*[Photo by woodleywonderworks.]*

The question came from a homeschool forum, though I’ve reworded it to avoid plagiarism:

My student is just starting first grade, but I’ve been looking ahead and wondering: How will we do big addition problems without using pencil and paper? I think it must have something to do with number bonds. For instance, how would you solve a problem like 27 + 35 mentally?

The purpose of number bonds is that students will be comfortable taking numbers apart and putting them back together in their heads. As they learn to work with numbers this way, students grow in understanding — some call it “number sense” — and develop a confidence about math that I often find lacking in children who simply follow the steps of an algorithm.

[“Algorithm” means a set of instructions for doing something, like a recipe. In this case, it means the standard, pencil and paper method for adding numbers: Write one number above the other, then start by adding the ones column and work towards the higher place values, carrying or “renaming” as needed.]

For the calculation you mention, I can think of three ways to take the numbers apart and put them back together. You can choose whichever method you like, or perhaps you might come up with another one yourself…

## Highest Place Value First, with Funny Numbers

This is probably how I would do this calculation with my kids. We add the tens first, then the ones.

20 + 30 = 50, 7 + 5 = 12.

Answer = “fifty-twelve.”

Then, after we chuckle at the funny number, we convert it to the equivalent and more familiar 62.

## Move the Pieces

We imagine each number as a “pile” of blocks or stones or whatever, and we can move the pieces from one pile to another to make the numbers easier to work with. 27 is close to 30, so we might imagine moving 3 of the pieces from the 35 pile over to the 27.

27 + 35 = 30 + 32 = 62.

OR we could move 5 from the first pile instead

27 + 35 = 22 + 40 = 62.

We do this more often with small numbers, when my kids are first learning the addition and subtraction facts within 20. Then I tend to get out of the habit of thinking this way — maybe because I can’t actually visualize the piles — but the trick works for any size of number.

Of course, this method easiest when the number of pieces moved is small, so you may want to reserve it for problems with somethingty-eight or somethingty-nine in them:

29 + 35 = 30 + 34 = 64.

## If Only, If Only…

I forgot this method in the first draft of this post, but John’s and Pseudonym’s comments below reminded me. We look for an easy-ish (but related) calculation that we would rather do, and then adjust the answer. I will often pose the problem as a question:

“If only the problem was 30 + 35, that would be easier. Could you do that one?”

When the student solves that calculation, then we see how we need to adjust it to get the original problem. In this case, we added too much, so we will have to take the extra bits back off.

“We were only supposed to add 27, and we added 30. How many extra is that? So our answer is 3 too big. How can we fix that?”

27 + 35 = 30 + 35 – 3 = 62.

## Add in Chunks

This is much faster to do mentally than to explain on paper. Again, we start with the big place value, in this case the tens. We break one of the numbers apart (it doesn’t matter which one) into whatever chunks seem easiest to keep track of as we work:

27 + 35 = 27 + (30 + 5)

…and 27 + 30 is 57, so now we have:

57 + 5 = 57 + (3 + 2)

…because adding 3 will get us up to a friendly multiple of ten:

57 + 3 + 2 = 60 + 2 = 62.

As in all the other methods, we would not write out the intermediate steps, just keep track of them mentally. The kids might point to the numbers and say the steps, if they want, but more often they just mutter under their breath. Sometimes they stare at the problem as if they don’t know what to do, but if I wait patiently, I find that their minds are working behind those blank eyes, and the answer comes out.

If I get too impatient and offer a hint, my children usually let me know: “Mom, be quiet. I’m thinking!!!” But by that point, I’ve broken their concentration, so they have to start the problem over. No wonder they get mad at me for it.

Also:

using 10s, 27+3=30…30+30=60…60+2=62

(kids good with open number line can solve like this)

adding on 27+30=57…57+3=60…60+2=62 or just 27+30=57…57+5=62

good with 5s, 25+35=60…60+2=62

over and back 30+35=65…65-3=62

the compensation can be more gradual 27+35=28++34=29+33=30+32=62

(stop when you get an easy one)

Etc!

