I am thrilled to have Chris Shore take over the newsletter this week to share about his experiences with Clotheslines. This week's newsletter has been translated into Spanish (bit.ly/TTMspanish46) by Ed Campos, Jr. Make sure you check it out and tell him thank you on Twitter.
A person’s ability to think mathematically has far more to do with their number sense than with their ability to study and memorize formulas and procedures. Number sense is the capacity to think flexibly, efficiently, and accurately about numbers. One of the great methods in teaching number sense is the open number line, which is a number line with no or few given benchmarks.
I’m currently working extensively with this open number line concept using an instructional tool know as the Clothesline. The Clothesline is a dynamic number line (string) on which students can actually move the values and benchmark (folded cards).
For example, let’s say we ask students to place the following three fractions: 1/2, 1/3, 1/4
Typically, students will use only 0 and 1 as benchmarks, then many will often inaccurately place the fractions in the following order:
This misconception offers a great opportunity to discuss how when something is divided into more and more equal parts, each part gets small. For example, 1/2 of a pizza is a bigger slice than 1/4 of the same size pizza.
Once the order of the fractions is correct, students are prompted to properly space the fractions. While students frequently find the proper spacing for one-half and one-fourth, one-third challenges them. Is it closer to the fourth or the half? There are several strategies for determining this, from converting to decimals (0.25, 0.33..., 0.50), finding a common denominator (3/12, 4/12, 6/12) or proportional reasoning (three thirds equals exactly one).
As a parent at home working with the number line with your child, you don't need to try to do the teacher’s job. Simply share aloud how you think about these numbers. We math teachers call these “number talks,” and it is one of the most effective ways you can support your child’s learning of mathematics.
If you are interested playing with the Clothesline or simply the open number line on paper, you can find free resources on clotheslinemath.com
First, I must say thank you to Kent Haines and Christopher Danielson for posting prompts like these on Twitter recently. This week's newsletter has been translated into Spanish by Ed Campos, Jr. Make sure you check it out and tell him thank you on Twitter.
Quick! Look at the image above and tell me how many eggs you see in the crate!
This is not only a fun prompt to try at home, it's also a great insight into how you group objects to count them more efficiently. Did you count the first long row, the second row with the individual egg on one side and the pair on the other, then move the third row and finish it off? Did you start with the fact that there egg crates hold 12 eggs, then subtract out the missing ones? Did you approach it differently? More importantly, how did your child approach it and did that differ from yours?
How about this one? Was it easier to count these? More difficult? Our brains work in mysterious and interesting ways. For me, I'm taking the group of four, group of two, and group of four, then combining them all. Do you see it?
I love bringing my kids to the counter and challenging them with these quick prompts, none of which take a lot of creativity and offer a lot of opportunity for thoughtful responses.
That is, until you go and start cracking eggs:
How many eggs do you see? What qualifies? How did you count?
This week's newsletter has been translated into Spanish by Ed Campos, Jr. Make sure you check it out and tell him thank you on Twitter. The link to share is bit.ly/TTMspanish44
Matt Lane, author of Power-Up: Unlocking the Hidden Mathematics in Video Games, grabbed ahold of the newsletter control board to talk about video games. Why would I want my kids to play video games? Is there value in them? Read below to find out.
With school out for summer, you've likely already made plans to keep your kids active, engaged in learning, and off of the couch. But if your kids are anything like me when I was a kid, they'd rather stay at home and play video games. Fortunately, there are opportunities for rich mathematical conversations that arise from these games, even the ones that aren't expressly designed to be educational.
Here are three reasons why it's worth engaging your kids in video games, especially when it comes to mathematics:
1. They're relevant. According to the Entertainment Software Association's 2017 report, Essential Facts About the Computer and Video Game Industry, 65% of households in the US have at least one person who plays video games for at least three hours per week. It's a multibillion dollar industry that already appeals to millions of people, kids included.
2. They have pedagogical value. In his 2011 book Mathematics Education for a New Era: Video Games as a Medium for Learning, mathematician Keith Devlin argues for using video games not only to teach basic arithmetic and numeracy, but also to help foster a a productive disposition, which the National Research Council defines as "A habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one's own efficacy."
3. They have a precedent. Did you know that Tetris has been linked to one of the Clay Mathematics Institute's Millenium Prize problems? Or that the origins of probability theory can be traced to a series of letters between mathematicians Pierre de Fermat and Blaise Pascal about a parlor game? Here's a simplified version of what they discussed (it's now known as The Problem of Points)
Suppose you and your child are playing a game where you both have an equal chance of winning each round. The first to win a certain number of rounds (say, 4) wins a prize. If the game is interrupted before either player can reach 4 wins, how should the price be fairly divided? To put it in concrete terms, what if you are winning 3 games to 1 when the game is interrupted? What if you're only winning 2 games to 1?
