Category Archives: Math Practice Strategies

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.

Penny Math for Families

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I can’t remember where the idea came from, but it was during my early homeschooling years that a friend told me about the concept, which I used with my first grade daughter and her siblings through the years. When we worked on math problems specifically, I would give her a penny for each correct answer. At the end of the week, we spent time making trades for coins of greater value. Eventually, the trading included paper money as well. The experience of handling money and working a little bit at a time with the trading allowed my children to know in practice what it meant to handle money. When their earnings accrued enough to buy a treat, that commercial transaction would also expand their engagement in handling money.

We later lived in Mexico, where the kids had the opportunity to handle a different currency, the peso. Again, opportunities to earn and spend allowed the kids to become familiar with money from a mathematical standpoint. One of the more disconcerting stories was the toy cash register that had a slider for an ATM card. When my bank card went missing, it was a bit of a nightmare, commercial lesson learned by the parents. Nevertheless, the kids engaged in handling small amounts of money in everyday life.

Today, it’s easy to jump to gift cards, whether store specific or the credit card variety. I think that we can lose sight of the value of handling coins, making change, and saving in little ways. In looking back, my oldest daughter especially remembers those penny math moments and their impact. She is now an elementary school teacher and has worked in similar fashion with students on money handling.

As a parent, I think that experiencing math in real life can be one of the greatest investments we can make in our kiddos as far as preparing them for future math courses. That doesn’t mean that they will be math whizzes, of course, and some of my kids have definitely struggled with areas such as Algebra. However, approaching math from avenues found in day to day life enables our kiddos to understand that math is integral. It doesn’t have to be the subject they love most, but it doesn’t have to be terrifying either. Look for the little ways that you can work counting, sorting, and exchanging into everyday life to promote a mathematical mindset that can be carried into the classroom in the future.

Practicing Patterns

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One of the things that I love about math is the patterns. They are, of course, everywhere. It might seem like repetition and drilling are contrary to the joy and wonder that come with seeing patterns, but repeated exposure to patterns is what makes them so cool. It’s always exciting to see that lightbulb go off when a student is able to see a pattern and make a connection. However, those observations don’t come easily to all students.

Repetition can provide the opportunity for students to make friends with patterns that don’t seem obvious at first. I think that teaching algorithms can be demonized, when there are students who need not only those methods but also memory tools for accessing those methods. It’s tough for those of us who easily make sense of numbers or words to understand a world in which those educational tools are confusing. It’s tough as a teacher to look at a problem that clearly jumps out at me in terms of solutions and to realize that my student is clueless as to where to begin.

One of the strategies that I have begun to implement with my math students who struggle is to provide several similar problems on the same work page or screen. Rather than working through one problem from start to finish, I work through all three at once, completing the first step in each example before moving on to the next steps. This allows my students to see the similarities in the problems, so that they can better observe how the starting actions for all of the problems are the same. I am beginning to see the algorithms stick. Of course, this is not where we will stay, but the letters and numbers on a page become a little easier to navigate, then we can tackle the application as we go.

It might seem like a basic approach, and I certainly would not have explored this path as easily in my pre-COVID classroom. The move to mostly digital instruction and paperless work has demanded creativity, which for me, includes ways to communicate through a screen, whether to my hybrid students in the classroom or my virtual students on a computer. Not being limited by paper has enabled me to be liberal in the space I use, which has also allowed for creativity in completing exercises.