Tag Archives: calculator

LEARNing: Quadratic Functions Investigation Zeros and Roots – #AskDon’tTell

What questions could we ask to help our learners investigate and “discover the rules” for the number of roots or zeros of a quadratic function?

Should we start by questioning their understanding vocabulary?  Can we questions our learners to connect roots, x-intercepts, and zeros?  Can we listen to the learners’ questions to take their path instead of our carefully scaffolded plan?

What questions should be asked to lead our learners to move them identifying no real roots,  1 real root, and 2 real roots graphically to identifying the number of roots using the equation and the discriminant?

Remember, we ask questions; we do not tell rules or definitions.  The art of questioning must be practiced and honed.

Want to explore the investigation? Here’s how:

  • Clicking on the screenshot should enable you to download the TI-Nspire document and open it if you have the TI-Nspire software on your computer.
  • Clicking on the Launch Player button should open a player file where you can interact with the document without having TI-Nspire software. (Be patient; it is a little slow to launch.)

I would love to hear your thoughts and feedback.  Also, what questions would you ask, and what questions do you hope your learners ask?


LEARNing: Linear Functions Investigation – #AskDon’tTell

I used to think I needed to carefully scaffold each algebra learning experience into a series of steps so that each learner would learn.  Now I think I should listen more and talk less.  What if I practiced inquiry and student-directed learning? What if I lead by following the learners questions? How might I set up opportunities for learners to explore and think PRIOR to my show-and-tell show?

What if I gave my learners the following TI-Nspire document and asked them to explore it for 10 minutes? What if I asked them to jot down observations, patterns, and questions  that come to them as they play with this document? What would they learn? What would they ask me? What should I be prepared to ask instead of answering their questions? Can I spend an entire learning episode answering questions with questions to facilitate student learning?

Want to explore the investigation? Here’s how:

  • Clicking on the screenshot should enable you to download the TI-Nspire document and open it if you have the TI-Nspire software on your computer.
  • Clicking on the Launch Player button should open a player file where you can interact with the document without having TI-Nspire software. (Be patient; it is a little slow to launch.)

I would love to hear your thoughts and feedback.  Also, what questions would you ask, and what questions do you hope your learners ask?


Improving Confidence, Skills, and Implementation

On Monday, July 17, Westminster again hosted 100 math and science teachers teachers for summer institutes to learn to use the TI-Nspire to integrate technology into classroom learning episodes.  This summer, the following sessions were offered:

I facilitated the Getting Started with the TI-Nspire in Algebra.  This session included several teacher-learners who wanted to take Getting Started with the Middle Grades Math (but the course did not make).  I had the opportunity to practice my skills in differentiating to accommodate all sixteen learners.

We started with a quick write using the following prompts: Why are you here, and what do you want to learn?  Overwhelmingly, these sixteen teachers wrote and spoke about relationships and improving their ability to engage their students in the learning process.  “I want to feel more confident about using this technology to teach my students.” They discussed feeling overwhelmed by the technology and implementing lessons with students.

How often do students feel exactly the same way?  Aren’t students looking for a teacher who knows their strengths and struggles?  How often do students feel overwhelmed by the content and implementing new skills and idea?

The curriculum – a binder of materials and activities – had approximately 10 activities per day. So the question…Go deep into some of the lessons or cover all 10 activities each day.  I chose to be selective about the number of activities and spend time asking questions to deepening knowledge, skills, and understanding.

As the teacher, I feel guilty about what I did not cover from the materials.  What if they need something that I did not teach them?

Isn’t this the same decision classroom teachers have to make every year, every week, every day?  Should we cover all of the learning targets or identify what is essential and teach for mastery? Are we seeking to expose our students to many topics, or are we striving to help them learn and retain core material?

The time we have with learners is limited.  We have to make some very important decisions about how to use this time.

TI-Nspire Day 2 – Round Robin

We ventured off track from “the plan” for day two.  There are nine National T3 instructors facilitating learning at our site.  We decided to have our teacher-learners change classes so that they could work with and learn from four additional T3 instructors.

The Middle Grades teacher-learners had the following learning opportunites on day 2:

  • Investigating Computer Algebra Systems with Paul Alves
  • Creating Sliders with Josh Mize
  • Data Collection with the CBR with Margaret Bambrick
  • TI-Nspire Presentation View with Alicia Page

I had the opportunity to facilitate the following learning:

Here’s feedback from one of our teacher-learners:

“Hey y’all,

I am so excited!  I gave myself homework, which was to recreate the document that Josh (TI instructor) taught us how to do today, without looking at my notes or the previous document.  I did it!  Change the leg lengths by increasing or decreasing the sliders and the figure changes shape.  It also calculates c (hypotenuse) by measuring, but then look at the second page and you can see where the c value is calculated using c = square root of (a^2 + b^2) and the two columns (one measured and one calculated) match each other.  Too hard for 6th grade but useful in 7th and 8th.


