Thursday, June 23, 2016

When Problem Solving, Precision, and Computation Meet: 5 Ideas to Ponder

I think it would be hard to find someone who would argue with the following statement:

All students need to learn how 
to compute efficiently and precisely.

I mean--I sure can't argue with it!  I think the discussion comes in when we talk about how we expect our students to get there. Take these practice pages for example.  I use them.  In fact, they are part of two different resources in my TpT store!  But what purpose do they have?  WHEN do I use them?
Want to see these resources?  Click HERE and HERE.
Here's what I believe to be true.

1.  Computation fluency comes as as result of carefully planned lessons that help immerse students in hands on, meaning-based learning.  That means students don't learn how to "cross out the top number and take away the bottom" unless they have spent a great deal of time learning how our base 10 system we can "trade" amounts from place to place, and so on.  I have heard the term "bare number" problems in reference to problems that are presented in numeric form with no "meaning" behind them, and I think we need to be mindful of how early we introduce these bare number tasks.  If you are familiar with CGI math (Here is a post about it if you have not!), you will know that CGI is based on the assumption that students can solve increasingly complex problems when presented in the form of a real world situation.  A child who may not understand the bare number task of 15 + 7 very well may be able to solve a problem (sometimes with physical manipulatives) that is worded in the form of a real problem such as, "Becky has 15 jelly beans.  If her mom gives her 7 more, how many does she have now?"  The simple act of MODELING with counters can help build this understanding.  This is especially true as the numbers get bigger and the math more complex--we need to build solid math understanding before asking students to do practice sheet after practice sheet.

2.  There is a time and a place for practice sheets--and it happens AFTER students know how to do the math!  Practice sheets are great for working on fluency, precision,and organization--but if a child is struggling with the math, handing them a page of six to ten double digit multiplication problems is probably not the best decision.  I also use sheets like this AFTER students know the math to use as an assessment tool.  I also use sheets like that AFTER students know the math as homework to practice that fluency and comprehension.  Because I believe students need to know the math, this may be a time when I don't send the same homework home with all students.  There is nothing more frustrating to a student and his/her family than to get homework home that they don't understand.  

3.  Math is not fill in the blank.  Providing students with the opportunities to explore numbers, play with numbers, and develop their understanding is key.  Doing it alone takes opportunities for learning away.  Consider doing computation work collaboratively.  Students can discuss, coach, share ideas, check for accuracy, and deepen their understanding.  Math doesn't have to be a solitary activity!  One of my favorite things to do is provide the opportunity to have students work together on activities that aren't fill in the blank--AND provides computation practice!  After all, PRECISION is key; I sure don't want my students to understand HOW to add, subtract, multiply, and divide--but not be able to do it precisely time and time again.  

4.  Another great way to combine problem solving and computation is to provide students with problem opportunities that have multiple solutions.  This task card is one example--there are multiple sets of numbers that will work.  Another way to provide this experience would be to provide students with a solution--for example 320--and then ask them to come up with as many different multiplication problems as they can that work.  This is a GREAT way for students to start to look for patterns in math...they may start with 32 x 10--and then start to experiment until they find other solutions.  

5.  Consider differentiation.  Many activities can be done by ALL students at THEIR level.  For example, one thing my students love is to roll dice to create problems where they practice their computation.  I'll often give students 5 minutes of "warm up" time where they grab dice and a partner or trio and "roll problems".  If I want them to do multiplication, some might roll enough dice to do 1 x 2 digit.  Others 1 x 3--or 2 x 2--or 2 x 4.  Similarly, some might use dice labeled 1-6 while others use dice with larger numbers.  Helping students know exactly what their "just right" practice level is takes some coaching but is worth it. (And having a nice big dice bucket is fun too!)  The cards pictured below are another example...there are three levels of cards.  All students in the class could be having the same practice experience--but at their own "just right" level.

Check out these cards and see if you can see how they provide HUGE amounts of computation practice along with much higher level thinking and the opportunity for great math talk with partners or small groups!
Think of the number sense and problem solving a task like this takes compared to "Solve 45 x 62."  Both work on multiplication.  Both expect precision and fluency.  But this is asking the brain to do FAR more math!  (I hate to break it to's also FUN!)  I absolutely love sitting back and watching students work on tasks like this--to hear their thinking, to see their strategies, to notice their misconceptions or areas where I could help them be more efficient.  I challenge you to find ways to give your students these opportunities--times when you can practice computation in a "non-fill-in-the-blank" way!  Sit back and watch and learn and see where to go next.
Earlier in the year I use the same type of cards for addition and subtraction practice...and my students LOVE them.  There are a ton of different ways to use both sets of cards..from intervention, to practice, to whole class challenges--and all meaningful ways to build computational fluency and precision.  Here they are if you want to check them out!  I'd love to know what you think about today's post...and any other ideas you have for raising the level of rigor with regards to computation!

