Tag Archives: 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]



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!

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?

Connect Abstract to Practical…Meaning to Mechanics

Square-foot gardening, heard of it?


Hold that thought for just a minute…We’ll come back to it I promise. 

I am conflicted about what I am teaching.  I want to say my team is conflicted about what we are teaching, but I think it is really me.  I worry that our teams are on pedagogy auto-pilot; we teach what we’ve been teaching the same way we’ve been teaching it, which would be fine IF they were learning it, but they are not – not everyone.  It is not relevant, and they know we are going to teach it to them again next year.  What is the point in learning it now?  (Totally over-simplified but it makes my point.  Forgive me, please.)

We don’t know how to find balance between by-hand skill and skill with technology.  It is such a hard question and so risky that many choose not to consider it.  We think “I learned it this way; this is the safe way; I understand this.  Since I don’t know what is right concerning technology, I choose to ‘do no harm’.”

I’m supposed to teach a unit on operations with polynomials.  I am searching for context.  Why do we need to learn to add (subtract, multiply, divide) polynomials?  Who cares?  Where is the application of this skill? 

Is the entire history of what can be taught in algebra based on what was possible to learn before technology teachnology?  I can and need to factor polynomials if I want to graph them.  Isn’t that the number one reason to teach factoring?  BUT, don’t I now have a calculator that will do that for me? 

Not only do I have a calculator that will graph any polynomial, I could have a calculator that will do the algebra for me too.  Interesting…Is it important to factor big-hairy polynomials or is it important to know what the factors tell me about the polynomial…to make a connection from the algebraic to the graphical to the numeric? 

Isn’t it a forest and tree thing?  If I can’t factor, I can’t see why I need to factor.  If I don’t see why I need to factor, why do I need to learn to factor? 

My current conflict is about adding polynomials.  If I don’t see why I need to add polynomials, why do I need to learn to add polynomials?  I want my learners to know how and why to add polynomials, but how much is enough by-hand?  How much time should be devoted to the mechanics of adding polynomials?  In the face of technology teachnology, shouldn’t we focus on meaning rather than mechanics?

For example, a great learner question is about adding like terms.  Why can’t you add x and x^2?  Why aren’t they like terms?  Aren’t they both x ’s?  If learners don’t understand the why will the ever care about the how? Do we stop to think about why you can’t add x and x^2?  Can we give an understandable explanation? 

I can use my calculator to show that you can add x ’s together; you can add x^2 ’s together, and you can’t add x ’s to x^2 ’s.  Boom.  Done.  But, this technology does not tell us why.  Why can’t you?  What reason – meaning – prevents these variables from being “like” terms?


This brings us back to square-foot gardening…Doesn’t this picture illustrate why x can’t be added to x^2?  Think about it…


Traditionally, each box in the picture is 4’x4’ and is subdivided into 1’ sections.  You plant in each section.  How many x ’s can you see?  How many x^2 ’s can you find?  Can you see why x can’t be added to x^2?  Can we use this image to give meaning to x and to x^2?  Aren’t units (shout out to all science teachers) critical to this understanding?

If we want art and design, meaning and service, reason and understanding, integrated studies and PBL, and so much more, don’t we need to connect the abstract to the practical?

Why would anyone garden this way?  Is square-foot gardening more or less efficient than gardening in rows?  Can you grow more or less in squares?  How can you efficiently irrigate this type of garden and recycle at the same time?  Would a 4’x4’ garden improve your heath, your lifestyle, feed the hungry?


Mel Bartholomew – Introducing Square Foot Gardening

Frequently Asked Square Foot Gardening Questions

SpinPost – PBL