If you're looking to spark some interesting mathematical conversation, the Problem of Points is a great place to start. And if you'd like explore the intersection between math an video games in more detail, consider checking out my new book! It's called Power-Up: Unlocking the Hidden Mathematics in Video Games.
Also, here are some recommended games to play with your kids, along with their ratings:
Portal 2 (E10+)
The Witness (E)
The Sims 4 (T)
Enjoy your summer! And if you ever want to chat more about math or video games, you can find me on Twitter (@mmmaaatttttt - that's matt cubed).
This week's newsletter has been translated into Spanish by Ed Campos, Jr. Make sure you check it out and tell him thank you on Twitter. The link to share is bit.ly/TTMspanish43
Edmund Harriss, author of Visions of the Universe, stopped by to talk about a seemingly simple idea that gets more complex every time we think about it. How does addition work, anyways? Come find out!
How many additions can you find with the same answer?
A good part of mathematics begins with counting, which can be thought of as just adding 1 several times to get each whole number in turn. The next step is to add in, addition. This is just repeated counting, beginning a pattern of repeated operations that we could explore in all sorts of ways, but instead we shall pause here and see what can be done with just addition. The next paragraph breaks down the exploration into occasionally agonising detail, but has lots of additional questions to dig even deeper.
From counting, we already have one way to get each number:
2 = 1+1
and so on
If instead we combine some of the additions at each stage, we get some new ways:
and so on
How else could we make four? 2+2 springs to mind, so just looking at 4 we now have 3 different ways to make it:
keeping just 2 numbers 1+3 is another possibility
Do we consider that the same as 1+3? Are there other ways? We have 1+1+1+1 which is the sum of four numbers, unless we allow 0 which would create all sorts of problems (why?) we cannot make any of the numbers to add smaller, so this is the only option. We also have 3+1, 2+2 and possibly 1+3 with two numbers. We could have 4 for just a single number. So with 1,2 and 4 we are missing 3. We can 2+1+1 as a way to add three, depending on your answer to the question of 3+1 and 1+3 this might be the only option, or you might also have 1+2+1 and 1+1+2 (why are there not other options?). This gives a complete list of possibilities for 4.
Exploring 4 above, you notice that the emphasis is on the questions more than the answers. For one thing, we are looking at all the addition questions that give a certain answer 4, the opposite of the first experience of mathematics. In addition, the question of whether 1+3 and 3+1 are the same does not have a right answer, but the answer given has consequences later. That answer gives two different answers to the motivating question "How many ways can you add to 4?". If we allow re-orderings to be different then there are eight, if we say they are the same then there are still five different ways.
Now think about a larger number, say 5. Can you write a list of all ways to make that? With some written down, challenge each other to make more, until you cannot find any more. You can also discuss how to count the number of ways to count orders. For example 1+1+2 has three orders as we saw above, but 1+2+3 to make 6 can be put into six different orders (can you find them?). For larger numbers still, you might want to try to find systems that give all options. For example if you choose 11, that can be made in 56 different ways (even not counting reorders). A good way to explore all options is with a dot pattern like this one for 4 (how these relate to the 5 ways of adding is left as a puzzle), these are called Young or Ferrers diagrams and have applications in a variety of places in mathematics and physics. You can see all the diagrams for 11 in my colouring book Visions of the Universe (Visions of Numberland in the UK), along with a lot of other visual mathematics to play with.
This week's newsletter has been translated into Spanish by Ed Campos, Jr. Make sure you check it out and tell him thank you on Twitter. The link to share is bit.ly/TTMspanish42
Based solely on the career statistics lines that you see above, would you rather have Player 1 or Player 2 on your team?
I intentionally took off the names so you could only rely on statistics. Which ones are the most important to you when looking to build a team?
Player 1 only had 137 home runs and 734 runs batted in, yet he had a .409 on base percentage. On top of that, he had almost half the at bats as Player 2, leading us to believe that he played for about half as many seasons.
Meanwhile, Player 2 only stole 50 bases and had a much lower batting average, yet he hit far more home runs and had substantially more runs batted in.
