Category Archives: Small Group Investigations

Radical Rummy


I received this file about 5 years ago at KCTM in Bowling Green.  Kari from WKU shared it.  I apologize I cannot remember her last name to give credit.


She actually used it to play a card game style activity.  I copied sets onto different colored cardstock and laminated, I have enough sets we usually do groups of 3 people.

I do this activity along with Go Fish for simplifying radicals. 

There are four different forms of each value.  Students use calculators to match cards with same value.  We create a poster as a whole class.  Then notice and wonder. 

I like how students develop their own understanding of rational exponents, negative exponents and radical forms.  It’s a great intro activity.


Card Tossing & Spiraling Curriculum #tmc14


Awesome session Mary and Alex!  Thank you. Thank you. Thank you.

The session focused on their experiences with Grade 10 Applied students ( Canada).  The entire course is activity based which allows students to not miss out on big ideas as they would in a traditional unit by unit aligned course.
Students have repeated opportunities to experience big ideas. The tasks are rich  with multiple entry points and different approaches to solving.  It’s a collaborative environment with accountable talk.  There are fewer disciplinary issues with increased engagement.

Each 6 weeks a mini – exam over entire course up to that point takes place.  Questions are in context and tied to activities they have completed.

We began with beads and pennies on our desks and this task… Cole has 2 smarties and 3 juju bed for $.18 while Noah has 4 smarties and 2 juju be for $.20.  They shared that systems are presented this way – no algebraic forms- for the first several weeks of class.  I, personally, can see how effective this strategy could be.

The next activity shared was Sum of Squares (he doesn’t refer to it as Pythagoras Theorem, yet – or did he say ever?)

Students are asked to cut all squares from side length 1 to side length 26.  Each square is labeled with side length, perimeter, area.  Then they build with them.

Basically students explore and eventually they focus on triangles formed with question, are there 3 you cannot make a triangle with?   Which combinations form different types of triangles. Begin looking at 3-4-5 triangle families, similar triangles (Kate suggested dilations here), discuss opposite side and adjacent sides, then give them a TRIG table and allow them to figure it out.

Compare side lengths with perimeter, or side length with areas.  The possibilities of math concepts are endless.
We ended the day with Card Tossing by collecting data, then using rates to make some predictions.

Video of Alex & Nathan picture below is only a screenshot.


@AlexOverwijk downed by @nathankraft 75 to 72

Each person in the room completed several trials of tossing our cards for 20 seconds.  We found our average rate of success, then determined who we thought might beat King Card Tosser.

Alex asked us to predict how long they needed to toss if he gave Nathan a 35 (?) card advantage so it would be super close and exciting.  Our prediction 38 seconds about 75 cards. Many ways of making the predictions were possible. Not to shabby, huh?

This task was fun, exciting, engaging.  Definitely on the to-do list.

This approach is definitely something I would like to consider, if administration will allow it!

Get to Know Your Students pt 2 #julychallenge Post 12


 Get to know your students, especially how they learn and think.

Taking my lead from this post, my intent is to consider how I can improve or implement the 14 ways discussed.  In my last post, I shared how important I feel it is to know our students as real people.  This one is to share #5things that impacted my classroom and helped me know how my students learn and think.

My 3 years with Kentucky Leadership Network and my experiences with #MTBoS have changed my mindset.  The work with KLN introduced me to a new set ideas and #MTBoS allowed me to explore with others and develop a new frame of reference as I seek to grow as an effective educator.

I cannot be grateful enough to all those who have challenged me and help me grow.  But as I think of the experiences that have opened my eyes to see better ways I can consider my students as learners, these are the ones that first come to my mind.  #5things for getting to know how my students think and learn…

Wait Time II
I learned about this routine from 75 Practical Strategies for Linking Assessment, Instruction and Learning (Keely, Tobey 2011).  A simple adjustment.  Yet it forced me to really listen to my students.  You can read more on a previous post, here.  Basically, it allows  the students AND teacher to process a student response.  We were all told in undergrad to wait 3 seconds after asking a question before calling on a student.  Some people actually think this deters the class flow.  I disagree. The idea with Wait Time II is to wait again, after the student response.  It allows the responder to consider what they said, the classmates to process what was said and the teacher to consider next steps, questions, etc.  A bit uncomfortable in the beginning, but once I explained the rationale to them, they got it, as did I.  Waiting and listening adds value to what students are saying.