I can also report an interesting story from Josh.  He says that he showed the Middle Grades teacher-learners several documents with sliders and then asked them which one they would like to create.  They said “none of them; it’s not what we teach.” So on the fly, he taught them to use sliders to illustrate the pythagorean theorem just as described above.  He was learning with his “students” to teach them what they wanted to learn.  Exciting!  Isn’t this how it is supposed to be?  Josh dropped his plan when it wasn’t going to work for his learners.  He taught how to use sliders to make math dynamic while meeting the needs of his learners.

When formatively assessed this morning, the Middle Grades teacher-learners could successfully work through the spiral activity showing they had acquired the essential skills of day 2 without marching through the standard curriculum.  Wow!

Age Estimation – Day One Lesson & Community Building

We want to try something different this year for the first day of Algebra I.  (We are hoping that other’s will join us too!)  Our learners will arrive with their new MacBooks. We want to use them immediately.  We think we are going to try the age estimation activity

The email request is shown below:

     We need you in pictures!  (and we need your permission to divulge your age to our students, if you’re game!)
     We are planning a multi-disciplinary/multi-grade lesson on graphing and numeracy that would help our students to start the year off using their new MacBooks.  We need volunteers who would be willing to tell us your true ages (as of August 11, 2011), knowing the students will discover these ages during implementation of the project. We will also be showing your picture; if you have a picture of yourself that you would like us to use, please email it to us.  If not, we will just use a photo on file.  We appreciate so much your willingness to participate, and we also appreciate your right to “pass on this one”.
     We usually use celebrities but thought it would be good community building to use our faces.  Jill piloted this activity with her 8th graders in May, and it was a big hit!  The kids suggested that we use faculty faces in August and the celebrities in May.  Think how great it would be for the kids to guess who we are (and how old we are – if they are smart, they will underestimate! – if not, what a teaching opportunity.)
     So, if you are game, please send us a photo and your age/date of birth.  Please?

Here is the presentation that we usually use:

In about 24 hours, we received 12 responses!  Enough to build the first lesson.  At the end of the week, 40% of the faculty responded to participate. Enough to build three lessons – one for 6th grade, one for 7th grade, and one for 8th grade.

Some of the GREAT replies include:

  • “I love the people I get to work with.”
  • “I will be 55 just like the speed limit!  If some kids says, “is that all??!!” feel free to smack ’em.”
  • “Well…a lady never tells…but I liked Gloria Steinum’s comment when she turned 60, “This is what 60 looks like.”  So, you can tell the kiddies that I am (almost) XX,  if you don’t think you will scare them to death.  :-)”
  • “I guess we cannot use that picture of Darlene if it is for student use, huh?!?!  I will send a picture for sure!  How much time do I have to go to glamor shots????”

And the pictures are great.  Can you image the face of a JH student when they see their principal like this?  How fun!

So, what’s the activity?

We are going to show you a series of photos, and you are to estimate each person’s age.  We want to know “Who is the best estimator (and who is the worst)?”   All you have to do is estimate the age of each person and enter it in the spreadsheet.  The learners guess the age of each person in the slideshow.

We think that we will use our teachers instead of the celebrities.  My 8th graders reported that they did not always know all of their teachers on the first day of school.  We hope that this lesson will help our learners connect with their teachers and build our community.

What about the math?

How will we determine the best estimator? the worst?  We will decide as a class.  Let the learners collaboratively develop their criteria to determine the best and the worst.

Usually, the first comment is to find the difference in the estimate and the actual age and sum the differences us – a great first thought.  The sign of the difference has meaning.

Did you over estimate or under estimate?  How does the sign of the difference help you decide if you over or under estimated? If your sum is the closest to zero, are you the best estimator?

How can we find how far off the estimates are but eliminate the direction of “off-ness?”   If we find the absolute value of the difference in the estimate and the actual age, will we find the best and worst estimators?

What would a graph of this data look like?  Which variable would be acceptable for the independent variable? Does it matter?

Why is there a positive correlation in the plot of the data?  What would be the line of best fit?  How do you know?

Can you determine from the graph if you over or under estimated?

What else can be learned?

There are several spreadsheet skills to introduce with this lesson.  The learners will generate a scatter plot and graph a function.  Mental math will be used.

