**IMPORTANT NOTE:** The moment alluded to below has arrived! *Which One Doesn’t Belong? *is now available from Stenhouse as a student book (awesome for home reading, too!) and a teacher guide.

As a result, I have removed all links to the version I was previously distributing free.

—

There are many shapes books available for reading with children. Most of them are very bad. I have complained about this for years. Now I have done something about it. Most shapes books—whether board books for babies and toddlers, or more sophisticated books for school-aged children—are full of misinformation and missed opportunities. As an example, there is nearly always one page for squares and a separate one for rectangles. There is almost never a square on the rectangles page. That’s a missed opportunity. Often, the text says that a rectangle has two short sides and two long sides. That’s misinformation. A square is a special rectangle, just as a child is a special person. After years of contemplation, I had a kernel of an idea the other night. The kids are back in school before I am, so I had some flex time available. One thing led to another and voilá. A better shapes book. (Links removed—see above note.)

## How to use this book

On every page are four shapes. The question is the same throughout the book—which one doesn’t belong? For example, which shape doesn’t belong in this set? If you are thinking, “It depends on how you look at it,” then you’ve got the idea.

- The bottom left shape doesn’t belong because it’s not shaded in.
- The top left shape doesn’t belong because it only has three sides, while the others have four.
- The top right doesn’t belong because it is the only square.
- The bottom right doesn’t belong because it’s the only one resting on a side.

Maybe you have different reasons for some of these. That’s great! The only measure of being *right* is whether your reason is *true*. **With an infant**, you can use this book like any other shapes book. Look at each page together. Point at each shape and talk about it as you snuggle. **With a young child**, ask which one doesn’t belong and why. Most pages in the book have at least one shape that a young child can identify as not belonging. Join the conversation by pointing out a different shape that doesn’t belong for some other reason. **With an older child**, challenge yourselves to find a reason for each of the 44 shapes in the book. There is no answer key. This is intentional–to encourage further discussion, and to encourage you to return to the book to try again. I have tested the file out on the Kindle app on my iPad, and it looks good. I made one printed copy and prefer it to the e-version because I can leave it out for browsing and we can touch the shapes without accidentally turning the page.

## The legal details

I owe thanks to Terry Wyberg at the University of Minnesota, who regularly plays the “Which one doesn’t belong?” game with numbers in professional development sessions; to Megan Franke at the University of California, Los Angeles, who adapted the old Sesame Street game “One of these things is not like the others?” and to my online colleagues including but limited to Justin Lanier, Megan Schmidt, Dave Peterson, Matt Enlow and Andy Rundquist.

# Some additional prompts

The following *Which One Doesn’t Belong? *prompts are yours to use in your classroom or home. If you’d like to share them more widely, please link people here. Thanks.

I love it!

Are you going to publish an ebook version, too? That should be easier to get going than a print book.

Denise: Maybe. In a sense, the ebook version is published right here, since any e-reader will open pdfs. I am thoroughly convinced that the best medium for children’s books is print. I suppose you could argue that having the e-version available on (e.g.) Amazon would mean wider circulation, and I’ll take that argument under advisement. Thanks for your support!Yes, wider circulation was what I had in mind. Especially since it takes time to get a book into print, but not so much for an ebook. I’ve shared this with my local teen math circle. Unfortunately, it’s only a virtual group — I wish we were still meeting in person, since it would be fun to hear the kids talk about the puzzles.

I had to laugh at your comment. All I could think was “same reason a physical book works better!” Thanks for the giggle 🙂 – Christopher’s neighbor

I knew almost without looking that you would make it so that there was at least one good reason to exclude each figure. For some reason, that reminds me of non-transitive dice : whether you can pick the “winning” die/dice is less clear-cut than it might at first appear to be.

At any rate, I’m sure this will be very useful and loads of fun. Maybe shake up a few people who assume that there’s always one right perspective, too. 🙂

Thank you so much!

Sent from my iPad

>

How timely! I am working with a group of kindergarten and first grade teachers. This will be a wonderful resource to engage students in deeper thinking about shapes. Also, looking forward to your book that I bought last fall. Keep up the awesome work!!

I really love this! One finicky question (not even sure whether you know the answer yet) – when you do eventually print, is there any chance you’d do a board book version, or are you just thinking paper pages at this point?

I am definitely considering a board book version!

Love it! One quick thought – I wonder if the pages that are red/green heavy might be tricky for colorblind kids (I had a couple in my class last year who might have had trouble with them). But I do love it! Honest!

Thanks, Lesley! I thought about this and paid attention to making sure (1) red/green was never the basis for a “difference” (as red/blue is p. 2), and (2) that the values contrasted, not just colors. I look forward to hearing from someone who knows better than I whether the contrast is enough without being able to distinguish the colors. Thanks for helping me think this through!

Look at websites like http://www.color-blindness.com/coblis-color-blindness-simulator/ to evaluate whether the value differences are sufficient.

This is brilliant! I just sent it on to my current kindergarten teammates and my former first grade teammates. I can’t wait to use this with my kinders. The conversations we will have are sure to be fascinating.

