The Line of Best Fit

I remember when I first started teaching 4 years ago, there were several topics that I just didn't feel like I was adequately prepared to teach.  Statistics was one of those topics, but especially Line of Best Fit.

Fast forward 4 years--- I LOVE STATS!! Seriously--we have the best time in my class during the time that we are learning about Statistics because I quickly figured out how much fun it could be.  I spent some time with our AP Statistics teacher and learned so much about Statistics!

In my Algebra 1B Class, we are about to start working on Linear Regression. As I was planning this unit with my intern, I ran across an activity that I have done before with TI-Nspires, but we don't have TI-Nspires in my classroom.

The basic idea in the activity is to look at a scatter plot, try to guess a "Line of Best Fit" using a movable line and its residuals to try and approximate the line of best fit.  Once you feel like you have a solid guess--compare it to the actual Regression Line, interpret, and make predictions.  So, in trying to figure out a way to modify this awesome activity, I remembered something I stumbled across in Grad School at Auburn--> Desmos Activity Builder!

If you've never looked at teacher.demos.com you definitely should!! It is SO awesome! You can search by topic to see if another teacher or Desmos has already created an activity OR you can create your own and share it with other teachers! Such a great technology resource.

I spent several days working on this between the rough draft, sending it to another teacher for feedback and the revised activity, but I think it will be well worth my time!

Here is the teacher link to my activity: https://teacher.desmos.com/activitybuilder/custom/563a42fb60dded490a7f410d

I made a handout for my students so that they would have a reference sheet to remember what we discovered after the activity is complete.  


I welcome any and all feedback you may have on this! I am always looking to improve activities for my students!

I am finally ready! :)

You know that it is THAT time again when you walk into Walmart and right in the front of the store there are huge displays of school supplies! 

Yep.. it is time for school to start back, and I am SO excited!!! :) Seriously, the weeks leading up to now, I have been in that "I am NOT ready to go back" mood.  But today.. I am excited.  I finally feel ready because today I sat at my desk for about 4 hours and completed my course outlines for the entire year--for all THREE preps! 

I am ready today because I have all these ideas floating around in my head about the first day of school! I am looking forward to meeting a new group of kids and learning with them this school year. 

My first year of teaching, I literally planned week to week. I felt like I was drowning at the end of a day. I could NOT have planned this far ahead then, but now I feel lost if I don't have my entire year mapped out from day one.  It is my road map. It lets me know what I have to do this week in order to get where I need to be next month. 


Now for those who want to know how I did this, it's pretty simple.  

1- I went through my course of study and wrote down all the major topics that made up the standards. Ex. Linear Functions & Arithmetic Sequences, Exponential Relationships, Quadratic Functions, Descriptive Statistics & Conditional Probability, etc.  

2-Once I had all my major topics, I put them in an order that I thought would flow well.  For example, I can't teach Quadratic Functions before Linear Functions.  Linear must come first. 

3-Now, I had to decide approximately how much time I needed to adequately address the standards within that group. Am I going to need to review some topics from previous grades? Is there a project that I need to leave time for? 

4-Once I knew how much time I needed, I mapped it all out to fit my 4 nine weeks of time that I have in this school year. Filled in all standards, resources, projects, etc. and then my course outline is complete. :) 

I am including one of my course outlines for this school year.  This is something that I adapt every year based upon reflections & notes that I keep from the previous school.  For example, if I planned 3 weeks for solving equations, but I only needed 2, then I can re-distribute that time elsewhere from the beginning. 

If you have any questions, please comment below :) 

Advanced Algebra 1 Course Outline (15-16)

Teaching in a Technological World

I am a child of the '90s.  I was grew up a time when we had bag phones --and we thought that was super cool! I remember when Promethean & SMART boards first came out, and it was a HUGE deal! When I got my first cell phone at the age of 15, I remember feeling really cool, but I had to be conscious of my minutes and texting was a definite NO because that cost more money.  And do NOT hit the internet button on your phone or you were going to get charged a lot for the data that you used.

This is a bag phone! 

Fast Forward 10 years since I first got a cell phone-- I don't leave the house without it! I am permanently attached to my iPhone or iPad at all times. I would prefer to text a question to a friend than to call them, and who keeps up with how many minutes they use? We use data like it's going out of style because social media has EXPLODED! My students are on Facebook, Twitter, Vine, Snapchat, Instagram, YouTube, Google Plus.. and probably something else that I have forgotten or haven't realized exists yet.

My point is that the world around us has changed so much in just the last 15 years. Our students LIVE technology every single day! The world around them is constantly changing... but our classroom's aren't. 

We need to bring the world where our students live INTO our classrooms!

"Technology is an inescapable fact of life of the world in which we live and should be embraced as a powerful tool for doing mathematics" (NCTM, 2014, pg. 82).

Technology has completely transformed our everyday lives, and it should transform our classrooms too! 

But HOW do we do that? 

In NCTM's Principles to Actions (2014), there is a comparison of productive & unproductive beliefs about the use of tools and technology in the classroom. 

1-Use technology to help students investigate mathematical ideas (NCTM, 2014).

