Monthly Archives: January 2013

Rigor, Relevance & GPA


A couple of weeks ago, our school participated in Instructional Rounds. Administration from our Central Office along with colleagues from nearby districts walked through a sample of our classrooms. Reading their comments / findings was interesting. 100% of teachers were doing something – working hard – teaching. 90% had agendas or I can… statements posted in their rooms. Students were doing what they were supposed to be doing – does this mean they were engaged or being compliant? 75% of questions being asked were in the bottom 2 levels of Blooms. That means 25% of what was observed was “rigorous”, I guess???

The following day we had PD – Rigor, Relevance & Congruent Tasks. We were asked to define Rigor…well, most of us felt Rigor was different for different students / tasks – but all agreed – it was just beyond the reach – some struggle may have to take place – not knowing something but making connections to what we can already do.

Using Etymology…where do these words come from?

Rigor “to be stiff” – does that mean, we may need to loosen up before we can actually accomplish a task? learn something?  It may not be accomplished on our first try???

Relevance – congruity, appropriateness, agreement, pertinent to the matter at hand – okay I get that – are the tasks I’m choosing meaningful and do they lead to the standard I have chosen?

The presenter mentioned a professor from graduate school – I shuttered – it was the same professor in my very first education class. I even shuttered aloud – uhhhhhhh and the presenter laughed.  Not good memories – I struggled in that course. It was not easy. But as I sat throughout the morning – I had vivid memories of lectures, activities and discussions. I remembered specific things from an Intro to Ed, Monday night-class 20+ years ago. Ask me about other Ed courses – I cannot say much. My last course before my semester of student teaching – we discussed a lot – student lead learning, research. But beyond those 2 courses in undergraduate, very little is recalled. But in both of the courses, my professors challenged me to think – they challenged my views / previous knowledge and experiences. I had a voice, they listened, they responded, usually with questions.  I may not have understood it then.  I certainly didn’t appreciate it then, but I now realize, I learned a lot under both professors.

In the past 3 years, I am realizing I am a guide. I am learning to listen to my students’ conversations. I am learning to ask questions beyond factual, yes/no levels. I can at least ask them what made them respond that way OR if they agree/disagree with a classmate.

Our discussions on rigor/relevance have really gotten me to thinking…wondering how I can improve, make learning worthwhile for my students.

A tweet from @AmberDCaldwell earlier this evening really resonated with me and my struggle to convince students,

I need all my students to read this! Student regrets getting high grades. A must read!! An A+ Student Regrets His Grades…

Again, a post from Emergent Math last month The Struggle for Productive Struggle – take time to read/listen to the NPR link he provides.

As I look through tasks from MARS, PARCC, Balanced Assessment, Illustrative Math – its obvious a classroom of 2 examples, practice these, check, quiz and move on to the next concept is not a prescription for success. I have wonderful students – but I don’t want them to do/think because I said so.

I want them to be able to think on their own, to feel challenged – yet without feeling a need to give up. I want them to feel comfortable asking questions, sharing their thoughts / ideas – even acknowledging their mistakes.
I want them to be able to listen to others’ ideas – and decide for themselves if they agree or not.
I want them to notice cool patterns in math – that its not just a bunch of worksheets and unrelated problems in a textbook.
I want them to recall things we discussed prior to this lesson because they developed an understanding deeper than just the surface.
I want them to be able to make those connections on their own (me only as a guide) and move forward.
I want them to be able to learn/be productive without me telling them when, what and how to do so.
I want them to value success because they’ve worked hard to earn it.

I want them to value learning – and realize “they are not defined by their GPA.”

Representing Polynomials FAL & Open Card Sorts


After an assessment last week, it seemed to me what I was doing wasn’t sticking for my students with polynomials.  So let’s just scrap plan A.  Plan B – I pulled out my Discovering Algebra book, came up with a box-building data collection that lead into the FAL I have linked  below.

Formative Assessment Lesson – Representing Polynomials

Thursday, students were given a 16 x 20 piece of grid paper and asked to cut out square corners and create a box with the largest volume possible.  We combined our data as a class.  Recording the corner size removed, length, width and height.  Students were asked to observe the data and respond I notice…  & I wonder… and that’s where our class began on Tuesday.

We shared out our responses, some adding ideas as we continued the discussion.  Work with our data on TI84s – we saw a connection between our constraints 0, 8, 10 and the graph of the regression equation.  This was not new, during the discussion, a question was brought up about what values would result in a volume of zero.  Students were able answer that with confidence and a reasonable explanation.

