This week's content is provided by Erick Lee, an educator who is sharing a simple game that you can use with your child(ren). It's perfect for the holidays, a ride in the car, or a chilly weekend chat.
The subtraction game is a common game that has many variants and is a great way to practice counting and develop number sense. It is played with just a few simple rules and does not need dice, cards or a playing board. It is ideal for playing around the dinner table. You only need a few object to use as counters. I’ve used toothpicks, pennies, sugar packets at a restaurant, or a pile of pebbles at the playground.
How to Play
Start with a pile of 13 of toothpicks (or any other collection of similar objects). Two players take turns removing 1, 2 or 3 toothpicks from the pile. The winner of the game is the person who takes the last toothpick. There is an elegant strategy that can be discovered to guarantee a win for the first player in this game*. Once you’ve discovered the strategy for this game, you can alter it. You can start with a different number of objects or allow a different number of objects to be removed each turn (a version of this game even made an appearance on the television show Survivor: Thailand with two teams removing 1, 2 or 3 flags from a starting group of 21 flags.
You could also alter the rules of this game so that the player who takes the last object loses instead of wins (this is called a misère game).
*In the subtraction game described above, the first player can always win by taking the number that leaves the other player with a pile whose size is a multiple of 4. So on the first turn, the first player should take one object leaving 12 (3 x 4). Whatever the second player removes from this pile of 12, the first player should remove enough objects to bring the pile to a size of 8 (2 x 4). Whatever your opponent takes from the final pile of 4, you take the rest and win.
The Counting Game
The counting game is a variation of the subtraction game that is played orally instead of with physical objects. In this version, two or more players take turns counting up from one. The first player says “1” and then each player on their turn can say the next one, two or three numbers counting on from where the preceding player left off. The person who says “13” is out. The game starts again at “1” with the remaining players. Keep playing additional rounds until there is a final winner. An example game with two players is below:
Player 1: “one”
Player 2: “two, three”
Player 1: “four, five, six”
Player 2: “seven”
Player 1: “eight, nine”
Player 2: “ten”
Player 1: “eleven, twelve”
Player 2: “thirteen”... player 2 is out!
Since no equipment is needed, this game can be played anywhere... on car trips, while waiting in a long line, or sitting around a campfire. It is also a great way to practicing counting in another language. Like the subtraction game, this game can be changed to finish at any number or to allow each player to continue with a different amount of numbers on their turn. For example, "Add at most 4; lose on 21".
The subtraction game and the counting game are part of a broader class of games known as "Nim Games" in which each player removes some objects form a pile or piles. The most common game of this type is called Nim and is played by two players. This game dates back to medieval times. In this game there are 3 piles of objects. No two piles should have the same number of objects at the start of the game. For example, a game might start with 3 objects in the first pile, 5 objects in the second pile and 7 objects in the third pile. On each player’s turn, they choose one pile and remove as many objects as they'd like (but at least one) from that pile. The player that takes the last object wins. The winning strategy for this game is a bit more complicated than the subtraction game and was not published until 1902 by Harvard University professor Charles L. Bouton.
If you like these games, try checking out some similar games:
Additional Nim Resources:
Nim Like Games
If you have any questions, please feel free to contact me on Twitter (@TableTalkMath) or leave a comment below.
Thank you for taking the time to improve math fluency for children, one table talk conversation at a time. For previous newsletters, please check out www.tabletalkmath.com/previous-newsletters
This week's content is provided by Pierre Tranchemontagne, an educator who has provided us with a fantastic resource for you to use with your child(ren). If you have not yet checked out his work, do it now!
How many dots?
How do you see them?
This image can be found on Number Talk Images- a website dedicated to collecting interesting images that spark "math-y" conversations. The questions above are a great way to start this conversation with children. There is obviously a fixed amount of dots in the image above; however, there are multiple ways of seeing them.
Did you see the dots organized vertically from left to right as 2-3-3-2?
Did you see 2 groups of 5 horizontally or vertically?
How about groups of 3 dots from bottom left to top right with one left over?
What about 3 groups of 4, minus 2 dots that were counted twice?
It is always fascinating to find out how other people “see” the image.
Here are a few more images from the website:
Using these types of images to prompt a Number Talk are a great way to spend time with children developing number sense concepts such as subitizing (recognizing small groups of items within the larger group), unitizing (treating a group of 4 objects, for example, as 1 unit), flexibility with computational strategies, efficiency and number facts, among others.
Here are some tips on using Number Talk Images at home:
Lastly, I am always interested in hearing about conversations prompted by an image from the website. How did it go? What did your child find interesting? What did you find interesting? Please share your experiences by sending them to email@example.com
How are you using images to have Number Talks at home? Comment below or find me on Twitter at @tabletalkmath to continue the conversation.
This week's content is provided by Brian Bushart, an incredible person and father of a budding 4.5 year-old mathematician. If you have not yet checked out his work, do it now!
I have a confession to make.
Promise not to laugh.
I still count on my fingers.
There. I said it. The guy who’s taught elementary school and developed middle school math curriculum counts on his fingers. Sure, I know how to do loads of math in my head, but that doesn’t stop me from regularly pulling out my fingers to help me out in my daily life. It’s like I was born to use them to do math.
Oh wait. I was!
Turns out, according to brain science, that even if we aren’t using fingers as we calculate, an area of the brain lights up that is actually “seeing” fingers! Not only that, but a child's early knowledge of fingers can be a better predictor of math performance than tests.
We’re born with our fingers, and we put them to use immediately making sense of the world around us: grabbing, touching, pulling. They are a tool in so many ways, including in mathematics, even as we get older.
That’s why my four and a half year old daughter and I talk about, count, and play with our fingers every chance we get. Do I want her to learn efficient computation strategies later in her life? Sure! But I also know the importance of a solid foundation of counting and quantity, especially super useful numbers like 5 and 10. Oh hey, I’ve got those right here on my hand!
One of my favorite games I like to play with my daughter is “How many fingers am I holding up?” The trick is that I’m holding my hand up against my chest so she can’t see my fingers, usually while I’m driving. She makes a guess - she’s got a 1 in 5 chance of being right - and then I pop up my fingers so she can check to see if her guess is correct.
At first she always had to count each of my fingers, but over time she’s learned to recognize 1, 2, or 3 fingers without counting them at all. In general, not just with fingers, she’s getting better at instantly recognizing those quantities without counting one by one. It was so cool to see her start doing this! As she gets more and more comfortable with 4 and 5, she’ll start to count those less and less and begin to recognize them right away as well.
Another game I like to play is Fast Five. I tell her, “Do you want me to show you 5 really fast?” She says yes. I hold my hand behind my back and say, “Ready...set...fast five!” Then I quickly pull my hand from behind my back with all five fingers up. She counts my fingers, and, wouldn’t you know it, it’s always five. She doesn’t know it yet, which is why we keep playing, but eventually she is just going to know that there are 5 fingers on a hand.
In the meantime, we’re having fun talking about numbers and she’s learning more and more about this really cool tool she’s got attached to her body that she carries with her everywhere. Before too long we’ll be moving on to Fast Ten and other games that use two hands. I can’t wait!
For older kids, I will hold up three fingers on one hand and two on another:
"How many different numbers can we create from these two?"
It seems simple, but gets more challenging as you learn about different operations. Try it; how many values can you create?
If you’d like to try out some fun activities with your child to help build knowledge of their fingers, check out this collection from YouCubed, an organization at Stanford University.
How are you doing this with your child(ren) at home? Feel free to drop a comment below and share. Also, remember to sign up for the FREE weekly newsletter!