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.

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.

Krypto – Developing and Expanding Number Sense

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Krypto Original Family Arithmetic Game
Krypto can be adapted for use with various age and ability levels, perfect for promoting number sense at home or in the classroom.

Mental math skill is an asset, but it can be intimidating to those who feel inadequate in their math abilities. Krypto is a great game for working on these skills, challenging to students, teachers, and parents alike. The good news is that Krypto can be adapted for those who aren’t ready for huge complications in their math days. You can definitely work your way up to the more challenging versions.

The card game is designed to use whole numbers 1 through 25. The deck is composed of 56 cards as follows:

3 each of cards 1-6

4 each of cards 7-10

2 each of cards 11-17

1 each of cards 18-25

As the game is designed, a round is played with a set of 6 cards dealt face up. The first five must be used in a math problem exactly one time each to produce the number on the 6th card, called the “Objective Card”. I call it the “Target” in my classroom when I conduct Krypto activities.

All players work on the same set of cards, with the first to find a viable solution calling out, “Krypto!” The game requires a limited amount of time be allotted for that individual to effectively explain their answer.

How I Use Krypto in the Classroom

I manage the deck, shuffle, and deal. I write the action numbers on the board and identify the target number as well. I remind my students that they must use all of the action numbers exactly one time. They cannot omit any numbers. They frequently want to know if there is a solution. I participate along with them and will let them know if I find a solution. If I struggle for more than a couple of minutes, I will allow them to continue working but will deal out a new set of options to keep things moving.

Adapting for Varied Needs

To reduce frustrations for students who struggle in math, I reduce the number of action cards to 4, so we use the first 4 cards for computation and the 5th as the target. At times, I will pull out numbers greater than 20.

I have successfully played the game with middle school students using the full deck and 6 cards at a time. With students with learning disabilities in math at the high school level, I use 5 cards. With younger students, you can adapt by removing double digit cards or by using only 3 or 4 cards total. If you are using fewer cards, you can also focus on whether a viable solution is available.

The rules of the game don’t include fractions, radicals, or negatives. However, you could definitely create your own variations.

No Krypto Deck? No Worries!

You can assemble your own Krypto deck from miscellaneous cards from other decks. This is a great way to repurpose incomplete decks. When this game was first introduced to me in 1988, we used blank flashcards to make our own decks. Index cards are very affordable, another perfect way to put this game together on your own. Another option? Write the numbers on pieces of paper or cardstock, and put them in a hat – draw them out instead of dealing them.

Number of the Day

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When I was a teenager, my mom implemented a dinnertime routine to keep the family engaged at the dinner table. I thought the goal was to get us prepared for college and spelling bees and such, but I later was told that it was to keep us involved and at the table! At any rate, each night was a different individual’s turn to pick a word out of the dictionary, which the others would attempt to spell and define.

What does this have to do with math? Well, it’s primarily about engagement at the table and developing simple activities that can be used in a set location such as the dinner table or the family vehicle. However, I have a math-y twist for you. Try working with the number of the day.

I use this as my daily warmup now with a different name, and I got the math idea from a presenter. I have put my own twist on it to help my students work on everything from math facts to seeing structure in expressions. I build my request around the number for the day. One week, I may ask students for addition problems that have the date as their value. Another, I may ask for any expressions. Still other days, I might request fractions or decimals. In my algebra classes, I’ve started using the date as a coefficient for a variable and asked for equivalent expressions. I’ve also used the date to create equations for students to solve.

How do you, as a parent, implement this? It is as simple as asking your youngster for something that equals today’s date. If you are terrified of math, as many parents claim to be, you can keep it simple. If you love math, you can dive into more complex options. You can adapt to the levels of different children. You might only expect addition or subtraction options from your little ones. Your older children, middle and high school level, might use square roots and absolute values.

Discuss, display, digest! You can discuss things you notice about a child’s response. “I notice that you subtracted or added 0.” You can use household items such as coins, game pieces, or other objects to verify. Have your child write their expressions. Put them on the fridge. If you are really daring, make your fridge your dry erase board! As you and the kiddos discuss and display, you can all digest the intrigue of numbers together!

Solve Me – The Power of Puzzles

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I have always loved number puzzles. My parents and I were recently conversing about math, and I stated that number puzzles in the Dell Pencil Puzzles and Word Games were my first priority. I may have been a nerdy kid, but those puzzles intrigued and engaged me. Now, in a digital age, a good number puzzle is valuable. Let me suggest Solve Me.

There are 3 types of puzzles, including Mobiles, Who Am I, and Mystery Grid.

Who Am I: These are number puzzles that give clues about the digits as well as the numbers, providing players with information to figure out the identity of the number. This is wonderful for working on place value, odd and even, and characteristics such as square numbers. This is just a sampling of some of the types of concepts, of course, but a great opportunity to reinforce in a puzzle. I have used similar number puzzles with middle schoolers in my past and use such puzzles with my high school math students who struggle due to learning disabilities in math.

Who Am I includes not only the Play mode, but also a Bingo mode and a Build mode. In the Play mode, the clues are given and the number is guessed. In Bingo mode, a grid of options is given, and players must identify the item that matches the clue, aiming to fill in a full row, column, or diagonal of correct responses. In Build mode, the player has the opportunity to work with these concepts to create a puzzle.

Mystery Grid involves correctly placing numbers in a grid such that each number appears exactly once per row and once per column. Clues may be given as to the sum, difference, product, or quotient of a group of numbers. The puzzles begin simply and move to very complicated grids that may involve a lot of position changing. Players can work from simple to complicated or dive right in at the hardest level. Parents can support their youngsters by helping them to backtrack if a puzzle is too hard. The Build mode allows a player to create their own puzzle, again allowing some reversed thinking about how the items must fit together.

Mobiles are excellent for supporting algebraic thought. At times, the mobiles display the full value of the mobile, and at other times, the mobiles are shown to balance without the total being known. Individuals must use algebraic thinking such as removing equal parts from both sides to determine the value of the unknown. The mobiles evolve in difficulty as the player works through the puzzles.

Greetings

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This blog has been brewing for a few years, and in a season of de-stressing and decompressing following the fall semester exams at my school, I have finally had the chance to choose a theme and dive in. I’m a math teacher, but I don’t view myself as a math expert; I love math for its patterns, and I love being able to follow a bunny trail that pops up in the midst of a math problem. The predictability in some areas, the open-endedness in others, I love it all. I meet my limits at times, but at others, I discover afresh things that have already been discovered.

As a mom of many children, I have spent lots of time on counting activities that crop up informally. There are many ways to make numbers familiar, and familiarity can minimize the fear, even put the magic into math. My articles will be designed to discuss ideas for instilling that magic in the early years and re-igniting it in later years, depending on your needs. No guarantees, of course, because every child is unique, one of a kind, but let us make the effort to make math less of a stress and more of a challenge to be embraced. Off we go!