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.

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.