Thursday, February 28, 2013

Fractions Day 16: Word Problems and Critical Thinking

Well . . . I promised a little update on the problem solving from the other day, so here goes!

What I often do toward the end of a unit is provide students with a class period or two to work collaboratively or on their own on an assortment of word problems.  Some of the problems are right at the level I would expect them to be able to do and others "push" and help them work on their "problem attack" and perseverance.  While they work, I can circulate and coach as they get started--but then my real goal is to be able to pull small groups and individuals to work on "issues" that have come up in recent lessons.

During this round of problem solving, I wanted to pull students who are still "iffy" on equivalent fractions, on comparing fractions, and on addition/subtraction with like denominators, so I got my collection of problems ready so the students would be focused while I pulled groups.

Here is one way I organize the problems:

This is a little cheap "tent" pocket chart that I think I got a few years ago at "The Bullseye Store"  It has those pockets on both sides.

So . . . on each of my two problem solving days, I did a 15 minute review minilesson to keep cycling back through those concepts covered over the last weeks and then I sent my students off to work in their math notebooks on the variety of problems.  I DO give a little "problem talk" about the problems (8 choices this time) so the students know who the problem is about (often students in the class) and what is happening.  In this collection there was a friendship bracelet problem (I KNEW which 3 girls would make a beeline for that one), a "travel team" soccer problem, a kitten problem, a candy store problem, a "Training for a 5K" problem . . . you get the picture.  Students often gravitate toward problems with topics that interest them no matter what the math content is!  I will also give a warning if I think the problem is particularly challenging or one that I call a good "warm up" which is the sign to those who are hesitant to TRY THAT ONE FIRST!

Drawing pictures to solve the problem . . . notice the use of highlighter to highlight key information.  This is the first class I have ever had that likes that as a strategy.

This student tried problem 6 and then decided to abandon it for now . . . we compare abandoning problems to abandoning books.  You shouldn't have to do it very often, but sometimes it is the best decision.

This student was stumped!  I spent some time talking with her and then found another student doing the same problem . . . 

Together they talked through it and came up with two strategies to try--both worked!  They were so excited!

The one thing that got me super excited during this time . . . I had just finished with a small group when one of my "fraction challenged" students came up to me with a question about her problem.

The problem states that Mara and her 3 friends ordered pizza and ate 2 1/2 of them.  They all ate the same amount--what fraction of a pizza did each girl eat?  She had drawn a picture and really felt like she had found a way for each girl to get the same amount--but she had no idea WHAT that amount was.  Check out her solution--brilliant, right?  She drew the 2 1/2 pizzas, assigned each girl a number (1-4) and started splitting the pizza up.  She knew each girl could get 2/4 but they would have to all split the remaining half of a pizza--so she split THAT into 4 pieces.  She couldn't quite figure out how much that added up to be.

I sat and listened to her reason through the problem and I noticed several other students starting to hover around (very typical . . . I tell them that smart learners tune in to others' lessons, so I encourage this!) so I asked if any of them thought they could help.

We threw her problem up on the Smartboard. . . 

Then asked for volunteers to go up with her and talk through it . . . this student tried to help her see that each of the girls WOULD get 2/4 of a pizza AND 1/8 of a pizza BUT. . . 

When it came time to do the math . . . 

Something less than perfect happened.  I felt my heart sink until I heard another little voice chime in and say  . . . 

"Hey!  Remember . . . when you add fractions you have to have the same denominator!  I think you should turn all the pieces into eighths."

. . . and so she did.  She came up and divided all the fourths into eighths and the THREE of them collectively shouted out "FIVE EIGHTHS!".  At this point, 8 students had gathered while the others worked.

High fives went around the group and they all rushed back to their notebooks to record their thinking.  


Common Core fractions?  CHECK
Standards for Mathematical Practice?  CHECK
Excitement about math?  PRICELESS

Thanks for stopping by . . . the unit is coming to a close!  Let's hope they are ready for their assessment on  Monday!

Wednesday, February 27, 2013

Fractions Day 15: Student Self Assessment and Reflection

I know I gave you all a fraction word problem teaser yesterday, but I'm going to hold off and include that in tomorrow's post.  Today what I want to share is something I did for about 15 minutes after math class was over.