Wednesday, June 22, 2016

What is "Structuring" in math class?

One thing my district has been really giving a lot of attention to in recent years is number sense-especially the concept of "structuring" numbers.  For those of you who aren't familiar with the term, I have shared below a small excerpt from the following article:

Structuring Numbers 1 to 20: Developing Facile Addition and Subtraction David Ellemor-Collins & Robert (Bob) Wright  (Click HERE to read more!)

"In early addition and subtraction in the range 1 to 20, students can progress from using strategies involving counting by ones to using more facile strategies that do not involve counting. Researchers recognise this progression to facile addition and subtraction as critical mathematical learning, yet many low-attaining students do not make the progression successfully. There is a pressing need to understand how low-attaining students can progress to facile addition and subtraction, and to design instruction that facilitates such progress."

What we know to be true is that there is plenty of resource to support explicit teaching of "number"--starting with subitizing, then moving to structuring numbers to 5, 10, and eventually 20.  What IS "structuring", you ask?  Structuring numbers refers to the ability a student has to combine and partition numbers without counting.  When you ask a child to list a "way to make 7", we want them to be able to instantly say, "6 and 1 or 5 and 2 or 4 and 3 or 7 and 0".  If they need to think about it or count, they aren't "facile" with their structuring and this foundation needs to be strengthened.

This may seem like an odd thing to work on in the intermediate grades when our curriculum is filled with fractions and geometry and huge numbers and division...but this understanding that numbers can be broken up into parts and can be combined is the foundation of all of these!  Don't we want our students to understand that 7/8 is the same as 2/8 and 5/8?  Or that 1.1 is 1.0 + 0.1...and 0.9 + 0.2?  Or that a right angle can be made from two angles that are 40 and 50 degrees?  Or that 1,000 is a combination of 400 and 600--and 200 and 800..and so on?  This is what is at the heart of structuring, and students who didn't "get it" with lower numbers are going to struggle with the more complex concepts.  This idea of structuring is the foundation for addition and subtraction and, eventually, algebra concepts so we have to make sure they are securely grounded in the concept.

If you want to see how your students handle this, do a quick interview!  Ask them the following questions...

"Tell me all the ways you know to make 5."

"Tell me all the ways you know to make 10."

"Tell me all the ways you know to make 13."

"Tell me all the ways you know to make 20."

What are you looking for?  Can they do it quickly and do they sense the "pattern"?  I love to notice which students are random with their answers and which students "get it"...that 5 is 5 and 0, 4 and 1, 3 and 2...some might even ask "Do you want me to do their 'other' fact?" or something similar (3 + 2 AND 2 + 3).  It tells you a lot!  Do they need their fingers?  Are they obviously counting in their head?  If they are, they may benefit from some intervention work with structuring.  This is certainly not something we would expect ALL intermediate students to need--this is a concept that should be incorporated into math instruction in the primary grades, but as you and I both know--sometimes students don't get it when it is first presented...for a whole myriad of reasons.

There is WAY more to structuring than I am covering here...but I thought I'd give you a little something to think about--and a place to get started if you want to read more.  If it looks like there is a great deal of interest, I can add in some more posts throughout the summer!
Using bead racks (rekenrek), linking/unifix cubes, or ten frames to show the "ten" and physically manipulating to make one more and one less, using base 10 blocks (usually down the road a bit), and other place value tools can be VERY helpful in letting students explore these ideas and really see and feel how numbers can be split and combined.  If you don't have some of these tools in your intervention box, grab some!  Hmmm...maybe I see another blog post coming up!