What Makes You Say That?
Making Thinking Visible, (Ritchhart, Church, Morrison, 2011)
A chat with Liz Durkin challenged me to consider ways I could implement these routines into my high school math classroom.  It was the question “What makes you say that?”  that helped me begin drawing out student thinking.  What were they seeing? What evidence supported their statement?  With this routine, I began learning new ways of seeing problems myself.  Students’ ideas, strategies and approaches are way more intuitive than my own.

Notice and Wonder
I was first introduced to Notice & Wonder with Max Ray’s Ignite talk sharing The Math Forum’s simple, yet impactful strategy.  You can read more in Powerful Problem Solving (2013) as well.  When I pose a problem, scenario, graph, students may not readily know where to start.  But they can tell me what they notice.  Its a starting point.  Everyone can share something.  When we listen to what others are saying, that ignites other ideas as well.  And they begin sharing their “I wonders” which are great transitions to explore more.  Its great.  Its simple.

This routine carries over to standardized tests as well.  Students shared how they didn’t know how to approach certain problems on ACT or their EOCs, but they looked at it, thought about what they noticed, connected it to something they knew and was able to at least make an educated guess. 

Friendly Class Starters
After reading What’s Math Got to Do with It? and completing the Jo Boaler How to Learn Math course last summer, I knew I needed to find ways to invite students to think differently about math in my classroom.  Some major a-ha’s and sad realizations as to why so many kids are down on math.  I began with things like Number Talks she presnted during one session.  Amazing how many different ways students can see / approach a single problem.  When I invited them to share their thinking, they owned the math.  This past year, I implemented Counting Circles, Estimation 180, Visual Patterns as well.  These resources were primarily used as bell ringers to get students in math mode. However, there were days it lead to deeper, richer discussions and I was flexible enough to go with it.  My students’ confidence began to grow.  Their number sense was developing.  They were sharing their reasoning without me asking them to.  I saw some big gains on benchmarking and standardized testing for several students and I attribute them to these “friendly” and accessible resources.

Small Groups and Discussions
When I completed my initial National Board Certification in 2002, I quickly realized small group discussions provided a definite means to seeing student thinking.  It was a chat last summer, that made me realize I needed to quit butting-in.  I would hear a misconception and jump to add my 2 cents rather than allowing them to reason out if they were correct or needed to adjust.  I was stealing their learning opportunities! Yikes.  I began listening more-offering questions rather than telling them the direction they should go.  It was frustrsting for some students.  They despised me answering their questions with questions.

5 Practices for Orchestrating Productive Mathematics Discussions (Smith & Stein, 2011) is a quick read that offers samples to incorporate into your classroom. The 5 practice provide structure to help you develop discussion based tasks rather than step-by-step inquiry lessons.

Another valuable resource for me are the Formative Assessment Lessons provided by Mathematics Assessment Project.  Most lessons follow a similar format to the #5pracs.  I used to struggle offering questions that would move learners forward.  Though some disagree with scripted lessons, this resource supported me with sample questions for specific student misconceptions.  As a rssult, I began asking better questions on my own.

Another aspect of the FALs is the way they suggest grouping students, not by ability, but similar thinking – whether it be similar misconceptions or approaches to a problem.  This supports what I have been reading this summer with Ilana Horn’s Strength in Numbers (2012).  She presents how social status in the classroom may actually hinder student learning and achievment.  I believe grouping students homogenously by approach and thinking puts them on equal playing fields to share and build their ideas. 

By observing student responses and listening to their discussion, I am able to select and sequence ideas for them to share that will allow more engagement from the class as a whole.  Students are able to listen and view strategies similar to their own, but also consider new approaches which in turn builds their own skill set and toolbox for thinking.


The common thread is to not to do all of the talking, but to sincerely listen to my students and their thinking.