Before this unit, I printed a list of the Common Core "learning targets" rewritten in kid-friendly language for my students.  We had looked at this list as our unit started and made little folders to keep them in.

Throughout the last few weeks, I have been giving the students exit slips and pulling them for extra help based on their results.  This is the method I typically use . . . we often go through 6-12 exit slips in a unit as I take the temperature of the group to know where to head with my large group lessons and who needs to be pulled for small group or one-on-one work.

Today we passed out all 7 of the exit slips we have done so far and we studied them and cross-checked them with the learning targets on the blue sheet.  As we discussed what each target meant, I showed an example on the Smartboard to make sure they knew what I was talking about and the students "self-assessed" with a plus to mean "I know it so well I could teach others." or a check to mean "I know it well enough to do it on my own." or a minus to mean "I am not feeling confident YET."  (YET is a big word with me . . . not "I can't do it!" but "I haven't learned how to do this YET.")

Students then looked at their exit slips and decided how they lined up with their plus, check, and minus marks.  Many students were shocked at some of their earlier exit slips and were excited to be able to say "I know how to do that NOW!"--which is exactly what we want!

When we finished reflecting, we packed them all up in the folder and put them away.  We will do this one more time on Friday with the remaining exit slips and then I will send the entire folder home for students to "teach" their families as part of their review for their summative assessment.  I ask them to show their parents their formative work, to try some practice problems at home and put them in the folder, and then have the parent sign the folder and send it back to school.  This really helps parents know what is expected, what the problems look like, how their child is doing, and what THEY can do to help.  It sure makes a difference; my students should be as ready for the summative assessment as they developmentally can be!

What do you think?  Are there other ways we can get students to be more accountable for their own learning?  How else can we communicate with parents so they know what is expected and what the level of rigor is?  I'd love to hear your ideas below!  Share your best ideas so we can all learn from each other. . .

Thanks for stopping by--word problem wonders tomorrow!

Tuesday, February 26, 2013

Fractions Day 14: Composing and Decomposing

Today was a math rotations day in grade 4 . . . I started the day with a minilesson and then we broke up into three math groups to rotate around the following centers:

--“teacher time”
--problem solving
--</>/= activity

Today’s minilesson focused on “composing” and “decomposing”  using unit fractions, and I was hopeful that it would go smoothly.  I was fairly confident based on the work we have done over the last few weeks that the students would easily manage this concept, and my students did not disappoint me.

We reviewed the idea of “unit fraction” and used the fraction bars in Kidspiration to model different ways to build fractions.  We talked about how you can build or “compose” new fractions with unit fractions (which I also call counting fractions…just like we count WHOLE numbers, we can count FRACTIONS too!). 

We then tied the fraction bars to a more traditional adding and subtracting notation—we modeled by counting unit fraction groups to add and by crossing out fractions to subtract.  Students seemed to be really understanding, so I felt ready to send them off to do our math rotations.

Here is our rotation board:

(I’ve flipped the top cards backward to hide the names).  Today’s groups were picked based on needs—I knew that I wanted to differentiate the “teacher time” rotation.  Some days my groups are mixed ability--it all depends on what I am trying to accomplish.  For most of the investigations we have done during this unit, my groups are entirely mixed by gender, ability, and so on.

So here were our rotations in  nutshell!

1.  The "teacher time" rotation was with me.  We worked on reviewing the material taught in the mini lesson by having students try different combinations of addition and subtraction in their notebooks.  For my strugglers, we started really slow and brought the Smartboard back up with the fraction bars so we could make the tie.  By the time they were finished, we had written all sorts of different equations with addition and subtraction with like denominators.  We drilled in the concept that we can only add and subtract fractions with equal parts--or else we need to do some extra work!  We showed how adding 1/2 and 1/4 cannot possibly be 2/6.  We worked to come up with many equations that equaled "5/8" such as ...
1/8 + 2/8 + 2/8 
7/8 - 2/8 
4/8 + 1/8 

and so on.  

