May 2022
Welcome to EFM's May Newsletter!
News
Math Games for Schools – EFM is developing collections of math games and puzzles for schools to use with their younger children. These resources are specifically designed to provide playful approaches to mastering math skills in school. Our first collection, Math Games for Tutoring, is a special subset of EFM games and puzzles that are quick and easy to introduce to students, and they provide focused skill building in more enjoyable and engaging ways than using flashcards or workbook pages.
Activities With Small Quantities
This month I am highlighting games and puzzles that involve small quantities. The first two can be played enjoyably in a very simplistic way that emphasizes playing around and getting lots of experience with these small quantities – without any need to analyze or think carefully about what is being done. All three have the potential, when the child is ready, to pose very interesting and challenging questions even though the numbers are quite small.
Mancala
This is a very old game, both in the US and around the world, with many names and variations. What’s lovely about this game is its simplicity along with the emphasis on basic quantities. As the child gets older, planning and strategy will also come into play.
As played in the US, the basic game consists of six pits in a row with a larger home hole at the end of the row for each player, as pictured below. Before play starts, a number of seeds, say three, is placed in all the pits for both players. While this is traditionally played on a piece of wood that has pits dug out of it, you can very easily play it on a piece of paper with circles and ovals drawn on it.
On a given turn, a player picks up all the seeds from one of their pits and “sows” those seeds counterclockwise by putting one seed in each pit in order until they run out of seeds. While sowing, if they come to their own home they put a seed there, but if they come to their opponent’s home they skip it.
When sowing a turn’s last seed, two special conditions can occur. If the last seed goes in their home, they get another turn. If their last seed is placed in one of their own pits that was empty, that seed, along with all the seeds from the opponent’s pit opposite that pit, are picked up and placed in their home, and the turn is over.
The game ends when all the pits for one player are empty. At that point, all the seeds in the opponent’s pits get put in the opponent’s home, and the game is over. The player with the most seeds in their home wins.
To make this game easier, reduce the number of pits to four or even three, and reduce the number of starting seeds to as low as two. To make it harder, you can increase the number of seeds to as many as six.
Treasure Maps
These are Jumping Julia Mats from Julia Robinson Math Festival.
The challenge with these puzzles is to start in the upper left hand corner of the grid and, after a series of horizontal or vertical jumps, end up in the lower right hand corner. Each jump can be up, down, left, or right, and the size of the jump must be the value of the square you are jumping from.
You can make these any rectangular size you please, though smaller than 4 by 4 might not be very interesting.
You can put the whole puzzle on one piece of paper. However, it is much more fun to put these on the floor and step from number to number. On the floor, put a large number on individual pieces of paper to be stepped on – that way, when the current puzzle is solved, you can shuffle the pieces of paper around and make a new puzzle with very little trouble.
One of the things I like most about these puzzles is that they are a great example of the problem solving technique of combining working forwards from the beginning and backwards from the ending. If you only look at possibilities going forward from the starting square, that can be a lot of possibilities to consider (especially for larger puzzles). Similarly, if you only work your way backward from the goal by asking the question of which squares can jump to it, you have a lot of possibilities to work through. However, if you combine the two approaches, that can often lead to a much more effective process.
Connect Islands
This is the game of Hashi or Hashiwokakero, and was published by Nikoli. It can be found these days in the Sunday New York Times. The puzzle consists of islands with numbers on them. The islands are to be connected to some of the other islands by horizontal or vertical bridges.
The total number of bridges to be constructed from an island is the number on that island. No more than two bridges can be placed between a given pair of islands. Bridges cannot cross each other or skip over islands. When done, it must be possible to get from any island to any other island using a series of bridges.
This simple logic puzzle provides lots of practice adding small numbers and seeing how numbers can be broken up into pieces. For example, if you know there are going to be 6 bridges from an island, and there are exactly three islands it can possibly connect with, then there must be 2 bridges to each of the three islands.
In the pictured puzzle, the island with 6 in it is the easiest place to start – you know that there must be two bridges in each of the three directions. Once that is done, the islands directly above and below it are easy to finish. The puzzle is quick to finish after that.
Making these puzzles is fairly simple. First, place some small circles on a grid. Next, imagine one or two bridges going between pairs of those circles so that all the circles are connected. Finally, put the total number of bridges you imagined inside each circle.
Although I haven’t tried it, I can imagine allowing up to three, or even four, bridges between two islands to practice with adding larger numbers. I would imagine that this would make the puzzles much harder to solve. Please let me know if you try it.
If you have any questions or comments, please send them my way. I would enjoy the opportunity to chat with you. Also, if you are interested in collaborating with us or supporting us in any fashion, I would love to talk with you about ways we can work together.
Chris Wright
May 18, 2022
Chris@EarlyFamilyMath.org
Facebook | Instagram
Early Family Math is a California 501(c)(3) nonprofit corporation, #87-4441486.