I thought I'd share a few games I use to help me when I'm either pulling students for an intervention group or want the students I have worked with to continue to practice.  Students LOVE these games...and they are a great way to EASILY teach a concept and provide a way for practice.  I will even send copies of the game home as replacement homework for these students.  The first one I use with students who are working on structuring to 10...and then I have a second version that goes to 20 when they are ready.  Make sure you don't rush it!  The game is called "Block Out" and the goal is for them to roll dice and look for combinations of numbers to "block out" each number, 1 to 10.  This forces them to really see how numbers can combine. I play it with them a LOT at the beginning to model my thinking.  You might here me say things like, "I got a 1, 3, and 4.  I know that 3 and 1 make four and 2 fours is 8." or "I got a 2, a 2, and a 6.  I know my answer will even because all my choices are even." (I might even ask my "partner" to help me come up with ALL the possible answers for that 1 + 2 is 4, 2 + 6 is 8, 2 + 2 + 6 is 10.  This models for them how they need to be flexible with their thinking!  Once they have really mastered "10", I bring out the manipulatives again and we explore the numbers between 10 and 20.  When they are ready for "bare number" tasks--we play "Block out 20"!
Another great structuring game is "4 in a Row".  This game is PERFECT to use with intervention groups or individual students who do not yet add 9, 10, or 11 fluently.  It can also be available for math workshop or stations for students to build fluency once they are independent.  Helping students “see” this is so important.  Consider using base 10 blocks, linking cubes, bead racks, or other “hands on” manipulatives where students can really see how adding 10 impacts the tens place…and adding 11 impacts both the tens and the ones.  Adding 9 is tricky for some students…showing them how
adding 10 and then “snapping off” a cube to show the “one less”.  Using bare numbers (no visual model or manipulatives) should really only happen after students have used these hands on materials to build their understanding. Too often we ask students to “memorize” their facts before they understand the structuring concepts involved! Use this game to TEACH, then let students continue to play it to practice fluency!

So there is a little background on "structuring"--and I can certainly blog about some of the more "hands on" materials and even some "make your own" ideas if people are interested.  Thanks for stopping by, and let me know if you would be interested in more information about this topic!

Monday, June 20, 2016

Monday Made It! Grad parties and more!

Happy Summer, everyone!

It has been the BUSIEST spring of my life with my baseball playing son finishing his senior year of high school!  I thought I'd share a little bit of what I have been working on in June in my few spare moments so I have hooked up with Tara at 4th Grade Frolics for my first "Monday Made It" of the summer!  Make sure to click the image below to take you to the link up to see all the other great ideas people are sharing this week!

The main thing I have been working on this spring has been survival--surviving the end of the school year (I finished June 9), finishing the baseball season (We finished last Tuesday with a loss in the State Semifinals), finishing my son's senior year/graduation, and FINALLY--his grad party on Saturday.  I thought I'd share a few things I made to make the day as special as I could!  It was a great day...although certainly bittersweet.

The party.  First o all, it seemed silly to spend $9 on a banner when I could make one, right?

Ummmm...several hours later and probably $5 worth of ink and card stock, world's LONGEST congratulations sign was born.
 I hung it on one wall of the garage with a bunch of photos...on the right I used red ribbon and strung all his school/formal pictures...from 3 months to 4K to senior pics! I mounted the pictures on black card stock and used a silver pen to write his age on each.  Everyone loved to see how he has changed in 18 years.  The other candids are some of my favorite shots from the last 18 years--many with friends.  I bought some cheap black mats online and just taped the photos to the back.
 I had to rent tables and chairs because we have none!  I used red and white (his high school AND college colors!) and these funky little dot garlands I found on clearance at Target to make the tables into a baseball theme.  More photos on each table, of course--as well as collages on little easels everywhere.
 My son is a "candy guy" so I made a candy buffet for his eat during the party and then had a stack of take out containers so they could take some "to go" as well!

 I didn't want to buy a cardboard box for his cards so I just tied a ribbon around a basket I already had and used a little chalkboard sign I found in the Dollar Spot.  I can't tie a bow but finally convinced myself that I just had to live with it.

With temperatures pushing 90 degrees and so much food that you should ALL have come, we had a great day.  I am so blessed that my son had an amazing high school experience with great friends and family.  It was quite a celebration.
Because I've been so busy, I haven't had much time for working on school things...but I am trying to get more intervention materials ready for the fall, so I did get one new game ready to roll.  Tomorrow I plan to blog a little about what "structuring" is and how important it is for our students.  This game is meant to help reinforce those concepts so stop back tomorrow and learn more about it!

Thanks as always to Tara for hosting this fun link up...and I am excited to be back blogging a bit more regularly now!  Happy summer to all of you--and for those of you not quite finished yet, the end is in sight!

Tuesday, May 31, 2016

8 Ways to Survive and Thrive at the End of the School Year!

One thing I have learned after teaching for this many years (25 now!) is that the more routines you can maintain at the end of the year--the better.  That doesn't mean that you don't throw a little fun into the game, but most students truly crave pattern and routine--and we need to make sure that we don't let our own end-of-year crazies ruin that structure for our students. I thought I'd offer up a few (ok--8!) tips for keeping things "cool" as the weather heats up!