Instead of graphing multiple lines by hand, let students use a graphing calculator or a graphing app to push past the tediousness of graphing and focus on the reasoning and sense making of the task (Dick and Hollebrands, 2011). By using technology to investigate ideas in our classrooms, we model for our students productive ways to use technology for future careers and pursuits.  Technology can also help students explore ideas that might not be possible without it. "Teachers need to recognize that mathematical action technology influences not only how they teach, but also what they are able to teach" (NCTM, 2014, pg. 84). 

2-Teachers need to be willing to learn new technologies.  

I think as teachers, we often get stuck in the rut of "this is how I was taught" and are afraid or unwilling to step out of the box.  We are also guilty of the "I learned this without technology" mentality, but the reality is that this technology wasn't around when we were in school--or if it was, it was too expensive! Maybe we would have a better understanding of some topics (like Geometry for me!) if we had the technology available to us! Explore options for Professional Development that focus on technology.  "Without well-designed professional development, teachers may feel uncomfortable about using tools and technology in their classrooms. However, once they understand the role of tools and technology as a support for student reasoning and sense making, teachers come to see that they allow opportunities to pose more challenging questions that focus on exploration and understanding" (NCTM, 2014, pg. 84-85).


These ideas are not new, but they are the first steps toward bringing your classroom into the 21st century, engaging your students in new ways, and transforming the way that you teach and help your students learn mathematics! :)

Resources: 
Dick, T. P. & Hollebrands, K. F. (Eds.). (2011). Focus in high school mathematics: Technology to support reasoning and sense making. Reston, VA: National Council of Teachers of Mathematics.

National Council of Teachers of Mathematics. (2014). Principles to Actions: Ensuring mathematical success for all. Reston, VA: National Council of Teachers of Mathematics.

Feeling Nspired!!

This past I have been at the first week of AMSTI Algebra year 1.  I have been working in an AMSTI school the past 3 years, but have never been able to get trained until now.  I have looked at SO many things in a new light this week! I have really enjoyed looking at problems from the perspective of my students. I am looking forward to working some of the problems I've worked/seen with my students this school year! :) 

One of the things they have really been working on with us is how to use technology effectively in our classrooms.  Specifically, we have been looking at the TI-Nspire CX.  I have used the TI-Nsprie CAS iPad app, and I have used the CX handlheld twice before this week. So, I will say that I have already learned a lot about how to work the Nspire CX.  I thought they would be really difficult, but I have found that the menu-based system is really very user friendly.  I think my students would actually feel MORE comfortable with the menu system of the Nspire CX than they would my TI-83s because you have to know what "button" to push in order to find what you need.  


While using the Nspires this week, I have been thinking a lot about how my students might use the calculator more efficiently with the menu based system of the Nspire than the button based system of the TI-83/84s.  I feel like right now, my students spend a lot of time memorizing which button to push in order to find what they are looking for. 

For example, if they want to find the intersection of a system, they press 2nd, Trace, Intersection, and then select their first curve, second curve, and then a guess.  
On the Nspire, they would just press menu, analyze graph, and then select intersection. They would then manually shade in their upper and lower bounds and the intersection is the result. Once they press the menu button, I think that they would understand why they would select "Analyze Graph" because that is what they are doing! 

If a student is transforming a figure on a geometry page, they select a transformation from the "Transformations Menu". It makes COMPLETE SENSE!! 
So here was the question that I asked myself... why are teachers afraid of trying new technology that may help our students be more successful in our classrooms? 
The only answer that I could come up with was that we as teachers are afraid of trying new technology or allowing our students to use new technology because we are uncomfortable with it. 

Here's the thing... technology isn't going anywhere! It is the world that we live in.  It is the world that our students live in.  We must adapt and bring their world into the classroom. If we do this, we can engage them in a brand new way! 


So, I personally can't WAIT to use the TI-Nspire CX in my classroom this year! What new technology are you going to use this year? 

Real World Statistics

I'm sure at some point in your learning career, you have been in a lecture/class about probability and statistics.  You've probably even looked at the probability of rolling a certain number on a 6-sided die.  

If your experience with statistics is anything like mine, a teacher at some point in your life said something like "if we were to roll this 10 times, what is the probability? What about 20 times? If we could roll it 100, 200, or 1000 times, what would it look like?"

So teachers wanted you to make conjectures about the probability after 100, 200, or 1000 trials, but told you it was not realistic for us to roll a dice 100, 200, or 1000 times in class.  

In this age of technology, it is VERY realistic for that to happen. We have apps on our iPad, Promethean Boards, or online that will roll a dice how ever many times we want AND it will keep up with the data and show you how the probability changes as the number of trials increases (Dick & Hollebrands, 2011).  

Statistics is all about REAL WORLD data. If is really happening outside of our classrooms, then we need to make phenomena observable inside our classrooms if possible.  Technology is a great way to do that!