Evaluating Statements About Length and Area


This lesson can be found same as title of the post.


This is one of six cards students discussed within small groups today. A student stated, “this is going to be a thinking day,” as they began removing the clips to start reviewing their cards. Most students would quickly come up with an always, sometimes or never true. However, to create their own examples or counterexamples to either justify or refute the statements was a struggle for some of them. Several groups had similar statements for this particular card. It was when a student asked, “do they have to be triangles?” that a turning point came for some.


Within our share out as a whole group, a student shared examples of reducing area, same perimeter and less perimeter. A question they wondered…can you reduce the area but increase the perimeter?

I really enjoy days like this, students are giving me the information, I am their scribe and I am slowly learning to let them determine if they agree or disagree with each others’ claims. I’m not even sure where the key is, that way I am actively having to listen to their arguments to determine if I agree or not. (Shout out to Max @Math Forum, I am listening to my students, not listening for the answer!) I go through the cards myself prior to the day of the lesson, just like I require them to do. But I am still closed minded in my own thinking at times. Why would you limit the example above to only triangles? Because that is what shape was presented on the card. However, does it state triangles only? Nope.

A task like this may drive some teachers crazy. Once you start considering different shapes, you begin to see what works for one, may not work for another. I had students cutting scrap paper, tracing patty paper, measuring side lengths…without me telling them to do it.

The classic question, a square and circle have equal perimeters, which has the larger area? I will do my best to share more reflections as we wind up tomorrow, if we wind up tomorrow…depending on their questions, discussions, claims and supporting evidence.

Quadrilateral Diagonals Properties


Over spring break, I was surfing online resources, searching for ideas and suggestions on how to plan and be more purposeful with the Mathematical Standards, which I have realized this year just how key these are to the success of CCSS. As I looked through Inside Mathematics , I ran across some PD training materials. I watched clips from Cathy Humphrey’s class. The Kite Task, an investigation of quadrilateral properties from seemed like a great activity to ease back on day 1 when we returned.

The task in short is for a kite company, who wishes to launch a new line of kites consisting of all types of qudrilaterals. The students are asked to devise a plan for how to cut/assemble the braces for each type of kite. They are only working with the diagonals in the investigation.

Rather than running copies and cutting out, I used my paper cutter to cut 1″ strips one color card-stock lengthwise and 1″strips width wise of a different collor (I didn’t realize how helpful this would be until later on). I created a strip to use as a guide on each strip, placed 7 holes equally spaced. Odd amount is best since they will be looking at bisectors some.
Each student would receive 2 of one color and 1 of another color.

Here are some snapshots of possible braces built.

For anyone who is having trouble visualizing, I’ve added some “sides” to the diagonals:

As we began the 2nd day of class, a few groups needed just a bit more time to wrap up their investigation. Using fist to five, I asked how many they still needed to determine. Most groups only 2 or 3, so I set the timer to keep us on track. I love days like this to walk around and just listen.

As I was questioning one of the groups, trying to ensure an absent student was on track, I asked the group’s members to “fill an order” – pick 2 sticks and construct the diagonals needed to brace…kite that was a rhombus, then another shape, etc to quiz them for understanding. AHA! Why couldn’t I use this as a formative assessment for the entire class?!?! Perfect.

When all groups had completed and debriefed a bit, I placed orders for kites and the students had to build the braces and pop up to show me for a quick assessment.


These pics were actually a geometrically defined kite. If you look closely, you can see a few wrong repsonses. To address these, I used extra sets of sticks to build a correct example and an incorrect example. To ask for suggestions why one was and the other was not correct. Why was one example actually a rhombus, allowing them to really compare/contrast the two figures.

Another great mistake I saw…when asked to create a rectangle, the top sketch is what I saw from about 6 students. Of course, my initial thought was, they dont understand the diagonals must be congruent.

Then I saw a student trace their shape in the air…second sketch. I literally saw their thinking. They had not used the sticks as diagonals. Clarified and corrected!

A post-it note quiz today, I built the braces, they had to tell me the quadrilateral name. A stop-light self assess, revealed most were confident, of the 10 yellows, 7 got all parts correct. The others missed 1, 2 or 3. All green students had each part correct.

