Tag Archives: TI-Nspire CAS

SMP-8: look for and express regularity in repeated reasoning #LL2LU

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We want every learner in our care to be able to say

I can look for and express regularity in repeated reasoning. (CCSS.MATH.PRACTICE.MP8)

But what if I can’t look for and express regularity in repeated reasoning yet? What if I need help? How might we make a pathway for success?

Level 4
I can attend to precision as I construct a viable argument to express regularity in repeated reasoning.

Level 3
I can look for and express regularity in repeated reasoning.

Level 2
I can identify and describe patterns and regularities, and I can begin to develop generalizations.

Level 1
I can notice and note what changes and what stays the same when performing calculations or interacting with geometric figures.

What do you notice? What changes? What stays the same?

Can we use CAS (computer algebra system) to help our students practice look for and express regularity in repeated reasoning?

What do we need to factor for the result to be (x-4)(x+4)?
What do we need to factor for the result to be (x-9)(x+9)?
What will the result be if we factor x²-121?
What will the result be if we factor x²-a2?

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We can also explore over what set of numbers we are factoring using the syntax we have been using. And what happens if we factor x²+1. (And then connect the result to the graph of y=x²+1.)

What happens if we factor over the set of real numbers?

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Or over the set of complex numbers? 

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What about expanding the square of a binomial? 

What changes? What stays the same? What will the result be if we expand (x+5)²?  Or (x+a)²?  Or (x-a)²? 

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What about expanding the cube of a binomial?  Or expanding (x+1)^n, or (x+y)^n?

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What if we are looking at powers of i?

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We can look for and express regularity in repeated reasoning when factoring the sum or difference of cubes. Or simplifying radicals. Or solving equations.

Through reflection and conversation, students make connections and begin to generalize results. What opportunities are you giving your students to look for and express regularity in repeated reasoning? What content are you teaching this week that you can #AskDontTell?

[Cross-posted on Easing the Hurry Syndrome]



Art of Questioning reflection – #NspiredatT3

My previous post shared the lesson plan for our 4 hour Art of Questioning session at T³. I want to share what actually happened, my reflections and what I learned.

We used Skype so that Grant Lichtman could join us and present with us.  I LOVED doing this.  Grant and I did this the day before in a shorter session for T³ instructors.

I did a better job (I hope) introducing Grant by reading from Step 1 – Art of Questioning of his book The Falconer: What We Wish We Had Learned in School.

Our educational systems have been constructed entirely around the goal of providing the correct answer to a question provided by an instructor or handed out on a standardized exam.  This system provides a form of valid comparison for the results of a group of students, and it provides a foundation of shared information amongst those who have followed a course of study.  Unfortunately, the real world, particularly the real world of the coming century, does not and will not work this way.  Our heroes are not defined by how well they answered canned questions or what they scored on their SATs precisely because these outcomes do not determine success in real-world situations.  The real revolution in education and training, if it comes, will be overtly switching our priority from the skills of giving answers to the skills of finding new questions.

Questions are waypoints on the path of wisdom.  Each question leads to one or more new questions or answers.  Sometimes answers are dead ends; they don’t lead anywhere.  Questions are never dead ends.  Every question has the inherent potential to lead to a new level of discovery, understanding, or creation, levels that can range from the trivial to the sublime. (Lichtman, 35 pag.)

Grant told two powerful stories of leveraging learner questions to facilitate learning. He made the great point that if you teach from student questions, you know someone in the room is interested in what is being discussed.

Then it got seriously interesting for us.  Grant facilitated an experience of questioning techniques while I drove a lesson (shown below) on the TI-Nspire.

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.)

Grant reviewed the “big 6” types of questions and transitioned to another type of question – “What if”.  Here’s what the exercise looked like when I finished following his directions.  Remember, he could not see the linear investigation, and I could not see him.


Grant then signed off so that we could roll up our sleeve and get to work experiencing learning through the art of questioning.  I opened the next section of this lesson by reading from Step 0: Preparation of The Falconer: What We Wish We Had Learned in School.

