Thursday, May 26, 2016

Dice games: Quick and fun mental math practice

I was looking through ThinkFun's games today (because they have so many awesome ones!), and found one called Math Dice (Amazon affiliate link here).  The idea is that you roll 2 twelve-sided dice and multiply them to get your target number, and then roll 3 six-sided dice.  You want to use addition, subtraction, multiplication, and division to get as close as you can to the target number.  They also have a version for younger kids (Amazon affiliate link here) that only uses addition and subtraction.

Unlike most other ThinkFun games, I don't think it makes sense to buy these, as all you really need are the dice.  I instead encourage you to buy an assortment of dice (here's an example via Amazon affiliate link) which you can use for many different games!

I wanted to note a few things about dice games:
  • Most can also be played with a deck of playing cards (like this), but there are subtle differences when playing with dice... previous roles don't effect your chances, you don't need a flat surface (rolling them in a tupperware container works well), etc.  
  • Look for games that have some strategy involved in addition to practicing mental math.  
  • Customize games to your child's skill level by modifying the allowed operations, number of dice, and type of dice. You can even invent a new game! 
  • Dice games are great for when you have just a few minutes to fill - a round goes by very quickly, and you can easily bring them with you anywhere. 
  • Sets of dice made for D&D and other role-playing games work really well for math games too!
  • Spend some time looking at the shapes of the various dice.  Why did they chose those numbers? Could you make a fair seven-sided die? Note that most of the ones in sets are Platonic Solids

Wednesday, May 25, 2016

Programming Resources for Elementary School (and Pre-K)

Today I've been looking at resources to teach elementary school students to code.  There are so many amazing options! The general idea is that kids drag and drop predefined blocks of code, usually to make an onscreen character do something, like jump or turn left.  This blog post by Vicki Davis on Edutopia, which was updated in 2015, has a pretty comprehensive list.  I don't yet have any strong opinions on which ones are best, so try out some of the free ones that fit with your kid's age and platform of choice.

A few ones that I want to highlight:

  • Code Studio's free online computer science fundamentals series is a full introductory curriculum for ages 4+.  It could be used at home or in a classroom.  It mostly consists of individually paced in browser videos and activities, but there are a few lesson plans for "unplugged lessons" along the way.
  • Scratch is the most well known introductory programming language for kids.  It allows them to program their own interactive stories, games, and animations, and share their creations with the online community. Scratch is free, works in a computer browser, and is aimed at ages 8 and up. Hopscotch is similar to Scratch, but made for iPad (and iPhone). For the slightly younger crowd, there's ScratchJr, an iPad and Android tablet app aimed at ages 5 - 7.  
  • There are a lot of board games that teach programming basics without any technology.  For ages 4+, there's Robot Turtles (Amazon affiliate link here) and Littlecodr (Amazon affiliate link here), and for ages 8/9+, there's Code Monkey Island (Amazon affiliate link here) and Code Master (Amazon affiliate link here).

Tuesday, May 24, 2016

Math in Context: Building and designing a house on a budget

This is a cool math project given to a 3rd grade class in Atlanta.  The challenge is to design a home with specific constraints that stays under budget.  The constraints are things like

  • "Your kitchen must be twice as big as your dining room"
  • "Two of the hallways must be parallel"
  • "You must have a garage which is 10% of your total square footage"
There are also details about the cost of land (per arce), the building cost (per square foot), the cost of paint (per wall), etc.

There's so much going on in this project! There's a lot of routine calculations, but in a manner that is obviously relevant, self-motivating, creative, and requires higher-level thinking skills. This project could be adapted to a wide range of grade levels, and kids could proudly present their finished creations to the class!

Anyway, here's the project sheet.  I didn't make it - a parent showed it to me.

Sunday, May 22, 2016

Long Division: Why do we care?