From experience, this lesson creates a loud conversational classroom.  Shock and awe at some of the ages and estimates!  Lots of laughter mixed with learning.  In May, the entire activity took approximately 30 minutes.  We hope that we can use the first 55 minute class period to learn and laugh together as we discuss our community while doing a little math.

Handicap Ramps: Connecting Ideas and Experiences to PBL – apply what you learn

I don’t often have the question “When are we going to use this?” launched at me.  Sometimes I wonder why?  Why aren’t my learners asking this question?  I often ask myself “When are they ever going to use this really?” when teaching Algebra I.  How can I better show our learners that algebra is used for many real purposes, not just on a test?

On September 14, 2010, I had the privilege of attending TEDxAtl where I heard Logan Smalley talk about creating a movement with Movement Turned Movie.  Logan introduced us to Darius Weems and his story Darius Goes West.  In the spring, Darius joined our 8th graders for their retreat – an amazing experience for all.

On July 19, we will host approximately 170 teachers from nine different states for a summer learning experience.  We’ve done this summer camp for teachers for several years.  Each year there is a teacher or two who will struggle to navigate our campus.  There are stairs everywhere.  We do have elevators, but they are not always in the most convenient places.

In Synergy, we problem-find and attempt to problem-solve based on observations of our environment and community.  Logan’s advocacy for wheelchair accessible spaces combined with accommodating teacher-learners with mobility problems has caused me to want to learn more about our campus and the ease of access to our spaces.

Where are our ramps and elevators?  What are the requirements and specifications for these ramps?  Are the requirements based on the angle of elevation or the ratio of the length of the ramp to the height of the ramp?  Is the angle of elevation connected to the ratio of length to height?  Isn’t this rise over run?

What can be learned by investigating the ramps on our campus? Does our learning have to be restricted to our campus?

  • Algebra?  (I think there must be slope, geometry, and right triangle trig at a minimum.)
  • Science? (I think mechanical advantage might come in to play here.)
  • Writing workshop?  (Do we need more ramps? Are there areas where a ramp is needed? How can we advocate for others?)
  • History?  (When and why did the Americans with Disabilities Act (ADA) become law?)
Here is a photo we took today at the entrance to Pressley where most of us enter to go to the dining hall.  If you look closely, you will see a meter stick on the ground near AS’s feet.  
In the latest version of the TI-Nspire CX operating system you can analyze a digital photograph.  It is a great way to use ratios and proportions along with unit conversion.  Can you predict how tall AS is based on the measurements and the scale?  (I was less than an inch off.)  Does our ramp fall within the ADA’s specifications?  
Let’s make sure the variables and measurements are defined clearly.  m=3.83 cm is the measurement of the meter stick on the screen of the Nspire.  rl=23.3 cm and rh=1.91 cm are the screen measurements for the ramp length and the ramp height, respectively.  ah=4.64 cm corresponds to AS’s height on the screen. 
Can you think of ways to use your environment to teach?  We should not be restricting learning to the four walls of our classrooms.  Can we find ways to show our young learners how their learning connects to their community and beyond?

Stopping Distance Reflection – Where did it take us?

We are studying quadratic functions.  We started with the Stopping Distances data to look at quadratic data visually.  Our hypothesis was that the distance required to stop while braking is proportional to the square of speed, d=k·v².  Many of our learners had trouble fitting the curve; they were hesitant to take a swing.

What if we read Peter Reynold’s The Dot in class?  What if we encouraged our learners to just try to find an equation, to see where it takes them?  What if we used the TI-Nspire Navigator to “frame” their first marks, to celebrate that they took a swing?

Here’s what @fencersz, a learner in DD’s 3rd period, said in a couple of tweets to me about the class:

@jgough I thought it was really interesting to look at math in a way that involved creative problem solving as opposed to just applying a set model to the problem, was surprised b/c I’ve never though about math like that b4 and would like 2 get better at this.

What if we did this for each other too?  DD just happened to come for part of my class while I was teaching Stopping Distances.  She liked it, but said she thought she could never teach it.  We agreed to team-teach the lesson in one of her classes.  The following is her reflection on teaching it alone for her other class.

I had the best time teaching my 5th period today.  I know it was not as smooth as yours, but I think it was pretty good for my first go-round.  I remembered things to say that you said; there were even things I sort of got lost in, yesterday, that I wasn’t planning to do, but understood what you did when I got to it and was able to show the kids.  I don’t think they had any clue that I was just learning what I was talking about when I gave my spiel on velocity being speed with direction to explain one reason it would be quadratic rather than exp.  The kids were asking great questions and making good connections.  They were also engaged.  Thank you for teaching this to my third period so I could learn it.  I know you felt horrible yesterday, but you trooped on over anyway.  What about you and me team teaching this lesson in 4th period PLC as our first lesson study (so we can improve it for next year) with the new TI-Nspire software?

