Life is No Fun When You’re a Remainder of One

It’s a joy to read stories written well in rhyme.  However, crafting a good story is hard enough, much less one that uses good rhyme. And writing a good math story in rhyme, even harder. Here’s a rhyming math picture book that’s definitely worth a read.

The Book

Written by Elinor J. Pinczes and illustrated by Bonnie MacKain, A Remainder of One has been a classroom favorite since its debut in 1995.

The story follows Joe, a left out bug from Squadron 25 (Note: There are 25 bugs in Squadron 25). The Queen bug doesn’t appreciate when an array of marching bugs is not complete. Squadron 25 begins marching in two lines, with poor Joe left as a remainder of one. Whether they march in 2 rows of 12 or 3 rows of 8, Joe is left out. In the end, Joe finds a way to fit in as Squadron 25 marches in a 5 by 5 array.


MacKain’s lyrical, linocut-style art provides a textured backdrop to engage the eye and complement Pinczes’ rhyming text. The whimsical bug characters in warm pastels invite the reader to join in the array-counting fun and route for Joe, the remainder of one.

The Math

Joe is one among 25. How many ways can 25 bugs be arranged? Here are the decompositions explored in the book:

2 rows of 12 plus 1 (Poor Joe)

3 rows of 8 plus 1 (Poor Joe)

4 rows of 6 plus 1 (Poor Joe)

5 rows of 5 (Proud Joe!)

Remainder of One provides opportunities to talk about division, remainders, multiples, and factors. For example, what if Joe was part of the 12th Squadron with 12 bugs, would Joe still be left out? Do the patterns with 25 bugs occur for every odd number or is there something special about 25?

Number theory explorations abound in this bug-filled book.

The Magic

As I mentioned before, reading rhyme is fun and not only that, like a song, rhyme stays with you. Siena (5) continued singing the refrain below the rest of the day.

“Hup, two, three, four!

We’re in the 25th Army Corps.

Queen’s count! Two, three!

We are the marching infantry.”

Also, Liam and Siena noticed a color pattern in the 25 array pictured below, that as a reader I did not. Reminding me to pay better attention to the pictures when reading picture books to the kids. After all, that is half the fun!



Lastly, after the 4 by 6 array was presented in the book, Liam covered up and pointed to parts of the illustration and said “The next one will work?” I asked him why. He proceeded to cover subsets of bugs with his hand explaining his method. He went so quickly, I had him repeat it.

Here is a summary of what he did. Think of the colored rectangles as moving boxes that move the bugs to different locations.


LIam's method 1
Move the two bugs in red rectangle to the red circles. Now the two left columns have 5 bugs each.
Screen Shot 2017-11-06 at 10.17.49 AM
Move the two bugs in the blue rectangle to the blue circles and two in the green rectangle to the green circles.

Here is what there is now: Joe + 4 columns of 5 bugs + a group of 4 bugs in upper right. Note: There are no bugs in the green and blue rectangles after the moves described in the caption under the picture above.

Now move the two right-most purple bugs according to the purple arrow.

Screen Shot 2017-11-06 at 10.25.17 AM

Joe fits in at the star.   Hooray! The 25th squadron makes a 5 by 5 array.

Lastly, as I was searching for this book, I came across this version that comes complete with bugs! How fun is that! Or you can add your own bugs like the ones here. Just be sure to order two packs since they are 14 bugs to a pack.


Have a magical math book you’d like share? Please go to the Shared booklist to find out how.  If you’d like to receive these magical math book posts each Monday, be sure to follow this blog in the side bar of this page.

Thanks and see you next Monday! #mathbookmagic

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