The other two groups were ready to move beyond this.  I started by adding in both improper fractions and mixed numbers for them and things seemed to be moving pretty smoothly.  I asked them to find many equations with the solution 1 3/4 and they handled that as well.  I then decided to work on true/false problems with them so I threw out problems such as 

5/8 + 2/8 < 3 x 1/3  True or False?


4 1/4 - 3 3/4 > 1/2  True or False?

The students handled the multiplication with ease--they have such good number sense that they realize that three "1/3's" is a whole--no computation required!

We then started playing more with improper fractions and the students tried writing problems to stump each other--tons of fun and very developmentally appropriate for each group.  

2.  The second rotation was a free explore station with fraction cards and <, >, and = cards.  I gave the students a number of different ways to interact with the cards...

--they could flip two cards over and decide which symbol would go between them
--they could flip 5 cards and put them in order smallest to largest
--they could flip two cards over and find 2 more cards that would fit between them
--they could come up with their own sequencing practice

And practice they did!  Each time I asked this group to rotate, they MOANED!  Can you believe?  They MOANED that they had to stop sequencing fractions!  I heard some fantastic discussion and some very creative thinking.  One group even decided to try to take a larger fraction card and find smaller ones that they could "decompose" the larger one into.  Really?  So much fun!  For those of you who have purchased the fraction sequencing set, this is one of the activities in that resource.  I will include it in the unit as well.

3.  The final rotation was a problem solving station.  Because I felt the students have enough number sense and addition/subtraction skills now, I was ready to send them off to try to tackle some word problems with fractions.  I will include these problems in my unit, but for those of you who aren't interested in purchasing the big unit in a few weeks (I promise I will work as hard as I can to get it done in the next 1-2 weeks), the set of word problems IS available in my store now.  I will put the link at the bottom of this post.  I let the students buddy up if they wanted.  I was super pleased with what I saw!  I will save one example of some great teamwork for another post!

Here is the teaser for the next post!  This was the "a ha" problem for one of my students today!

So, I hope you get a taste for one way I organize my math time . . . many of my lessons have been very "investigation" oriented--today was a more "guided math" approach which allowed me to really dig in and see how far certain students can be pushed.  Tomorrow--more problem solving and small group pulling based on the 3 exit slips I have given this week.  I have a few petunias who need a little more work with equivalent fractions and comparing fractions--so while the class has rotations tomorrow, I will be pulling small groups instead of having a "teacher time" rotation.  

Thanks for stopping by!  Stay tuned . . .

Monday, February 25, 2013

Fractions Day 13: Benchmark Fractions

It is getting close to the end of our fractions unit . . . and I am trying to reflect on what more I need to accomplish!  I am really seeing students (for the most part) having an "ease" with fractions, with equivalence, and with some of the "big picture" ideas.  I know I want to dig into some problem solving with addition, subtraction, multiplication, and division, and I want to throw some more "traditional" exit slips at them to see how well they do on more computation-based tasks as well.

Today I asked the students to solve some quick equivalency problems (3/4 = ? /16) and so on.  17 of my students aced it, 3 of them missed one (fact mistakes . . . AYE!  So many fact mistakes!!!), and 2 of them seem to still really not understand the concept.  Tomorrow--I will pull the 2 and go back to some paper folding to try to connect to the computation and see where they are missing the boat.

So . . .  today!  What did we do?  I flashed the following direction on the Smartboard:

I split the students into four color groups (based on the color of their chart paper--totally randomly handed out colored chips to assign groups).  Each piece of chart paper had a stack of papers on it.

I told the students to move to their station and to get to work.  They looked at me with puzzled looks.  No one said a word, but their little faces said, "LADY--you are LOSING it.  What do you want us to DO?"

I smiled and shrugged.  I told them true problem solvers would just dig in.

And so they did.

This activity is one of the ones I have in my Hands On Fractions set . . . and I will include it as a part of my fraction unit based on these blog posts.  You could most certainly create this yourself!  What the students found at their station:

--a set of cards labeled "Close to 0", "Close to 1/2", "Close to 1" and so on 
--a set of fraction cards
--6 blank "my fraction" cards

Immediately, the students started sorting and organizing.  As always, I loved sitting back and eavesdropping and spying.  It was delightful to see that I now have VERY few students who just sit back and need encouragement to jump in.  Our hard work at getting everyone involved and feeling safe has paid off!  These were big groups--5 or 6 students--so they really needed to listen to each other.  For the first 10 minutes, all I did was observe.