Model learning.  Encourage others.  Try new things.  Collaborate.  See where it takes you.

Connections: Questions, Photographs, Algebra Graphs, Perspective, Environment

Several of my tribe are participating in the 3six5 project (365 days, 365 points of view) which has caused us to wonder about this experience for education.  Here’s the invitation I received:

Dear Ms. Gough, Tara, Sarah, and Whit:

There are many stories to tell in the world of education and many voices that need to be heard.  We have started a new project, edu180atl, designed to share stories of learning and highlight voices from the Atlanta community.  During the 2011-2012 school year, we hope to have 180 different learners (students, parents, educators, etc.) participate in this project.  The purpose of edu180atl is to nurture and encourage the spirits of those who love to learn, to connect learners across disciplines and settings, and to deepen the national conversation about education by enabling parents, students, and educators to share stories of what they are learning every day.  In order to test the feasibility of such a project, we will be piloting a similar experience (edu180atlbeta) during the month of April.

We would like to invite you to be one of the founding writers for this project.

If you are interested in participating, please complete our beta signup form (link: http://bit.ly/edu180altsignup) and select two Monday-Friday dates (your first and second choice) during the moth of April when you will be able to submit a reflection based on the prompt:  what did you learn today? We will respond with a confirmation of the date and further instructions for submitting your reflection.  We ask that your reflection be no more than 250 words (1 typewritten page), and if you choose, you may also submit a photograph or a short video (no more than 90 seconds) to go along with your submission.

We look forward to this project and how you will consider participating.

**  Due to a limited number of days in April, we may not be able to accomodate every learner who wishes to participate.

One of our Synergy 8 team, Whit Weinmann (aka @runningwitty), started us off on April 1 with a challenge to realize connections.  In Synergy 8 we believe in the value of prototyping and the power of feedback.

I spent the weekend working with my writing team in Dallas planning a professional development experience for teacher-learners.  I thought I might practice answering the question: what did you learn today?  I also want to ask for your feedback while I practice and prepare for my edu180atlbeta day on April 11.


The theme of the day has been the art of questioning. My learning revolves around trying to ask better questions, questions that are open-ended enough to promote risk-taking, differentiation, and finding connections.

How can we leverage the technology in our hands help learners apply what we are teaching and see the beauty of the world through their learning?

Imagine learning about linear functions, perspective, design, and environment simultaneously.  Do we think helping our learners see that the patterns we teach actually model things they see will help them find relevance and pique their interest?  For our artists, would it be fun to insert their art into our technology and model the shapes and structures that exist in our community?

Can we take a photograph of the bridge to the Summer Camp and find the algebra in the photo?  What can we learn?  Do we need a lesson on perspective to gain an understanding of why these lines are not parallel in the photo but are in reality?

Can there be an entire theme for integrated learning around our bridge and the environment where it sits?  Who comprises the team that would spend just even one day using this bridge as the catalyst for learning?

Where are our resources to integrate and blend curriculum to support learning? How can we model lifelong learning side-by-side with the young learners in our care?  Will we?

I learned I am willing to try. It is worth the risk.  Will you join me?



It is interesting to me that I could write about what I learned today in 250 words, but I’m having trouble describing myself in 140 characters.  I think it is good to know this today while I have time to think and revise.

I’d love your thoughts and feedback.

Turnpikes, Toll Roads, Express Lanes

Atlanta:  Traffic, traffic, and more traffic…

Coming Soon! Peach Pass available in Spring 2011. 
I-85 Express Lanes in Atlanta open in Summer 2011. 

View the Peach Pass video to see lots of accessible math connected to a real community issue. Learn more about the I-85 Express Lanes.

  • What’s the difference between an express lane, a toll road, and a turnpike?
  • Are you charged by the mile or by the minute?
  • Why is the target speed 45 miles per hour?  What is the target speed for other express lanes?
  • How will the Peach Pass know when I should pay (because I have less than 3 people in my vehicle) and when I can ride toll free?
  • What is the mathematical model that determines the toll?  We know it is positively correlated.  Will the model be linear, exponential, or some other type of function?
  • What will the revenue generated by the Peach Pass be used for and who controls these monies?
  • Are the Peach Pass and other E-ZPass-type cards cost effective or just convenient?
  • How do the other locations listed in the video charge for the use of their express lanes?  How do other states collect this money?  Utah, for example, uses an ExpressPass.

