Math in the Real World: Student Created Memes

One of my overarching goals for my geometry classes is for students to realize we can find geometry nearly everywhere we look in our everyday lives.  Yes, most of my students can identify the corners of their walls as right angles, they understand the ceiling and floor represent planes, and they are cool with the idea that the rails on train tracks are parallel.  But what I want for students is for them to really see the geometry all around us and for them to begin to see things geometrically even when they're not trying to.

To help develop this habit of mind, I assign students the challenge of creating a meme.  Students must take a picture (searching in Google is not allowed) of something they see in their lives that relates to the geometry concepts we have learned about.  I give students a quick intro to the website addtext.com and set them free.  Students submit their memes on our class Padlet and are able to see the work of their classmates.

I plan to do this same assignment once a quarter.  Here are a few of my favorites from the first quarter.

Algebraic Reasoning with Mobiles

This summer I had the opportunity to teach other teachers through the South Dakota Counts program.  I enjoyed working with teachers from around the state and learned a lot from them.

I especially like the resources that CAMSE kicks out each year.  One of my favorite resources that I gained exposure to is Math Mobiles.  These puzzle type challenges do a great job of developing algebraic and logical reasoning.

I did some searching for where I could my hands on some more of these, and thanks to a blogger by the name of JFairbanks , I found a site that not only had pre-made challenges, it also allows you to create your own.

Solve Me Mobiles

Here is an example (I recommend you check out the site and play on there):

Students are to find the value of each shape, assuming that the mobile is balanced and adds to the total weight in the top circle.

I'm excited to try some of these out on my students!

The NEED for High School Math Teachers to Be Integrating Statistics Into Our Classes

I attended a class last week @ South Dakota State that focused on deepening the understand of what the CCSS expect students to know in the realm of statistics.  The class was great for two reasons.  First, my statistics content knowledge needed some dusting off -- this class provided the cleaning.  Second, the class has really made me think hard about how our department can do a better job of integrating these standards into the lessons and activities that we're already using.

But most of all, I really started thinking about the NEED for students to be exposed to lessons in statistics.  The greatest need is for students to be made aware that they need to think critically when presented with statistics found in the real world.  I get the feeling that a majority of adults don't understand the different ways statistics can be manipulated to show different angles of the same story.  Just yesterday, I saw this video on CNN of an interview between CNN's Don Lemon and Sheriff David Clarke.  

Source: http://www.cnn.com/2016/07/18/us/wisconsin-sheriff-david-clarke-i-predicted-this/index.html

I'm not here to pick sides on this issue.  I don't follow politics enough to shed light on the data that Lemon is referring to.  I don't know if the President was lying about it, nor am I implying that he was.  I am simply using this four line discussion as evidence that there is a lot of gray area in the realm of statistics.  

What I do know is that there ways that statistics can manipulate the truth about what is really happening.  Students need to understand how things like sample size and random sampling can have an effect on data.  They need to analyze different data sets that hold relevance in their world, whether it be things such as shoe size, height, or average sleep time.  Students need to understand why the median is resistant to outliers.  They need to understand how we determine if a data point can be considered an outlier.  They need to see how graphs can be misleading.

All of these skills (and others) are needed in order for our students to be wise consumers of statistics.   I am afraid that many people hear statistics on the news or read statistics found in newspapers, but never really consider the process of how the statistic was created or how the statistic was represented and what message it is intended to support.

Geometry Field Trip

This semester, we are having our geometry students complete a floor plan project.  Students were given a blueprint of (part of) a house and were required to find the area of the floor and the walls in each room.  We made some of the rooms a bit irregular to draw in more concepts than simply finding the area of a rectangle.

To increase the real-life element of the project, we also asked students to find prices of flooring and paint and calculate what it would cost to pay for the carpet / tile / vinyl and the paint.  Students needed to provide price quotes and samples of colors they would choose.

We didn't want students to simply to and look online for prices.  Instead, we took a field trip to Barrett's Design Center for one class period.  Students were able to see the various flooring and paint options, talk with the interior designers, and ask questions about our project.

