Math Tic Tac Toe

Kids have been playing tic tac toe for years and this is a format that can be used to reinforce any skill from basic facts and fractions to algebra and order of operations.

Make 9 to 12 cards with math problems on them. Make a tic tac toe board with the answers to the problems in each of the nine spaces. Laminate (and color code if desired) the cards and board.

The cards are spread out face down and players decide who will be "X" and who will be "O". Players draw a card and do the problem on the card on their scratch paper. They mark the answer on the board with their mark using a write on/wipe off pen or different colored discs. The first to get three in a row wins the round. Extra cards that don't have answers on the board, make the game more lively. The ability to win is based on chance as well as skill.

The Finger Counting Problem

I was intrigued by this problem posted on the computer forum last night and I used it with my groups this morning.

count 1, 2, 3, ... on your fingers, starting with your thumb and going back and forth on your hand. On what finger will you reach 1704? 2476? 3121? When counting, be sure not to "double count" on any fingers.

To keep solutions consistent, let P be your pinky finger, R be your ring finger, M be your middle finger, I be your index finger, and T be your thumb. Use these variables when you write your solutions.

I had students trace their hands and write the numbers on the fingers they would count on, telling them they should be looking for a pattern. On each finger a clear pattern emerged which helped them to answer what fingers the larger number would land on. Then we divided the larger number by 5 (the number of fingers on the hand) and found that the remainder also helps identify which finger the number will land on. From there, we went on to looking at patterns in charts and continuing them and also writing them as an algebraic expression.

Comparing Fractions

Students enjoy this game which we play in a group of three or four. While playing, I observed that a number of students were able to move from the algorithm to the conceptual in the process. I teach students to compare fractions by multiplying on the diagonal. In this new district, they use the shoelace method, which is a take off on the diagonal method. As kids played the game, I saw them using the algorithm as well as visualizing and analyzing. They were able to acquire a familiarity with the fractions and fractions concepts that sometimes takes a lot longer to get.

Each student has a small white board, marker, and a pair of dice. The dice can be shared with the group if necessary, but it works better if everyone has their own set. Players roll the dice and write their two numbers as a fraction with the smaller number rolled as the numerator and the larger number rolled as the denominator. The fraction is then written in lowest terms. Students then look at everyone's fraction and announce by turn if they want to keep their fraction or roll again. Those who wish to, roll the dice a second time and write their new fractin based on their second roll. After that, the group works together to arrange the fractions from least to greatest. The person with the greatest fraction gets three points(if there are four players they get four points) the next fraction gets one less point on down to the lowest fraction which gets one point. Students keep track of their own score. The game is over when the time period is up or when someone reaches 21 points.

Variations on this game include:
giving the highest points for the smallest fraction
putting the larger number as the numerator requiring the students to change the improper fraction to a mixed number

Lessons Learned From A Wheelchair

In the spring of 2006, I had surgery on my left foot. My foot was in a heavy boot and I could barely walk with crutches so the obvious solution was to use a wheelchair at school. I have often referred to the experience of teaching from a wheelchair as one of the the most difficult challenges I have ever faced. Thankfully, I had a student teacher at the time who covered more than half of my classes. But there were a few classes I was left with while she worked in another room.

My regular practice is to be up on my feet, constantly moving around the room or teaching from the board, pointing and writing. In the wheelchair, I could barely reach enough of the board to write on and standing, even with my leg resting on a chair, was quite painful. So several times, when going over math problems and teaching new concepts, I would ask for a volunteer to go to the board to do the writing while I talked.

There was no shortage of volunteers and I wanted to keep it that way. I tried very hard to verbally describe what I wanted my volunteer to write, while explaining the math procedure to the class. It felt like it was a painfully slow process, and at first I worried that those poor students must be bored to tears. But really, the opposite was happening. They were sharing in the responsibility for their own learning, and I had slowed down enough that it was easier for them to grasp and retain the concepts I was trying to teach.

I had listened to various lecturers make the point that teachers needed to allow more lag time after questions and when explaining new concepts, and now out of necessity, more lag time was built in. The students were responding beautifully to my slowing down and learning was increased.

I am happy to say that I no longer need a wheelchair. It was a tool that served its purpose at the time, but I was thrilled to send it back. I have tried to incorporate what I have learned from that time into my daily instruction. I ask a question, and instead of calling on someone right away or answering it myself if there are no volunteers, I take a long sip of water. Sometimes after posing a problem, I take a walk around the classroom before I call on anyone, telling them to just think about it until I get back to the front of the room.

I also look for ways to get students to share the responsibility for their own learning. I try to emphasize how they can do this and encourage them when they do. Sometimes it is having them do a self-correct, help a peer, or make an informative presentation to the class. I never would have thought that being in a wheelchair would help me in any way to be a better teacher, but it certainly did.

