Open Equations - Getting Students To Solve Equations Creatively

I ended my last blog post wishing that my students had the chance to solve equations creatively. I think I have found a structure (heavily influenced by Chapter 5 of the book Thinking Mathematically) that will give students this opportunity.

I am going to call these open equations. Here’s how they work:

I project an equation, such as a + 5 = b + 8, on the board.

I ask students to spend a few minutes finding at least three sets of values for a and b that make this equation true.

Then I give the students an additional few minutes to compare their answers and look for any patterns in the responses.

Then I lead a number talk with the whole class as we discuss possible solutions and any patterns that emerge.

I like this format for several reasons:

  • The right side of the equation is an expression instead of a numerical value. This is intentional. I want students to stop thinking of the equals sign as the place where the answer goes and start thinking of it as a symbol that connects two equivalent expressions. I want kids to think “Whatever I make a and b, I need to make sure that a + 5 has the same value as b + 8."
  • The fact that this is a two-variable equation gets kids away from finding the answer and towards finding some answers. I think this is an important distinction because it leads kids to look for patterns within their answers, something that students rarely do when solving a problem like 2x + 3 = 15. In this problem, students might think “Wow, it looks like a is always 3 greater than b. Why is that?" 
  • This practice of finding pairs of values that make the solution true will be quite valuable when students begin learning about functions. Graphing a linear function just becomes another way of representing all the possible solutions for an equation. Which is exactly what the graph of a function is!
  • In this format, the way students look at the expressions a + 5 and b + 8 is different than the way that students often look at the expression 2x + 3 in the problem 2x + 3 = 15. In the latter problem, students often view that expression as a series of steps that need to be undone. They go operation hunting in order to solve the equation without considering that 2x + 3 is an object in and of itself. I want kids to think of a + 5 and 2x + 3 as things. That’s how I view them and how I think other fluent algebraic thinkers see them as well. Fluent readers see sentences as things as well as seeing each component of a sentence on its own. Fluent writers know how they can manipulate the components of a sentence and still maintain its original meaning. I want my students to be fluent algebraic thinkers. I want them to see the forest and the trees.

I’ve given one simple equation above, but there are a lot of other examples that I think could help students with any number of math topics.

Are students simplifying their expressions incorrectly? Give them the open equation a - 7 + 10 = b and ask students why b is always 3 greater than a.

Are students having trouble combining like terms? Give them an open equation like a + a + a + a = b + b + b. Students will pretty quickly move from adding to multiplying a by 4 and b by 3. If you want, there is a bonus lesson about proportionality in this equation!

Want to explore the distributive property?  Give students the open equation 2(a + 5) = 2b + 10 and ask them to find three pairs of answers. Why are the values of a and b always the same? Could I rewrite the equation as 2(a + 5) = 2a + 10? Can any students write another open equation where a and b are always equal?

Most importantly, I think that this format becomes more successful the more frequently it is used. I use them as warm-ups and like to think of it as a slow-motion number string, but you could work through several equations in a row if the need arises.

I have many, many more ideas for how open equations can be used as a gateway to solving equations. And I will be writing a lot more about those ideas and fleshing them out online and in my classroom.

But the purpose of open equations is not to get students to be better at solving equations. The purpose of open equations is to get students to understand equations. Once they understand how equations relate two equivalent quantities, then they can build their own strategies to solve problems like 2x + 3 = 15. And when we move from informal to formal strategies, they will have a basis for understanding those formal strategies and why they work.

Let me know what you think!

Stop the Operation-Hunting

I’d like to talk about an incredibly common mistake that students make when simplifying expressions. And I’d like to talk about why I completely empathize with students and see why they must be so flabbergasted when this problem is marked wrong.

First, the mistake.


The amazing thing about this type of mistake is how insistent my students are that they have not made a mistake at all. For the most part, my students feel ill at ease when simplifying expressions, but this problem feels like sturdier ground to them. And yet it is wrong! How can this be?

I think that students’ confusion comes from a very understandable place: order of operations. When most students learn about order of operations, they learn it as a process of operation-hunting. They look for each operation in turn and then grab the number to its left and right in order to evaluate the problem. Normally, it looks something like this:

Why do you multiply 5 and 4? Because those are the numbers connected by the multiplication sign, of course! Ditto for the subtraction and addition signs.

