# Balancing Act: Understanding Equality

Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. ย CCSSM 1.OA.7

That’s right, understanding the true meaning of the equal sign–it does not mean ‘the answer is coming’–is a 1st-grade common core standard. See this blog post for more about the 1st-grade standard. But what about your 3rd graders? Or 4th or 5th graders? That standard wasn’t around when they were in 1st grade.

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I made this little activity for my 5th-grade/3rd-grade peer tutoring group, and I introduced it by just showing them the orange strip (no dominoes) and asking what the equal sign means.

The most common response was (very confidently, I might add) that we would “add the first two numbers to get an answer and then add the last number on.” Not surprising. Also not right.

I explained to the kids that the equal sign is like a balance, and then I held out my arms to the side with my flat palms facing up to illustrate. I told them that it simply means that whatever is on one side of the balance has to equal what’s on the other side.

If I have 3 + 2 on this side, how much is the side worth?ย Five. Right!ย So if I have 3 + 2 on this side and 4 on the other side, do they balance? No. (I showed with my arms how the 5 side would be ‘heavier’) So what would we have to add to the 4 side to make the two sides balance? One! So, if I have 2 + 3 on this side and 4 + 1 on the other side, they balance? Yes. How do you know? Because 2 + 3 is 5 and 4 + 1 is 5. Oh! Good thinking.ย

We did a few more examples like that with one-digit numbers, and then I introduced the activity pictured below. It was actually one of my 5th-grade tutors who realized she could place the orange strip at the top of the paper and put the computation underneath. Then you just move the strip down the paper as you do additional problems (see the second picture). The dominoes are used to make 2-digit numbers, and the empty space can move around. Notice that underneath we practiced some of the alternate strategies the kids are learning in their number talks.

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It was really cute watching the 5th graders teach their 3rd-grade tutees this very difficult concept. Grab your copy of the mat here. Just copy it on colored cardstock and cut the strips apart to use. I’d love to hear your comments!

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1. Love the focus on the equal sign! So very important. Have you read “Thinking Mathematically” by Carpenter, et al.? Lots of good research support there for the work you’re doing.

I am not sure I understand the place-value dominoes, though. Dots usually signal quantity for me, but I need to ignore quantity to some extent when I am thinking about place value (which is a numeration idea). What do the dominoes get us that cards with number on them would not for this activity?

1. Donna Boucher says:

I will definitely check out that book! To be honest, the dominoes were just sort of a random number generator. As we were working with them, though, we talked about them in terms of place value. For example, on the first domino pictured showing 6 and 3, we discussed that the 6 was 6 tens, or 60, and the 3 was 3 ones. or 3.

1. Dwayne Takeda says:

One more thing… I don’t teach elementary school; I’m in middle school.
Is teaching the addition of two digit numbers horizontally standard? To my mind, it makes more sense to get all those tens digits in the tens column and all the units digits in the ones column. Just a thought.

1. Donna Boucher says:

Horizontal is becoming more common because it promotes strategies such as those you see in the picture–combining tens and then ones. These strategies lead to the standard algorithm, which is vertical. But through working horizontally, students better understand why the digits must line up when working vertically, rather than just memorizing it as part of the process.

2. Dwayne Takeda says:

I agree with this. There aren’t 44 dots; there are 8. There’s no place value assumed on dominoes like there is with written numbers.

1. Donna Boucher says:

The dots do not represent ones, so they are not counting the dots. Some dots represent tens while other represent ones. It’s similar to having students roll two number cubes and use one numeral represent the tens and the other represent the ones.

2. TheElementary MathManiac says:

Love this! You could totally do this with kids of almost any age. With 1st and 2nd graders you could use the quantit of dots on the dominoes and you could also vary the difficulty of the numbers by using double six or double nine dominoes. Thanks for sharing!

1. Donna Boucher says:

Yep, absolutely!

3. Teaching in the Tongass says:

I just found your blog and am enjoying reading your posts! I live in Juneau and a few years ago we had a math consultant work with our district (Nancy Norman) and I learned so many of these same strategies from her (and she turned me on to all of the books you have listed, plus Fosnot’s Context for Learning)! Unfortunately, we went the way of Math in Focus, and it’s just not the same. Anyway, I’m happy to see there are more people like Nancy spreading the math gospel!

Teaching in the Tongass

1. Donna Boucher says:

It sounds like Nancy and I would get along just fine! Hopefully, you can find a way to sneak some of the ‘good stuff’ into your Math in Focus curriculum. ๐

4. Jill Sapoznick says:

I am wondering if students get confused by the quantity of dots on the dominoes and how they are thinking about those dots in relation to place value but I think it could be a great way to show 17 made with two dominoes on the left side of the equal sign and then a different pair of dominoes that make 17 to say true or false and why. Another way then would be the way shown in the post with a missing addend and the students have to draw what the domino could look like for the missing domino.

1. Donna Boucher says:

These were older students using this activity, and we discussed the two numbers on the dominoes as representing tens and ones. It definitely requires an understanding of place value, but no more so than knowing that in the number 54 the 5 represents 5 tens. I like your ideas for variations!