Overall, our field trip was very successful.  A load of thanks goes to Angie and Cindy for helping us and allowing us to come and visit.  And how cool was their little wrap up speech advocating for math!

(I used to do a project similar to this one where students were asked to remodel their bedroom and one other room in their own house.  They needed to then calculate the actual cost based on what types of materials they would purchase.  I never had the chance to integrate a field trip into the project!)

Better Questions

I know, I know.  Week 3 of the 2016 Blogging Initiative has long come and gone.  It's nearly the end of week 4!  I apologize for the tardiness.

I love the topic of "Better Questions" not only because the types of questions you ask can really dictate the type of classroom and style of teaching you have, but also because I need to continue to grow in the area of questioning.  Blogging about questioning will help me re-focus on the types of questions I'm asking my students.

This past fall, I presented a training module to HS / MS math teachers that focused on Effective Questioning.  The module brings to light the four levels of questions defined in NCTM's Principles to Actions (2014).

Question Type		Description
1	Gathering Information	Students recall facts, definitions, or procedures.
2	Probing Thinking	Students explain, elaborate, or clarify their thinking, including articulating the steps in solution methods or the completion of a task.
3	Making the Mathematics Visible	Students discuss mathematical structures and make connections among mathematical ideas and relationships.
4	Encouraging reflection and justification	Students reveal deeper understanding of their reasoning and actions, including making an argument for the validity of their work.

I've set the goal of asking more level 3 and 4 in my classroom, including on assessments.  These types of questions are sometimes tricky to create and grade.  Here is an example of a question that I asked my algebra 2 students.

I consider this a level 3 question due to asking students to make connections between the factors, solutions, and zeros on the graph.

Some example of student work:

Student 1: Excellent.

Student 2: This student did not understand the connection between the zeros of the graph and solutions of the equation.

My Favorite: Linear Regression & Movies

Week 2 of the #MTBoS challenge: "My Favorite".

There are many lessons and activities that I have that could qualify as "My Favorite", but since I haven't blogged about this one yet, I'll go with it.

For the past couple of years, I used an activity I found on Yummy Math to help apply linear regression to a real world context.  I modify the activity just a bit by changing some of the movies to some of my favorites.  (If you haven't check out Yummy Math yet, I'd recommend it.  A lot of the site is free, but I'd recommend paying the $20 per year for full access.  The best thing about Yummy Math is that they're always updating their activities.)

Using Desmos, it's very quick and easy to create regression equations.  Students are asked to find some of the data from their own favorite movies, which completely draws them into the activity.

There is a lot of opportunity for rich discussion in the activity.  Some examples of really great discussions I've had w/ this activity:

• Why are some movies way above or below the line of best fit?  What would cause that sort of behavior?
• How does inflation / release dates play a role in this data?
• Why do Disney animated movies tend to be below the line of best fit?
• Why are sequels typically below the line of best fit?
• Are most of the top-grossing films of all time (Titantic, Avatar, and now Star War 7) above the line of best fit?
• When looking at a series of films (ex: Fast & the Furious), why do some fall above the line while others fall below?

Lastly, I love the activity because it leaves a bit of a cliffhanger.  Each year I do this, I wait and see what is the #1 movie in America during the past weekend and include that as part of my data set.  This past year, it was Hotel Transylvania 2.  At the end of the activity, I have students write down their prediction for the total gross amount for that movie.  Of course, there is no right answer at the time since the movie is still in theaters.  So we wait and occasionally check on it throughout the semester.  When the dust settles, we look back and see which student was closest to correct and award a prize.  