Expressing Your Math in Writing

This summer I signed up on Yahoo! Answers forum. On a regular basis I get on the computer and answer questions that other people are asking. I frequently go to the "mathematics" and "homework help" sections. As I write this, I currently have 2461 points on that forum, a distinction that places me just 39 points away from being a level 4 member. About 31% of my answers have been voted or designated as "best answers".

It has been a rewarding experience. When I help a person with their homework, I try to explain the steps, definitions, and processes of how to do the assignment, rather than just telling the answer. This is something I do in the classroom all the time, but now I am doing it in writing. That's not so different from what we sometimes ask our students to do--express their math processes in writing. The avatar shown above is my avatar from Yahoo! Answers.

Here is a question that was recently asked:

Find the average......................?
The 45 students in a class each recorded the number of whole minutes, x, spent doing experiments on Monday. The results are
total x= 2230

Find the mean number of minutes the students spent doing experiments on Monday.

^that one i can do, but its the next one i dont know

two new students joined the class and reported that they spent 37 minutes and 30 minutes respectively.
calculate the new mean including these two students.

Here is my answer:

To find the mean or average, divide the sum of the data by the number of individual data entries -- so in the first problem 2230/45 = 49.5 or 50 when rounded.

In the second problem add the two new data values to the previous sum (2230 + 37 + 30= 2297) and divide by the number of data values (45+2=47). So 2297 / 47 = 48.8 or 49 when rounded.



Math and Writing

Sometimes kids complain when we ask them to write in math class. They feel that writing is not and should not be part of math. But teachers reply that mathematicians need to know how to write -- after all what good is all of their math if they can't communicate their findings? Participating in the Yahoo! Answers forum has helped me sharpen my own writing skills as they specifically pertain to math.

Playing Cards

I sometimes tease the students in my classes when I let them play games that when their parents ask what they did in school, I'm hoping that they don't reply, "We rolled dice and played cards today!"

But cards and dice are useful tools for math classes. They are inexpensive and readily available. The game that we play using a regular deck of cards is one that involves adding, multiplying and dividing. I have seen my students' mental math abilities improve as well as the basic multiplication and division skills.

The game is best played by at least three people and no more than five. Everyone needs to have their scrap paper and pencil ready.

Face cards are worth ten, aces are worth one, and jokers can be used optionally as 100. All the other cards are worth the value that is on them.

The shuffled deck is handed to the first player. Whenever a player gets the deck, they have the choice of cutting it or taking their cards off the top. The player takes three cards and places them face up and one card which is placed face down. Then the deck is passed on to the next player. The student multiplies the values of the three cards together then divides the product by the value of the card that is face down. This is the number of points which they get for that round. They keep a running tally until the period is over. The deck is passed around the group until the cards are used up, then the cards are shuffled again. Each player keeps track of their own score.

This game provides a great opportunity for pointing out the pattern when multiplying or dividing a number by 10 or 100. Some students don't pick up on that unless it is pointed out several times. Then they are enabled to do more math mentally and to also come up with accurate estimates.

My students are now suggesting variations on this game that could make the points earned in each round a bit higher. They have suggested multiplying four cards together before dividing. We might be trying that soon!

Target 200

Target 200 is a game that I have used in my math classes for so many years that I forgot where I first learned about it. It is great for computation skills and for getting students to strategize.

Students play with a partner, although a third player can be added if necessary. Students use individual white boards or scrap paper to keep track of their score. Players decide who will roll first by rolling two dice. Whoever gets the highest number goes first.

On the first turn, the player rolls the two dice. They can add, subtract, multiply or divide the two numbers that they get. Then they write their answer on the white board. For example, if they roll a 6 and a 3, they could write 18(product), 9 (sum), 3 (difference) or 2 (quotient). I would probably take 18. Then the second player rolls the dice and uses the two numbers to get a number they write down.

On the next turn, the player rolls the dice and again adds, subtracts, multiplies or divides the numbers. They then take this answer and may add, subtact, multiply or divide with the number they have on the white board. So if I rolled a 5 and a 2 on my second roll, I would take 10 (the product) and multiply the 18 on my board by it giving me a total of 180.

The game continues like this with players taking turns. The winner is the first person to reach exactly 200.

Often, both players will be close to 200, but will go over and under it several times until one gets the exact amount needed. Seldom, but sometimes, a person can make 200 in just two rolls by 10 x 20 or 8 x 25. It is all a matter of luck.

We have come up with a number of variations to this game. You can use a negative number die (although a red die can be designated as a negative number die) and play Target -200. Or use two 1 to 10 dice to play Target 2000. My husband came home from school today (where he teaches the same students I taught last year) and said that he had them play Target 199!

Students enjoy playing this game from time to time and it helps keep their computation skills sharp!

Put the Coffee Pot on the Table!

