November 2021
Welcome to EFM's November Newsletter!
News
Support: Last August we received a $1000 gift. We are very pleased to announce that a supporter wanted to match that and generously gave us $1500! The math question that only you can answer is: What is the next term in the sequence $1000, $1500, ____ ? The IRS would like me to tell you that while EFM is a nonprofit, it is not currently a tax-exempt organization - we hope to achieve that status near the start of 2022.
Early Family Math is an all-volunteer collaboration of people who feel passionately about early math education. The support we receive from many people and organizations means a great deal to us! If you are interested in collaborating or supporting EFM in any way, please contact us and let's talk about working together to further this important cause!
New Activities to Enjoy!
The Activities this month come from Peter Liljedahl's excellent recent book Building Thinking Classrooms in Mathematics. Although this book is aimed more at school classrooms than home living rooms, it has some good ideas for families to think about. Two ideas, using rich math tasks and playing with math in small groups of 2 to 4, are natural for families using EFM. He also advocates using vertical, non-permanent surfaces for children to play with their ideas on. If you don't have a whiteboard, go ahead and use whiteboard markers on windows or any other vertical surface those markers can easily be cleaned from. He also gives detailed guidance on how to handle math questions and discussions productively.
As originally given, the first and third puzzles in this newsletter are simple answer-the-question activities. However, if you enter into them with a spirit of playfulness and exploration, all three puzzles provide opportunities for asking new questions that lead to interesting mathematical paths to travel!
Chapters 3 — Neighboring Numbers
The original version of this puzzle challenges your child to fill in the boxes in the first (leftmost) diagram with the numbers from 1 to 10 so that no connecting boxes have consecutive numbers. Answering that challenge provides some good playing around, but it can also be just the start of a fun mathematical journey.
These puzzles create lots of playful and deeper thinking about the sequence of the first few numbers.
At first, you may decide that "connecting" means that the two boxes share a side. However, you can also decide to use the more challenging definition that "connecting" means sharing a side or a corner.
Vary this by trying different arrangements of boxes and ranges of numbers (a few are shown). Why is it impossible to solve for some box arrangement? Instead of using boxes, you can use an Island Hopping arrangement with circles connected by line segments, as in this diagram.
Another variation is to reverse the rules and challenge your child to fill in the diagram so that consecutive numbers are always connected by a side. Sometimes, some starting squares will work while others do not - explore why that is.
Chapter 3 – 5 — Where Did These Answers Come From?
Use the numbers from 1 through 10 with addition and subtraction. Without showing your child, separate the numbers into pairs. For each pair, use either operation and produce an answer. Finally, show your child the list of answers and challenge them to figure out pairings of numbers that produce them.
These puzzles are easy to create. As an example, pair up (2, 8), (1, 10), (4, 7), (3, 5), and (6, 9). Next, use addition or subtraction to calculate 2 + 8 = 10, 10 - 1 = 9, 4 + 7 = 11, 5 - 3 = 2, and 9 + 6 = 15. So, the set of answers to give your child would be 10, 9, 11, 2, and 15. Remember to be open to your child finding correct answers that are not the same as yours!
At first, let your child play with the numbers and don't be concerned about being analytical or solving the problem quickly. If this proves to be too hard, use a smaller range of numbers, say 1 to 6 or even 1 to 4.
At first blush, there may seem to be an overwhelming number of possibilities to consider. Help your child find some of the ways to make the search easier. For the range 1 to 10, answers that are 10 or larger must come from adding. An answer like 15 can only come from 7 + 8, 6 + 9, or 5 + 10, so that reduces the possibilities considerably. Answers of 1 or 2 can only come from subtraction.
A more sophisticated observation, for an older child, is that if all the problems used addition, then the ten numbers would add up to 55. Each time a subtraction is used instead of an addition, the total is lowered by twice the number being subtracted. For example, if 5 - 3 = 2 is used instead of 5 + 3 = 8, then 2 = (5 + 3) - 2 x 3 - the adding result of 8 is lowered by 2 x 3. In this example, the sum of all the answers is 10 + 9 + 11 + 2 + 15 = 47 = 55 - 8. Because 8 = 2 x 4, the sum of all the numbers being subtracted is 4 This means either there is a single subtraction of 4, or two subtractions using 1 and 3. Pretty cool!
To increase the difficulty, you can use a larger range of numbers and you can include multiplication or division. Another option, as is the case for many of these puzzles, is to have your child create these puzzles for you and then help you solve them.
Chapters 5 — Biggest Product From a Split
This puzzle has the potential to take your child on a fun mathematical journey whose end may surprise them.
Start with any number, say 12. Take the product of any group of numbers that add up to 12. The challenge is to find the group of numbers that gives the largest product.
Be playful! Your child may quickly come up with 12 = 6 + 6 and 6 x 6 = 36 - write this down. Another quick possibility is 12 = 1 + ... + 1, which produces a product of 1 - write this down as the smallest possible product, which is pretty interesting on its own Play around with this a while, write things down, and see if any patterns emerge.
If nothing jumps out from the experiments, consider using powerful problem solving strategies - Do Examples and Do Simpler Versions Of The Problem. For this, the two strategies are the same. Replace 12 by small numbers starting at 1 and make a table of the results and how you got them. Some trends should emerge. Using any 1's is always a bad idea. Using a 4 is the same as two 2's, so use 2's to keep things simple. If you have a number larger than 3, replace it with 2's and 3's. Combining those observations, the question becomes: What is the best way to use 2's and 3's? The final observation is that, surprisingly, a pair of 3's (3 x 3 = 9) is better than three 2's (2 x 2 x 2 = 8).
That's it! You have solved the puzzle! For 12 the largest answer is 3 x 3 x 3 x 3 = 81. Now that your child knows how to do 12, try out these ideas on other numbers.
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
November 18, 2021
chris@kitchentablemath.com