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More Mental Math Strategies

After a recent post on addition strategies, I had a reader email me and ask about subtraction strategies. She was working with her daughter on homework, and while she understood better how to help with addition after reading the post, she was a little perplexed about how subtraction would look.  She also wondered about addition with larger numbers.

Here are some of the problems she posed to me and my responses. Remember, the whole point about strategies is that there are multiple ways to solve any given problem. The ways I’m discussing are not the right ways, and you might have others.

The first problem was 860 – 389. My reaction to this problem was that I would solve it by adding up to 860 from 389. Understanding the relationship between addition and subtraction is a powerful strategy.  Here’s how I did it:

You could also jump hundreds first (389 + 400 = 789), then jump to the next hundred (789 + 11 = 800) and then jump the last 60. That’s just not my preferred method–you do what feels right to you!

The next one was fun and called for a new strategy I haven’t yet discussed–compensation. The problem was 312 + 498.  Now, because I noticed how close 498 was to 500, I thought of using compensation. Basically, with compensation, you change one of the numbers to a friendly number, but then you have to change the other number as well to compensate for the fact that you changed the first number. Since I added 2 to 498 to make 500, I had to subtract 2 from 312 to compensate. So in my head, I’m thinking 500 + 310.

Now, take a few minutes to think about how that works. It might help to sketch them on an open number line.  Can you use compensation for the following problems?

495 + 187

380 + 442

311 + 292

Hmmmm, I wonder if compensation works for subtraction? Well, it does, but it works a little differently. Say I want to subtract 495 – 187. Can I add 5 to 495 and subtract 5 from 187 and get the right answer. I’ll give you a minute…

Didn’t work, did it? For subtraction, you have to change both numbers in the same way. So if I add 5 to 495, I also have to add 5 to 187. Now I’m subtracting 500 – 192. At this point, I’d shift into jumping-up mode. I’d jump from 192 to 200 and from 200 to 500. Cool beans…two strategies on one problem!

That’s enough mental gymnastics for tonight. I’d love to hear your comments!

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  1. I absolutely love these strategies. I teach third grade and I teach using these but I catch a lot of flack from traditionalists. They are concerned students won’t use the algorithm. Everything is about theat and I am more about conceptual understanding and good number sense before moving on to the algorithm. What is your take on that?

    1. Sometimes we are educating parents along with the students! Many parents are very resistant to strategies like this, so we just have to help them understand exactly what you said–this builds conceptual understanding and number sense, which means that when they do use the traditional algorithm, they’ll understand it better.

    2. I think the problem with this is that our kids should be taught the basic math strategies before we start trying to teach them any other algorithm. While it might be fun and exciting to an educator because they might be able to flex their mathematical muscles, it is a disservice to the kids if the parents aren’t able to help their kids with their homework. Kids should be able to learn the simple structure of math first like 34-14=20 without having to make it more complicated than it really is.

  2. Hi Donna, I have found and used several strategies from your blog and shared with other teachers. I like this method of subtraction! I discussed one with my coteachers at our meeting today using the Magic 9. Have you heard of that? I just posted about it on my blog, check it out on http://www.sanders6thgrade.blogspot.com
    I will keep checking your blog for all your wonderful tips and ideas! Thanks!!

  3. I once heard subtraction is distance on a number line. It’s an interesting way to think about it.

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