The excitement of learning, the compelling personal drive to take one more step on the path towards wisdom, comes when we try to solve a problem we want to solve, when we want to solve, when we see a challenge and say yes, I can meet it.  Great teachers lead us just far enough down a path so we can challenge for ourselves. They provide us just enough insight so we can work toward a solution that makes us, makes me want to jump up and shout out the solution to the world, makes me want to step to the next higher level. Great teachers somehow make us want to ask the questions that they want us to answer, overcome the challenge that they, because they are our teacher, believe we need to overcome. (Lichtman, 20 pag.)

Wow! Worth repeating:

Great teachers lead us just far enough down a path so we can challenge for ourselves.

So, how do we do this?  Sam accepted the challenge of modeling this type of facilitation of learning by leading a lesson.  We wanted the participants to experience the investigation, question generation, and learning. Sam chose to use the EllipseInvest.tns file show below.


Sam employed the 3-12-3 protocol:

      • 3 minutes: Independent investigation of the EllipseInvest.TNS file.
      • 12 minutes: Work with a partner to share questions, convert closed questions to open questions, and generate additional questions. Partners should identify their top 2-3 questions.
      • 3 minutes: Use the TI-Nspire Navigator to collect each student’s top question.

Facilitate a class discussion by responding to student questions encouraging responses from students as well as the teacher.

It was awesome!  Sam knew that he was going to administer a formative assessment next.  As his peer observer, I could see his effort and questioning to guide the discussion through the participant questions to the essential outcomes of the lesson. Another point from The Falconer that is worth repeating:

Great teachers somehow make us want to ask the questions that they want us to answer, overcome the challenge that they, because they are our teacher, believe we need to overcome.

Sam then used EllipseForm.tns to model the leveled formative assessment idea.


Experientially, our participants could make their own determination of the value of this type of formative assessment.  Continuing his questioning technique, Sam prompted the participants to identify why the questions were at the given level.  Could they see a leveling up in the questions?  What did it take to move from one level to another?  The discussion was excellent, and Sam received strong feedback about his assessment design.  Yay!

We were at about the 2-hour mark in our 4 hour workshop.  I asked our participants if they could stand a 4-minute Ignite talk on assessment to set the stage for the next 2 hours of work and then we would take a break.

We resumed after the break by watching Dan Heath: How to Find Bright Spots.

Leveled assessments provide the opportunity to bright spot the work of every learner.  They come in the door saying I can do this, Ms. Gough; will you help me level up? Let’s take the challenge of highlighting what learners can do rather than what they cannot.

For the next hour, Sam and I watched, listened, and coached as participants worked to designed a leveled assessment on a topic of their choosing.  We displayed an example through the projector as a point of reference. Our participants asked for the template of the table of specifications.  All files linked on my previous post are .pdfs.  The table of specifications as a Word doc is shared below.

How great that our participants asked for a usable resource! Learn and share!

I was so pleased with the engagement and the collaboration of our participants.  There were so many good questions.  I challenged our participants to share their ideas – in any form by dropping their files in my Dropbox.  I’ve promised to zip all files shared in my Dropbox by Tuesday and share this on this post.

This morning I noticed this tweet…

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I love having this type of feedback!


Lichtman, Grant, and Sunzi. The Falconer: What We Wish We Had Learned in School. New York: IUniverse, 2008. Print.

The Art of Questioning: Leading Learners to Level Up – #NspiredatT3

Today, we are delivering a ticked, 4-hour session on the Art of Questioning at the Teachers Teaching with Technology International Conference in Philadelphia. What follows in the intended lesson plan for the session.  We are so excited that Grant will be joining us via Skype!

The Art of Questioning: Leading Learners to Level Up

“Questions are the waypoints on the path of wisdom.” ~ Grant Lichtman. This session will focus on the art of questioning and using the TI-Nspire™ Navigator™ System as a formative assessment tool. Work on becoming a falconer…leading your learners to level up through questions rather than lectures. Come prepared to develop formative assessment strategies and documents to share with students to help them calibrate their understanding and decode their struggles. Be prepared to share your assessments with others for feedback and suggestions

(10 min) Introductions – Jill reads from Step 1
(20 min) Grant discusses the Art of Questioning – The Falconer
(10 min) Grant and Jill use Linear TNS file to model What if…questioning
(40 min) Sam facilitates EllipseInvest.tns and 3-12-3 protocol for questioning
(30 min) Leveled Assessment using Navigator and discussion pedagogy
(10 min) Jill’s Ignite talk on Assessment
(15 min) Break
(15 min) Dan Heath’s Bright Spot video and quotes
(60 min) Work time – participants develop a leveled assessment
(20 min) Share and feedback session
(10 min) Conclusion and Challenge

As a system…

As a system…

Drop your files: http://www.dropitto.me/jplgough
Upload password: droptojill

On Tuesday, Jill will zip all files and link them to this blog post.

