Sunday, February 19, 2017

Fraction Concepts Day 2! Conceptions, Misconceptions, and Mathematical Language

teaching fractions
Well, things got even more interesting as my fraction unit unfolded on day 2!  For those of you who read my first fraction post, I discussed how we used paper folding to get our fraction concepts unit "launched", and we ran out of time to finish the investigation.  I gave them another 25 minutes today to work on folding their different fractional amounts and then asked them to spend some time writing in their math notebook on any of the following:

What did you notice?

What was challenging?

What was easy?

What patterns did you discover?

fraction unit
First I gave the students "alone" time to write and then let them go compare notes with their partner from the investigation...



fractions
Not a great photo...not great handwriting...sorry!
What really stood out to me was the very NEBULOUS (vocabulary word!) math language I was hearing.  A phrase that kept coming up was "small fractions" as in "It was harder to fold the small fractions."

So, being the pesky and annoying teacher that I am, I presented this question to the class:
fractions
Tee hee...I love to ask questions like this!
I didn't let the students talk . . . I just said that I was hearing MANY students use this phrase, so I thought we had better made sure we all knew what it meant.  I asked each student to write the answer to that question on a post it note and come slap it on the easel.
mathematical thinking
And--seriously--I only wish I had taken some photos of the notes up close.  SERIOUSLY!  I had everything from . . .

64

to

1/100th

to

"When you make it smaller"

to

"Small fractions are like tenths"

At this point, any concerns I had about backing up this far and moving this slowly were GONE!  I read off the notes to the students and then asked what we had learned about what a "small" fraction is and one of my delightful yet very shy students called out, "ABsolutely nothing!" and we all laughed hysterically.

So where do we go from here?  I reminded the students about how we have been talking about the Common Core and how important it is to use precise mathematical language.  I asked if "small fraction" was precise enough that anyone else would know what we meant. . . and we had complete agreement that it was not.  I asked them if, on our 4, 3, 2 , 1 rubric we were evaluate how clearly we communicated, we agreed that all our post its were 1's with a few 2's sprinkled in . . . until "T" asked, "If there really isn't an answer, how could someone get a 4 on this?"  Have I mentioned how much I love fourth graders?  He honestly asked--and honestly wanted to know!  I told him I had to think about it. . . always a good strategy for the tricky questions!  Don't answer too soon!

So I set the students to work on a little activity to set them up for tomorrow's lesson...I asked them to work completely alone to answer the following question:

fraction misconceptions
If it is too small to read, the question asks "Is this shape divided into fourths?  Explain your thinking."

I let them work for a few minutes and then on their way out to recess made them commit to either a "yes" or a "no" response . . . and it will be our kickoff "debate" tomorrow!  Remember, one key element of the Standards for Mathematical Practice is the ability to explain one's own work and to critique the reasoning of others . . . so tune in tomorrow to see how it all shakes out!

After they worked for a few minutes we gathered back together to talk about T's question . . . is it possible to get a 4 on a question with no answer?   Maybe I should save that for another blog post . . . this one is already plenty long--but I told him that yes, I do believe you can get a 4 on a question with no answer.  I reminded him that we are stressing explaining our thinking, and we could certainly do that with this question.  I modeled something sort of like this (this was done aloud, not in writing . . . I'm just trying to give you the gist of it)

"I know that there are different ways to look at the idea of being small.  Some people might think the numbers should be small--like in 1/2.  Some people might look at the size of the pieces to determine what a "small" fraction is.  Still others might think a small fraction is one where you only look at one piece of a whole--like 1/3 is small where 2/3 is big.  So for the question "What is a small fraction?", it is very important to understand that there might be more than one way to answer it."

Would I expect a fourth grader to be able to answer like this?  Maybe not today . . . but now that it has been modeled--perhaps.  We must push their thinking, push their comfort level, and push them into the rigor that is required.  If we make it meaningful, they can do it!  Thanks for sticking with me again--sorry these are getting so long!  Stay tuned to see how our debate works out . . . I'll give you a hint--the vote was 9 "yes" to 13 "no"!


***UPDATE***
This blog post is now a part of my comprehensive fraction unit available by clicking the image below.  Hundreds of teachers have now used it to change the way they teach fractions!  



Saturday, February 18, 2017

Fraction Folding--discovery learning

hands on fraction activities
Today we kicked off our fraction unit, and I think I am going to really try to do a lot of blogging about it over the next few weeks--because I will be immersed in it AND because it is such a critical component of the Common Core for intermediate grades.  I think it is vital that we dialogue about ways to help students build their understanding of fractions, so I invite you to share along with me as I "trace" the path of our unit as it unfolds.  I'll try to be clear--but you know how I tend to get wordy!  I'll try to include lots of photos and work samples as I go, and I am hopeful that the rest of you will share great ideas and resources that have been successful for you.