The coffee pot on the table story, that I tell every year to help kids with solving algebraic equations, is one that my 8th grade algebra teacher told when I was a student in her class. Patricia Bassett was my teacher, and though I don't remember a lot of what went on way back then, I remember her telling this story. So I have used the same approach in teaching solving equations and it has been very successful.

First I teach simple equations like 4x = 20 or n/4=3. I explain the basic principles of solving by using the opposite operation on both sides of the equal sign. After a practice worksheet or two, I tell this story when the class is ready to move on to more complex equations. I usually write the complex equation on the board--something like: 4x +7 =27 Then I tell the class the coffee pot on the table story which I refer back to numerous times throughout the algebra course. I am relatively proficient at storytelling and I can embellish it to enhance student enjoyment.

The Coffee Pot on the Table Story


Once upon a time there was a man who was new to this country. There wasn't much he could do to earn a living, but he got a job at a factory as a "gofer". His supervisor told him that one of his tasks would be to make the coffee for coffee break. He stressed that this was an important part of the day for the workers. The man was a bit nervous about this and told his boss that he had never made coffee before and didn't know how.

The supervisor replied "Oh it's easy to make coffee. I'll show you how." The supervisor then proceeded to show him the steps to follow when making coffee. He explained that first you take the coffee pot off the table and go over to the sink and put water in it up to the line. Then take it to the coffee machine where you put the coffee in the filter, pour the water in the back, put the pot on the burner and turn the machine on. And lo and behold the coffee drips into the coffeepot!

The next day the man came to the break room to make the coffe. He followed the steps exactly: he took the coffee pot off the table, went to the sink and filled it to the line with water, took it to the coffee machine and put the coffee in the filter, poured the water in the back, put the pot on the burner and turned the machine on. and sure enough, the coffee was made.

So every day the man would make the coffee for the factory's coffee break. (When telling this story, I usually manage to repeat the coffee-making steps several times until the kids' eyes start to roll! It's just fun!)

Well things were going pretty well for the man. He was settling in to his job and performed his duties faithfully. The workers appreciated that the coffee was always made on time and available in the breakroom. But then it happened--the day came that the man's worst nightmare came true.

It was a challenge to this poor man's ability. On that fateful day, the man walked into the breakroom and the coffee pot was not on the table! What was he going to do? The workers all depended on him, and now he was afraid he would let them down. He decided not to panic and run screaming out of the factory. He took a deep breath and tried to think.

And then an optimistic expression came accross his face--he had an idea! Holding his head up and his shoulders back, he looked around the room. He saw the coffee pot on the counter by the sink. He boldly walked over to the sink, picked up the coffee pot, carried it back to the table and placed it on the table. Then he turned around and walked out of the room.

A few minutes later, the man walked into the breakroom. The coffee pot was on the table and there were no problems. Now the man could make the coffee! He picked up the coffee pot from the table, went to the sink and filled it to the line, took it to the coffee maker where he put the coffee in the filter, poured the water in the back, put the pot on the burner and turned on the machine. Everything was alright!

And do you know what the moral of the story is? The moral of the story is: If there is something you don't know how to do, change it to something you do know how to do.

That is a principle that is used over and over again in algebra and other branches of mathematics as well. Now that students know how to solve simple equations, I teach them to solve more complex equations by changing it to a simple equation. This is something they already know how to do. To change the equation, they follow the algebra rule that what you do on one side of the equal sign, you must do to the other side.

So they learn to change 4x +7 =27 to 4x =20. That is something they know how to do--they have mastered simple equations and they change complex equations to simple equations. And they do! In addition to that, they go home and tell their families the corny story that Mrs. B told during math class!

Move The Ladder Over Before You Climb Up It!

I hadn't heard this expression until last year. I was reminding my class about plotting points on a coordinate grid. You have to go over (on the x axis ) and then go up (the y-axis). I can't begin to count how many times over the years that students have become confused and plotted points backwards thus messing up their graphs.

One of the bright young scholars contributed what his older brother had told him. He said "You have to move the ladder over before you climb up it!" I don't know where his brother had heard it, but he's right. I now refer to the illustration of needing to take down something like the clock high up on the classroom wall. The ladder is by the door. It would be ridiculous to climb the ladder and then have it moved over to where you need to use it. First, you move the ladder, then climb up it to reach the clock.

So when students are plotting coordinate points, I just call out "You have to move the ladder over before you can climb up it!" and graphs come out the way they should.

My Own Blog

I read so many other people's blogs that I guess it is about time I get my own! As a middle school math teacher, one of my annual goals is to develop more math games, simualtions and mathematical models and use them in my instructional goups. Over the years I have collected stories, games and a variety of visual representations for making math instruction more clear and I am now attempting to catalog them here in this format. So Everything Math by Marsha is a collection of my thoughts and reflections in the ongoing journey of being a middle school math teacher.