Our participants have been asked to bring their laptops so that we can build assessments using the TI-Nspire software.  We intend for participants to leave with an assessment ready for use next week.

Design for Learning and Inquiry – PD Day #T3Learns #nspiredatT3

The tribe of T³ Instructors gathers the day before the 2013 T³™ International Conference to learn together. I am honored to be part of the team that has been asked to facilitate a PD experience for my colleagues and friends.  Our topic: TI-Nspire Advanced Authoring – Designing for Learning and Inquiry.

Below is the description as it was sent to the T³ Instructors and our tentative agenda. (Note: The links in the agenda below are password protected and available only for face-to-face professional development participants.)

TI-Nspire Advanced Authoring – Designing for Learning and Inquiry

Ruth Casey, Jill Gough, Sam Gough, Jeff McCalla

From the PD signup document:
Design For Learning – 120 minutes
This workshop was successfully piloted last summer.  The focus was on the effective design of activities, using TI-Nspire to encourage student learning.  This session will give you an overview of the goals of the PD offering, and the instructional approach taken in the workshop.


Materials Needed:

  • Diagnostic Assessment sent via email to participants by Monday, March 4, 2013
  • TI-Nspire Navigator for Networked Computers
  • Chart paper and Markers for Storyboarding

(15 min) Introductions
Review Diagnostic Assessment Results
(30 min) Introduce the Anatomy of a Document
Anatomy of a Document and investigation
(60 min) Introduce Storyboarding
Instructor modeled storyboard and design of a lesson and Storyboard a concept using principles of  “Anatomy of a Document”
(15 min) Share the Design for Learning and Inquiry Google site

We encourage the idea of Storyboarding prior to launching in to designing with TI-Nspire. We are inspired by Garr Reynolds and Presentation Zen.  In particular we are going to try to avoid creating Nspire documents that are slideuments. For more information, please read “Slideuments” and the catch-22 for conference speakers. Can we begin to see ourselves as designers of learning experiences?

We want our participants to learn to design a one-page TI-Nspire document that promotes student investigation, learning, and inquiry.  Our goal is to discuss – experientially – the essential learnings for the summer workshop. We know we can’t do justice to a 2-day workshop in 2 hours.  We planned to go deep into one activity rather than cover the entire agenda at a rapid pace.

These essential learnings are:

At the end of this workshop, participants should be able to say:

  • I can exercise the ideas of restraint and simplicity when designing learning investigations.
    • I can identify what is important and remove what is not important.
    • I can design where less is more visually – I can include only what is necessary to promote inquiry and investigation.
    • I can design documents that are engaging and prompt questions and inquiry from the learner.
  • I can storyboard a learning investigation prior to beginning to design to streamline the concept and balance the information to be learned.
    • I can explain the goal of the activity and outline the expected learning outcomes.
    • I can design a variety of dynamic constructions that are controlled by different inputs including points, sliders, and stored variables.
    • I can design documents with a variety of outputs, which use color and strings to support opportunities for  visual connections.
  • I can create TI-Nspire documents to promote student investigation and inquiry.
    • I can enhance documents with conditional statements to make information appear and disappear as needed to enhance a lesson.
    • I can apply TI-Nspire construction tools: geometry tools, scatterplots, data capture, etc. to create the investigation.
    • I can use free points, restricted points, sliders, stored variables, etc. to control the actions in the document.
    • I can use color, text boxes, strings, etc. as inputs and outputs to connect ideas and promote questions.

Art of Questioning – PD Day #T3Learns #nspiredatT3

The tribe of T³ Instructors gathers the day before the 2013 T³™ International Conference to learn together. I am honored to be part of the team that has been asked to facilitate a PD experience for my colleagues and friends.  Our topic: The Art of Questioning.

