Friday, November 17, 2017

Understanding Standard Deviation

Long story short, I was presented an opportunity to teach standard deviation and I jumped at the chance. There are times as a coach that I wish I were back in the classroom. I do miss working with students daily. Anyway, here's what I did.

To start, I wanted students to think about the center of a data set. I asked them to individually consider each of the following and provide a typical number that best represents the situation.
I intentionally avoided using the words average, mean or center. Sometimes when I use a "mathy" word students hesitate. I'm not sure why, but I have my theories. I know I get better responses when I use the word typical so I went with that.

As we discussed each one, I wanted to draw out some important characteristics: Minimum value? Maximum value? and Are the possible values close in range or spread out?
I used the first situation as a contrast to the others to bring out variability. There are 24 hours in a day and this doesn't vary given you are talking about a day on Earth. The other situations have some variation in the data. That's what we are talking about today: variation in a set of data.

Going to the Movies
The Justice League movie is releasing next week and we are very excited about getting to see the movie. Imagine you are in the theater and settling in to watch the movie. There are people all around you and you begin to wonder...
  • How old are they?
  • Are they all the same age? If not, what is the typical age?
  • How close are the other ages to that typical age?
Here's what we got from that question.
The typical age is 26 years old with many people between the ages of 16 and 36. (Okay, I had to prompt them to think beyond the age of 30. At first it was 5 yrs old to 30 years old.)

Okay, so I've got them thinking about measure of center and variability. They've even put some values to that variability. Now, let's collect some real data.

Estimating Time
So here's why I wanted to teach this lesson. I really, really, really enjoy playing with this site and like how its simplicity captures students' attention.

Ever waited to go on vacation and it seems like time slows down but when you are on vacation the time goes by quickly? While the passing of time is constant, our estimation of how much time has passed varies. I wonder how close are we at estimating 10 seconds? Let's find out.

Each student was asked to estimate 10 seconds and record the amount of time to the nearest tenth on a sticky note. (Because this is a small school, I collected data earlier that morning from a variety of teachers and students. This allowed us to have enough data to analyze.) We built a histogram from the data. Great opportunity to assess student understanding of values within each interval.

The conversation started broad by asking what they noticed about the data and if it helped to answer our question about how close we are at estimating 10 seconds. I directed the conversation towards getting closer to standard deviation (without saying standard deviation, yet).
The students were developing a good sense of center and the values in relation to the center. Students talked about the estimates being 1.5 seconds less than 10 and 1.5 seconds more than 10.

This 1.5 you are using to describe how far away the values are from the center was a number mathematicians wanted to calculate. It's not the full range of the data but a value that can tell us how far some of the data is from the center. This number is called the "standard deviation".

Illustrative Math Task
Students were given the Illustrative Math Task Understanding the Standard Deviation. I used Part 3 as the formative assessment to gauge student understanding.

Up Next
The next lesson is to calculate the standard deviation and work on interpreting it. I like the EngageNY lesson Algebra 1 Module 2 Lesson 6 Interpreting the Standard Deviation.

In case you were wondering, because, let's face it, I was wondering, how close were the estimates for mean and standard deviation? The mean was 10.4 seconds and the standard deviation was 1.9 seconds.

Monday, November 6, 2017

Reflections on NCCTM17

The North Carolina Council of Teachers of Mathematics has their annual Leadership and Math Conferences this past week.It was a whirlwind three days with fabulous speakers including Jennifer Bay-Williams, Peg Smith, Juli Dixon, William McCallum, Joliegh Honey, Jennifer Wilson, ....  The list goes on. Anyway, I always enjoy the conference and wanted to share a few highlights.

First, I had the privilege to present with one of my teachers, Kim Clark. We presented Factoring Using the Area Model. It is very rewarding to work with a teacher who pushes you to grow because she wants to be better. I have learned a lot from her over the years and was thrilled to get to present this session with her.

This is a slide from our presentation. Did you know that the diagonals of the area model have the same product? Using that fact, we asked participants to find the possible area models. This question is the lynch pin of the investigation we did with students to help them develop a strategy for factoring when a is not 1. It's pretty cool and effective.

Second highlight was hearing Jennifer Bay-Williams talk about Becoming Fluent in Developing Procedural Fluency. This is an area of growth and learning for me. I am an avid supporter and believer that conceptual understanding is critical to students understanding and making sense of mathematics. What I am learning is that it takes specific and intentional instruction to help students develop that understanding into procedural fluency. I am also learning how to better define fluency as more than just quick.

The last highlight I'd like to share was the session from Dr. Valerie Faulkner on Opportunity, Equity & Agency: How do our grouping practices mediate student sense of mathematical identity? She makes a very convincing argument about how grouping in education (high and low students) just doesn't make a lot of sense. The statement I have been using to summarize her talk is one she made.

"We are surprisingly bad at evaluating what a person can and will do even if we have a vested interest in doing so." 

We should stop trying to separate students using some abstract high and low characteristic of student ability. We must begin to talk about students in regards to what they know and when they need to work on.

As I shared, the three days were a whirlwind. I am excited about new friendships developed and being able to connect with teachers from across the state. Maybe I'll even start blogging more often. Haaaa.....Haaaaa.....Haaaa. No promises but always a goal.