Title	Opening Gross ($ in millions)	Total Gross ($ in millions)	Opening Date
The Avengers	207.438708	623.279547	5/4/12
The Hunger Games: Catching Fire	158.074286	424.668047	11/22/13
Harry Potter & the Deathly Hallows Part 2	169.189427	381.011219	7/15/11
Jurassic World	208.806270	650.493056	6/12/15
The Sandlot	4.918712	32.434006	4/9/93
Toy Story 3	110.307189	415.004880	6/18/10
Dumb & Dumber	16.363442	127.175374	12/16/94
Frozen	67.391326	400.738009	11/22/13
Avengers 2: Age of Ultron	191.271109	458.924272	5/1/15
Major League	8.836265	49.797148	4/7/89
Hotel Transylvania 2	48.464322		9/25/15

Vikings / Seahawks win probability

Growing up in Minnesota, it's a state law that you must cheer for the local professional sports teams (assuming that you follow sports).  Being the law abiding citizen that I am, I've had to endure numerous heart breaks and losing seasons from my beloved Vikings and Timberwolves.  (I'm also a closet bandwagon Twins fan, but don't know enough about hockey to follow the Wild.)

Today's Vikings loss was as gut-wrenching as any since the 1998 NFC title game.  As I browsed through ESPN tonight, I did happen to stumble onto this crazy Win Probability graph.  I've seen these graphs before; the best ones are when teams have miracle-like wins, such as today's game.

I'm thinking I can use this graph down the road when learning about probability and / or graphs of functions.

For the article that describes some of the swings, click here.  It hurts to be a Vikings fan.

Desmos Marble Slides: Parabolas

Desmos just released their newest set of activities on Tuesday, aptly named "Marbleslides".  I decided to use the last day before Christmas vacation to let my algebra 2 students give the Parabola challenges a try.  I forced students to work as pairs instead of having them work independently.  I have found that my room is very quiet when students work on Desmos activities by themselves.  I wanted more collaboration and thinking out loud; I wanted more noise.

It appears Desmos hit a home run with their latest release.  More than once I had students arguing about whose turn it was to run the controls.  Many students left class with a new addiction.

Desmos Activity Builder Part 2

In case you missed my first post 5 short weeks ago about Desmos Activity Builder, please feel free to check it out.

It appears as though Santa came early because this was on my wish list.  Great new via Twitter today:

We are now able to copy someone else's activity and customize it to make it fit our specific needs.

For example, we used this activity on rotations by Andrew Stadel in geometry class a few weeks ago.  I now have the ability to duplicate the activity and make small tweaks to fit the needs of my specific students.  I don't have to try to re-create the magic that Andrew already created, yet I now have the ability to put my personal spin on these activities.

One final thought:
This open sharing of resources that is taking place is so powerful.  I love it that the people at Desmos and anyone who shares their wonderful creations through Desmos are not concerned about getting paid or someone else copying their ideas.  Sites like teacherspayteachers.com have some great resources, but it bothers me when teachers aren't willing to share resources with each other without getting something ($$$) in return.  The folks sharing via Desmos are willing to share resources for others to use and potentially for others to improve.  It's such a different mindset when compared to those teachers willing to share but only if they get something in return.

Happy customizing in activity builder!

Using Desmos to find the quadratic model of a real-world image

Desmos is at it again...

Since I started using Desmos more regularly this summer, I have found that it is gradually replacing a number of my other math software options.  The latest instance happened when my algebra 2 students were finding examples of things that take the form of a parabola.

In the past, I have had students find an example of a parabola in their everyday lives and snap a picture of it.  I've seen images of everything from the McDonald's logo to a water fountain to a mug shot of someone's jaw line.  Then I would have students upload their images into Geometer's Sketchpad and construct a number of points along the parabola.  Here is an example...

After that, I would have the students enter the ordered pairs into a TI-84+ or TI-Nspire graphing calculator, find the regression equation, and go back to Sketchpad and graph the function on top of the image.  Again, an example:

Some years I would have students explore around with vertex form a bit to try to fit a quadratic model before we found the regression equation.  But it was a painfully long process to bounce back and forth between Sketchpad and graphing calculator.

This year, I decided to direct students to Desmos to compute the regression equation.  After all, we had used Desmos to find linear regression equations earlier in the year and all students had access to Desmos w/ our 1:1 laptops.  (Not all students would have a TI-84 or Nspire to use in years past.)

It turns out the students had a great idea. Why use Sketchpad to plot the points and graph the regression function?  Why not just use Desmos?  I loved the idea and it really simplified the process.

Students simply inserted their images into Desmos, plotted points on the parabola, created a table with their ordered pairs, and found the quadratic regression equation (& graph) all in one place.