We did a little speed dating to use properties to solve problems. As I listened to their approaches, most everyone seemed on track. Overall, I was very pleased with the results of the lesson.

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.



Formative Assessment Lessons


Its been 3 weeks since I’ve blogged.  Not because I didn’t want to – but life has just been head over heels busy.  The week following my last entry – I presented at KCTM – Literacy in Math Class.  I’ll blog about it soon.

In Kentucky, I was introduced to Formative Assessment Lessons about a year and a half ago.  I remember the first one I tried was not so successful.  But the more I learned, the more I realized, there was some good things embedded within these lessons.   At our last KLN meeting, we were asked to discuss our experiences with the FALs.  I hadn’t realized I had actually used as many as I have until we started running through the list.

My students can find some level of success as well as being challenged on the other end.  I observe student success with these lessons.  They are formatted in such a way, I am able to listen to student discussions, considering their ideas and able to pose questions that will foster more discussions.

Part of my session on literacy was to give students opportunities to talk, share and ask questions about their thinking.  Within the FALs, students are given either a problem solving task OR a conceptual development task.

In all lessons I’ve used, students respond to a given task as a pre-assessment, after completing the lesson, class discussions, they are given the opportunity to revisit the same or a similar task.

In the problem solving tasks, students are put into groups homogenously (based on similar approaches to solving a problem or even similar misconceptions/mistakes – not necessarily ability).  This allows students moving in the right direction to continue; while my time can be targeted to smaller groups of students, using questioning to guide their thinking, discussions.  Each group is given sample responses, and asked to think about the student’s reasoning – why they approached the problem as they did.  This gives the group an overview to see multiple ways to consider and opportuinties to critique the reasoning of others.

In the concept development tasks, students are usually given a task/questions and card sorts/matching activities.  Instructions will almost always require students to verbalize their reasoning, then their partners must either explain the reasoning in their own words OR why they disagree with their partner.  I feel verbalizing their thinking is a key component of literacy – helping them work through their own understanding but also listening to ideas of others, in a small group setting.  Many lessons offer extension suggestions if needed.

To complete the lesson, there is often a plenary discussion to wrap up, solidify the concepts.  Its very important to really listen to students – in some lessons, you are encouraged to scribe student comments/ideas with their names for ownership in the discussion.  White boards are a common component – seeking student responses – sharing different responses – asking questions – if others agree, disagree or have something to add to someone’s comments.

I am sharing about FALs because today, I left my geometry classes feeling good – that students were given an opportunity to think, discuss, share and learn – clear up some misconceptions.  I am looking forward to our whole-class discussion on Monday and the follow-up assessment!  Though there are still some mistakes – I think the sharing out will add/deepen to students’ understanding.

Representing and Combining Transformations was the lesson students worked on today.  I paired students based on similar responses on their pre-assessment.  I really enjoyed “sitting back” and listening to their discussions.  The particular task, they were given 6 different graphs with an L-shape and 8 different transformation cards.  They were asked to connect 2 shape graphs with a card describing the relationship between the two.

I’ll be honest in questioning the need for the transparency graphs – but after observing students, these were a key learning tool for most of them.  When they asked for help, I encouraged them to use their graphs to “see” what happens, then use what they noticed and apply it to their shapes.  I also found allowing students to place a push-pin at the center of a rotation was very beneficial to their understanding.  To observe how using different centers of rotation will affect the movement of the shape.

Recently, a colleague decided to try a FAL – Forming Quadratics with an Algebra II class.  In our last PLC, my colleague shared pros/cons observed during the lesson and that all but only a couple of students had improved / were very successful on the post-assessment.

FALs are idealy used about 2/3 the way through a corresponding unit of study.  This allows the teacher to view misconceptions and clear those up before finishing the unit.  Most lessons consist of a 10-15 minute pre-assessment, 1 hour for lesson/discussion (this can vary depending on students), 10-15 minute follow-up assessment.

Each lesson is aligned to 8 Mathematical Practices and outlines which CCS is addressed.