For more context, and in honor of this being the 42nd newsletter, Player 1 is the stat line of Jackie Robinson, the man famous for breaking Major League Baseball's color barrier. Not only did he put up this impressive career line of stats, he did so while breaking into the league when African American athletes were not welcomed--or allowed--on the same field as their white counterparts. He was the Rookie of the Year during his rookie season and also earned an MVP award. His number, 42, is the only number to be retired across all of baseball.
Player 2 is the stat line of Ernie Banks, a Hall of Fame player who spent 19 seasons with the Chicago Cubs. Banks came into the league just a few seasons after Robinson and endured many of the same issues. While he took second place in the Rookie of the Year voting, he did manage to earn two MVPs for this performances.
Now that you have this information, does it change your decision? Which player do you go with and what is your mathematical justification?
Maybe you and your child(ren) aren't into baseball, and that's just fine. Here are some other sports that might interest you at the table:
Premier League soccer
Right now is a great time to be a fan of sports, with finals here and seasons in full swing. If you sit down to enjoy the game, turn it into a learning opportunity as well!
This week's newsletter has been translated into Spanish by Ed Campos, Jr. Make sure you check it out and tell him thank you on Twitter. The link to share is bit.ly/TTMspanish41
Joshua Zagorski is a father of two wonderful boys and, like many of us, likes to engage in math-based conversations at home, even if it's while preparing lunches in the morning. He has come aboard the helm of the newsletter this week to share with all of us. Take a look and try it out with your own child(ren):
It was the end of spring break and it seemed my wife and I had forgotten to purchase new water bottles for our sons’ (K & 2nd grade) lunches on the first Monday back to school.
When I realized the situation we were left with in our cabinet (see the picture above), I decided to have a little fun with my sons. After calling them down to help pick out items for their lunch, I asked them the following question:
Which water bottle do you want in your lunch tomorrow?
Immediate arguing ensued over the taller water bottle. I could not help but laugh and follow up with:
Why do you both want that water bottle?
Both boys agreed on one thing, “it was the bigger one.” The math teacher in me continued to prod with questions:
How do you know it is larger?
After hearing a third question from myself, my second grader knew I was up to something and asked a question back:
Which bottle holds more water?
I was half tempted to rip off the labels at this point but decided that may be taking it too far. Instead, I grabbed the bottles and put them up on a higher counter. My sons and I briefly discussed ways we could tell which bottle held more water without reading the label. My older son guided my younger son and insisted the best way to tell would be to pour both of the bottles into two different plastic cups. The cup or cups that had more water would “win.”
This was the point of the conversation where I knew he understood the basic concept of volume. On this day, we did not complete the experiment to its capacity and instead explored both labels. Even after reading 10 oz on both bottles I watched as both my sons worked through the idea that different shapes could hold the same volume.
This five to ten minute conversation was an essential piece to their math journey. Without using fancy math words, or being in “math class,” my sons were exposed to an authentic math activity. They experienced comparing three dimensional shapes (K math standard) and its attributes. The fact competition and debate were involved helped enhance the discussion.
Have an older student and like the above image? I would love to explore the question:
Which bottle costs more to produce?
Everyday life experiences can lead to impactful math conversations.
This week's newsletter is also available in Spanish, thanks to Ed Campos, Jr. Make sure you tell him thanks and share the link with friends: bit.ly/TTMspanish40
Math tends to come with some shortcuts, easy ways to remember things, or "tricks" to abide by. This week, I brought in Tina Cardone, author of Nix The Tricks and all-around amazing person, to share her thoughts on the matter:
Have you seen people using their fingers to multiply by nine? There’s this trick where if you put down the finger corresponding to the other factor, you can read the product off your remaining fingers. Fingers to the left represent the tens place digit and fingers to the right represent the ones place digit.
If you or your child haven’t seen this trick before take some time to play and verify that it really works. Then ask kids’ favorite question - why?
Ask your child to do some observing and look for reasons behind the patterns. It might help to write down the numbers to be able to see a pattern.
9 * 1 = 9
9 * 2 = 18
9 * 3 = 27
9 * 4 = 36
9 * 5 = 45
9 * 6 = 54
9 * 7 = 63
9 * 8 = 72
9 * 9 = 81
9 * 10 = 90
The big idea here is that we have ten fingers and a base ten number system (every time you count to ten you start over in the ones place). Using the idea of "nine is one less ten" we can figure out that every multiple of nine will be that many less than the decade number (multiple of ten).
For example: 10 x 3 = 30 so 9 x 3 = 30 - 3 = 20 + 7
Putting down your third finger leaves two fingers to the left (twenty is the decade number less than thirty) and 10 - 3 fingers to the right (you have ten fingers! Your fingers show you the seven.)