All groups worked to organize their "close to" cards first. . . 

This group started with a linear number line until one student suggested they made two rows to fit it all on the chart paper.

This group immediately put them in order from smallest to largest--something I thought would be obvious. I was wrong!  Two of the four groups did NOT put them in order from smallest to largest!  One went largest to smallest and the other went . . .  ummmmm . . . "creative"?

As the teams began to sort their cards into the correct area, I noticed a pattern with their cooperation.  Each team seemed to be functioning well and work was being done.  HOWEVER . . . looking closely, I realized that really each team was functioning like 5 individuals sitting next to each other.  It was like parallel play--for math!  Each student was taking a card and announcing its correct placement and then grabbing for another card--no discussion, no questioning.

I decided it was time to intervene; the whole point of this activity was to really discuss why each fraction belonged where it did--these were not all easy answers!  I asked the students to stop their work and explained to them what I noticed.  I asked them to take a few minutes to talk with their team about how they could improve their teamwork, and then we shared ideas.  I sent them back to work and saw MUCH better work!

I saw students pointing.  I heard students debating.  I heard really cool stuff that we have been working on for the last few weeks!  Things like . . . 

"It HAS to be larger than one because the numerator is larger than the denominator!"


"It is closer to 1/2 because 16/30 is only one thirtieth away from 15/30 which is equivalent to 1/2.  That's way closer than to one whole."


"4/9 is really less than half because 1/2 of nine would be four and a half.  This one only has 4 ninths--so it is less than half."

Pretty neat, eh?

So after they had worked for quite a while, I noticed some teams breaking out the "my fraction" cards and making their own.  Can you believe--not a single student asked me for a single direction all day?

I was so glad that I put these in the lesson. . .  it was a great way to differentiate (not all groups got 6 of them made) and I could see how some students worked to try to get one in each category and so on.  When I felt it was time to move on, I had the students rotate from group to group and study their work.  Rules?  Critique but do not touch.  Be ready to share.

And it should be no real shock to you that my students quickly picked out the ones they saw that were different from what they had done.  I sat back and listened to the really nifty discussions and "proofs" they were using.

When they had seen all 4 groups, I flashed photos on the Smartboard from each group and we talked about "issues" we had seen.  Students from different groups defended their answers, explained that they thought they were right but when they saw other groups with different answers, they thought more and now wanted to change their answers, and then we had a few we had to tackle more in depth.

This student wasn't pleased that the 14/10 was in the "close to 1" instead of "close to 1 1/2".  He did a great job explaining that the extra 4/10 was only 1/10 away from 5/10.  The purple group agreed and said they would move it.  Another student commented on the fact that a few posters had the groups kind of blending together and that it was hard to tell which fraction went with what.  She then smugly added, "That's why OUR group drew lines on our paper to show that."  I acknowledged how important it is to organize work and work precisely--but also let them students know that I had cut the paper and it was a tight fit!  When we were finished, I sent the groups back to make any last minute changes and to glue down their work.

So . . . all in all, I think it was a valuable experience.  Tomorrow I am giving an "entrance" slip asking students to write down three fractions that are close to 1/2, close to 1, and close to 1 1/2.  We'll see how they do!

I hope you found this interesting--and maybe will give it a try!  For those of you who are following along and are interested in this unit, please let me know what you would like included.  Here is what I am thinking:

--The actual blog post (maybe reworded a bit) with clear learning targets as tied to the Common Core
--Any journal prompts ready to be printed, cut, and glued into math notebooks
--Direction cards for tasks (if used)
--The fraction number cards or other things actually USED for the investigation (even if they are already included in fraction sequencing set I sell . . .)
--Exit slips to match some of the learning targets

What else would you like?  Thanks so much for your support. . .  you don't know how your kind words have been "filling my bucket" over the last weeks!

Sunday, February 24, 2013

Fractions Day 12--Equivalent Fractions Example/Counterexample

Happy Sunday, Everyone!