This is awesome (as expected coming from you) and I love the idea of finding a reason for each shape!! I can’t wait to use them with my classes and have them practice “talking math”.

Thank you. I can’t wait for the board book

my idea for another page:

friendly curves from polar coordinate graphing

By the way,

Joshua, I love these very much and am inspired. Thanks for sharing them!Another potential extra page:

recursion fractals

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What a relief to hear the words “it depends how you look at it.” Sometimes I think phobias about being Wrong come from that big red X splashed across math papers. When I practice the art of looking at things from multiple perspectives I can feel my whole field of knowing expanding. When I can treat a wrong answer as an opportunity to see where I can expand my understanding, profound growth happens. I haven’t looked at the PDF yet, but if it’s anything like your blogs I am sure it will knock my socks off.

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Love this!! Thank you! I’m adding to my Kindle and sharing with my daughter’s preschool teachers!!

This is AWESOME! Just posted on my Facebook page!

Thank you for this! As a math educator, I actually try to avoid shapes books with my kids, and always cringe at all the misinformation when they manage to get me to read one. Can’t wait to print yours and share it with my boys!

I love this! Thanks for making it available. Hope you are published soon!

Love this book! I spent some time looking at it with my 2 year old and 4 year old last night and have also shared it with a class of first and fifth graders this morning. Amazing! Definitely the only “shape book” I can use with such a wide range of ages. This really got my students talking about attributes of shapes! I would love to feature this book in my Monday Math Literature series on my blog!

Tara: Great to hear that the age range is working out for you! Please do feature in your Monday Math Literature series, and shoot a link back to me, OK?Pingback: Student Created Shape Books | Educational Aspirations - Matt Coaty

Christopher – I’m excited to share this book with early childhood educators! I have looked over dozens of shape books and found so many disappointments for the same reasons you mentioned.

I look forward to checking out your site!

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Reblogged this on How To Math and commented:

I am so excited to repost this blog. Christopher Danielson who writes Talking Math with Your Kids has created a super shape book that is accessible for all ages. What’s great about the book is that there are no right or wrong answers. This book is all about explaining and justifying your thinking. I can’t wait to share this resource with my kindergarten teachers who are just about to begin their geometry units!

When I first read the post I was sitting next to my own kindergartener (the one who I take home every night) and I thought, “Hey, I’ll try this out on her.” We scrolled through each page and had a great conversation around why we chose each shape. It was interesting, when I disagreed with her, choosing a different shape for a different reason, she was pretty willing to go along with my idea. I asked her, “Who’s right?” and she quickly said, “You must be, I guess!” It took a few pages of convincing her that we could both be right and by the end it was a bit of a game to see just how different our thinking could be. I especially appreciate the developmentally specific prompts given in the post so families and teachers can use the book with varying age groups. I’ve included some additional supplemental pages on my Downloads page if you want to add to the book.

When I printed the PDF the pages came out, with a border, to be a 7 3/4 square so I tried to size my supplemental pages to fit the originals.

Please take the time to read more posts from Talking Math with Your Kids. So many goodies!

I am so glad I saw this! Thanks for the pdf, too! I must reblog this super idea!

Reblogged this on Teachezwell Blog and commented:

Which one doesn’t belong? What if EVERY answer were correct? This is a fascinating post with a free download of Christopher’s book on shapes. Intriguing and useful for many grade levels.

Thanks a lot for the shapes book!

Our five year-old discovered it by chance after I had printed it out for a rainy day. Went through it twice and I didn’t have the heart to stop him for explanations most of the time as he was apparently enjoying to just look at the shapes and picking one that he thought stood out.

One discussion was fun, though. He picked a regular shape (five corners? six corners? something like that) among more twisted looking ones and said that one stood out because the others were art. Why are the others art? Answer: Because one doesn’t see them as often. Now I keep thinking about his definition of art 🙂

Another observation I found quite interesting: After he picked one standing out because of its colour, he ignored that criterion on the next page where he could have used it again and came up with another one.

In any case, that book unexpectedly gave me some brain food, too. So thanks again!

My favourite conversation so far:

on one page, there was a shape for which neither my son nor I could find a reason for it to be the odd one out. I suggested the following: it doesn’t belong because it is the only one for which we can’t find a reason it doesn’t belong. He thought for a moment and said, excitedly: no, that can’t work! If that is the reason, then we have a reason that it doesn’t belong, so then property isn’t true! But if it isn’t true, then we still haven’t found a reason, so it is true. That’s like “all englishmen are liars” (referring to a snippet from Number Devil where Bertrand Russell is musing about self-referential statements).

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One more idea for a page:

Numbers

I printed a couple pages and had a nice conversation with my toddler. Also, I’m teaching linear algebra this semester and am going to adapt this idea and have my students give, say, four matrices, and then for three or four of the matrices, explain why (using concepts from the class) it doesn’t belong.

So, your basic idea works with three-year-olds *and* engineering majors…!

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Christopher — I am so looking forward to edition 2: now with pexagons edition! 🙂

Elizabeth (@cheesemonkeysf)

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Love it! Great conversations with the kids.

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I can’t download the pdf from the link! I’d love to read the book. Any suggestions?

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