Now, it is important to learn to calculate statistical measures by hand--YES! But, I think it is more important to understand the meaning behind those statistical measures. Most of the time, real world data isn't going to be sets of 10 points that are easily calculated.  They are probably going to be messy since random data rarely spits out nice neat integer points.   By using technology to calculate statistical measures, students are able to focus on the reasoning, sense making, and WHY we do what we do--instead of focusing on the calculations (Dick & Hollebrands, 2011).   

So, next time you plan a statistics unit, you might want to see how you can incorporate technology in a way that it transforms your teaching and allows your students to take mathematical ACTION instead of just using it for just conveyance.  

Resources: 
Dick, T. P. & Hollebrands, K. F. (Eds.). (2011). Focus in high school mathematics: Technology to support reasoning and sense making. Reston, VA: National Council of Teachers of Mathematics.

The Calculator Dilemma?

A few weeks ago, I wrote about The Technology "Crutch".  Today, I am looking at the use of calculators specifically.

It seems that most teachers are either advocates of calculator use in the classroom, or they are very much against it.  I have met very few math teachers who don't have a real opinion on the matter. 

The teachers that are against the use of calculators in the classroom usually are of the opinion that it hinders their learning of basic mathematical skills or it is a "crutch" (NCTM, 2014).  

However, if you look at all the research that has been done on the use of calculators in the mathematics classroom, all of it points to the idea that calculators do NOT hinder a student's learning in the mathematics classroom.  

In a research brief on calculators put out by NCTM in 2011, they examined 3 articles that synthesized research on calculators from 3 different periods along with an additional 50 studies. All of this spanned 40 years of research (Ronau et all, 2011).  They found that students who used calculators performed better and had a better attitude towards mathematics than those who did not (Ronau et all, 2011).  All of these studies pointed in this direction, however, people are still against the use of calculators.  Here is what the authors of the brief had to say on this point: "Few areas in mathematics education technology have had such focused attention with such consistent results, yet the issue of whether the use of calculators is a positive addition to the mathematics education classroom is still questioned by many areas of the mathematics community, as evidenced by continually repeated studies of the same topic" (Ronau et all, 2011, pg. 2). 

I think that calculators allow students to explore problems that are much more difficult than what they could do by hand.  It allows students to extend their knowledge to problems that may be out of their reach when computing by hand.  

For example, in a calculus class, there are some integrals we cannot compute by hand in Calculus 1, but their calculator can still find the area under that curve that we cannot calculate its integral by hand.  Should we exclude these problems from a Calculus 1 course? Or should we allow our students to explore why they can't compute by hand,but still find the value of the integral on their calculators?


What are your thoughts on calculator use in the classroom?  

References: 
National Council of Teachers of Mathematics. (2014). Principles to Actions: Ensuring mathematical success for all. Reston, VA: National Council of Teachers of Mathematics.

Ronau, R., Rakes, C., Bush, S., Driskell, S. Niess, M., Pugalee, D. (2011). Technology Research Brief: Using calculators for teaching and learning mathematics.(King, Karen, Ed.) Reston, VA: National Council of Teachers of Mathematics. 

Spreadsheets in the Classroom

Lately I have been doing some work with using spreadsheets to explore mathematical concepts.  In working with these spreadsheets in my graduate classes, I have been thinking about how I could use spreadsheets in my Algebra 1 courses to help my students explore different concepts.

In Algebra 1, we often give our students problems that look like this: 

And we ask the question: What is the function that created this table? 

Usually, students will start with identifying the y-intercept of the function (if it is given) and then identify the slope. So in this case, the function is easily identified as y=2x+2

But what if the y-intercpet wasn't given, or the function isn't linear? Then identifying the function might be much more difficult--especially with all the hand computations required.  You have to compute each value in the table by hand to make sure that it matches with the function you identified.

What if you got an example like this: 

As math teachers, I think that we forget how easy it is for our students to get lost in the journey to a solution.  Basic computations with functions are easy for us--almost second nature sometimes.  But for our students, they are still mastering the idea of functions, so they may get caught up in the computations and lose the focus of the task--Identifying the function.

Spreadsheets can easily put the focus back on the computations.  We could put the table into Microsoft Excel or Numbers and then students could put their proposed function into the spreadsheet as a formula.  

What are the benefits of inputing the function into the spreadsheet? 

1-Puts the focus back on to identifying the function instead of focusing on computations of proposed functions.  Students could also make errors in computations and then either falsely reject a good function or falsely accept a bad function.  

2- Students see immediate results. By using spreadsheets, students are able to see the immediate result of their work.  If it is wrong, they can quickly modify and try again.  When students have to do the computations by hand, they have less time to work problems, but this way they can work more examples.  

3-Allows students to see connections between symbolic & numeric representations.  Focus in High School Mathematics: Technology to Support Reasoning and Sense Making (NCTM, 2009), they talk about the importance of connecting symbolic, numeric, and graphic representations of functions.  Spreadsheets are a very easy way to connect two of those.  

Please share any thoughts below! :) 

Resources: 

Dick, T. P. & Hollebrands, K. F. (Eds.). (2011). Focus in high school mathematics: Technology to support reasoning and sense making. Reston, VA: National Council of Teachers of Mathematics.