I woke up to a big surprise--my Skittles "Freebie" was featured in the TpT newsletter!  It has already had over 6,000 hits today!  Crazy!

Today I want to share an activity that I did last week that I feel was a great review of equivalent fractions AND a great opportunity for students to practice their problem solving and their mathematical language.

Here's what we did!

First, I prepared six "posters" with a common fraction on each (If you need a different number of groups, just make a different number of posters . . .  I wanted 3 or 4 students per group).  I made a 1/2 sheet direction sheet for each and placed them around the room.

I reviewed the directions with my students and made sure everyone was clear.  In a nutshell . . .
  • They would get about 4 minutes per rotation.
  • They needed to write 3 equivalent fractions on their poster ("examples") and one COUNTERexample.  As a team, they needed to decide how to create a counter example they felt might "trick" the next group.
  • After their 4 minutes (or so--I watched the groups and moved them when it seemed like they were all ready to go), I rotated them to the next poster.
  • When students get to their next rotation, they need to hunt and find the counterexample and neatly "X" it off.  They then need to work together to find three NEW equivalent fractions and one new counterexample. . . no repeats allowed!
  • After 4 minutes, we rotated again and repeated the process.

The students worked REALLY well together . . . did lots of pointing, used lots of scratch paper, and got super excited about finding counterexamples they thought would fool their classmates.  Overheard . . .

"If we make the numerator and denominator only one off from the original, it might fool them!"

"I think the last group made a mistake with this one--I can prove it!"

"This one can't be equivalent to 3/4 because it isn't even close to 1/2!"

As you can see, students had to really apply the understanding they have been building over the last few days . . . and I think it paid off.  When I gave them a more "traditional" exit slip the next day, only two students didn't show excellent understanding of the concept!  

When we were finished, I put them together and hung them up on our fractions board . . .

Later that day I noticed students stopping by to check it out and to see what other fractions students wrote down.  I think this activity could have lots of other applications in math!  

Thanks for stopping by--and have a great Sunday!

Friday, February 22, 2013

Five on Friday! 5 Things to Inspire You

I have had some pretty inspiring moments this week . . . and without going into detail, I'd encourage you all to find a time over the next week or so to try to inspire someone in your life . . . whether at home or at school or in your "real world"!

Here are 5 things that have inspired me this week!

1.  This is a photo taken of a plaque on my bookshelf at school.  It was given to me by a coworker several years ago when we went through a pretty tough time with our son with autism.  I look at it often but really try to take moments to think about it when I've had a tough day.  It was a kind of rough week at school, so I read and reread this one yesterday.  I love it.  I really really love it.  Some days it is harder to believe . . . this I know to be true.

2.  This video clip on YouTube.  Have you seen Kid President yet?  I haven't heard of this guy yet--but if you need a pep talk . . . this is 3 minutes well spent.  It has gotten over 10 million views in a short period of time.  There is a reason.  Watch it.  Share it.  Live it.

3.  My blog followers.  I know--cheesy, right?  Seriously though . . . some of the comments you have all made this week have really made me think.  Whether your comment be about my homework post or about my fraction posts or whatever--I really and truly value that you took the time to interact with me in this techie 21st century way!  Isn't it bizarre to feel "listened to" by complete strangers?  Thanks for being an active part of my blog.  Really.

4.  My coworkers.  You know, with as crazy as everything is getting in education, it is crystal clear to me how much we are going to need to lean on one another.  I heard a brilliant man speak the other night and he talked about faith--faith in the world . . . faith in ourselves . . .faith in others.  And he talked about how he surrounds himself with people that will raise him up when his faith waivers.  That was so meaningful to me.  I hope I can be that person who "lifts" the others in my life up when they are struggling.  Sigh.

5.  My students.  Even though the weather changes have made them  . . .  ummmm . . . "busy" this week, I am continually amazed at their willingness to work for me--to answer unanswerable questions, to go back and try something one more time, to push themselves just a little bit more.  I see them being kind to one another (usually), caring toward one another (most of the time), and genuinely enthusiastic about learning.  I had several glimpses this week at just how fragile some of them are, and I was inspired to remember how vitally important I am to them and their physical, emotional, and educational safety. It's a huge responsibility--and I hope I never forget that!