The FAL pre-assessment confirmed my students weren’t quite ready for the full blown lesson.  With discussion of rigor and relevance the past few days, I wanted to offer students something engaging but not so over their head, it was a flop.

I backed up and did a bit of prep work yesterday – with the following discussions in class:

FALreppoly FALreppoly2

Following with whiteboards / sharing for this slide from FAL:  FALreppoly3

and a simple practice set to ensure they were on track.   FALreppoly4


We began class today with a quick check of the 6 practice – with a focus on similarities / differences.  Noting the double root of #5.

Prior to the actual FAL, I decided to use the same equations and graphs they were to match during the FAL, except I would have them do a card sort.  Originally, I had planned to ask them to sort cards into 2 groups.  While pondering how I could make it better, I recalled a colleague sharing ideas about open card sorts from a John Antonetti training she had attended.  So, this is what I did.

I told students I wanted them to sort the 11 equations – any way they wanted – they just needed to be able to share out their reasoning behind their choices.  After a few moments, I called on different groups and we looked at their sorts.  I should have snapped pics / documented their responses.  I was amazed – not that they did it – but how well they did it.   The things they were looking at – were much better than my original idea to sort in to 2 groups.  Students were asking students – why they put one in one group instead of another. Pausing after we had the cards sorted on the board – giving other opportunity to look others’ groups…some were obvious, others were not.   I even had groups who had the exact same sorts, but with completely different reasoning.  Wow.



At some point we began talking about “What does that tell us about the graph?”  Almost everyone was engaged and comments added to the discussion.  Next we went on to the graphs to sort.  Again, any way they wanted…just be ready to share reasons.


Most of the sorts were better than ANYTHING I would have suggested.  My eyes were opened – I could see their thinking.  And others did as well – it was obvious in the eye brows raised and head nods.  In both classes, there was one equation that never seemed to “fit in” the other sorts – but students were confident suggesting it belonged to a particular graph (& they were correct).

When I realized the sharing took more time than I had planned – I ran copies of the equations and graphs to send home with students and asked them to match on their own.  My plan is to put them back in their pairs for the actual pairing of the FAL.  They also had blank graphs for any without a match.


I learned so much listening to my students today…  I am looking forward to the assessment of this standard.

I didn’t feel like I taught anything today…

…but I did feel like my students left with a better understanding…because I chose to step aside and give them the opportunity to share their thinking…

It was a great day.



Lesson Planner Resource #made4math


While trying to catch up on my reader – I ran across Simplifying Radicals post on using Google Docs to create lesson plans.  She had an update at the top and suggested reading a comment made by another reader.

I briefly went over to Common Curriculum to check it out.  I’ve created an account, watched a few of the videos, looked through suggestions by other users and played around with lesson plans.  I like it.  I think I will eventually like it a lot.

Basically, you set up your schedule – what you’re teaching; edit a template – creating category “planning boxes” that you will use often.  You can add / delete any given day.  Within the template – hover over the settings for the box and choose “Show on Class Website” for items you wish for students to have access to on the class website it automatically creates for you.  The standards box is automatically included – you can search either Math or ELA CCSS by keywords or standard #.





Uploading files / linking to online resources / videos is VERY simple.  Currently files are quickly loaded from your computer, google drive, evernote, box and it seems dropbox will be added soon.

I created 2 separate resources boxes in my planner – Student – to include links to online resources / files for students to have access to on the class website; and another Teacher – to link resources I need.   This is one I just experimented – a drag and drop.  It displays the photo or file name.boxmax


I attempt to keep a class blog – but sometimes get behind keeping assignments / resources up to date.  What I think I’m going to LOVE about this site – as I’ve already mentioned, you can choose which planning boxes you want to show on the class website – then either give students site address OR post link on current blog, etc.

Website View

Website View

Excited about this new resource – Thanks to Debbie Hurtado for posting on Simplifying Radicals for the heads up!


See, Think, Wonder #makthinkvis


For our next Making Thinking Visible chat, we were asked to read Chapter 3 and implement the first routine presented – See Think Wonder (STW) pg 55.  I realized late Wednesday evening students were scheduled off for a staff PD day on Friday.  I scrambled wondering how I could incorporate this strategy in a meaningful way.  We had worked with parallel lines / transversals and the angle relationships created.  My goal was for students to look for ways to prove lines parallel.  How could I use STW to get this accomplished?

