No, that’s not a typo in the title. This post is about a book that explores situations represented by equations such as 1+1=5 and 1+1=0. And I’m not talking about modular arithmetic with that second example.
I’ll let the first pages of this Monday’s Math Book Magic selection demonstrate how these unlikely additions add up.
The picture book 1+1=5 and Other Unlikely Additions by David LaRochelle was published by Sterling in 2010.
Brenda Sexton’s wacky digital illustrations provide crisp images of a variety of objects for children to count and/or add.
Here is the first page.
What do you notice? What do you wonder?
Do you see how one plus one might equal three? And no worries if you’re answer is not yet. I’ll post that page turn below.
In addition to 1+1=2, this book has twelve more 1+1 equations for children to puzzle over. When would 1+1=14? Could 1+1 = hundreds? Oh yeah, remember that 1+1=0, can you think of a situation for that one?
Let me be clear, this book is NOT trying to confuse children into thinking that the addition statement 1+1=3 is true! Of course the expression 1 + 1 has one and only one value, 2. In order for the equation 1+1=3 to make sense, we need to add in some units.
Ok, here’s that page turn I promised.
For this unlikely addition, what we are really talking about is:
1 (unicorn) + 1 (goat)= 3 (horns), OR
1 (unicorn horn) plus 2 (goat horns) equals at total of 3 (horns)
The items in the parentheses are units. To get the 1+1=3 equation to work out, the 1(goat) in that first equation is really a composed unit of 2 goat horns (as goats have 2 horns).
The rest of the book follows the same format. First, an equation for 1+1 is offered along with an illustration to support the equation. Then on the next page the units of measure are added into the equation to clarify.
Whether children are figuring out how the equations work for themselves or checking the answer after the page turn, there are opportunities to count and add.
In addition to counting and adding, a key mathematical concept involved in these unusual additions is that of the unit or as Danielson describes, answering the question what counts as one?.
Below is a scenario and explanation related to this idea of unit as an answer to the question of what counts as 1?.
While grocery shopping last week, I told my children they could each pick out one yogurt for dessert that night.
Liam scurried up behind me with a huge grin. “Mom, I got my ONE yogurt.”
Unfortunately, he knew that he would be trading in those 8 containers of yogurt for 1 container. [By the way, this interaction happened to be after reading this book. I’m curious as to whether the book inspired his creative use of ONE.]
Below are some different quantities of yogurt:
1 carton of yogurt
1 six-pack of yogurt
1 four-pack of yogurt
1 6-ounce container of yogurt
1 bite of yogurt
All yogurt. All 1. However, since all of these ones are relative to different units, the quantities are different. What counts as 1 depends on what unit you are talking about.
The idea of unit is key to understanding both place value and fractions. For example, in the number 1131.1, each 1 represents a different quantity. On the face of it, they are all 1, but the value each 1 represents is different. From left to right, 1 thousand, 1 hundred, 1 one, and 1 tenth. Since all of these ones are relative to different units, the quantities are different.
“[Our] whole number system depends on being able to change what we count as 1. Our whole number system depends on being able to change units.” [Danielson, TED-ED video. If you have 4 minutes to think more about units, you might watch this video.]
I read this book at the same time to my daughter (5) and my son (7 1/2). They both had very different reactions.
Before I even opened the book, he exclaims: “What? That’s wrong. 1+1 is 2 not 5.”
As I flipped through the first three or so situations, he wasn’t buying it.
“That’s not right.
“No. It’s not.”
Having a lot of experience in first grade with addition, Liam had difficultly getting over the false equations (e.g., 1+1=3) on the pages even after the units of measure were added in (e.g., 1 unicorn +1 goat = 3 horns).
But the more we flipped through the book and he counted and checked, the more interested he became.
“Oh, I see what they are doing. It is all different ways.”
“They add the words. …So you know what you’re talking about.”
I thought this was a great takeaway. The units “let you know what you are talking about.” He was right that the unit-free equations made no sense. But add on the units and Liam was able to see how 1 isn’t always 1 and how 1+1 could make 3.
I was hesitant at first sharing this book with Siena. She’s still making sense of addition and I didn’t want to confuse her. Unlike Liam, she can’t offer answers to basic additions like 2+3 and 6+8.
About five pages into the book she asked, “Why do they all say mysteries?” (Love this way she has for describing questions.)
I replied, “They’re wondering what you think.”
“Oh,” she replied.
Since she has less experience with addition facts, she didn’t have in her mind that there is one answer to 1+1. But she also didn’t think that there should be many answers. Later I came to find out that she thought that there should be exactly two answers. Here is her reasoning:
“1+1 should only equal 2 or 11.”
Here’s her explanation for a sum of 2:
“When it is one plus one it equals two ones.“
I found it interesting how she denoted the unit (ones) after the two, but not after each of the ones. It made me wonder and want to learn more about how understandings of units develop in children both from educational research studies and from listening for this in future conversations with Siena. An example of something I am wondering about is whether Siena thinks about all whole numbers as collections of ones?
Here is her explanation for a sum of 11:
1+1=11 because “one and one together is 11, as a number.”
Her response provided some insight into how she was currently making sense of addition and double digit numbers. I’m curious to chat with her more about those ones in 11 in the coming months as she gains more experience with counting, adding and begins to learn about place value in kindergarten.
If I had given into my nervousness and not shared this book with Siena, I would have never heard this wonderful sense she was making about number and her thoughts about how she thought 1+1 could be two different things.
That being said, if I had a child that was in the throws of working on single digit addition in school, I would probably wait to share this book with them. But that is just my opinion. Perhaps you have a different one (and if you do, please share in the comments or email me at firstname.lastname@example.org and I’ll love to discuss more!).
1+1=5 and Other Unlikely Additions by David LaRochelle is a great book for exploring units, addition, and counting with children. I hadn’t seen any book quite like it before. That being said, just a few weeks ago I did come across a tweet by @jnathanedmonds that reminded me of this book. Jonathon Edmonds suggested to take a counting book you already have and remove the unit. I love how Edmond’s idea provides another way to engage children in specifying what and how they are counting. I think you could even add a post-it on this page with 3______? Lots of counting and unit naming opportunities here.
I am off to get my post-its. But before I sign off, I have 1 more equation I need some help with.
1 blog+ 1 community= MANY matchbookmagic ideas.
I’d love it if you could send some of your mathbookmagic ideas my way! Here are 1+1+1=3 ways you can do it.
- Tweet me at @KellyDarkeMath using #mathbookmagic. Title, author’s last name and why is it magical?
- Go to Contact page and follow the directions there.
- Email me at email@example.com with the following info:
- Book Title
- Grades/ages you have used the book with; and
- A bit about why you find the book magical