Category Archives: Games

Revisiting Multiplication Tic-Tac-Toe: Common Factors and Multiples

After reading about Ultimate Tic Tac Toe, I was inspired to post a bit more about Multiplication Tic-Tac-Toe. Multiplication TTT offers similar constraints to Ultimate tic-tac-toe, but the constraints are tied to the common factors. Check out my earlier post for rules and notes about classroom use.

One main difference though, is that you can claim more than one square at a time. Below I have shared a filled out board, color coded by the amount of numbers you claim with that move. For example, if you place a token on 6 and 4 you in fact get 4 positions on the Tic-Tac-Toe board as 24 shows up 4 times. In the image below, blue represents “quadruple plays”, yellow’s represent “triple plays”, oranges represent “double plays”, and white represents “single plays.”

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Here is a google doc of the Color Coded Multiplication Tic-Tac-Toe Board.

Recently I introduced the game to my 3rd and 4th graders. I mentioned you could analyze the game based on how many positions each number occupies. Then, the other day I was walking down the hallway and saw a multiplication tic-tac-toe board on the floor next to their cubbies. The cool thing was, the student had begun to color code the board herself (I have a feeling I know which student it was). This kind of investigation could naturally lead students towards questions related to common factors and multiples.

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One of these days, when I have the time, I’d like to figure out the optimal strategy for the game. If anyone out there discovers it, please let me know.

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Filed under Elementary, Games, Mathematical Investigations, Unsolved Problem

Multiplication Tic Tac Toe

Analytic presentation of all possible Tic-Tac-Toe games

I adapted this tic-tac-toe game from one I saw in a workshop claiming to be a multiplication tic-tac-toe game. It was similar to this multiplication tic-tac-toe game. It was fun enough, but it had no resemblance to a tic-tac-toe board.

I think my version has what it takes to be considered a “tiny math game.” All you really need to play this is paper, pencil, and two tokens (which could be two pieces of paper).

Embedded in the game is practice with multiplication facts, common multiples, and some good old fashion tic-tac-toe strategy, with a twist.

Instructions:

-Make a big Tic-Tac-Toe board, then make a tic-tac-toe board in each of the 9 squares.

-Now fill in the 9 squares with the multiples of 1-9 (see example below).

-Write the numbers 1-9 underneath your board.

-First player places two tokens (pennies in this case) each on one of the 9 numbers at the bottom. Multiplies these together, and places an “X” anywhere that multiple is found on the tic-tac-toe board. (in the example below player one has chosen 6 and 4, and has placed an X on all four “24s”)

-After the first move, players take turns choosing to move only one of the pennies to select their multiple to “X” or “O.” (For example 2nd player could move the “6” to a “3” and put an “O” over every 12 on the board)

-If you win a small tic-tac-toe game, you win that square on the larger board. The goal is to get Tic-Tac-Toe on the large board, by getting three of these smaller boards in a row.

-Also, I like to say any “Cat’s game” is a wildcard spot once it is completely filled in. This allows for player one and two to win simultaneously, which I like because I’m a sucker for win-win situations. 🙂

Notes For the Classroom:

I usually start by handing out this blank Multiplication Tic-Tac-Toe board. Students then take a few minutes to fill out the multiples of one in the top left corner, then multiples of 2 in the top middle, multiples of 3 in top right corner, etc (See example above).

First player places two tokens (pennies in this case) each on one of the 9 numbers. They multiply those two numbers together, and put an “X” anywhere that multiple is found on the tic-tac-toe board. In the example above, the first player has chosen to place the two tokens on 6 and 4, allowing her to place an “X” on the four places where “24.” I call these “quadruple plays,” and tell students to try to get as many of those as possible. Sometimes I have them color code the quadruple plays, and triple plays (like 36 or 9) so they are aware of them. This introduces and/or reenforces the concept of common multiples and factors. Check out my follow up post for more information on this.

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Filed under Elementary, Games

100 Game Theory

This is a great game to introduce game theory at a young age while simultaneously assessing place value and addition concepts.

Race to 100

How to Play:

Two players start from 0 and alternatively add a number from 1 to 10 to the sum. The player who reaches 100 wins.

I usually give students a hundreds chart to help keep track of where they are, as well as a “Race to 100” handout with tables to record what number each person said. Also once they discover that saying “89” guarantees they will be able to say 100, I have them star that number. I ask them to think if there are any other numbers that guarantee a win.

A 3rd grader shows the winning strategy visually on a hundreds chart. He put an X on the numbers he wants to say, and then shaded in the numbers his opponent "him" would be able to say.

A 3rd grader illustrates the winning strategy visually on a hundreds chart. He put an X on the numbers he wants to say, and then shaded in the numbers his opponent “him” would be able to say.

Extensions:

How would the strategy change if it were a race to 99?

How about if we change the numbers that you can add?

This is part of the family of games commonly known as Nim, NRICH has a thorough article and resources for these games called “Meet the Nim Family.”

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Filed under Elementary, Games

Number Bowling

This is a student favorite, simple to start, differentiated, with various levels of success. I’ve done this with grades 2 and up, but my 5th-6th graders have gotten the most out of it.

Most importantly students learn to be very clear in writing their expressions. This activity naturally leads to a discussion about order of operations and mathematical “grammar.” Also, in their pursuit of the strike, students often ask to be introduced to new operations.

Instructions: Begin by rolling a die three times and recording these numbers as your “1st bowl.” You may cross out (knock down) any number that you can write an equation for using those three numbers each only once. For example, if I rolled a 6, 6 and 5, I could knock down the number four by writing: 5=6-6+5, or knock down 1 by writing 1=(6÷6)^5. Notice I used all three numbers, but each only once.

The goal is to knock as many “pins” (numbers) as you can. Knocking all the pins down on your first bowl is called a “Strike.” If you can’t think of any other equations, you may bowl again and try for a “Spare.”

Notes on Implementation:

I’ve created this Number Bowling handout for students to keep track of their games. I’ve also experimented with keeping score, and I think three frames is a good length time for a game. This can get complicated though, since scoring bowling is foreign and not straightforward to many students.

This year, after a few games, I had students write down their “favorite” equations on a notecard. We used these equations during a “strategy session” where we came up with tricks to help knock down more pins. The “tricks” can all be described as using an operation to change a number, or two numbers, into another. For example, 3 can be changed into 6 by using factorial, (3!=6). Some more advanced tricks include using square root, and/or the floor and ceiling function (rounding up or down to nearest integer).

Extensions:

Last year students wondered whether a strike was possible for every combinations. We chose the brute force method of proof :). First students had to figure out how many unique outcomes were possible with 3 dice. For many I assisted them by having them look at this pdf. Obviously rolling 1,6,6 is the same as rolling 6,6,1.  Then whenever a student achieved a strike, we crossed that off the list.

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Filed under Games, Middle School