Follow this link for an example (image shown below).

Better yet, using the sliders in Desmos is a breeze.  Simply insert an image of anything parabolic and have students fit the model using vertex form.

Desmos Activity Builder

The team at Desmos has really hit the ball out of the park when it comes to user-friendly math technology.  If you haven't checked out the Desmos Activity Builder at teacher.desmos.com, you're missing out.

I've recently created some inquiry-based activities for my algebra 2 students.  The best part is that there are math educators much smarter than me working on creating activities as well.  Big names such as Michael Fenton and Andrew Stadel are sharing their ideas, and I'm able to use them!

My most recent activities for algebra 2:

Transformations of Functions

Absolute Value Graphs & Transformations

Quadratic Functions in Vertex Form

My most recent activities for geometry:

Equations of Lines - Part 1

Equations of Lines - Part 2

Again, it's great when giants like Andrew Stadel roll out activities (such as this one on rotations) right as we're working on transformations in geometry.

Keep up the great work, Desmos!

Un-structuring Word Problems

This year, I have the opportunity to teach two sections of advanced algebra 2.  My colleague, Todd, teaches the other five sections of advanced algebra 2.  Todd is an excellent educator and someone I have already learned a lot from.

We are both fans of Dan Meyer's TED talk and love the idea of un-structuring our math problems as much as possible.  In an attempt to force our students to think about what information is necessary to solve a problem, we've begun to modify a question or two each assignment in this way.


Here's a question found in a typical systems of equations lesson:

(c) Pearson Education 2015 Algebra 2

We give our students this problem instead:

*1.       You are considering renting a car from two different rental companies.  Write a function that shows the cost of renting from each company.  Which one should you choose?  Explain.

Our students are aware they do not have enough information to complete the task.  Not yet, anyway.
If we assigned this problem on a Monday, students would be expected to think about what information is important and necessary in order to correctly answer the problem.  On Tuesday, to start class, we give students 1 minute to ask anything they want about the problem.   The problem is then part of their homework for Tuesday night and is expected to finished by Wednesday.

It has been interesting to hear some of the questions students ask on problems like this one.  Often times, students ask for the necessary information very quickly and don't ask for anything unnecessary.  There have been times, however, that we get some interesting questions about things that have no effect on the problem.

Student Centered Advanced Algebra 2 Activities

I love it when students are active learners and are collaborating with one another.  I really love it when technology is involved.  The past two days of Advanced Algebra 2 have been a buffet of fun for me.

We have begun our unit on functions.  After a day of discussing topics such as mapping diagrams, function notation, domain and range, vertical line test, etc, we turned our attention to representing and modeling relationships with graphs.

On Wednesday, students analyzed distance - time graphs with a heavy thanks to this MARS MAP activity.  Students worked in groups of two to sort and match the graph, table, and story.

On Thursday, students completed the Desmos Water Line activity.  I have students reflect after doing Desmos activities, and here is what some of my students had to say today.

"It was a fun and simple way of looking at something that can be very complex."

"It made me think, and it was fun to create my own glass."

"It changed the way how I look at how water glasses fill up."



Responses like that are GLORY in its finest form.

I told students that I would take the most creative glasses and post pictures of them on Twitter.  We are hoping that @Desmos will retweet our post.  Time will tell if we were successful!

Here are some of the more creative glasses we had.

E.P. - 2nd Hour

O.D. - 2nd Hour

C.A. - 3rd Hour

Geometry Vocab Activity

In today's lesson, we borrowed an idea from Andrew Shauver and had our students complete an introductory vocabulary exploration.

Students had to come to class with introductory vocabulary words already defined (using their web resources and / or textbook).  Then they worked in groups to generate a "group definition".  They entered their "group definition" into a Google Form.  After all responses were gathered, we came up with a whole class definition and clarified any syntax and notations that went along with the terms.

Students work in table groups to complete the introductory vocabulary exploration.

In addition, we used all of the group's definitions to create a wordle for each term.  (It was very simple to do... go to the Google Form response sheet, copy all of the text for any term, and paste into wordle!)  We were pleased with what was created and decided to print some for our walls.

