Think About It!: The Equals Sign

Ask yourself and answer these three questions:

• What do you understand the equals sign to represent?
• What do you teach your students about the equals sign?
• What do your students understand about what the equals sign represents?

8 + 3 = 11

12 - 5 = 7

33 x 10 = 330

Most people understand the equals sign to show where to put an answer. When they look at the above number sentences they will likely interpret the left-hand side as the problem and the right-hand side as the answer.

A few weeks ago, I started having my classes work on basic number sense (since they still don't have it in 4th grade--a future "Think About It! discussion!). The very first thing we talked about was the equals sign. I asked them what they knew about it and all of them told me something along the lines of "it means the answer" or "it means 'is'; where you put the answer." I was a little taken aback and suddenly realized why they struggle with very basic problems such as 12 + __ = 20 or 7 x __ = 140. (Do your students struggle with those too?)

I have seen several recent studies that involve asking students the same questions about the equals sign. Let's, for instance, use 4 + 7 = __ + 2. On problems like these students would oftentimes add 4 and 7 and place the sum in the blank so it reads 4 + 7 = 11 + 2, completely ignoring the 2. I'm sure you as an adult (and teacher) look at that and instantly understand how the number sentence and thinking is wrong. Unfortunately, most students will not.

The equals sign indicates equality. It is placed between two things of equal value. Think of it as a balance. So when you say 8 + 3 = 11, you are saying that 8 + 3 has the same value as 11.

Can you see how knowing this in elementary school will help students better understand algebraic equations?? When students are learning about evaluating algebraic equations such as 3x + 7 = 28 they MUST understand that both sides are equal or they will never understand how to solve for x. The time to teach this is NOT in middle school but in the early grades.  

Students should also understand that one side of the equal sign does not always have to have a single number. The equals sign can show two expressions that have the same value, like 9 x 4 = 12 x 3.

I want to address one more misconception--the running equals sign. Here is a word problem and example:

Sally had 4 marbles. She cleaned her room and found 3 more marbles. Her friend then gave her 8 marbles. How many marbles does Sally have now?

Number sentence: 4 + 3 = 7 + 8 = 15

How is that wrong? Remember, both sides have to be equal. The correct way to write it would either include multiple number sentences or parentheses:

(4 + 3) + 8 = 15

OR

4 + 3 = 7

7 + 8 = 15

I wouldn't say repairing this misconception is a quick fix, but it is definitely doable. (And, of course, it would be better if instruction about the equals sign is correct from the beginning.)

• You can start by having a discussion about it with your kiddos. Bring in a balance to help you explain (for instance, put two pennies on one side, then add 6 more to it. Record that as 2 + 6. Then add pennies to the other side (8) and let your students witness how the trays become balanced. Record it as 2 + 6 = 8.)
• Another activity would be to have students use equality cards. You can make some simply by putting a number sentence with one missing number on one card and have the students match it to a number sentence in the same fact family with a different number missing (e.g., 6 + __ = 8 on one card and 8 = __ + 2 on the other.) Look for a free Equality Card activity in my Teachers Notebook store!
• You could also make a mat with an equals sign and have students place cards of equal value on either side of the symbol. The cards could contain pictorial models, standard form, expanded form, etc.

Sidenote: I'm not sure if you are familiar with the National Library of Virtual Manipulatives--if you aren't, you NEED to be! It is exactly what it states. The site has every manipulative imaginable in virtual form for FREE. It also has activities to go along with each.

To those that teach the higher grades--check out this balance that provides visual representation of algebraic equations:

Awesome, right?!?

Even if you don't teach the higher grades I encourage you to check out the algebraic equation scale and ponder about how easy it would be for our current students who do not understand an equals sign.

Think about it...

I always enjoy incorporating literature into math class. I have not actually read this book but it looks like a cute little story about creating equal sides. Check it out by clicking on the picture.

Thanks for tuning in. I WOULD LOVE TO HEAR YOUR THOUGHTS! THROW 'EM AT ME! :D

Also, I would LOVE to tell you to look for these posts on a certain day and time but I can't right now--life is not that predictable! Installment number 2 will be before school starts back though! :)