So a couple of days or weeks later, you ask your students to simplify 5x - 7 + 10. They start their operation hunt. First they find the implied multiplication of 5 and x, but since that can’t be simplified, they move on. Then they find the subtraction sign, but since 5x and 7 aren’t like terms, they can’t be subtracted. So all they have left is addition! Grab the number on the left and right and evaluate! Simple as pie.

This is why kids are so baffled when I mark this answer wrong. It feels like a repudiation of their well-earned knowledge about the order of operations. And it is! At least, it’s a partial repudiation. It tells students that they cannot simply hop from operation to operation, excusing one’s dear Aunt Sally until the final answer has been derived. And this tension between order of operations and the fundamental properties of math is hardly addressed.

In the past, I have tried to remediate this issue by drawing boxes around the problem like so:

This doesn’t really make sense to students. First of all, since when can you put a box around part of an expression? The“- 7” doesn’t look like a valid math expression and neither does “+ 10". It's a fundamentally different way of looking at the problem than the way that I've taught them to look at 5*4 - 7 + 10. But they go along with it, I suppose, because I am big and loud and insist that this is the cool new way to think about simplifying expressions. Forget what I said in late August! It’s early September, for God’s sake!

But really, does this make any sense? Would it work with the aforementioned order of operations problem? What would you do if you saw one of your students with this work on their paper?

The curse of operation-hunting is the problem, and it must be stopped. We need students to learn the order of operations, of course, but we need to spend much more time working with kids to figure out when the order of operations can be overruled, and why.

We need kids thinking creatively about simplifying expressions. 

But how in the world do we do that?

This post is the second in a series.

The Two Types of Math Mistakes

I want to talk about the two types of math mistakes I see in my class, and why I get so excited by the first type and so depressed by the second type.

First, the mistakes. 

Problem 1: Big Horn Ranch raises 100 horses on 150 acres of pasture. Jefferson Ranch raises 75 horses on 125 acres of pasture

Problem 2:  x + x + 4 = 22

The first mistake is an error in reasoning. It's incredibly common when students are attempting a new or unfamiliar problem. Students try to apply their previous, incomplete understanding of math to a new situation and find that they don't have the tools or the understanding to get the right answer.

This mistake is exciting because it represents the beginning of a conversation. This student reasoned his way to an answer that satisfied him. All I need to do is create some dissonance within his mental model of the problem in order to get him to re-evaluate the problem. Maybe I give another more extreme set of acres and horses that are also 50 apart, such as 60 acres and 10 horses, and ask if this ranch also has the same amount of acres per horse. What about 51 acres and 1 horse? My next step depends on the student and the setting, but at least it's building on some prior understanding of the scenario.

Mistake #2 is a different animal entirely. This is not a mistake that my student reasoned her way into. This is a mistake made by someone with no understanding of equations who is desperately trying to recall some long-forgotten rule about subtracting x from both sides.

More distressingly, this is clearly a student with a lot of experience solving equations. There is no chance that this student would have tried subtracting x from 18 if she were a true novice at solving equations. And if you asked her to explain why she subtracted x from 18, she would likely have nothing meaningful to explain about preserving equality or finding the value of x. She would probably say something about "getting rid of the extra x's" and look at me terrified that I was going to ask her a follow-up question. Even her final answer of x = 17x shows that she has no consistent understanding of what a variable is and how it can be manipulated in an equation.

The second mistake depresses me because it is a mistake that I helped to create. I have taught this girl how to solve equations for weeks. And she has emerged from that experience with a worse understanding of equations than when she began. If I had given this to her on the first day of school, she probably would have at least tried to guess-and-check her way to an answer. And that would have been so much better! At least that would show that she understands the purpose of the exercise.

But no. Somewhere along the way, I helped to break something inside her head. I pushed her up the ladder of abstraction too quickly, and now she's swinging in the breeze. And fixing that issue becomes a twofold challenge. First she must unlearn before she can even begin to learn.

Almost all the mistakes the students make when solving equations are this second type of mistake. And that's something I'm going to try to change.

This post is the first in a series.

Open Number Sentences: Is this _____ actually useful?

Michael and I can’t stop gushing about the Project Z resources and how they have sharpened our thinking about teaching and learning integers.