It’s great to see examples of how to help students be more fluent with their mental math. Too often, we emphasize the memorization of facts but neglect to take students to the next level–applying that knowledge along with understanding of math concepts to fluently manipulate numbers.

Mental math is crucial to computational development!

The problem with addition in your head is carrying. So one other approach is to avoid carrying by using a balanced representation.

Suppose that you could represent the numbers -1 to -9. I draw them with an underline, 1 to 9. You should also come up with a way to pronounce them, such as adding the syllable “buh” to the front.

The first thing you do is mentally convert any digit which is greater than 5 into this form. So 27 + 35 becomes 33 + 35, which clearly equals 62.

Once you have done this, most additions will not require carrying until the final stage when you convert your number back to standard form. So, for example:

38 + 26 = 42 + 34 = 76 = 64

Uhm… the underlines didn’t show up in that post. Let me try with bold:

Suppose that you could represent the numbers -1 to -9. I draw them with an underline, but I’m going to use bold here:

1to9. You should also come up with a way to pronounce them, such as adding the syllable βbuhβ to the front.The first thing you do is mentally convert any digit which is greater than 5 into this form. So 27 + 35 becomes 3

3+ 35, which clearly equals 62.Once you have done this, most additions will not require carrying until the final stage when you convert your number back to standard form. So, for example:

38 + 26 = 4

2+ 34= 76= 64Thank you all for the comments!

Pseudonym’s negative number approach and John’s over-and-back method reminded me of another way we handle mental math at our house — the “If Only” technique:

“Oh, if only this problem had been an easy one! If only it was 30+35 instead of 27+35…”

We do the easy problem, and then we adjust it to make whatever the original calculation was supposed to be. In this case, we added too much, so we have to take off the extra at the end.

This is interesting and helpful. Thank you for sharing.

FYI, even if you paraphrase, copying an idea from another source is considered plagiarism if you don’t identify the specific source.

I’m not sure about that definition of plagiarism, since ideas can’t be copyrighted — only the expression of them. I guess it depends on how close the “paraphrase” is and on how “original” the idea was. If I’m a fantasy writer, for instance, I can’t use hobbits in my story, but I certainly can use elves.

At any rate, in this article, I morphed a specific question from one person on one forum into a more general question that can stand for all the young homeschoolers who ask similar things on all the math forums I’ve participated in. I don’t think it matters which forum sparked the post. Does it?

The primary “idea” here is not in the question, but in the answer. Yet even in that, you could probably find similar ideas taught by

anyonewho is trying to explain how to do mental math.I discovered your blog not too long ago and I just wanted to say how much I LOVE your blog! The reactions you get from your kids remind me of my daughter’s reactions. We’ve been learning math mostly through games and life explorations. I’ve been wondering how best to take it to the next level when I found your blog. Thank you for sharing so many wonderful ideas! I’ll be checking in often. π

Hi, Min! Thank you for the encouragement. I’m glad you enjoy my posts. You may also like my PUFM series, where I discuss various mental math thinking strategies. I written about addition and subtraction so far, and the next post — whenever I get around to writing it — will include mental math strategies for multiplication.

Some plagiarism is legal, schools just don’t allow it (but you didn’t write this for a grade, now did you?), and includes ideas as well as specific phrasings. Copyright infringement is illegal, but, like you said, ideas can’t be copyrighted, so what you did was fine. Besides, the idea was so basic that even in schools it wouldn’t be plagiarism, normally speaking (it was a question, for one, and a very basic one at that). Even if you’d quoted the person verbatim, I don’t think any court would have convicted you for copyright infringement even, but I am not a lawyer. This blog post was educational, and there are educational exceptions to copyright law (part of “fair use”).

http://en.wikipedia.org/wiki/Fair_use

Hi Denise,

This is the first time I happened to see this blog. But I really appreciate your effort. I enjoyed reading this post, and I have already added your blog to the list of my favorite Math sites π

Thanks for this post, and I expect to find more interesting stuff. I plan to check in soon π