Growth Mindset & the Smarter Balanced Assessment

I was first exposed to Carol Dweck's research on fixed and growth mindsets a few years ago when I was completing some of my graduate coursework.  For those of you who haven't heard about mindsets, you can find a lot of resources online (here and here to name a few).

Dweck's colleague, Jo Boaler, has completed some further research on mindsets and the implications in the classroom.   If you're really interested in learning about mindsets in a mathematics classroom, I would recommend taking Stanford's free online class (sign up here).

In Dweck's Self-Theories book, she discusses the reaction of students with a helpless mindset (fixed mindset) when they experience failure.  In short, when a student with a fixed mindset experiences failure they begin to fall into a helpless state of mind.  They quickly lose confidence in their abilities and lose perspective on the successes they had achieved in the past.  Moreover, many students who had fallen into the helpless mindset "abandoned or became incapable of deploying the effective strategies in their repertoire." (pg. 9)

After hearing that research, I grew concerned about students who have a fixed mindset and their ability to successfully complete the Smarter Balanced assessment.  The Smarter Balanced Assessment is a computer adaptive test.  Students are given grade-level questions for the first two-thirds of the test.  Then the computer software selects remaining one-third of the questions based on the student responses up to that point.  If a student is answering questions correctly, then the software can select questions from a higher grade level.  Conversely, if a student is answering questions incorrectly, then the software can select questions from a lower grade level.

Theoretically, this type of testing system makes sense.  It's designed to find the "boundary" of each students' achievement level.  My concern lies with the students with fixed mindsets.  For those students, the "boundary" is like an electric fence.  The test is designed for students to eventually touch the fence.  Once a student with a fixed mindset does touch it, the electrical current sends shocks throughout their body.  They lose ability to focus on the problems and are more focused on how much the fence shocked them.

Dweck summarizes her research by saying "...the helpless response is a reaction to failure that carries negative implications for the self and that impairs students' ability to use their minds effectively."  I fear the computer adaptive test could indirectly lead to major negative consequences for students with a fixed mindset.

What can we (teachers) do to help?

I think the best way to help combat this potential problem is to teach students about fixed and growth mindsets.  I am planning to introduce these ideas during the first week of school in all of my classes.  More to come as to what that will look like...



Dweck, Carol S. Self-theories: Their Role in Motivation, Personality, and Development. Philadelphia, PA: Psychology, 1999. Print.

TIE Conference 2015

I am currently at the TIE Conference and learning some great new ideas.  One thing I just found out about was the ability to post to my blog by simply writing an email.  (Thanks to the Monday AM Keynote Ramsey Musallam!)

I believe I set this up correctly and am testing this newly discovered featured out.  I guess if you’re reading this, it worked!

How tall is the light pole?

In early March, Mother Nature was very cooperative with our lesson plans and we had a couple of 60 degree days.  We were in the middle of our similarity unit and wanted to get our geometry students outside for an activity.  My co-teacher and I had both done activities involving indirect measurement before in our previous schools, but we had both used clinometers and right triangle trigonometry as the vehicle.

This year, we decided to use mirrors and model our activity similar to the one shown here.

We modeled our lesson after Mr. Chuck Pack's lesson (from The Teaching Channel).  He does a much better job of explaining the activity on video than I could in writing.

The students loved getting outside and the day was a success.  Most importantly, students demonstrated strong understanding of the concept of indirect measurement and experienced it firsthand.

3.11.15 - Brookings HS

The indirect measurement activity using mirrors worked very well.

The one thing I really loved about Mr. Pack's activity that I had never thought of doing was collecting the data and tie in some data analysis / statistics into the day.  It's inevitable that at least one group make some major errors in their measurements or calculations.  Outliers and the effect an outlier has on the mean is begging to be discussed.  TI-Nspire does a very nice job of displaying the data from our three classes in the scatterplot. Great idea, Mr. Pack!

Next year, we're planning to improve our activity in two ways.  