One of the most eye-opening pieces of their work is a set of videos of 1st and 2nd graders solving math problems such as ___ + 5 = 3 and 2 - ____ = 6. It’s amazing to listen to little kids talk through the exact same ideas that our 7th and 8th graders struggle to understand.

One amazing moment occurred in the second video on this page.  Violet, a 2nd grader, correctly answered -4 to the problem 2 - ___ = 6. She giggles nervously at her answer, but when the teacher prompts her to explain her reasoning, she (adorably) says:

“Because it always goes the other - a negative number to me, when you’re adding or subtracting it, it goes the other way than it usually goes with a positive number.”

That is basically a summary of big ideas 2 and 3 from my previous post. In fact, her phrasing may even be easier to understand than my own!

But I think a great deal of her success is due to the format of the question. I am going totally off of intuition here, but I don’t think she would have gotten the correct answer as easily if she had been given the problem 2 - (-4) = ____.

When students are given a standard problem such as 2 - (-4) = _____, I think two thoughts run through their heads:

This problem is different than 2 - 4 because one of the numbers is negative

Subtraction makes numbers go down

Except in their heads, I think it sounds a lot more like

This problem is differ-  Subtraction makes numbers go down!!!!!!!!!

There is an enormous imbalance between the new, dissonant math problem and the old, well-worn grooves that addition and subtraction have made in students’ minds. This is why integer addition and subtraction are so much harder to teach than multiplication and division.

With multiplication and division, you are just adding a rule about negativity on top of the existing structure of multiplication and division facts that kids already know. But with addition and subtraction, what is happening in the problem is the precise opposite of what students have seen for years and years. Imagine the nightmare it would be if multiplication and division worked this way as well: Imagine explaining that 10 * (-5) = -2

But with the open number sentences, the problem directly confronts students’ preconceptions about addition and subtraction. Look at a problem like 9 + ____ = 4. This problem insists that addition can make a number smaller. It’s up to the student to figure out which number has that effect.

This creates a subtle but important change in the sequence of teaching integers. Typically in my classroom, I present the following idea:

Adding a negative number makes the answer smaller

But the open number sentence turns this single idea into a two-parter.

Part 1) Sometimes addition makes the answer smaller

Part 2) This happens when you add a negative number

I like the open number sentences such as 9 + ____ = 4 because it drops kids right into the space between Part One and Part Two. They first have to grapple with the idea that addition can make an answer smaller. After all, it’s right there in front of them!

Then they have to decide what sort of number would have that effect. Students as young as Violet with an intuitive understanding of negative meaning “opposite,” so it stands to reason that they would gravitate toward negative numbers as an answer.

This is the part of my post where I want to disclaim again: I am 100% speculating based on my personal experience and intuition about teaching integers. I have no idea if these sorts of problems have a meaningful effect on the way kids think about integers. So I will be seeking out research on this topic. If I don’t find any, I might have to make some of my own...


Integer Arithmetic - Contexts Aren't Enough, So Which One Should We Use?

Michael, the Project Z research you found on the ways students view integers has shaken my preconceptions as much as it has shaken yours. They have led me to take a big step back and try to look at this unit more holistically.

As an algebra teacher, I think that integer addition and subtraction boils down to three big, interconnected ideas:

  1. Addition and subtraction are opposite operations, or inverse operations. That means that they have the opposite effect when operating upon two numbers
  2. Positive and negative numbers are opposites, or additive inverses. That means that they have the opposite effect when added to a number
  3. Subtracting a number has the same effect as adding its opposite

Everything in my teaching of integers is aimed at these three big ideas. And this research has really helped me think through the ways that students come to understand these ideas.

In your last post you did yeoman’s work trying to connect each of CGI’s addition and subtraction problem types to word problems involving integers.  The issue, as you discovered, is that not every problem type lends itself to easy interpretation using a real-world context. This is also discussed in one of the Project Z presentations, in which the authors state:

"These problems involve context, and when we set out to think about integers, we looked at contexts. But interestingly, we found that when we gave students contexts, such as owing money or increasing or decreasing elevation, they generally avoided using negative numbers. I can talk about a debt as negative dollars or a loss of yards in football as negative yards, but when was the last time you watched a football game and someone said, “Wow, that guy just gained negative 3 yards?.”