There is some prep-work involved, so don’t print a FAL and expect to use it immediately.  I use card-stock for the card sorts (each type of card gets its own color) – if you laminate them, maybe they will last longer.  Also, when it calls for a poster of student work, I don’t want them to glue pieces on a poster – then I’ll have to make an entirely new set next time.  I want to reuse them.

  Today, I had students add a post-it note with their initials and I snapped a pic of their cards.  They can create an answer key on paper as well.

I would love to hear about others’ experiences with FALs – ways they’re using them in their classrooms!

Round Robin – Parabola Patterns #made4math 8/6/12


So… since last Monday – I’ve met my new administration.  Finished up final work in my room (148).  Found out I was being moved from the 9th grade wing to grades 10-12.  I no longer am in my comfort zone of Algebra I but will be venturing to the world of Algebra 2 and Geometry.  I emptied my classroom – I have A LOT of stuff – helped my retiring friend remove things from her classroom; my mom helped me clean my new room and finally, I have things in my new room.  There’s still a lot of organizing to go – but at least its presentable (somewhat anyway).

I haven’t had time to sit and even read through the standards for Geometry & Algebra 2 until today – while sitting in our district wide PD.  The highlight of my day – I not only had the highest attendance percentage in my building – but in the entire district…yep, that was kind of cool – especially receiving $200 for classroom materials (will be following @troystein’s suggestions on some tools for flipping instruction!)  – it may not get me ALL that I want / need – but it is a starting spot.  I want to believe it was because the kiddos loved the math – but the truth is – I had an amazing group of students who simply enjoy school – so I really could take no credit in it.

So, I am very much cheating on #Made4Math today (sorry @druinok for the late post) – I am just linking an activity I have really found success with through the years.  As I sit here, typing – it feels like I’ve shared this before – so I apologize if I have – its the best I can do this week.

I have used Round Robins with groups from size 3 – 8 students.  Each student places their name on the page – to ensure they get it back at the end of the activity.  Everyone in the group follows the instructions in the first box.  Pass the page to the person to their left, and everyone will move down one box and follow the instructions given.  Again, pass the page to the person to their left, move down one box and complete the given instructions.  Students will continue this format until they have completely filled in the “worksheet.”

Because the worksheets are designed with a variety of examples / number values – you may not be working with the same problem as the one you just completed, so everyone else’s work / responses relies on your response – its imperative you give your best effort.  I see students work harder on this activity than others – because their work matters to the next person.  Students have to stay on task – since others in the circle will have to wait on them if they are off task.

After the group has completed their pages – they are passed back to the owner.  Within the group – they must discuss patterns they recognized and try to develop “rules” they can use in future / similar problems.

Parabola Patterns Round Robin / Jigsaw -Addresses transformations on quadratic functions – vertical & horizontal shifts; narrow/wide & reflection; mutliple changes.   This is by no means my idea – I simply typed up a new set of problems.  These can be easily modified for any function transformations.  The original activities I received came out of Jefferson County Public Schools in Kentucky.

If you modify to another concept – please share your other ideas!

Geo-board Investigations


I was clearing out some files this weekend and ran across this packet from a presentation at KCTM in 2002.  I had just completed my initial National Board Certification earlier that spring (still didn’t know if I had certified yet) and thought these lessons were worth sharing.

I’m not sure if you’ll be able to read the first two pages – orginal files are long gone and just by happenstance I rance across this packet.  Reading through it – its almost like I was “blogging” 10 years ago – but it reminds how important reflection on your lesson will always be – how much you can learn about teaching by pausing to think about student thinking/responses.  Whether you use actual geo-boards, paper/pencil or modify to – maybe they will give you some ideas for your classroom.

Geo-board Investigations

  • Parallel & Perpendicular Investigation – use rectangle properties to find relationship with slopes
  • Amusement Park – distance between 2 points (I hate using distance formula and often allow students to find slope triangle, then apply Pythagorean Theorem)
  • Midpoint Investigation
  • Midsegment Investigation

*I used the reinforcement tabs for students to write coordinates/label points on geo-boards.  BUT don’t let them peel and stick…just leave on paper and drop over the geo-board tab.