Again, knowing that the CCSS places a great deal of importance on fractional understanding, our district math team made a decision to build in two fraction units into our year--this one and another one later which will work to tie together decimals, fractions, and a more sophisticated level of understanding.

Today I started by asking students to reflect on what they already know about fractions and to rate their overall confidence (using our 4  3  2  1 ) scale (Click here to revisit earlier blog post about this!) and the results were quite amazing!  In a ten minute writing time, I got to witness a number of misconceptions, poorly explained reasoning, and a bunch of "3's" and "4's" in confidence!  Good thing I had planned on starting slowly!  Today we started our new math notebooks, and I explained to the students that we are "raising the rigor" one level more (poor things have heard this all year) and we are going to work hard to use our new notebooks to both record our thinking, our practicing, and our new learning.  I am going to try a version of interactive notebooks, knowing that I cannot do it in true form as some teachers do . . . I'm just not quite there yet.  I'll blog more about this later as it unfolds!  For today, we started the section we called "Fraction Concepts" and even talked about what the word "concept" means.  Fascinating!  I told the students that our job through this unit would be to determine some things we could determine to be "true" about fractions and that we would be working our way through a number of these "truths" during the unit.

Today's "truth" involved ensuring that students understand that fractions represent equal parts of something (I didn't really want to use the term "whole" yet--I don't really like to treat fractions of "objects" and "sets" differently until I have to!) and that we would be spending some time creating equal parts.

I put the students into pairs (I love popsicle stick picking!) and gave them each 3 minutes to find a classroom object that was either a square, rectangle, or circle and was bigger than a deck of cards and smaller than a book.  Each team was then assigned a color of paper and the following task:

With your partner, trace and cut out your shape.  You will need many of these as the investigation unfolds.  Your job is to find ways to divide your shape into equal parts. . . first in two equal parts, then three, and so on.  


fraction unit
I made sure we had some circles, some rectangles, and some squares...

I then showed the students the following charts--each is labeled "halves", "thirds", "fourths", and so on.  As they discovered a way to fold their shape, they were asked to put it on the correct chart. . . and I was on the prowl for work that was not accurate and precise.
teaching fractions
Voila!  Posters are taped to the ground with deliberate "aisles" so they don't get trampled on!

fractions
Each poster is labeled with the word for the fractional part--I will use the words and different symbols interchangeably throughout the unit.  NOTE:  The chart does NOT say "1/3" because we were not identifying 1 out of 3 parts.

Students dug into their work and did need some frequent reminders about using straight edges, working carefully, and so on (I know--shocking!), but the were very engaged and thoughtful.  I heard some pretty nifty stuff like:

"Hey--as you try to get more pieces, the pieces get smaller!"

and 

"Man--it's a lot harder to fold the odd numbered pieces!"

and

"If you fold it in half and then in half again, each half gets cut in half!"

fraction unit
Tracing...
fraction activities
Using a straightedge for precision...
fraction lessons
Circles are the toughest as this team found!

As the period unfolded (and it became clear that this was going to be a TWO DAY investigation!), our charts began to fill up and things moved a little faster.
fraction lessons
The charts started to fill...first halves and thirds, then fourths and eighths...we'll see how it "unfolds" tomorrow!  Got to love a little math humor, right?
Tomorrow we will finish up our investigation and then ask ourselves if we have discovered any new "truths" to record in our notebooks.  Although I am pretty confident that my students would have been able to answer correctly if given a sheet of "shaded fractions" to identify, today's activity showed me that many students are missing some very critical understandings about equal parts and about patterns that arise when dividing shapes.  I'm pretty sure that the work we will do over the next few days is going to be very important to build a foundation for the more advanced skills coming.  If you aren't familiar with what the CCSS requires from students regarding fractions, I would encourage you to dig in and follow along with us as we try to construct meaning over the next few weeks!

***UPDATE***
This blog post is now a part of my comprehensive fraction unit available by clicking the image below.  Hundreds of teachers have now used it to change the way they teach fractions!  



Wednesday, February 15, 2017

Studying Dialogue to Improve Reading AND Writing

teaching dialogue
Today is my day to post over at Upper Elementary Snapshots, and I hope you'll head over to read about a lesson I did last week to help slow down my readers and get them thinking more deeply about the books they read.  I am excited to see if our dialogue studies transfer to their writing next week--so stay tuned!  Want to learn more about what we did?  Just click the image above and check it out!

Monday, January 30, 2017

How To Keep The Class Learning While You Pull Small Groups

teaching math
This week we are starting our "big multiplying" unit.  Not facts--but "big" multiplying.  Our standards state that our students must be able to multiply up to 1 x 4 and 2 x 2 digit multiplication, and for some of my students--I know this is going to be a biggie.

