Lesson 5.1: Writing Equal Equations

What is an equation?

An equation is a number sentence with an equals sign. All of these are equations:

3 + 5 = 8

9 = 3 × 3

2 + 4 = 2 × 3

An equation is "balanced" or "equal" if the amounts on both sides of the equals sign are the same.

Children get used to equations being written in just the first format of the examples above, and they may not recognize the other equations as being true. Indeed, the older they are, the more likely they are to believe that you can't have an equation like 9 = 3 × 3 or 2 + 4 = 2 × 3 (K-1 children are the most flexible, grade 4-5 are the least flexible). That's a problem, because just as they get the most used to the first kind of equation (and the most convinced that you can't have the second or third kinds), is also when they meet up with algebra, and all of a sudden there are lots of equations of the second and third kind.

It seems to be that it's easier for children to learn how to use equations for algebra if they are exposed to a lot of different kinds equal equations starting n the early grades. This idea and this issue is the impetus behind standards like these two at the first grade level:

• CCSS.Math.Content.1.OA.D.7 Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.
• CCSS.Math.Content.1.OA.D.8 Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = _ – 3, 6 + 6 = _.

What is an un-equal equation?

When you're doing mental arithmetic, you often string together steps. For instance, to multiply 7 by 6, you might start by multiplying 7 by 5 and then adding on another 7. If you were writing this down as notes to yourself, you might write something like:

7 × 5 = 35 + 7 = 42

That's a great way to figure out the problem, but when you write it down that way, you get an un-equal (un-balanced) equation. Look at the equations in the number sentence above:

7 × 5 = 35 + 7

35 + 7 = 42

The second equation is fine: 35 + 7 = 42 is a true equation--the sides are the same amount

The first equation has a problem. 7 × 5 is the same amount as 35, and 35 + 7 is the same as 42, so that equation is saying 35 = 42

7 × 5 = 35 + 7
 \    /        \    /
   35  =     42

Those aren't the same amount, so that part of the equation isn't equal and balanced. It's like the difference between formal and non-formal language when you're writing and speaking. It's fine and useful to string things together in your memory when you're doing mental math problems, just like informal language is great when you're speaking out loud to a single person. When you're writing down your math ideas, you need to use equal equations, just like when you write a paper for school or give a speech to a large group you have to use more careful sentences.

Using the word "same" to describe what an equals sign does is often helpful. The word "equal" is generally tied to children's experiences with equals signs, whereas the word "same" is more flexible, and helps children who have a narrow idea of what the equals sign does be more flexible.

How can you fix an un-equal equation?

Most of the un-equal equations you write are probably of the mental math kind--ones where you are stringing ideas together. There are two easy ways I know to fix that:

Solution 1: break it down into smaller steps and equations:

With the equation:

7 × 5 = 35 + 7 = 42

If you just break down the two steps and write them separately:

7 × 5 = 35
and
35 + 7 = 42

you'll probably find that they are now equal and balanced!

Ideally, you should be quickly checking these equations to make sure that the amount on the left and the amount on the right are always the same, but breaking it down to a single equation for each arithmetic step will often solve the problem.

To use the writing metaphor--one way to fix a comma splice is by breaking it into two simnpler sentences, and one way to fix a broken equation is to split it into smaller equations.

Solution 2: include all of the pieces at each step, even if they are just place holders:

With the equation:

7 × 5 = 35 + 7 = 42

The big problem is that you were planning to add the extra 7 on all along, but you didn't write it down until the middle. If you write it down from the very beginning, even though you're not ready to use it yet, that will make the equation equal and balanced:

7 × 5 + 7 = 35 + 7 = 42
 \    /      |       \     /       |
   35  + 7 =     42    = 42
       \   /            |          |
         42          42       42

If you follow my reasoning going down, you'll see that the expresssions:
7 × 5 + 7, 35 + 7 and 42
are all equal to 42, so the equation is equal and balanced.

In the writing metaphor, this is like fixing a comma splice by adding an and or a but or change part of the sentence to a clause. Adding the missing ideas into each step is a more sophisticated way to fix a broken equation. It has the disadvantage of being longer and more complicated, and it has the advantage of being similar to equations children will encounter in later grades.

Both of these ways of fixing the equation work fine. There are other ways that work fine too (but are harder to put down in a typed document (even the not-very-good v's I used up above were hard to put in and make work)).

Teachers should:

• Model correct ways of writing equations (and try to avoid writing unequal and unbalanced equations)
• Ask children questions that use different forms of equations.

What are questions teachers can ask in grades 1-3 that use different forms of equations?

The Common Core Math Standards call for two (moderately common and well known) types of problem and experiences that use equation forms that are different from the standard _ + _ = __.

True/False equations

I have most often seen these done in a number talk type setting. The teacher shares a few equations, some of which are true and some of which are false. For example the teacher might ask which of these are true or false and why? (These would be an introductory set of true-false equations).

6 = 6

6 = 6 + 0

8 = 4 + 4

4 + 4 = 3 + 5

4 + 3 = 1 + 5

4 + 3 = 7 + 2

This is a set that starts easier and gets harder.

6 = 6 is fairly easy to accept, even though it's an equation in a different format than you are used to. In part because there are no operations to the right of the equals sign.

The next easiest would be 6 = 6 + 0. It's a different format--the addition is to the right of the "=" sign, but adding 0 is a pretty small step from just a number on both sides.

It's important to establish that you can have an equation where the operation (addition or subtraction) is on the right, and a number is on the left. Because 4 + 4 is a double it's a little easier to put together and think of as a unit.

The next two equations (4 + 4 = 3 + 5 and 4 + 3 = 1 + 5) help to establish that you compare the value of the left side to the value of the right side of the equals sign, and those values can be either the same or not the same.

The last equation (4 + 3 = 7 + 2) is the trickiest, because many children focus in on just the first number to the right of the equals sign, and would accept the equation as true because 4 + 3 = 7.

The goal of true-false equations is for students to learn what makes an equation equal--that the values on both sides of the equals sign are the same.

When reading true-false equations such as 6 = 6+0 out loud, it can be useful to read it using the word "same" in place of the equals sign:

is 6 the same as 6+0?

This helps children pay attention to the balance meaning of the equals sign.

Missing number equations

These are (also) often conducted in a math talk type discussion. Children mentally try to solve problems of these sorts and discuss how they got the answers. This can be followed by individual practice. Some missing number equations that might be posed in grades 1-3 are:

It's a good idea to include several varieties of ways of writing equations and of asking missing number equations.

What you need to know:

• How to write down equations with all of the equals signs being equal. You should be able to take a mental math description that a child gives and figure out how to write it down in a way that keeps all of the equals signs correct.
• True/False equations: what they are, what ones are easy, what ones are hard.
• Missing number equations: what different formats there are (you should be able to write down a good variety of problems including some easier and some harder problems).