When I searched for images of parallel lines in architecture, I ran across a picture of a building in Australia and a picture of the Illusion as well.  You can find more here Cafe Wall Illusion.

My plan was to use the optical illusion – the placement of the black and white blocks causes one to think the lines are getting closer / farther apart.  However, as I flipped through my book, I saw the routine of Zoom In and wondered if I could combine the 2 somehow.  And here is what I did:

Zoom In – Ask Students what they see, pretty standard – black rectangle.  So many ideas (some silly) of what this could actually be part of…


Slide 2 was a little more interesting, alternating black / white rectangles with several things they thought it could be a part of – keyboard, referee’s shirt, prisoner uniform, zebra…


Slide 3 eliminated some of their predictions…I did have a person actually state a building. Hmm.  I think they must have seen it before.


When I revealed the final picture – it was fun listening to their comments.  One was very perplexed “Why would anyone want their building to look that way?”  It is found in Melbourne, Australia.


After a few moments of sharing / discussion – students were comfortable.  As I shared with students that we were going to do a thinking routine called See Think Wonder – I tried to explain each step.  This is the slide I shared with students:


I went through each step, allowing time for students to record what they saw, what they thought and anything they wonder (a question they could investigate/answer).    We then shared our responses.  After the first 2 statements, I paused and revisited what we were to do for each step.  We agreed the statements would actually go to “Think.”  Here were responses:


After sharing, students were given a copy of the Cafe Wall Illusion – but not allowed to use rulers/protractors to measure anything.  You can see from the snapshots, several chose to use patty paper.

Student A traced the lines to show they were actually straight, then translated the copied lines over the originals to show they were parallel.


Student B traced the edges of the rectangles, then translated to different levels to show the lines were equidistant at all parts, thus parallel.


Student C over-layed tape, traced edges at 2 different levels, then peeled the tape and matched them up…IMG01093

One student used the pink line on the notebook paper and overlaid it to show the lines were actually straight and several traced the rectangles onto patty paper and translated to others to show congruence.

It felt a bit contrived – I’m not sure what level of thinking was achieved, but I will use See Think Wonder again.  It was a good start to model the 3 steps of the routine.  Following the activity, students could be asked – if I only had 2 lines – how could I prove they are parallel?

In discussion some responses:

  • to extend the lines to see if they ever intersect (student knows the definition);
  • measure the distant between the lines at different points (again, student understands they are equidistant;
  • draw a line perpendicular to one line, extend it, if its perpendicular to the other line, then the 2 lines are parallel (yep a student came up with that one)
  • and finally, cut both lines with a transversal, measure/compare the angles to see if the relationships exist (the understand converses/working backwards to prove).

What I appreciated about STW – I didn’t tell students what question to answer or even how to answer it.  They created their own question and chose a way to answer it.  The only problem with this – they may not wonder/choose “the question” I’m wanted them to investigate/answer.  In the end, if you can get students to make a connection with the content, give them opportunities to notice/wonder, allow them to come up with their own questions – they’ll be interested in finding the answer…

Linear Equations Card Match #made4math


Let me first say – I did NOT create this set of cards.  I received them in a session at KCM about 3 years ago.  Kudos to whomever they belong. 

 I was looking for resources to use during my RTI and ran across a box I had used in the past. 

LinearEquationsMatch – the file of the cards.

You can do several different sorts with them.  POINTS-SLOPE, POINTS-EQUATION, GRAPHS-EQUATIONS, etc.

2013-01-14 11.08.59

I have each complete set on different colors of cardstock, so I can have several sets out at once, but none of them get shuffled.

Midpoint – on a different day than Distance


In years past, I’ve usually taught Midpoint and Distance on the same day or at least on consecutive days.  After a reminder of some brain research last fall – how our brains store information by similarities but retrieves information by differences – I decided to try things in split them up this semester – hoping to lessen the confusion students often face (do I add or subtract with midpoint/distance formulas?).  Again, this confusion stems from teaching a procedure without paying close attention to in-depth student understanding.

I chose to introduce “Midpoint with Coordinates” the same day we were working with segments, bisectors, midpoints of segments.  No bells/whistles here – just the basics


I gave students a grid index card and the points A(2, 1) B(8, 11) and C(8, 1) to attach to their INB RPH.  Simply starting with locating the midpoint of the AC and BC.  But also asking them to compare/contrast the coordinates of ACE and BCF each time.

