Wednesday, June 24, 2015

Beyond Pizza and Pies Book Study Part 3

Welcome to week 3 of our book study on Beyond Pizzas and Pies.  For more information this summer's book studies, check out this post.  If you need more convincing about how much this book has to offer, check out this short introduction video!

I can't believe we are over half way done this book!  I feel like it is a fairly quick read yet I have some great information to take back to the classroom with me in the fall.

The posting schedule for Beyond Pizzas & Pies (all Wednesdays)
June 10th Chapters 1&2
June 17th Chapters 3&4
June 24th Chapters 5&6
July 1st Chapters 7&8

Chapter 5: Is 1/2 Always Greater Than 1/3? 

"Our research suggests that many students did not understand that, within a context, the size of the wholes must be determined before comparing fractional quantities." 

This quote from chapter 5 really sums up what I have seen over and over again with students and teachers. When I learned about fractions in elementary school, I can't recall a single time where we solved a problem about fractions where the wholes were not the same.  Even now, when companies send me various curriculum samples, many of them do not contain any problems in which the wholes can vary.  I think a lot of the issue with kids not understanding that the size of the wholes must be determined is that they don't get a chance to see a lot of problems where the size of the whole might vary.   

If your curriculum is lacking in this area, the activities suggested in this chapter will go a long ways toward helping your students uncover this important fraction concept  The pattern block activity described in 5.2 and in the video clips is one I have done with students many times and it is a great way to bring forward a variety of big ideas around fractions  I also love the ideas presented in 5.2 around using 6, 12, and 24 packs of soda or water.  This is definitely somethings I will be thinking about adding to my repertoire for next year.  

In my school, we have a long standing tradition of introducing the ideas of the size of the wholes with Hershey bars.  A good teacher friend of mine came up with this many years ago when we were co-teaching a fifth grade.  Now I do this lesson in grade four.  I get two paper bags and some candy bars.  One bag contains regular sized Hershey bars (the ones with 12 small rectangles).  The other bag contains the smaller Hershey bars.  Here you can either use the minis or the small ones with 4 rectangles that often come in an 8 pack at drug stores and supermarkets.  I then hold one bag and have another person hold the other bag.  Then I ask, "Who would like a chocolate bar?  Mrs. ____ and I have chocolate bars in these bags.  I will be giving out whole chocolate bars and Mrs.  _____ will be giving out a half of a chocolate bar.  Please raise your hand if you want a whole chocolate bar and I will come give you one."  Most or all of the kids choose the whole chocolate bar which of course means they get one of the small ones.  If no students choose the half, I will "run out" of whole bars and one kid will have to settle for half of one.  When it is reveled that the kids who is getting half actually gets more, many kids are shocked but I think it leaves a big impression about making sure you notice the size of the whole.  

Does your curriculum include lessons about the size of the whole or ask questions where the two wholes vary in size?  

Chapter 6: How Come 1/5 is Not Equal to .15

I was shocked by the statistics presented in the "What's the Research" section of this chapter.  It really scared me that so many students had no concept of how fractions and decimals are related.  The fact that almost half of the sixth graders in the study couldn't write 4.5 as a fraction really made me question the way fractions and decimals are taught.  In most cases in a given year, I cover fractions first and then kind of glide into decimals after.  I try to make sure I am showing how they are related and do many activities that involve fractions and decimals during our decimal unit but I am thinking about how I can make the work with fractions and decimals even more explicit.  It also brings up the question about which topic to teach first and how much of one topic (fractions or decimals) I should teach before I start using both simultaneously,  These are good questions to ponder as I begin planning for next year.

In previous years, we had probability units in grades 4 and 5 where a lot of work was done with fractions and decimals between 0 and 1.  Now that we have moved to using the Common Core standards, the probability work has shifted toward grade 7 and we no longer do these units in grades 4 and 5.  I want to make sure my students are still getting some of these opportunities to work with fractions and decimals together.  I like how many of the activities suggested in this chapter are game based.  I am going to make sure we have a set of math stations next year that really work on this fraction and decimal connection.

Two activities I added a few years ago when we stopped doing the probability unit that I think can be really helpful and are very similar to the double number line lesson presented in this chapter are the Meter Stick Number Line Lesson and the 100 Bead String Number Line Lesson.  

How do you make sure your students have a good understanding of how fractions and decimals relate?  

Looking forward to reading your responses in the comments below!


  1. I love the Hershey bar activity--powerful! Just pinned it. I enjoy the perspective you are able to give from your experiences with students and fellow teachers at different grade levels. As a former fifth grade teacher, I surprisingly had many kids come in knowing procedures for "doing" math with fractions and decimals, yet they had no understanding of what a fraction/decimal represented. One of the best explorations for helping the kids understand how fractions and decimals (and percentages) were related was with a dollar and a hundred grid. My soon to retire teaching partner shared it with me. Kids understood a dollar can be made up of 4 quarters (1/4) and this was related to how the amount is written .25 (and how if you have a quarter it is 25% of a dollar). We used money and shaded the hundred grid to represent different fractions of a dollar. We also did it thinking about dimes (.10, 10%, 1/10) of a dollar. Students could later transition into the thinking of 4.5 as four wholes and a half (like four dollars and 50 cents). It seemed to really help the kids understand the relationship that exist between fractions, decimals, and percentages. We explored this each of the many years I taught fifth grade. Thanks so much for sharing this book and your thoughts!


    1. Hi Sarah,
      Thanks for sharing your ideas! I think your ideas about connecting money, decimals and fractions could benefit a lot of kids! I am glad you have had such a great teaching partner! My favorite teaching partner / mentor retired a few years ago and I miss her so much! It is amazing how co teaching can make you a better educator!

  2. I am really enjoying reading your posts - especially all of the connections that you are making to your own experience with students! Developing deep understanding of fractions is a complex endeavor. Students need exposure to many examples and counter-examples, as well as ample opportunities to engage in discourse, constructive struggle, reflection, and revision.

    1. Hi Julie! Thanks for stopping by! I have been enjoying your book and am looking forward to reading Beyond Invert and Multiply. I like your point about counter-examples. Sometimes I get so caught up in teaching and giving examples, I forget how powerful counter examples can be. Thanks for the reminder!

  3. The more I read this book, the more I feel as if I have been a horrible math teacher when it comes to fractions. I am guilty of being concerned with them knowing 1/2 is bigger than 1/3, but the author breaks it apart like the example with 1/2 the chapters and 1/3 of the chapters. If it is the same book then yes 1/2 is more but if one book has 6 chapter 1/2 is 3 and the other book has 9 chapter well that is 3 too, which now means they are the same. Wow! I am definitely changing how I teach fractions. Let's not even start the decimals, I fought all year with my 4th and 5th grade 1/2 is not 1.2 and so on. I am so glad to subscribe to your blog and find these books, because by the time I finish these 2 books and a book on math talks. My math students are not going to know what hit them!

    1. Hi Lisa. I know how you feel! When I first read A Focus on Fractions, I realized not only was I not teaching fractions very well, but I myself did not have a good conceptual understanding of fractions. All I ever learned were procedures and I didn't have the ability to explain why they worked to my students. I was asked on a pre-test to describe 5 different ways to compare fractions and I only knew 1. I am so glad that educators have taken the time to write books and share their knowledge with the rest of us. I love hearing your enthusiasm for math and for improving your teaching practices!

  4. I so excited to be your newest follower! Stop by anytime!