The Pennsylvania Turnpike is the oldest turnpike in our country.  Beginning in January, 2011 there was a rate increase; cash tolls increased 10% while E-ZPass tolls increased by 3%.  Is there a savings to use the E-ZPass, or is it just for convenience?  Since there is a Pennsylvania Toll/Mileage calculator, we can investigate the cost to drive on the PA turnpike.  Would this help indicate a reasonable rate for driving on any toll road or express lane?

To see if there is a pattern to the cost, I chose to collect data entering the Pennsylvania Turnpike at Interchange 57-Pittsburgh and then vary the exiting interchange for a class 1 vehicle with 2 axels.  I wonder what the toll rate for an 18-wheeler would be compared to my passenger vehicle.  My learners have many choices.  They may choose to start at any entry point on the turnpike and vary their exiting interchanges.  I suppose they could vary both the enter and exit interchanges.  They could also change the type of vehicle to investigate the charges and the rates for different size vehicles.

Is there a pattern to the data?  It the relationship linear, exponential, logistic?

To see the relationship between the data, we graph.

Cash toll charged vs. miles driven on the PA Turnpike:

E-ZPass toll charged vs. miles driven on the PA Turnpike:

To compare the two data sets, graph on the same grid.

More questions:

  1. What are the mathematical models that could represent these data sets?
  2. What are the meaning of the slopes of these lines?
  3. Is it cost efficient to purchase the E-ZPass?
  4. Is there a relationship between the E-ZPass toll charged and the Cash toll charged?
  5. What is the mathematical model that could represent these data?
  6. What is the meaning of the slope of this line?

Which leads to more questions:

  • How does the rate charged by the PA Turnpike compare to the rates of other turnpikes?
  • How does the rate charged by a turnpike compare with the charge on a toll road or express lane?
  • From $0.60 to $6.00 is a pretty big swing in cost to use the 16 miles of the I-85 express lane in Atlanta.  How will traffic volume be determined since tolls go up when traffic volume increases and the toll is lowered when traffic volume decreases?
  • How do the toll roads, turnpikes, and express lanes in other countries compare to our toll roads, turnpikes, and express lanes?  How do they compare in cost, in speed, and in access?

Can our learners aquire the needed content through a problem or project based approach?  Will they find the content more interesting and engaging?

As we learn more about problem-based learning and project-based learning, would this be type of lesson help learners see the application of content? … the blending of content? … the relevance of content?

I think so.  Are we willing to experiment?… to learn by doing?

Calculator is to Arithmetic as Spell Checker is to Spelling???

Is using a calculator for math comparable to using a word processor for English? Is the calculator an arithmetic checker like the word processor is a spell or grammar checker?

My friend Jeff makes a good point about technology integration (advances) in English. Students that use a word processor must still proof their writing. Do we worry that kids won’t learn to write because of spell check or grammar check? Or, do we think that because of these tools they are free to concentrate on ideas, organization, voice, word choice, sentence fluency, conventions, and presentation? (Okay, conventions have to do with grammar and spelling, but I’m making a point here.)

The grammar or spell checker does not always catch “there” when I mean “their”. A calculator will not catch that I meant (-2)^2 when I entered -2^2. The calculator does not know that I mean 1+(6+4)/2 when I enter 1+6+4/2.

As Peyton pointed out in the Writing Workshop meeting, MS Word will not alert you to your error in writing “warmest retards” when you meant to write “warmest regards”. Your calculator will not alert you to an entry error; it will not know that you entered -3.75 when you wanted -3.57.

The spell checker automatically corrects some of my incorrect spelling. When I type “recieve” it automatically changes it to “receive”. When I type “calcualtor”, the word automatically changes to “calculator”. When I write “I never here anyone…”, the grammar checker alerts me to check my spelling or word choice. The Nspire calculator will autocorrect a little bit, but it assumes what you mean. For example, if you open parentheses, it will close them. However, you must make sure that it closes where you intend to end the grouping.

This is an interesting place for me in my thinking. I know that it has to do with age appropriate learning. I believe that young children should learn their numbers and arithmetic just like they learn their letters and words. I believe that junior high students should learn how to graph and solve equations by hand, graphically, with tables and spreadsheets, and with technology.

How much more could we learn about algebra, calculus, and statistics if we used technology to accommodate 8th graders that struggle to compute? Don’t our students need to spend more time on data gathering, mathematical modeling, and interpreting graphs and less time on mechanics.

Have you seen Conrad Wolfram’s Ted talk Teaching kids real math with computers?  What do you think?


Just look at any TED talk by Hans Rosling to see examples of how critical the analysis and synthesis of mathematical information is to our future.