see the graphical representation, and make a relationship with the factored form of the equation.  The calculator was a great tool for helping students make conjectures about roots of a given quadratic and then quickly test their predictions. It gave them confidence in their factoring and understanding of quadratics.

Knowing that there are both TI and Casio users out there, I have made a quick video that demonstrates how to find the roots of a quadratic with both the Casio Prizm and the TI-84+ CE. Hopefully this will be helpful to you.

A Picture’s Worth More Than Memorization – Prizm Pictures

DispCap7With all the talk of learning math in context and making mathematics “real-world” and relevant to students lives so that they see and connect what they are learning to a purpose, it is often amazing to me how often there is no attempt to help make those connections. Especially as students move into more abstract concepts. From personal teaching experience, whenever I could bring in something – a physical object, a picture, a story, a technology simulation (loved Sketchpad!) that helped provide my students with a connection to the concepts we were learning, the more my studentDispCap4s engaged and remembered. And more importantly – were able to apply it to different situations because they hadn’t just memorized isolated skills/steps, but learned in context, and therefore had a connection that they could pull from easily.

One way to bring in context and relevancy is to use pictures of real things that can then be used to apply mathematical concepts, or used to solve interesting problems. If you have technology, where students can actually use the pictures with plotting points or graphs, then it becomes a powerful learning tool. There are lots of ways to do this – I for one used Sketchpad as much as possible in the classroom. What I have found as I work with schools throughout the country is that access to technology with math students on a consistent basis is very limited – laptops shared with 20 teachers, or one computer lab shared with multiple departments.  But – graphing calculators are something a majority of DispCap3students do have access to, and Casio Prizm is one that has built in pictures that can be used in a variety of ways and in a variety of math subjects.  Hands-on, real-world, and easily accessible. DispCap6

Here’s the basics way to access the built-in pictures in the Casio Prizm:

  1. Choose “I” from the menu (or arrow down to “I”). Hit EXE.
  2. Click EXE to open the CASIO folder
  3. Use the arrow key to choose the g3b or g3p folders.  Hit EXE.
  4. Use the arrow keys to scroll through the picture choices to find one that fits – there are so many choices!  Hit EXE
  5. Picture appears, and voila – start exploring!  Choose OPTION button to begin.

Obviously, there’s more to it than this – go here to find out more, but what I wanted to share in this post is the built-in pictures already in DispCap2existence on the Casio Prizm that could create some powerful context and applications for mathematical concepts you might be exploring with your students. Bring the math into students world.

Here’s link to the Casio Prizm Quick Start Guide that is also very helpful.


Pi Can Explain Practically Everything

Happy Pi Day 2016! My last post gave you some links and suggestions for celebrating the day. Today, on Pi Day, I want to share a YouTube video I found that does a fun job showing 3 applications of Pi.  It was made for last years’ magical Pi Day (3/14/15) but that’s okay – still pertinent!


And lest we forget – Happy Birthday Albert Einstein!! stock-photo-21045934-ussr-postage-stamp-albert-einstein

Pi Day 2016 – 3.1416 Rounding out to a good year!

In celebration of Monday’s Pi Day, which is a pretty cool one this year, since 3/14/16, or 3.1416 is Pi rounded to the nearest 10,000th, I am devoting this post to Pi.  I will probably do another one on Monday, but for those teachers out there who want some ideas for things to do with their students Monday, a few days advance notice is always a good thing!

There are Pi Day activities everywhere, so I thought I would share a few ideas, links, and an activity from Fostering Geometric Thinking with Casio Technology (Dr. Sonja Goerdt) to support those of you looking for things to do in the classroom.

Here is a couple things I did with my students (middle school and high school alike!)2016-03-10_15-05-29

  1. Have students bring in different circular food objects (i.e. Little Debbie’s cakes, Moon Pies, actual pies, cookies of all sorts, crackers, etc.).  We would take this variety of circles, find the diameter and circumference (using string) and then calculate ratios to “confirm” the magical number Pi. Then we’d eat!
  2. Have a contest of who could recite the digits of Pi.  Winner was the one who could go the furthest – always fun.  This does require students to prepare ahead of time, so if you want to do this, make sure you tell students who are interested to get studying.

Here are some links to activities I’ve found just searching the web.  There are a lot out there.