Wednesday, September 28, 2016

Floating Down the River

In an effort to support teachers across the state with the implementation of the revised NC Math 1, 2, and 3 standards, the K-12 Mathematics team are hosting weekly webinars. Each Thursday focuses on a different course: Math 1, Math 2, Math 3, and Math Leaders. Find out more here.

The sessions present a math task (if you register early you will get it in advance) and frames the discussion on standards, implementation, anticipating student misconceptions, and connections.

This past month I was able to participate in the Math 1 and Math 2 sessions. The Math 1 session was on Functions and we were given the Floating Down the River task. (Which I tweaked a little. You can find the original version here.) I really like this problem especially to discuss the key features of the functions and interpreting them in context.

This is one of my SOAP BOX concerns: the difference between F-IF.4 and F-IF.7. When we discuss functions it quickly becomes about the "families of functions." You know the ones I am talking about - linear, quadratic, exponential, etc.). With these functions we are able to use the symbolic representation and determine key features. For example, rewrite a quadratic into vertex form and identify the vertex. This is F-IF.7.

So what is F.IF.4? It doesn't seem like it should be the same thing. IMHO - it is not. While F-IF.7 focuses on those classical function families, F-IF.4 is broader. It includes all functions. This includes functions I refer to as functions that tell a story. These functions may be represented symbolically, often by a piecewise function, which is well beyond the focus of Math 1. However, it is not unreasonable for students to reason with and interpret the key features using a table or a graph.

That's why I like the Floating Down the River task. Students are given multiple tables of values and a simple question is posed. I would expect students to do what we did during the session and graph the values. (Further commentary could be given on whether depth is positive or negative but I'll leave that up to you to decide.) Now, there are opportunities for students to discuss intervals of increase/decrease, maximum/minimums, average rate of change, and intercepts. They also have to compare and connect the events. Such as, when the water is shallow the speed increases. Can students also recognize that the distance function becomes steeper? Will they recognize why that would be occur?

There is great potential in the task which is why I am encouraging my teachers and sharing with you. Try it and see what students do with it. I guarantee learning will take place. Also, consider joining me and others at the next Math 1 webinar.

Friday, August 26, 2016

Starting a New School Year!!!

The 2016-2017 school year begins next week and we are busy getting ready. North Carolina revised the standards for Math 1, 2 and 3 so the past few weeks have been focused on getting the curriculum materials aligned. Math 1 has some changes but Math 2 and 3 had a lot.

I've been getting requests for our pacing guides and outcomes. They have been updated on the respective pages (Math 1 and Math 2).

For Math 3, a major concern is to make sure students do not experience gaps in concepts due to the movement of some standards out of Math 3 and into Math 2. For example, complex solutions to quadratic equations is now in Math 2. This year is a transition year in Math 3 and won't look the same as last year or next year. Therefore, we are not revising anything as much as we are "tweaking" some things. If you have questions about Math 3 please contact me and I'll share what we are doing. If there is enough interest I may post it here.

Okay, time to get started with the new year!!! Thanks for checking in.

Thursday, May 19, 2016

NCFE Review Solutions

There have been multiple requests for answer keys for the NCFE review sets. Well, I have finally had some time to sit and work on them. I have posted the solution sets for Math 2 and Math 3 right now. My goal is to work on Math 1 in the next few days and also get it posted. Yeah!! Math 1 is also done!

There may be mistakes. Actually, I would be surprised if there weren't any mistakes. So, if you find one (or two or three or....), please let me know.

Mistake #1
Math 1: Review 4: Problem 2 -- the area of the triangle should be 13 unit squares not 26.

Saturday, May 14, 2016

Math 1 EOC Review Materials

Your students have spent all semester or possibly all year studying the Math 1 standards. Now they must take the Math 1 EOC. In an effort to provide review material for teachers, my colleague and I put together some resources.

We designed the resources around the conceptual categories: Number & Quantity, Algebra, Functions, Geometry, and Statistics. We also did a Linear & Exponential folder. In each folder you will find Resource Description. In this document we have listed the resources along with the alignment to the standards, type of resource, recommendation about calculator use, and some instructional suggestions. Our intent was to provide some variety for review along with opportunities for class discussion and strategies for solving problems.

There are two other folders. One is General Resources which provides information about the specifics of the Math 1 EOC. The other is Comprehensive Resources. This folder has some calculator inactive tasks, midterms we have used in our district, the released EOC and the mini quiz review sheets.

I hope you find some of this beneficial and I wish your students the very best on the exam.

Here is the link to the entire Math 1 EOC Review Resources. I will also post on the Math 1 page.

Tuesday, March 1, 2016


Contact Information
Wow!!! The traffic to this website has exploded overnight. More and more people are visiting every day. To help in communication, I have placed a contact form on the left hand side. This will allow you to email me directly with comments or requests.

EOC and NCFE Reviews
Several of you have requested answer keys to the review resources. I'm not ignoring your request. I just don't have them. However, I am working on them. I hope to have them posted and revised before the testing in May/June.

Curriculum Maps
These are coming. I added two more in Math 1 today and should add more to Math 2 soon. As the maps are added so are the resources noted in the maps.