1)  We used meter sticks and yard sticks to measure our distances along the ground.  Next year, we will hopefully have 100-ft tape measures.  It will be interesting to see if our data has less variation.

2) We are planning to use clinometers as well next year after our trigonometry unit.  We can then potentially use our mirror data and compare it to the clinometer data.  

If anyone has ideas of some statistical analysis type activities we can do to extend the lesson next year, please reply below.  I am especially interested in stats connections to the CCSS.

Golden Ratio & Apple Cores

@BHSGeomety students recently studied the Golden Ratio and Golden Rectangles with a number of different activities.  For example, in one activity students found lengths on different measurements on their face and bodies and calculated to see if any of their ratios were "golden".

The activity I want to share about goes a little something like this...

1.  Give each student (or group of students) an apple.  Have the student cut the apple horizontally so that the cross section is a circle and it exposes the core.

2.  As you can see, the inside of the core has a pentagram shape.  This is one of many instances of the golden ratio found in the natural world.  We had students take a picture of their apple cores and import them into Geometer's Sketchpad.  Our document camera was extremely efficient at doing this.

3.  Using a regular pentagon custom tool in GSP, students constructed the pentagon around their apple core.

4.  Students then compared each other's apple core to see which had the most "golden" core.

This is a fun 10-minute activity that integrates technology and is very hands-on and student centered.  Students were interested to see what each other's cores looked like.  We actually used four different types of apples in our experiment.  No one type of apple seemed to be any more or less golden than the others.

Here are a few more examples:

Student Created Videos

New year, new goals.

While doing some planning for our new semester, Mr. Huntimer found an activity on myOER that we decided to use to begin the second semester with our geometry classes.

Working in groups of two or three, students were given the task of finding real world objects that model a variety of 2-D and 3-D shapes.  Students were to either take a picture or a video clip of the object.  Then they were to create a video that fused all of the pictures or videos together.  As part of the video, students were required to define the object that they had found.

We allowed students to make a lot of decisions regarding their videos.  Students could define their objects using a number of different methods.  They could define objects as they recorded the video clips, they could record voice narration and align it to their pictures, or they could insert text into the video.  We had students use all three methods.

Students could create their videos around theme or concept.  We had one group use the theme of "shape hunters" and another use the theme of "mathletes".

We also allowed students to use whatever type of video editing software they'd like.  Our laptops have MovieMaker installed, so that served as our default video editor for those who had no preference.  Mr. Huntimer and I were rookies working with the software and found it very user friendly and easy to implement.  Some groups chose to use their iPhones and iMovie.  Another group used a third video editing software that we had never heard of before.  Our only stipulation was that students needed to submit their videos in mp4 or mov format so that we could view them from any computer (via Edmodo).

Overall, the task went very well.  We had some bumps in the road the first time through.  The biggest challenge was we had a number of groups submit their MovieMaker projects before they had converted them to mp4 format.  We simply had to ask those groups to convert and re-submit the videos.  A majority of the videos were similar in that they contained the minimum amount of effort / work needed to complete the task.  A few groups forgot to include all of the required information, thus costing them a few points.  Meanwhile, a few groups produced excellent videos that really caught our attention.

The content of the geometry in the task was really quite elementary, but the task did align to a CCSS ( G-MG.1).  The bigger skill that students practiced was collaboration and technology use.  We asked students to reflect on what they had learned in doing this project and many said they had never or rarely used software to create a video.  We were very happy with the technology application component.

I leave you with two links.  The first is a link to the task via myOER and Mr. Zachary Feldman (click here).

The second is a link to one of our best student videos.   The three gentlemen who created this video have agreed to share it.
Student Video

New Year's Resolutions

A couple of quick New Year's Resolutions for my classroom and @BHSGeometry...

1.  Focus on highlighting the standards of mathematical practice.  We use many of them every day in class.  Are the students aware of this?


2.  Less telling, more doing.  Force the students to do the heavy lifting in class.  This is always a goal of mine, and always a challenge to meet.


3.  More student self-reflection.  The more students reflect, the more that gets transferring into long-term memory.  Google forms, exit tickets, possibly student blogs... we'll see where 2015 takes us!