Well, damn. So at best, these contexts can get our students part of the way toward a comprehensive understanding of integers. The rest of the battle, which is a subject for a later post, probably has something to do with number lines and open questions like 5 + ___ = 2.

Still, there is at least some value to finding a context that builds a basic understanding of how negative and positive numbers interrelate. And I think I have found one. Or at least, I have found a context that seems very promising.

But first, a game.

You are in a hot air balloon. Sort of. This hot air balloon is different from normal hot air balloons. It is a lawn chair is held in the air by a series of small balloons, each of which can raise your lawn chair by 1 foot. It is also held down by several sandbags, each of which lower the height of the lawn chair by 1 foot.  For the sake of consistency, let’s start the balloon at a starting height that we will call 0, and let’s say that your lawn chair currently has 5 balloons and 5 sandbags attached.

Your opponent also has a lawn chair held up by 5 balloons and held down by 5 sandbags, also starting at a height of 0. Your goal is to raise your lawn chair up to a height of 10 feet above the starting position. To do this, you and your opponent take turns drawing cards. You do what it says on the card and change the height of the lawn chair accordingly. The cards look like this:

But there are also some wild cards that are a bit more complicated:

The first person to raise her lawn chair to 10 feet above starting height wins!

You have a player token and a vertical number line to keep track of your progress. You can use 2-color tokens to represent balloons and sandbags, or any other manipulative you wish.

I have no idea if the mechanics of this game make it fun, or whether ten feet is too easy/too hard to achieve. That part I can tweak later. This game has the main thing I want, which is students grappling with the effect of adding and removing (adding and subtracting) balloons and sandbags (positive and negative numbers). The formal symbols, as always, can wait. I am trying to build a conceptual framework first.  

So after students play this game a time or two, I would give them the following set of questions:

1)  Go through your deck of cards and pick out the three cards you think are most helpful to your chances of winning the game

List those cards below.

Why did you pick these three cards? Explain

2) Now go through your deck of cards and pick out the three cards that are most harmful to your chances of winning the game.

List those cards below.

Why did you pick these three cards? Explain.

3)  Now go through your cards and pick out two cards that had no effect on your lawn chair’s height.

List those cards below.

Why did you pick these two cards? Explain.

N.B. I asked students to list 3 cards in questions 1 and 2 even though there are 4 cards that are the most helpful: Add 4 balloons, Remove 4 sandbags, Add 3 balloons, Remove 3 sandbags. I do this intentionally with the hope that students will get different answers and end up debating whether removing 3 sandbags is “the same” as adding 3 balloons. Again, no formal symbols yet. Just laying groundwork.

The third question gets to the big concept number 2: Positive and negative numbers are additive inverses. I want students to realize that sandbags and balloons cancel each other out.

From here, I have lots of ideas. Here are a few:

  • Play a second version of the game, but every card has two instructions. For example “Add 3 balloons and add 4 sandbags,” or “Add 1 balloon and remove 3 sandbags.” After the game, have students rank the cards from most helpful to least helpful
  • Ask a series of questions such as “Mr. Haines wants to add more sandbags to his lawn chair, but he doesn’t want his lawn chair to go down in height. How can he accomplish this?
  • Ask a series of questions such as “My lawn chair is at a height of 4. I want to get it down to a height of -2. What are some ways I could do that? List as many as possible”

Then, at some point, we transition to problems such as 4 + (-7) = ____ and 5 + ____ = 2. But maybe, just maybe, the students will come to understand those ideas more quickly because of the significant time they spent analyzing balloons and sandbags. How, you say? I have no idea. I haven’t tried it yet. But I think it’s at least worth investigating.

In order to make this transition, you need a lot more than one day of connection. It’s not as simple as Monday: Sandbags and Tuesday: Negative Numbers. Students need to make their way up and down the ladder of abstraction until they know where the rungs are without looking down. Give students a problem such as “add 3 balloons and 4 sandbags” and ask them how you could represent it with math symbols. Give students a problem like “-2 + 5 = ___ ” and ask them to write a word problem about hot air balloons to match it.

Not everyone is going to reach 100% fluency with this model. As I mentioned above, students can and will avoid negative numbers whenever possible. But this might be a good starting point for a unit on integers.

