Category Archives: The Integer Wars

Integer Wars – The Battle is “Real” – Othello for Building Number Sense

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One of my favorite board games is Othello. As a fan of solitaire types of games, I found as a teenager that I could spend a decent amount of time with Othello because of its strategy building. I love puzzles and simple but challenging games. This is a fave.

Othello is easy to start, easy to play, and can be as challenging as two players want to make it. The basic premise is opposite colors on the chips, black and white, which can be flipped when a player places a chip such that their color is on opposite sides in a vertical, horizontal, or diagonal line. As strategic sense is developed, an individual may look for key positions that will facilitate multiple flips on their turn. Simple but engaging, for sure.

The winner is the individual who ends the game with the most pieces on the board, which is a great place to look at opposites and differences. Who wins, and by how much? This is my basic question in the classroom when we look at integer operations that otherwise confuse struggling students.

For example, 6-4 is easy enough for my students to calculate as 2. However, 4-6 throws them off, even if they have encountered such problems in the past. My ongoing questioning strategy is to ask, which team wins? Team negative or team positive?

But wait, aren’t they both positives? I help my students to understand that subtracting 6 is the same as putting a negative 6 in the problem, which seems to be accepted the majority of the time. In other words, subtraction has the same effect as adding the opposite.

The team approach really depends on the difference model of subtraction. Students often struggle because they see subtraction only as taking away. Spending some time visiting the difference model and building this approach to subtraction is well worth the time, and Othello is a great option for introducing or reinforcing the idea.

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Integer Wars: The Battle for Understanding Integers is Real

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The pun is there in the title for those who see it. Integers form a subset of the real number system that can cause tremendous grief for Algebra teachers and students alike. Integer War is a game that many math teachers use to develop fluency with the addition and subtraction of integers, but the challenges can continue beyond these games of practice.

One of the first challenges is an understanding of positive verses negative, and the use of two-colored tiles or counters is helpful for tackling this. Red and black are great options as these colors often denote being in a negative or positive zone financially. However, by the time our students reach Algebra I, we often don’t feel like we can justify the time to go back and build visual models. When I’m working in a general education setting, I often defer to a simple team model when we are adding integers. Team Positive and Team Negative are my go-to descriptors. If the values are on the same team, the team grows, and the team sign stays the same. If the values are on opposite teams, the teams face off and cancel each other out. The team with an excess of players wins by that excess amount. It’s not as complicated when I’m in the moment. Which team wins? By how much? That tells us sign and value.

While this may seem silly, it is pretty effective for directing student thought as to what sign the sum will have and what the numeral will be.

Examples:

-3 + -6 Both values are on team negative. Team negative has 3 members and 6 more members. The net value is -9.

-5 + 12 The values are on opposite teams. Team positive wins by 7. The net value is 7.

4 + -9 The values are on opposite teams. Team negative wins by 5. The net value is -5.

If both values are on team positive, students typically don’t struggle. If the problem is set up as subtraction, I often rephrase it as adding the opposite. This way, we are looking at the addition model and are able to keep this team mentality on hand for quick reference. I don’t solve the problem for the student but rather provide the direction to help the student figure out the answer.

If you are looking for models to help support your student or students, check out my bundle, “Visual Supports for Operations with Integers“, at Teachers Pay Teachers.