Monday, February 4, 2013

Fractions Day 5: The "Whole" Story

I know I left a few of you hanging last time . . . sorry about that!  So today we are...
It was a FASCINATING discussion--try it with your students!  Which is larger--1/2 of a grape or 1/2 of a strawberry?  You should have HEARD the ideas they had!  You should have heard the LOGIC they tried to flip out as "truth"!  We had all sorts of conversations about things such as:
What is a "typical" grape?

Does it matter that grapes are round and that strawberries are different on the top and bottom?

ARE grapes round? 

 . . .  and the discussion went around and around until I pulled out THIS PHOTO that I had PURPOSEFULLY saved on my phone from earlier this fall. . .

That's right--that is a GRAPE on the left!  The kids were in AWE.  We talked about the idea of a "benchmark" object to help us know the relative size of things and how "big" grape to one person might not be a big grape to someone else.  This all, however, was fine and good but didn't get us to our main discussion--that a fractional amount is totally and completely dependent on the size of the "whole".  I then asked them who ate more pizza last night--me, eating 2/3 of a pizza or my husband eating 1/2 of a pizza.  They immediately declared me the winner . . .  until I sat quietly looking pensive.  A few seconds later someone chimed out, "Wait!  Were the pizzas the same size?" and we were off to the races.

This leads me to today's investigation . . . today I reminded the students about grapes and pizzas and we reviewed some of the "truths" about fractions that we had learned last week--that fractions refer to equal pieces, that the top number (numerator) refers to the number of pieces we are working with (NOT shading!), and the bottom number (denominator) refers to the number of groups total.   I then passed out 6, 3" x 4.5" rectangles of paper to each student and asked them to remember what they know about folding fractions to create models of "one whole", halves, thirds, fourths, sixths, and eighths.  Believe it or not, this was still not easy for some of them!  I could really tell which students were beginning to develop some better fraction number sense.  When I saw them struggling to get six and eight equal parts, I finally broke down and asked the question:

"What did you do to make fourths?"  

The students talked at their tables and we shared our thinking.  Overall, most of them were NOT randomly trying to make four equal parts--they made halves and then used the halves to make fourths.

"What did we just discuss that could help you in your quest to make sixths and eighths?"  A flurry of activity started and I waited for them to finish.

One of my learning targets for today was to make sure I stressed the concept of "unit fractions"--that a "whole" can be divided into smaller, equal parts that we can "count" with...just like we can count with whole numbers.  We added a definition of unit fractions to our interactive notebooks and then I asked the students to glue their "wholes" down and record number sentences reflecting how they can be decomposed (this is a BIG concept in the Core, people--they need to be able to see how fractions are made of other fractions . . . today was the first step in developing that understanding).  Here's what we did:

First I threw some fraction bars under the document camera and modeled how the pieces worked together to make "wholes"

Then I asked what would happen if I separated each "whole" into its respective "parts" . . .

I then modeled how you could write a "number sentence" to match what had just been modeled.  I needed them to understand that those "unit fractions" can be counted to make a whole--and that a "whole" can be made up of any number of equal parts.

You see where I was going?  I wanted them to understand that 8 "eighths" makes up a "whole" (This is multiplication of fractions, by the way!) and that 4 "fourths" makes up a "whole".  Over time, we will be able to get more sophisticated with this, but this was where I wanted to start.  Overall, the students seemed to get this concept--though I am sure that I will certainly get some 1/2 + 1/2 = 2/4 along the way.  I'm a realist.