1.  Check your stress at the door.  I know it's are trying to get through your curriculum, you have report cards looming, you have 18 IEP's and 42 meetings.  None of this is your students' fault--so try to take a deep breath as you walk through your classroom door in the morning and keep that room a sanctuary for your students--and YOU!
2.  Keep things as academic as you can--but toss in a little fun!  As I finish our required math series (and even on some of the days before we finish), I like to throw in extra problem solving--but I like to keep it seasonal and fun!  I use word problems with an "end of year" flavor to keep students interested and offer up fun challenges like my Thinker Task: Amusement Park Challenge to keep students working hard--without realizing they are!  Throw in a fun STEM activity or science lab.  
 ...and even more "Tip 2"!
3.  Keep reading aloud!  As the year wraps up, I actually read aloud to my students MORE often.  Whether it be a great novel (I finish the year with "Wonder" and time it so that our last day of school is Auggie's last day!), or fun picture books--students love to be read to and multiple, short bursts of read alouds can help cure the wiggles and keep students calm.

4.  Connect with your students!  As the days begin to wind down, some students sense the impending separation--and we sometimes even see students acting up more and more.  Some call it spring fever--but I think for many students is the anxiety of the impending change and separation from the place they have spent the better part of 180 days...connecting with you and the other students in the class.  Change is exciting for some students--but can be excruciating for others.  Remember to keep connecting with your students in those last days.  Greet each one as they walk through the door in the morning and wish them well as they leave.  Ask them about their evenings and their sporting games.  Tell them about YOUR evening; help them feel valued and "noticed" until the very end.

5.  Throw a little controlled fun into your day!  Finish math early and give them 20 minutes of game time.  Do a project.  Read outside on the playground equipment.  Use sidewalk chalk and do math review on the pavement.  Buddy up with another class for some partner reading.  Do a team building activity.  Working some of this "controlled fun" keeps students happy--and you sane!  
6.  Be sensitive to students who may be moving over the summer.  Talking a lot about the activities of the next grade in your school (or--in my class--at the next school the students move to for fifth grade) can be heartbreaking for students who are moving and won't be able to enjoy them.  Instead, consider focusing on things that are more universal--like how much more independent they will be in the next grade, how they will be able to read even more exciting books, and so on.  We take a "transition field trip" to visit our new school and it was VERY hard on my students who aren't going to be attending it next year.  I worked hard to try to make those two gentlemen feel special by really bragging up how ready they are for new challenges, and so on.

7.  Let your students work together.  Collaboration is a life skill that needs to be practiced.  Let them do their math work together and try to find errors and correct them.  Let them buddy read.  Let them read their writing to each other and look for ways to revise and edit it.  They WANT to talk--let them do it productively!  Some of the same activities that work at the beginning of the year can also be great at the end of the year with students' new skills and maturity!

8.  Smile.  A lot.  You all deserve it.

Sunday, May 22, 2016

Stress Free Teaching Tips and Giveaway!

I'm so excited for today's post! I know how stressful it can be as a teacher. You work hard all.year.long helping students do their very best! That's why this post is for you! I'm going to share a few tips I've used to help me unwind during the summer months that will hopefully help you out as well. Even better? I'm giving away a great prize as well! But the best part? I've joined some of my other blogging friends, and they are ALSO giving away some fun treats! Read on to learn about my tips for your stress free summer, to enter my giveaway, and to find the links to my friends' giveaways too. :)
So what are some of my stress free summer tips?
Are you a teacher? Are you stressed? Going into the summer months, we want to share a few tips to help you de-stress and unwind this summer. Plus we have a GIVEAWAY! Click through to see how you can win some great items! But act quickly because this deal ends on May 28, 2016.1.  Remember that our "summers off" gives us the chance to recharge at a different pace; we all work throughout the summer--whether we are reading new materials, organizing our digital and physical resources, coming up with new lessons, learning new get the picture.  Remember to pace yourself and do a little bit of work throughout the summer so you don't feel swamped at the start of the school year.

2.  The second part of the first tip--give yourself SOME time completely away from school!  Whether you take the month of July off...or never work after 2 pm...or only work on Mondays and Tuesdays--let yourself recharge for chunks of time throughout the summer.  We have a VERY stressful job and we owe it to our students to come back refreshed.