Because of that, in my planning this week I am planning ahead to be ready to free myself up to pull small groups as much as possible.  I know that some students already know the standard algorithm--and others are still not really even confident with arrays.  This is going to take some navigating and planning!  If you are interested in any of the resources pictured in this post, simply click the image to learn more about it.

Here are a few of my "rules to live by" when faced with this type of situation.  After all, if I'm going to be working with a small group--I want to do way more than just keep the other students "busy".  Right?

1.  Know where your students are and where they need to be.  Make sure you are clear on your content--and have a plan for assessing students formally and informally throughout your teaching so you can really target your instruction to exactly what they need.  I'm not a huge fan of pretesting (I think it's tough to make decisions about concepts based on one or two problems) but I am a HUGE fan of formative assessment along the way.  If a student can demonstrate mastery, they may need SOME work to build fluency--but their time is better spent doing other things.  I use my formative assessment resources all the time to take quick snapshots of progress.

2.  So what if they need that "something else"?  I love to immerse my students in challenging, open-ended tasks.  For this unit, I am presenting my Thinker Task Valentine Challenge to the class as one of these options.  Notice that I said TO THE CLASS.  I make sure ALL students have access to these quality problems.  Some students may not get as far into the challenge as others--but we so often "dumb down" our instruction for our struggling students and don't give them access to rigorous and meaningful tasks.  I might encourage them to use a calculator...to work in teams...or to use the easier version (I love that these have 2 levels for just this reason!).  Again--if I am pulling small groups, there will be time for students to do other work--and this is a great way for students to ALL have a common task.
Valentine problem solving
3.  Building fluency is another great thing to do when not in teacher-led groups.  Playing games can be a great thing--as long as the students are working on games that are "just right" for them.  If students are already fluent with facts, their games should involve some strategy.  If they AREN'T fluent, make sure they actually are ready for the skill on the test.  There is nothing worse than asking a child to play a game to build fluency and for them to not have the strategies to do it--instead, they spend time practicing wrong!  I certainly don't want students who don't know their multiplication facts to practice them incorrectly because I need them to be independent.  I'll have to find another skill for them to do independently.  For the next two weeks, I am making these multiplication fact fluency games available--each at two levels of challenge.  They aren't right for ALL of my students, but they are for MOST.
multiplication game
4.  Another great thing for students to work on when they aren't with you is word problems.  Whether you have them try them alone or work in partners--problem solving is NEVER a bad use of time!  Try to find problems that will be engaging (these all have a February or Valentine's Day theme) and are at a variety of levels.  I keep mine cut apart and in a pocket chart on my wall and try to put the easier ones toward the top.  Many of these also have an "extra" component so students can tackle that piece if it is a good fit for them.   As I transition between groups, I'll do some laps around the classroom checking student choices and doing some coaching along the way--but they really do a nice job of coaching each other!
Valentine problem solving
5.   One more option is to provide the class with a meaningful "warm up" problem or set of problems.  By starting off a class like this, you can make sure students understand the task and that they can be productive while you work to pull other groups.
Valentine printables
So...as you are planning for instruction you know might be difficult where you might need to do some focused attention, think about what kind of MEANINGFUL activities you can provide for your students.  I can almost guarantee--if you give them engaging things to do, the management concerns all but disappear and you are free to work your magic!

Want to pin this for later?  Here you go!

Saturday, January 28, 2017

More Fraction Number Lines: More Critiquing Reasoning!

fraction number lines
If you missed my post the other day about using number lines to improve fractional reasoning, you might want to take a peek at it before digging into this post!  Just CLICK HERE if you missed it...because I want you to know that the lesson featured today was not the FIRST time my students had worked with this number lines....and the lesson is one that should come after they have had some other experiences using number lines and having conversations about them.

So let's dig in!  

Step one--present students with a number line task.  As I mentioned the other day, even a number line task can be simplistic and obvious.  I am partial to a more "open" number line--where partitions are not already drawn.  When you include those partitions, you've done so much of the thinking FOR the students!  Remember, I always ask students to THINK before they pick up their pencil so they have a starting point.
teaching fractions
As students worked, I walked around and checked out their work and hunted for misconceptions.  I would make note of how they were organizing their work or strategies they used that I thought might be worth talking about.  In my last post, the next step involved pairing up and having discussions together and coming to consensus.  Today?  Not so much!
standards for mathematical practice
My next step was to ask students to come up and place a blue dot on this "class number line" to show where they had marked in in their math journal.  I reminded them to NOT change their answer based on what they see--because they need to trust their gut!
fraction number lines
By the time all the dots were up, it was time to have a discussion where we "critiqued the reasoning" and defended our thinking.  We had to use math language and vocabulary with our sharing--so I heard things like...