The one thing that REALLY shocked me happened way back in the dividing of the rectangles. . .  some of my students clearly had some major misconceptions about what "equal parts" means.  For example, one of my math rock stars (really--one of my BEST thinkers) drew this to represent thirds:

Yup.  I was shocked too.  I've never had someone do that before!  I've had circles divided incorrectly, but not rectangles!  So, being the pushy, "what ELSE don't they know?" kind of teacher, I threw a few things up on the board. . . 
and I asked, "Which of these are thirds?"  I gave them time to talk and then we voted.  The little numbers in red show how many of my 22 students thought each shape WAS in thirds.  Interesting, eh?  So . . . I have a few little petunias trying to find ways to prove/disprove these tomorrow--but the biggest reminder for ME was to never assume that students really understand.  When we always present them with pre-divided circles and fraction bars, we don't always encourage the kind of reasoning needed to decide whether or not equal parts really exist.  So I pulled out an exit slip I had planned on doing later in the week so I could check my theory . . . check out some of these misconceptions...sorry they are so blurry!
Trying to show 3/5!
Also trying to show 3/5!
Showing 11/12!
So, know I know where I need to head tomorrow.  I need to dig a little deeper on the misconceptions they have about dividing certain shapes.  (And a little review on working "precisely" will accompany it, as you can see!)  I'm not quite sure yet how to tackle this one--but I think I may ask the students to try to prove it on their own without giving them resources yet!  We'll see what they come up with!  Next steps?  If I give them a fraction like 1/4, can they draw the "whole"--and how many versions of that "whole" can they find?  It should be interesting!  Hope I haven't scared too many of you away yet . . .  but be careful what questions you ask--because you might find out what you DIDN'T want to know--and you will need to act upon it!  Thanks for stopping back for day 5!

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!  


  1. This is great! When I did my lesson about this, I asked mine if they'd rather have a half or a whole candy bar. Of course they all said whole. I gave them all a mini Hershey bar and said, "here's your whole. I'll just enjoy my half." And I pulled out an XL King Size Hershey bar. They all gasped. ;-)
    ideas by jivey

    1. Oooh...yours sounds yummier! Thanks for stopping by again! :)

  2. I totally relate to those last lines of your post - be careful what you ask - it is always hard to find out just how much they don't really understand or the misconceptions they are carrying around!

    1. Exactly! But we do need to ask, don't we?!!?!? We'll see how tomorrow goes.

    2. I also wanted to say that you do have me excited to teach fractions (gasp). I hope you don`t mind but I am pinning your post to my math board to remind me how excited I am!

      Looking From Third to Fourth

    3. That's great! I am glad you are enjoying it! I was thinking of trying to put this all together in a unit, but I'm not sure how well it would be received...this isn't EASY teaching! :)

  3. Thank you for sharing your lesson. Love seeing the examples from your students as well. This reminds us that while we often think the kids 'get it', we need to check in often with them.

    Fractions can be so tricky for the kids. I love the paper folding activity. Will keep this in mind when we start our unit (we are on Geometry!).


    1. Thanks, Nancy. Sometimes I think these lessons are more insightful for ME than for the students! Hopefully by the end of the unit we'll have things all straightened out in their minds!

  4. Thanks so much for these lessons. I wish I could somehow put them all into a powerpoint, not only for my students, but for me!!! These posts are great teaching tools. It is hard for me to teach fractions the common core way, but your blog is helping. Thanks, Pam

    1. Thanks for your kind words, Pam. I am working on the unit...I hope it will help you! Thanks for following along.

  5. I think that your "rock star" student was "thinking outside the..." um, circle. Who says you can't divide a rectangle that way and attain three equal areas? Although he/she didn't quite draw it correctly (the "Y" would need to fork one-third of the way up, then each angle proceed to its corner), but I would not say that he or she didn't "get it." Maybe the teacher needs to think more open-mindedly and accept more creative solutions. (Remember that when cutting a rectangle at angles, you can always subdivide it further at right angles to check that one side equally exchanges area with another.) THAT is the true essence of teaching!

    1. Thank you so much for taking the time to reply. As you obviously have an understanding of math, you will--of course--realize that the explanation or "proof" is key in showing mathematical understanding. This child's explanation said that the pieces "looked" the same--there was no math involved. I'm sorry that you felt the need to be critical of the teaching when, if you were an avid reader of mine, you would know that open mindedness is not something I tend to lack. Perhaps a sentence needs to be added to the post to clarify. Remember, also, that this is fourth grade and the students need to have the math needed to "prove" such arguments.