3.  Challenge yourself to try something new this summer--something NON school related.  Read a book that you have been hearing about.  Try growing tomatoes.  Take an art class or go to a painting night.  Rent a tandem bike.  ANYTHING that is new for you--and find someone fun to share the experience with!

4.  Reconnect.  As the school year goes on, I know my family does less together.  I don't talk with my friends and neighbors as often.  I go into survival mode.  Summer is the time to rebuild those lost connections.  Invite people over for a fire.  Take a walk through the neighborhood.  Slow yourself down.

5.  Treat yourself.  Whether you get yourself a pedicure, buy a new pair of fun sandals, splurge at the bookstore, sleep in until 9:00, or buy that giant treat at the bakery--summer is the time to splurge a little bit!

6.  Make yourself a little bit healthier.  Try yoga.  Reduce your soda intake (GULP!).  Exercise more or differently.  Try to add more vegetables into your diet.  Sleep more.  This "tip" is put in ESPECIALLY for me--because this summer I am setting some major goals in this area!

If we could all take some time to do these six things this summer, I think we will be a little more prepared and "centered" as we tackle our next group of students so we can give them our very best!

Whether you are still teaching and could use some last minute resources--or are already done and want to start thinking ahead for next year, who couldn't use a Teachers Pay Teachers gift card and some free resources from my store?  While I'm not asking you to sign up to follow me in any way, I certainly would appreciate you following me in any of the ways listed here: Newsletter (you get a FREEBIE when you sign up!), Pinterest, Teachers Pay Teachers, Facebook, Instagram, Twitter, Google+. I appreciate the follow, but remember - for this giveaway it is not necessary.  Check out the links in the upper corner of my blog or check the rafflecopter for ways to enter.  That's it! Please enter the rafflecopter below for your chance to win.  
a Rafflecopter giveaway

  As awesome as my giveaway is, I'm excited to share that my fellow blogging buddies are doing their own stress free teacher giveaways too! Click on each link below to check them out. 

  Amy from Teaching in Blue Jeans 
  Heather from HoJo's Teaching Adventures 
  Jennie from JD's Rockin Readers 
  Jodi from Clutter Free Classroom Kalena from Teaching Made Practical 
  Meg from The Teacher Studio 
  Tammy from The Owl Teacher   

Sunday, May 15, 2016

Increasing Independence with Book Clubs!

Check out my post over at Upper Elementary Snapshots today!  Check out some of my best ideas for planning and implementing book clubs with your intermediate students.  I'd love to hear what you think--and to hear YOUR ideas about using book clubs!   Just click the picture above and check out today's post!

Tuesday, May 10, 2016

Decimal Sequencing and Critiquing Reasoning

"A well-crafted argument/critique requires a thoughtful and logical progression of mathematically sound statements and supporting evidence."
Pretty powerful words when you think about it, right?  Let's look at a few of the phrases:


thoughtful and logical

mathematically sound

supporting evidence

Pretty sophisticated stuff for fourth graders, right?

Not so fast.  Consider this simple activity.  We are deep into our studies of decimals through the hundredth place, and I really wanted to see if my students had fully grasped some of the misconceptions that we had been seeing in recent days--for example, that 5.4 and 5.40 have two different values.  I put my students into groups of three and handed each group a small collection of decimal cards.  I asked them to practice reading the decimals aloud (checking to make sure everyone in the team was confident with this) and then asked them to have a discussion about what whole number each was closest to.  I reminded them to use clear math language--and as I circulated, I listened for words like "tenths", "hundredths", and to hear if students were using what they know about halfway points between whole numbers.  Overall, things seemed pretty stellar!

I decided to push things a little bit farther by showing them these "category cards"--but instead of just using the "Close to 0", "Close to 1", and other whole number cards, I added in cards like "Close to 1 1/2", "Close to 2 1/2", and so on.  I asked the students to continue their discussion with their decimal cards by deciding if their decimals were ACTUALLY closer to those halfway points.  Now things got a little interesting!
After my students were convinced they had things right--I took them down the hall where I placed all the category headings in order and had the students place their decimals in the correct category.  I started down at the low end and called for all decimals close to 0...then all those close to 1/2...and so on.
When all the cards were placed, we worked to sequence the decimals in each category--and had some great discussions about  place value and how we KNOW what order is correct. 
As the process unfolded, we found some decimals that we couldn't get agree on.  Time to critique reasoning!  At this point, I really stressed the point that in order to truly "critique", students needed to have a clear point, use correct and mathematically sound language, and be able to give examples and evidence that would make their point clear to other students. 
 This became particularly true when we tried to place this one!  Where should 1.25 go--is it closer to 1 or closer to 1 1/2?  The debate raged--we had "Team Rounding" who was convinced that because the "5" made it "round up", that this card should be placed closer to the 1.5.  The other team felt this wasn't meant to be rounded--and that the number should go exactly halfway between the two.  
By the time the debate was over, people were satisfied (most anyway!), that this arrangement worked--and that unless we are TOLD to round, we have to follow the directions clearly.  And, in this case, they were told to put the cards in the category that the numbers were closest to--and for a few numbers they could NOT be placed in any one category but had to fit between them.  Not bad for a 25 minute math warm up, right?