Below is the description as it was sent to the T³ Instructors and our tentative agenda.

The Art of Questioning – 120 minutes

This session will focus on the art of questioning.  Strategies will be discussed to assist learners to “level up” through questions rather than lectures.  Come prepared to develop documents to help students calibrate their understanding, and share your assessments with others for feedback and suggestions.


  • Introduce a protocol for facilitating learners to ask their own questions
  • Introduce leveled assessment
  • Participants develop a leveled assessment and share it with other attendees


Jill Gough, Sam Gough, Grant Lichtman, Jeff McCalla

Materials Needed:

  • TI-Nspire Navigator for Networked Computers

Agenda 2:

(10 min)            Introduce facilitators
(20 min)           Learn with Grant Lichtman via Skype
(30 min)           Art of Questioning using EllipseInvest.tns
(15 min)            Break
(20 min)           Introduce Leveled Assessment
(30 min)           Participants develop a product
(10 min)           Share session

Grant is going to share two powerful stories of teaching and learning through the art of questioning. He has also agree to take us through an exercise from The Falconer to illustrate the possibilities when asking quality questions.  <EXCITING!>

Jeff, Sam, and I will model methods of using TI-Nspire’s interactive capabilities to give rise to more student questions by putting our participants in the seat of the students.  We will share student-learner reactions, challenges, and successes. The overarching idea is posted at Practice seeking questions – #AskDon’tTell and a sample story is shared in the post Ellipse Investigation – #AskDon’t Tell.

We will recommend and read from several books (listed below) that have helped us frame our work. Our goal is to facilitate a workshop where the participants learn by doing.

From an earlier blog post:

  1. Read, read, read…Currently high on my list:
    1. Grant Lichtman‘s The Falconer: What We Wish We Had Learned in School,
    2. John Barell’s Developing More Curious Minds, and
    3. Dan Rothstein’s Make Just One Change: Teach Students to Ask Their Own Questions.
  2. Practice, practice, practice…Remove the scaffolding:
    1. Watch Dan Meyer: Math class needs a makeover and try it.
    2. Stop creating slideuments.  If your TI-Nspire document, your PowerPoint presentation, or your worksheet has multiple pages, slides, or steps, eliminate lots! Create space for questions, investigation, and thinking.
    3. Use Gamestorming games to develop techniques for learning to ask questions. I like Brainwriting3-12-3, and others.
  3. Risk, reflect, revise:
    1. Try it – more than once. One trial does not make an experiment.  Celebrate even small successes.
    2. Have strong wait time, and have questions in your “back pocket” if prompting is needed.
    3. Seek feedback from a trusted colleague. Engage in peer observations to help you see from another perspective.

Ellipse Investigation – #AskDon’tTell

One more learning investigation from our experiment – see Circle Investigation – #AskDon’tTell and Parabola Investigation – #AskDon’t Tell.

Sam designed the following ellipse investigation (EllipseInvest.tns).


Look at the continued improvement in the learners’ questions:

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Sam notes that his learners are beginning to ask more mathematically relevant questions. Isn’t this exciting? Can you imagine learning math this way?

I’m pretty sure there is a hyperbola lesson coming soon.

Parabola Investigation – #AskDon’tTell

Continuing our experiment to lead learning by following and responding to learner questions, Sam designed the following parabola investigation (ParabolaInvest.tns) for his learners. (Note: Sam teaches in a 1:1 MacBook program, so these files are designed to be viewed and investigated in computer view rather than handheld view.)


Here are the questions generated by Sam’s student-learners.


Wow! What an improvement in the number and the quality of questions.

The importance of collecting many questions is critical.  If we answer questions in the order they are asked, we might not get to the most interesting or most critical questions.

Sam reported that he is very pleased with the learners’ questions and their engagement with the mathematics.  He says he is encouraged to continue this method of teaching because of the change in participation and interest from his learners. It is fun to teach and  facilitate learning this way! He says:

Kids should interact with these graphs instead of memorizing facts.  I want to continue offering investigations where they control points and identify patterns. It’s a great way to learn!

He is busy working on an Ellipse Investigation, and he is already planning a Hyperbola Investigation too.