October 2022
Welcome to EFM's October Newsletter!
News
Mobile App for Android Phones As announced last month, our free mobile app for Android phones is now available. Simply go to the Google Play Store, enter "Early Family Math," and enjoy EFM games, puzzles, and storybooks on your phone with your family! We hope to have a Spanish version available in a month or two.
Feedback We are always looking for ways we can improve. Given that only two people responded to our survey, I would guess our readers are not fans of doing surveys. No problem! Please just drop me a note by email and let me know your feelings about the newsletter, our games and puzzles, our storybooks, or any other part of our program that you have thoughts about that you would be willing to share. I promise we will listen carefully to your ideas.
Math Puzzles
1 + 3 = 22; 1 + 3 + 5 = 32; 1 + 3 + 5 + 7 = 42. The sum of the first odd numbers is always a square! Amazing! Why does it happen? I could show you some algebra; however, this picture gives a beautiful wordless solution to that puzzle.
Solving puzzles is at the core of doing mathematics. Seeing a beautiful pattern unfold is magical. As soon as you witness a new pattern, the natural question – Why does this happen? – leads to an intriguing puzzle to solve.
As a 12-year-old, I noticed that 10 x 10 = 102 – 02, 9 x 11 = 102 – 12, 8 x 12 = 102 – 22, and so on – what a magnificent pattern! Why did it work?
We should want all children (and adults) to experience math with this sense of exploration and beauty. Instead of routine, tiresome math exercises, most math homework should be reframed as interesting puzzles. While there is a place for routine exercises, they should be used as sparingly as possible.
Puzzles and games are more engaging than routine exercises, and they inspire students to go more deeply into the math. EFM offers a wide array of puzzles, such as Sum Groups, Number Scramble, Magic Triangles, Magic Squares, Magic Designs, Revealing Products, Four 4’s, Parentheses Puzzles, Turning the Tables, Letter Substitution Puzzles, Limited Calculators, and Cross Products.
Introduce Puzzles With Games
For a child, puzzles may seem daunting and abstract. Not every child will initially feel the pull to solve a Number Scramble or Magic Triangle puzzle.
Playing games together with children has many wonderful qualities. An important one is that games introduce very young children to puzzles and problem solving in an enthusiastic and playful atmosphere. When looking for a good move in a game, a child is solving a math puzzle without even realizing it. With repeated experiences of playing with math with others who are also having fun, children will pick up the pleasure of solving puzzles.
Keep Puzzle Solving Social
For a young child, it is essential to make puzzle solving a social activity. Math suffers because it is often seen and experienced as an activity done alone. While older children and adults can choose to enjoy working on puzzles by themselves, giving a young child a puzzle book or electronic app to do by themselves reinforces this isolated view of math for them at a time when social connections are essential.
To counteract this, almost all puzzles EFM selects are ones that an adult can easily create for the child. This fosters engagement and understanding in the adult, and it increases the likelihood that the child and adult will play at solving the puzzles together.
Puzzle Sites – Old and New
Supplying math puzzles is nothing new for this newsletter, but it has been a while since the last ones. In July 2021, I discussed the puzzle ideas from Open Middle. Later, in November, I described some puzzles from Peter Liljedahl's excellent bookBuilding Thinking Classrooms in Mathematics.In December, I gave several examples of puzzles Julia Robinson Math Festival uses for their math festivals.
This month I would like to introduce the 26 puzzles of Play With Your Math.
Play With Your Math has elegantly presented math puzzles on their website, and they also have PDFs for posters of their puzzles to put up on walls. In addition to some standard puzzles, such as Chicken McNuggets, Four 4’s, Going Up Stairs 1 and 2 at a Time, and Squares Within Squares, they have many less familiar ones. Their puzzles often have easy entry points that allow anyone to get started, and then provide natural paths that invite further exploration deeper into the puzzles.
Most of these puzzles are from the middle to the upper end of EFM skills. I have selected a few of their puzzles to talk about here. My descriptions of the puzzles won’t be nearly as engaging as theirs. You should go explore all of them!
Puzzle 1: Find a way to split 25 into a set of numbers that sum to 25 and which produce as large a product as possible. For example, 25 = 10 + 10 + 5 and 10 x 10 x 5 = 500. However, 25 = 5 + 5 + 5 + 5 + 5 and 5 x 5 x 5 x 5 x 5 x 5 = 3125 is larger. Look for the strategy for the best split of any number by first looking at how to split up smaller numbers (often a good problem solving strategy).
Puzzle 3: Chicken McNuggets. Which numbers can’t you make by adding up some 6’s, 9’s, and 20’s? What is the largest one that can’t be created? What happens when you change 6, 9, 20 to other sets of two or three numbers?
Puzzle 4: Which numbers can be represented as the sum of consecutive numbers? These are calledTrapezoidal Numbers. You can do a lot of exploring of this at a basic addition level. The final answer is very satisfying; however, deriving the reason for the final answer requires a bit more sophistication. You can also explore otherFigurate Numberssuch asTriangular Numbers.
Puzzle 7: In how many ways can you climb n steps if you always climb them one or two steps at a time? If you start listing out what happens for the first few values of n, you will see the Fibonacci Numbers – why?
Puzzle 9: Four 4’s. Starting at 1, how many numbers can you create using four 4’s? For the numbers you can create, how many ways can you do it? For example, 1 = (4 / 4) x (4 / 4) and 2 = (4 / 4) + (4 / 4). This is fun to play with once your child can do the four basic operations. It can become more involved as they learn more math, but it’s good for a lot of fun even in those earlier days.
Puzzle 11: Star Polygons. Connect every 2nd (or 3rd or 4th or ...) dot for n evenly spaced dots around a circle. Without doing any numerical math, you can have a lot of fun exploring this and making lovely drawings. Understanding the final answer does require knowing about relatively prime numbers.
Puzzle 16: Self-aware Numbers. Every digit in a self-away number refers to how many digits of that sort there are in the number. For example, 1210 is self aware because it has 1 0, 2 1’s, 1 2, and 0 3’s. The question asks if there are any ten-digit self-aware numbers; however, you can have fun exploring for all such numbers and discussing your results. For example, there are no 1- or 2-digit self-aware numbers. Although this looks like it involves understanding multi-digit numbers, it doesn’t really. This is a fun logic puzzle that is very engaging.
Puzzle 21: Squares within Squares. If you want to split one square into a collection of squares (not necessarily of the same size), for which numbers of squares is this possible? This can be investigated and enjoyed with lots of experimentation. Eventually, to get the final answer, an organized approach will be needed.
Puzzle 24: Plus – Minus. If you use any choice of +’s and -’s between the numbers, for which sets of numbers from 1 to n can you create 0? For example, 0 = 1 + 2 – 3 and 0 = 1 – 2 – 3 + 4. Can you do it when n = 5, n = 50, or n = 100? Suppose you replace 0 with other numbers – what happens then? This can lead to some good discussions about even and odd numbers.
Puzzle 25: Trains. Suppose you want to make a train of length 4 out of existing trains of various lengths. How many ways can you do it? How many ways can you do it if none of the trains is smaller than length 2? What happens if you replace 4 with other numbers? Welcome to the number theory world ofpartitionsandcompositions. You can do some advanced math with this. However, a child who only knows how to add can also have lots of fun doing additions and looking for patterns.
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
October 18, 2022
Chris@EarlyFamilyMath.org
Twitter | Facebook | Instagram
Early Family Math is a California 501(c)(3) nonprofit corporation, #87-4441486.