These cards are a part of my Decimal Sequencing Activities.
I have similar activities for FRACTIONS and LARGE NUMBERS and a BUNDLE OF ALL THREE as well!

Wednesday, May 4, 2016

Teaching Tips for Concept Sorts!

Have you used concept sorts in your classroom?  If you have followed my blog for a while, you know that I use concept sorts regularly--and have posted a number of times about different ways I use them.  Click HERE and HERE for two of those posts!

Today I want to talk about a geometry sort I did the other day to "take the temperature" of my class after a few days of instruction about points, lines, segments, and rays and the terms "intersecting", "parallel", and "perpendicular".  Although I wasn't surprised at all the misconceptions, I am ALWAYS surprised at how many NEW misconceptions I discover!  If you are looking for a way to get much deeper math talk going on in your classrooms, concept sorts are truly an amazing (and easy to prepare!) way to force students to share their thinking and have great mathematical discourse.  Check out some pics from this week's sort!  I'd love to hear from any of you using sorts--and tell me if you think you've noticed an improvement in YOUR students' mathematical thinking!

 My students can get their sorts cut in just a few minutes now...they divide and conquer by each taking a sheet, then cutting the 8 cards in half, stacking them, then cutting them apart.  Voila!
 This is have to model (and if you have a group doing this well, try doing a fishbowl so the students can see this in action).  Students, by nature, want to just grab a stack of cards and put them into categories.  The assertive students take over, the passive ones sit back helplessly, and NO learning happens.  Show them how to take one card at a time to discuss it, interact with it--and make sure that all students are a part of it!  Watch for this as you teach...if you see students rushing, slow them down--even if they don't get through the entire stack.
 As you walk around, listen for the dialogue and interject when needed..."Is it possible for two segments to be perpendicular if they don't touch?" (YES!) or "Can two lines intersect if they don't cross?" (YES!).  Try to ask questions that get them thinking without TELLING them!  

 This is huge--and a piece of mathematics instruction we often overlook.  For this sort, I started off by NOT giving my students the category headers.  I wanted to see HOW they organized their cards on their own....and it allowed me to plan some instruction.  For example, one group made 14 categories--some of which only had one card!  We had a great talk about what makes good categories and then I handed out the four header cards.  Students then worked to reshuffle their groups to align to the new categories.  I also called groups over to look at this group's work because they were keeping everything all lined up rather than in piles all over the floor.  It makes it much easier to compare cards when they are neatly arranged.
Some groups are pretty convinced that they have everything correct, and if I want the other groups to have a little more time, I ask them to use the blank cards to try to make more examples for each category that are NOT found on any of the other cards.  This asks them to really apply what they know and to work together to do so.
 After my groups have their categories, we do a gallery walk where they travel as a team to check out the other groups and have some math discussions about what they see.  They look for similarities and differences and work to use that specific math vocabulary as they explain their thinking.
 As they worked, I asked my students to put a yellow mark on the cards where they struggled to come to agreement.  As they did their gallery walks, they looked to see if other groups had the same struggles!  After the gallery walk, I collected these cards for our class discussion.  One by one, I put these tricky cards under the document camera and we had "debates" as a class and looked for ways to prove our thinking...from lining up rulers to check for intersections to revisiting definitions. (If lines go on forever, don't they intersect even if the drawing doesn't show it?  YEP!).  This was a great time for me to hit home some of the big concepts I needed them to have--and for them to hear the thinking of others.
So...there was a LOT of math thinking that took place during this 45 minute lesson!  Hope this gave you some ideas about what you could try in YOUR class!  Let me know how it goes!  Want to see a few of my concept sort resources?  Here are a few of them! (Check my store for the custom category "Concept Sorts" for more!)