Or maybe not! But I'll play the role of context optimist for the time being. It's more fun to argue.

Use Red Pen - My First Video

This is the first video I've ever made about teaching. Basically, about a year ago I was complaining at a lunchtime PD session about the 14,563rd time someone has told me to use green pen. I decided that instead of inflicting my rant upon my colleagues again (or god forbid, my wife), I would rant into a lens. And here is the result.

To be clear, this video is as much a reminder to myself as it is advice for anyone else. I struggle to make sure my comments and questions on graded work are worth students' time. I want to help my students with my assessment system, and when I get lazy I look at my red pen and remember to try harder next time.

Teaching Integers - Common Contexts

Howdy folks!

This is the first in a series of posts from myself and Michael Pershan about teaching integer operations. In this series, Michael and I are going to discuss the many contexts and strategies that teachers use to teach integers. We will then try to categorize integer problems into several problem types and analyze how these problem types are explained using these different contexts and strategies. Hopefully along the way we will actually learn how to teach integers, but no promises on that count!

Integers feel like a topic in middle school math that is awash in metaphors, visual strategies, and quick tricks. It can be hard for a teacher to figure out which contexts and strategies will provide students with the best foundation for a conceptual understanding of integers. Specifically, some of the metaphors for integer addition become quite confusing when they are used to explain integer subtraction.

Personally, I have always felt like I get my students about 80% of the way there with my various analogies and number line strategies, but I’ve never finished an integer unit with the feeling that I had nailed it. I am hoping that by deeply investigating integers, I will find something useful to use in my own classroom.

But first: The four major contexts I have found for teaching integers.


Elevation problems often place the student in the mind of a rock climber, scaling cliffs that stretch from deep canyons below sea level to high altitudes. Other elevation problems discuss submarines diving beneath the sea and helicopters rising up above the ocean. At other times, students are asked to compare the heights of various mountains with the depths of various ravines and trenches.

One particularly evocative elevation context involves a hot air balloon basket that is held up by balloons and weighed down by sandbags. If the balloons and sandbags equal each other, the basket remains at its baseline height (zero, for the purposes of the problem). Adding a balloon causes the basket to rise one foot, while adding a sandbag causes the basket to fall one foot. Removing a balloon causes the basket to fall, while removing a sandbag causes the basket to rise.


Temperature is a common context that teacher use to introduce students to integers, in part because negative temperatures are some of the few negative numbers that students encounter outside of the math classroom. This prior knowledge is  less substantial in states like Alabama, where we are more used to temperatures with three digits than those with one. Regardless, most students have some prior knowledge about temperature and know that 23 degrees below 0 is colder than 8 degrees below 0.

Word problems involving temperature often discuss changes in the temperature from one time of day to another, or compare temperatures between cities to find how much colder one place is from another.

More abstractly, some teachers use the invented context of “hot and cold cubes” which are fictional cubes that either cause or lower the temperature of water by one degree. This context lends itself to student investigations with manipulatives.


Money problems are common because their application to the real world is so strong and resonant. Every child has some prior knowledge about earning, spending, and owing money. Even though they may not have formal experience with debt, they are familiar enough with borrowing money from their parents to understand the idea.

Money is also useful because money can be earned and spent, and debts can be created and forgiven. These varied scenarios seem to provide a contextual support for many types of integer problems. Some teachers allow students to create a budget for themselves and keep track of the debits and credits to their savings over time.

Piles and Holes

James Tanton uses the metaphor of piles of sand and holes to teach integers. Piles represent positive numbers, while holes represent negative numbers. This image helps understand integer addition such as 5 + (-3), since every pile can be used to fill in a hole. Also, piles and holes can be removed, which can represent subtraction.

I have found a couple of examples of teachers using James’s analogy, but it definitely does not feel as widespread as the other three contexts. I include it because I think it would be valuable to compare the most common contexts with a context that is new to most people.

Other Contexts

I have found a couple of other contexts, such as “good things” and “bad things” or golf scores above and under par, which were less common or relied on more specific prior knowledge. I can’t imagine a lot of my students knowing much about how golf is scored, for example.

So there you have it! If you think of any other contexts that might be useful, please add them in the comments.