Finally, asking them to locate G, the midpoint of AB.  Walking around the room, it was quite fun watching the various strategies.  The great thing was asking students to share their different strategies.  One used rise/run, several “counted diagonals” from A and B until they got to the middle, one used the midpoints of AC and BC and traced up from E / over from F until he found where G was located.  After discussing methods using the graph, a student stated “I just added my x’s then divide by 2 and added my y’s then divide by 2.”  When discussing how the coordinates were alike/different, a student asked “Isn’t that, what C____ did? Just averaging the x’s and averaging the y’s?”

So, I never actually gave them the “Midpoint Formula.”  Awesome.  Of course, we went on to practice the skill a few times.  I also chose 8 questions from Key Curriculum’s Discovering Geometry (did I mention, I *LOVE* this book?!?!? And have since the mid-90’s!)  – that required a  bit more thinking beyond skill/drill.  Two questions that led to some great discussion today was:

Find two points on segment AB that divide the segment into three congruent parts.  A(0,0) and B(9,6).  Explain your method.

Describe a way to find points that divide a segment into fourths.

But in class, I offered another – what about if I need to divide it into fifths?  Students worked individually, pair-share – then class discussion.  Quite different approaches.  I loved it.

What was even better, a student asked, “But  the examples we’ve used all have an end point at the origin.  Will it still work if the endpoint is not at the origin?”  Aaahhhhhhhhhhhh! That’s music to my ears!  Wow. Wow.  I love it.  I love it.  I love it.

This is a nice little open question to share with your students.  It definitely allowed me to see student understanding of the task by their work / responses / discussion.

Developing Definitions


I’m back!  Nearly 2 months? Yikes. Some fellow teachers on Twitter were committing to blogging once each week.  I think  that’s reasonable – besides, usually my best reflection comes during the moments I blog.  Reflection – seems to be the first thing I push aside when I just don’t have the time.  Yet, its the most valuable use of my time.

I’m sharing some successes from Kagan Geometry (one of my favorites by the way).

I was going to be out for a number of days due to being seated on the jury for a trial (give me 100+ teenagers over the courtroom anyday!).  I wanted to leave something productive.  I did short videos (<10 minutes) filling out certain pages in the INBs in addition to other activities.

The first Kagan activity was for vocabulary.  Each strip of paper included examples and counter-examples for each term.  I modified from the round-table recording it suggested.  Students were asked to pair up (a new partner for each new term) and develop their own definitions.  I loved it simply because most were terms students had previously been exposed to in middle school.

When I returned to the classroom, I ran through all I had left during my absences to address any concerns/questions.  Several students commneted how they liked (appreciated) doing the definitions this way.  Their comments ranged from – ‘You actually had to think about the terms; Talking with someone about it really helped you process what it was before writing it down;  The pictures of examples / nonexamples really helped understand the word better.’

Yesterday, we developed more definitions about angles.  When I told them what we were doing – they were excited about the activity.  Listening to the conversations – I was very happy with their discussion / questions / specifics they included in their definitions.

I remember several times in the past doing examples / non-examples, especially when using Frayer Models.  I believe taking it out of my hands/mouth and giving them the opportunity to work in pairs really enhanced their understanding of the terms.  Even when discussing HW  today – they used appropriately terminology.  Yeah!

Another Kagan activity I used as a LHP activity

from Kagan Geometry

from Kagan Geometry

– very similar to Everybody Is a Genius’  Blind Draw.  Students were placed into small groups and given 12 cards with written directions.  Person 1 chose a card, read the directions, gave others time to think and draw a diagram with labels.  The reader confirms/coaches/praises others’ work.  A new person chose a new card and the rounds continued until all cards had been used.  One thing I appreciated about this – another card asked students to draw a ray from E through M.  This allowed students to realize differences in very similar diagrams.

Again, when I returned to the classroom, students shared how this activity was different from anything they’d done before, saying it was both challenging but helpful in that it helped to clarify certain misconceptions they had; especially with labeling the diagrams.

I have learned the Kagan strategies help students develop and process concepts.  There are “game like” activities where students must find their match and discuss.  Visual, Auditory, Kinesthetic – something for everyone.  Its not an end all – be all resource.  But the amount of HW / practice is minimal when I’ve used these strategies correctly.  I am a firm believer that they help start a strong foundation to build upon.  Hey – if students are smiling and laughing while “doing definitions” – its gotta be good.