# The Evolution of a Number Bond

__Number bonds__ help children develop an understanding of part/whole thinking and lead to working flexibly with numbers. Let’s take that idea a step further and make a case for just how important the concept of composing and decomposing is.

According to the CCSSM, Kinder kiddos should “fluently add and subtract within 5 (K.OA.5)” and “for any number from 1 to 9, find the number that makes 10 when added to the given number (K.OA.4)”. For more information on how to develop those two skills, be sure to check out the other __blog post__.

So let’s look at this example, which shows *one way* a 1st Grade child might determine the sum of 8 and 5 (“add and subtract within 20…use strategies such as…making 10” 1.OA.6) using the skills he learned in Kindergarten. The student knows they want 2 more to put with the 8 to make 10 (K.OA.4), and they know they can decompose 5 into 2 and 3 (K.OA.5). So now they are thinking of 10 + 3, or 13.

Let’s move on to 2nd Grade, where students are expected to “add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction (2.NBT.5)”. Now our student is faced with finding the sum of 37 and 25, which would normally be thought of as a “regrouping” problem. This student is still looking to make 10s, so they know they need 3 more to make 37 into 40. And notice that they are really still splitting 5, even though now it’s actually 25. After splitting, they’ve got an easy mental problem in 40 + 22.

Not sold yet? Let’s move on to fractions. In 4th grade, students must “decompose a fraction into a sum of fractions with the same denominator in more than one way (4.NF.3b)” and “add and subtract mixed numbers with like denominators (4.NF.3c)”. Let’s see how our number bond knowledge can help here. We split the 4/5 into 2/5 and 2/5 (4.NF.3b) to create a whole out of the 3/5, resulting in 1 2/5 (4.NF.3c). Seems a bit easier than finding an improper fraction and then introducing a totally different procedure for converting it to a mixed number.

How about measurement? Elapsed time maybe (3.MD.1)? Doesn’t this actually mirror what we do when we calculate elapsed time mentally?

Or adding and converting measurement units (4.MD.2)?

Notice that all of these examples follow the same format for recording the “splitting” process and that it all starts in Kindergarten! Which brings me to another point–the power of vertical conversations. Be sure you have a process in place to discuss math instruction across grade levels. If there’s not a formal process (vertical teams, professional learning communities, etc.), start one or just begin dialoguing with your peers on other grade levels.

I’d love to hear your comments, as well as other applications of composing/decomposing that you have used or might have thought of based on these examples.

Are you ready to get started with number bonds? ย Grab a set of __number bond cards__!

Your post was so eye-opening for me! Thank you so much for sharing!

Alex

theschoolpotato.blogspot.com

My pleasure, Alex!

Nice post Donna! I hadn’t thought of doing number bonds with fractions, time, and measurement. Wow, you make math so easy to understand and teach!

Thanks so much!! ๐

I love the way this post walks through the evolution of this skill. It really helps show how important it is to a students’ long term math understanding. Can’t wait to share this post with my colleagues.

Tara

The Math Maniac

Thanks, Tara! I think it’s so important to see the vertical view of any skill we teach.

I am a K teacher and this post was a wonderful refresher of how everything fits together. Your statement about the importance of a vertical team was right on a target.

Thanks, Sandi! Kinder teachers rock, and what you teach is so important!

Thank you for clearly illustrating ways that numbers can be composed/decomposed. I love that not only are you able to “see” math, you are able to explain it to others. This is one of the many reasons that your blog is one of my favorites!

Julie

TheHipTeacher

What a neat compliment, Julie! Thanks so much! ๐

I’ve known this was important, but not really why–this is amazing! Now I can explain to parents why their students need to know this. (And my colleagues too!)

Awesome!! It always helps to have parents on board, especially when the math is a bit out of their comfort zone.

As always…another amazing strategy you’ve shared to share with kiddos.

And…another mental slap to my forehead…now…why didn’t I think of that!

Thanks for sharing!

It’s not something I came up with on my own, trust me! These examples are a compilation of ones I’ve seen at different workshops over the years. ๐

I have just been introduced to number bonds and I had to google what they even were. I teach fourth and so far I haven’t had any students that know what they are. I do find importance in them because as an adult, so many things I “just did” make sense of what I did. This blog reminded me that I really need to introduce this to my students and develop them number sense in them. BTW my own daughter is in 2nd grade and she has had some experience so I do know they will come.

Right! It’s definitely a different way of thinking and new to a lot of people. Glad to hear that you’re daughter is learning this way in school. ๐

I have to say that although I appreciate your detailed explanation and examples, I don’t think this should be the only way that math is taught. I don’t always think in base ten. I don’t see any reason why we can’t teach students how to borrow or regroup. Sorry, but I’m not sold!! I strongly believe we need to teach children many different strategies to add & subtract. For years, we have used those terms. I don’t know why we are all of a sudden feeling the need to change language to compose and decompose. There are many students struggling with this concept now in the classroom and we’re not teaching them any other way to solve. Bigger Bottom, Better Borrow works just fine for me!

We absolutely DO teach the standard algorithm as well. We find, however, that when we teach it after experience with strategies such as those described in this post, students understand it better. I know when I learned to add and subtract, teachers did not do a very good job explaining why I “went next door and borrowed”. We were merely taught a series of steps and we performed them with little understanding. I think we have moved to a place that’s exactly what you believe in–exposing students to many different strategies, in addition to and including the standard algorithm.

Thank GOD for your website!! Have a 4th grade son with homework on this and I was stumped and he “couldnt remember”….again, but this explained it perfectly! Thank You

Josh – Quad cities, il

Thank you so much. I really enjoyed your post – you explained it so clearly!

Thanks! Glad to hear it made sense. ๐

This is my first year in 2nd grade and I have such a greater understanding of this whole concept after finding you! And attending a workshop in my district (First Steps in Math)… I do wonder how does this work for the student after doing mental math then we introduce this do they get confused?? I felt that I stressed so many mental ways with my class earlier so they would have a stronger number sense than I did and we just god done with regrouping/borrowing and I wonder does this conflict with the previous skills or will they use the one that works??

I love this post!!

Thank you

Chelsea

Kickin It Whole Brain In Texas

Hi, Chelsea! Mental math strategies help students build strong number sense and lead to a better understanding of the standard algorithm. The hope is that students will choose an appropriate strategy for the situation. For example, if I’m faced with 203 – 97, I could do a whole bunch of fancy regrouping, but it might be just as easy to count up. It’s all about flexibility!

I really appreciate the way you extended the number bonds to fractions, measurement, etc.

Great number sense building through decomposing!

Powerful! I never considered the validity of this idea as it has always been a mental struggle for me. But seeing it here makes it seem so much easier!

Enjoyed your article. I am going to discuss it at my next math vertical team meeting. It so clearly shows the importance of number bonds as a foundation skill that moves throughout the grades in math.

So glad i stumbled across this! Great illustration on how the bonds are applied outside of the simple math structure ๐ Thank you!

You’re welcome!

I tutor a dyslexic. She cannot learn this way. It is way too hard for her. I’ve been using clothesline math to help her see this through her lens.

I like the way you have demonstrated how important to teach and learn number bonds. I never thought this far to even to extend fractions and more! Thank you for sharing the way we can extend number bonds. Great way of building through decomposing!, Thank you.