So . . . as we head into this weekend, tell me PLEASE--what inspired YOU this week?  Something big?  Something small?  Share it with us!  You never know when passing on inspiration can make a world of difference.  Just like Kid President says--"We are all on the same team!"  Let's make this world a more awesome place.

Thursday, February 21, 2013

Fractions Day 11: Where'd You Get THAT Number Line?


Hello everyone--and thanks for coming back for more fraction adventures!  Today's post is really a combination of classroom "happenings"; I don't know if you've noticed, but sometimes something I plan as a quick warm up turns into something just a teeeeeny bit more involved!  That was the case with my warm up the other day.  It seemed simple enough.  Note:  I purposely did NOT put any points or any other references on the number line.  See below . . .

teaching number lines

Now, if you've been reading days 1-10, you know that I like to torture my students with odd questions and bizarre situations and obtuse tasks.  Watching them struggle makes me beam with pride.  Seriously!  My students will take on a task like this with no questions asked any more!  I did get a few comments . . .

"Is this one of those 'There are lots of right answers' types of problems?"

"Can we move the 1/2 to 'where it supposed to be'?"  (I loved that one...shrugged and informed them that it IS where it's supposed to be Thank. You. Very. Much.)

"Do they have to be fraction numbers or can they be 'real' numbers?"  (Love how fractions are fake numbers . . .)

So off they went to their teams.  I asked them to work for 10 minutes or so with their team and to come up with ONE united solution.  They put their heads together and I sat back and listened to all sorts of amazing discussions.  There was lots of pencil tapping, fingers used as "spacing guides", and even a few rulers pulled out to try to prove points!

mathematical thinking

After they had worked, I had all of the groups come up to the Smartboard and share their ideas . . .

fraction number lines

We talked about how similar and different the different solutions were.  I was surprised at how uniform they were, to be honest.  Pretty much all groups plopped a "0" and a "1" on either side of the 1/2--with the 0 being as far to the left as they could go.  Different groups then did varying degrees of division . . . some counted 0, 1/4, 1/2, 3/4, and so on.  Others put in eighths and a few added "estimated" fractions like 1/3.  But every single group put the zero "as far over as they could".  They were all very insistent that there be EQUAL spacing between the 0, the 1/2, and their 1.  Most groups also had room to add the 1 1/2.  I was intrigued.

I was still surprised that no one really thought out of the box, so I asked if they wanted to see MY solution.  They couldn't believe it because, of course, I NEVER show them how I would do it!  You could just tell they were thinking, "What is she DOING?"  Always keep them on their toes, that's what I always say.  Because I was at the Smartboard, I have no photos, so here is my recreation of my first attempt:

fraction lessons

I threw this bad boy up there and the students were silent.  Someone finally chimed in and said, "You didn't fill up the line."  Someone else said, "Why didn't you put the zero at the end?"  I spoke nary a word.  I waited.  The kids complained.  They talked.  Finally I heard, 

"Wait, guys.  The number line has arrows on the end.  It goes on forever in both directions.  There isn't an end."

Are these not your favorite moments?  Those unplanned, unscripted, "I don't know where this is going and it's a little scary for me as a teacher" moments?  That comment was the catalyst for the next part of the discussion where we talked about spacing on a number line, equal parts, and so on.  Not being able to leave well enough alone, I then threw this one out there:

critiques reasoning of others

Now this one really irritated them.  

"That's IT?  That's ALL you are putting on it?"

Again, I shrug.  I sent them back to their teams to find other numbers that would work on MY number line.  I only gave them a little while to work, but when I started hearing things like . . . 

"We need to find other fractions that are equivalent to 1/2."


"There isn't much space between 1/2 and 5/8."

I knew we were making progress.  So--step outside the box with YOUR little people.  Look at things a new way.  Give them LESS information and see what they can figure out!  The world's problems aren't going to be solved by people who can pick from among four bubbles, folks!  We must be helping them construct meaning. Stay tuned for tomorrow when I tell you how we tried an example/counterexample warm up. . . and it was so much fun!  Thanks for joining me!

This blog post and sequencing activity 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!