  1. Pi Day This site lists a million digits of Pi, and then, if you click on the links to the right, you can search the digits of Pi (for special sequences, like your birthdate), Pi puzzle (New York Times), or Einstein Rap.  There are lots of other links, so explore away.
  2. Exploratorium has a whole list of lessons/activities that explore Pi in many ways. One is the search digits one as well. Another one I think sounds very interesting is the Tossing Pi (scroll down the list to find this) – calculating Pi tossing toothpicks. Kids would love that!
  3. Project Mathematics This has videos you can choose about the history of Pi, uses of Pi, people explaining what they think Pi is.  Might be good to warm up your class with.
  4. Joy of Pi –  This page has lot of links to interesting articles about Pi, history, etc.  Lots of resources.
  5. Live Science – Has a video and other resources.
  6. Edutopia –  Lessons and activities for elementary students.
  7. Education World –  Lessons and activities around Pi – multi-grade level.
  8. Teachπ.org –  Lots of everything about Pi – books, activities, history, etc.

Finally, we have a fun graphing calculator (Prizm) activity that helps students discover Pi by exploring the circumference/diameter ratio using the picture of clock faces of multiple sizes.  It’s a great activity, especially if you want to integrate technology.  I’ve attached a PDF of the lesson, Investigation 4.3 Investigating Pi.  The activity includes sample answers that go through the calculator actions as well. Download PDF here: Investigating Pi FGTwCT Fostering Geometric Thinking – Investigation 4.3

Have fun planning for Pi Day!

#CCSS Attending to Precision – Mathematical Practice #6

Precision in words and actions is an important skill.  It helps communicate ideas and understanding. Without precise language and processes, miscommunication, misunderstanding, confusion, and chaos rule. Obviously, in the bigger scheme of things, lack of precision can be dangerous. For example, if a civil engineer designing a bridge is not precise in their measures and calculations, bridge collapse and death are possible. One of the things educators need to do is foster this skill of precision in our instructional practice. Which is why helping students “attend to precision”, is one of the 8 Mathematical Practices in the Common Core State Standards. Teachers should be cultivating precision in their classroom.

What does this mean, to “attend to precision”, in the context of a math classroom?  Here is how the practice is defined in the Common Core:stock-photo-58636092-triangle

Math Practice #6: Attend to Precision

Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. 

If you look at some of the words/phrases I have highlighted above, you will note that precision focuses on communication, using definitions and symbols accurately and appropriately, labeling and identifying quantities carefully, giving explanations based on facts and definitions. In short – understanding the words and symbols being used in mathematics in order to communicate with others. Being precise has to do with helping others understand what you are doing and saying. It is not enough for the student to understand a mathematical concept, they have to be able to help others understand. Why is this so important? Because, the Common Core Standards are designed to help students become “college and career ready”, and in college and in careers, people must communicate with others to accomplish tasks and solve problems. They must be able to precisely explain what they mean, what they have developed, what they want others to do – which requires common language and clear explanations.  In other words, precision.

So, let’s go back to what this means in the context of the classroom. Teachers, no matter what grade – preK through college – should be using the correct language of mathematics and expecting their students to also use this language precisely and appropriately.  They should expect students to explain their thinking using that language, whether verbally or in writing, as students progress through the grades. I know students always groan when they hear “show your work”, but it is important.  And not just their work, but the why of their work. If students are working with measures, then their solutions and explanations should include units of measure. They should use vocabulary and definitions to explain their thinking. The more we have students talking and communicating with math, right from the beginning, the more confident and precise they will become. As teachers, we need to model this as well, by making an effort to use proper mathematical language and symbols, as appropriate for your students, and helping students do the same.

Below is a chart, based on work I did this summer with teachers exploring the Mathematical Practices, that gives some student outcomes aligned to teacher actions that may be helpful as you think about ways to help students “attend to precision” in your classroom.

Students should be able to…… Teachers support this by….
Use correct math vocabulary Teaching vocabulary, (with visuals, if appropriate), and using precise mathetical vocabulary consistently
Know and use definitions appropriately Teaching definitions and modeling using these consistently and intentionally
Communicate/explain their thinking using words and symbols, both written and verbally Encouraging classroom discourse; use think aloud strategies; establishing a culture of inquiry and communication
Record and label their work Providing exemplar for what precision looks like; setting expectations, modeling expectations and providing consistency
Choose and use appropriate mathematical symbols when solving problems or explaining Teaching appropriate symbols and their meanings and using/modeling these consistently