"The dots on the far left can't be accurate because they would really be past 0 and into negative numbers."

and

"I think [pointing to a dot] this isn't possible because there is no way that you can keep equal parts."

and

"It was more clear to me when I put in all the whole numbers so I could find where they 1/3 really should go."
fraction lesson
Some students needed some convincing--and we had frequent "turn and talks" throughout to get EVERYONE involved in discussion the ideas people shared.   It was great to see some light bulbs REALLY go off!  After our discussion, I sent the students back to their own notebooks to study what they had done and to "revise" their thinking if necessary.  SUPER powerful stuff!

This was a very nice and quick warm up for our lesson--and the process can be used not just with number lines but with ANY problem type!  Stay tuned for another post coming soon with more fraction fun!

Want to see the resource that these number line problems are from?  Lots of different problems and options...
Also available bundled with whole numbers to 1,000 and whole numbers to 1,000,000.
Want to pin this for later?  Here you go!
fraction number lines

Thursday, January 26, 2017

Fraction Number Sense and Math Discourse Part 1

fraction number lines
It seems like I have picked a few topics lately that just don't fit into one blog post!  I wanted to share two different "warm ups" I did with fraction number lines--and two different types of "math talk" that resulted from them.  Give these two lessons a try and see what you think!  I went kind of "photo journalism" style because I thought the pictures helped tell the story!  Watch not just for the important fraction understanding--but the immersion in the Standards for Mathematical Process as well!

First of all...if you have followed me for long, you know I love number lines--especially number lines that think "outside the box"--even number lines can be rote and low level--so we want to watch for that!

The other day I gave my students this one...it had the 0 and the 2 marked--and asked student to identify what number they felt that "dot" was showing.  My first step is always to ask them to THINK before they even pick up their pencil.  While they think, I remind them to consider what they know and can tell from just looking at it.  I really think slowing them down before they start writing can lead to deeper understanding and reduce careless errors.  Plus--for those students who ARE slower processors...not having to watch 20 other students get to work feverishly while they sit is SO refreshing and validating for them.  After some think time, they were off!
teaching fractions
 Some asked if they could use rulers...I simply said, "If you think it will help..."
Standards for Mathematical Practice
 I noticed that some students seemed to be putting fractions on their number lines WITHOUT adding the whole numbers first.  I asked, "Are you sure 1/4 goes there?  How do you know?"  Their answers told me a ton about their level of understanding.  Some had already visualized where the "1" went--others were simply putting it "where it looked right".
number lines and number sense
 As with all my number line work, I want them to be able to explain their reasoning both in writing and orally.  As students were working to finish, I had students begin to write their ideas down so they were ready to buddy up.
critiquing reasoning
 When we were all in a good place (in other words, essentially finished), I put students in pairs and trios and gave them the following direction:

You must all come to consensus about what fraction you will assign your dot.  When you are all in agreement (and this took SOME debate in some groups!), you will mark it on the white copy and start explaining your thinking.  When you present, I will call on whoever I want so make sure everyone is accountable for the information and explanation.

(or something to that effect)
teaching fractions
 I circulated and listened and coached and looked for misconceptions and mistakes.  I also looked to see the variety of strategies shown so I could have a variety of ideas to share under the projector.
math talk
For this particular problem, we had some debate.  About half the groups believed the dot to be at 3/4 while the other groups had a variety of answers.  One by one, they presented their solution and defended their logic.  Along the way, there were a few "a-ha's"...and a few stubborn souls who stood their ground despite very good arguments from others!
math discourse
 At one point, we put two different solutions up and had students try to determine which one they felt was "more right"...and which explanations seemed most plausible.
fraction number line
 We also showcased different strategies that seemed to help some groups.  This group used different colors for each step along the way--and other groups agreed that this seemed to make their explanation easier to follow.
standards for mathematical practice
 Finally, we ended up grabbing a spare number line and doing some folding to find those midway points and really "prove" what the 3/4 teams were proposing.  We had a great discussion...and SO much learning happened--and the students did it ALL!  Consider trying the strategy:

1.  Think time
2.  Independent time
3.  Partner/consensus time
4.  Whole group sharing and debating time

Watch how engaged your students will be...and let me know how it goes!
fraction lesson ideas
Looking for the fraction number line resource pictured?
I also have two other number line resources and a bundle of all three...see what you think!  I use them ALL year long.
Want to pin this post for later?  Here you go! And watch for several more fraction posts coming soon!

Looking for an entire fraction UNIT to get students thinking and talking?  Check this one out! 
(the number lines are not a part of this resource)