Try it out for some other multiples of 9. Get it now?
Every time I see a math trick, first I verify that it really works and then I ask why. Asking “why?” is what math is all about! Once you’ve figured out the reasons behind a trick, you can rename it a strategy. There are lots of tricks that people use in math class without understanding the math - so many that I wrote a book about them called Nix the Tricks. If you’re interested in turning more tricks into strategies, check out the free download.
Thank you to Ed Campos for translating this week's newsletter into Spanish. Check out the link and make sure to thank Ed: bit.ly/TTM38spanish
Growing up, I was awful at Origami. If it meant folding a paper into a cool design so I could pass a note during class, that was one thing. Sitting down in an art class and folding paper for a purpose, though, was a major challenge. Recently, Nalini Joshi shared a tweet about Mathigon and I just had to share it with you.
When you go to mathigon.org/origami, there are a ton of beautiful creations that are laid out for you to try... FOR FREE. They start out rather simple (well, relatively) as you can see in the image above. If you click on any of the images on the site, you can download the "net" and print it out. The net of a 3D object is a diagram of all facets (or faces) laid out flat. This makes life a little easier when trying to fold that thin sheet of paper into something beautiful that you and your child can be proud of.
When you finish with some of the first few shapes, you need to try at least one of these:
There are more, and they get even more complex. So, here's what you need to do: find a printer and run off a few copies of some of the more basic nets. Then, pick one of the "Archimedean Solids" and run off a few copies of it. I say a few copies because there's a good chance that the first attempt is not going to be a success... and that's alright.
Here are some things to keep in mind when trying Origami at home:
This is designed to be fun. If it isn't fun, stop. Come back to it later. Maybe start with something more basic, then gradually move up to more complex designs.
There is no way (that I know of) to cheat. Use whatever resources you can to get to the final product.
It's OK to not be perfect! Oh, so your design doesn't look like the impeccable model in the picture? Totally fine. Maybe it's something that you laugh about, how far off it is from what "it's supposed to look like" before moving on to something else.
There is a lot of math involved. Included on the website is a two page explanation of the math behind Origami and it's work checking out. If it's too intense, that's fine. Just know that some beautiful mathematics pops up from the art.
This week's newsletter is also translated into Spanish from Ed Campos. Give him a follow on Twitter and share the link with your friends: bit.ly/TTM37spanish
You probably already know this, but the Internet is an amazing (and intimidating) place to be. For kids looking to feed their curiosity, online games are especially popular. This week, I wanted to share my favorite place to go for high quality games: mathmunch.org/games
No matter how old your child is, and no matter your level of math comfort, there is a game for you and the family to play. With 34 games currently shared on the webpage, you are certain to have a good time and can take comfort in the fact that the team at Math Munch have vetted the games for value and relevance.
With that said, there are some pointers that I recommend:
Playing the game with your child. Opening up the browser and walking away has much less value than sitting down and taking the time to talk through strategies, share new ideas, and bond over the successes and failures.
Limiting the time on the game. Have you ever played a game for so long that you lost track of the clock? No? Just me? OK, well let me tell you that it's easy to do, especially when you're focused. If you and your child set a time limit, you can pace yourselves and ensure a meaningful interaction.
Having fun! Games are intended to be fun, and they're more fun when playing alongside someone you love. If a game gets too frustrating or overwhelming, talk through it, then move on. Put the device away. Come back to it later. There's no need to create tension in something designed to bring people together.
This time of year, the weather starts playing tricks on us. One day, it's snowing; the other, warm and sunny. What gives?!
The above tweet from a weatherman in Maine, originally retweeted into my feed by Tracy Zager, demonstrates an incredible question based on weather patterns.
Why doesn't 43 degrees Fahrenheit feel half as warm as 86 degrees Fahrenheit? Would it be different in Celsius?
Here's another example with a bit better shorts-and-flip-flop outlook (sorry if you're in a place where it isn't warm and sunny):
What is the difference in the high temperature and low temperature on Sunday, April 23rd?
Why are the highs expected to fluctuate so much, but the lows remain fairly consistent?
Does the wind speed have an effect on how warm it feels outside?
Yes, these questions can be answered with the help of science, and that's perfectly fine. Here's my challenge to you as a parent or teacher: take a screenshot of the 7 (or 10) day forecast in your area and ask your child(ren) what questions they have. Pick one or two, and work to answer them. Then, when you're done, send me the image on Twitter or email.