Mental Math: Addition

[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.