Long division is a contentious subject in math education.  Students struggle with it, and most people fail to see its relevance in an age when you can just ask Siri.  I wanted to give some reasons why long division is worth teaching and mastering.  What you'll hopefully see is that it is absolutely crucial for us to have a general procedure which takes a division problem and gives us a quotient and remainder.   You might have seen this written as \(19÷3 = 6 \; R \;1\), but remember that \(19÷3\) is the same as \( \frac{19}{3}\) and \(6 \;R\; 1\) really means \(6 + \frac{1}{3}\) in this situation.  Another way to write it is \(19 = 3*6 + 1\), where we multiplied both sides of the equation by 3 to get rid of the fractions.

If I manage to convince you, and you're wondering how to teach it, take a look at this blog post. Hopefully, you can use what I wrote here to help motivate everyone involved in the learning process.


Long division is the first big example of an algorithm. An algorithm is a step-by-step description of how to do something.  You can conveniently reuse algorithms again and again in similar situations, share them with others, and teach them to a computer. Once we have an algorithm for how to complete a particular type of task, we no longer have to think about it so much.  Even if we don't always use it, it's there in our back pocket, and gives us confidence that we know how to solve any problem that comes our way.  Algorithms are fundamental to the closely related fields of math and computer science, and when kids learn some coding in elementary school (here are some resources for that), they are more prepared for understanding long division as an algorithm.

Decimal Expansions

Long division connects fractions, division, and decimals. We can use long division to write 5/7 as a decimal... we just have to keep going until it terminates or repeats.  In other words, it gives us an algorithm for finding the decimal representations of fractions.  Moreover, long division explains why all rational numbers have decimal representations that terminate or repeat.   Becoming comfortable with rational numbers and all of their forms is one of the most challenging and important tasks of middle school mathematics.

Other Bases

We write numbers in base 10.  So 1068 means \(1*10x^3 + 0*10^2 + 6*10 + 8\).  But we could also use powers of another number, like 7.  How would we write 1068 in base 7? Well, we use the division algorithm to find that \(\frac{1068}{7} = 152 + \frac{4}{7}\), or equivalently that \(1068 = 7*152 + 4\).  Next, we divide the quotient 152 by 7 to get a new quotient 21 and a new remainder 5, yielding \(152 = 7*21 + 5\).  Continuing in this way, we find that \(21 = 7*3 + 0\) and \(3 = 7*0 + 3\).  So 
$$1068 = 7*152 + 4 = 7(7*21 + 5) + 4 = 7(7(7*3) + 5) + 4 = 3*7^3 + 5*7 + 4.$$
This tells us that \((1068)_{10} = (3054)_7\), where the subscripts tell us the base the number is written in.  

Polynomial Division

The algorithm of long division can be applied to polynomials as well as positive integers.  Just as long division allows us to rewrite 17/5 as 3 + 2/5, with 3 as the quotient and 2 as the remainder, it allows us to rewrite $$\frac{x^4 - 5x^2 + 6x^2 - 18}{x^3 - 3x^2} \; \; \text{as } \; \; (x - 2) - \frac{18}{x^3 - 3x^2},$$ with \(x - 2\) as the quotient and -18 as the remainder.  This becomes useful in calculus, as its the first step to integrating a rational function where the degree of the numerator is at least as big as the degree of the denominator.

Modular Arithmetic

Modular arithmetic is born out of the idea that we can create a new number system by only caring out the remainder when we divide by n.  So, if we're looking at remainders after dividing by 7, for example, 2*4 = 1, which means that 4 = 1/2.  Crazy, right? Modular arithmetic is fundamental to many areas of mathematics and computer science, including cryptography! And it's a great topic to explore in math circles for upper elementary and above, as it's interesting, useful, and reinforces many of the main concepts of elementary and middle school mathematics.  

Continued Fractions

Expressing fractions in terms of quotient and remainder allow one to make continued fractions: $$\frac{7}{10} = \frac{1}{\frac{10}{7}} = \frac{1}{1 + \frac{3}{7}} = \frac{1}{1 + \frac{1}{\frac{7}{3}}} = \frac{1}{1 + \frac{1}{2 + \frac{1}{3}}}.$$  You can even write irrational numbers as (infinite) continued fractions.  Take, for example, the golden ratio \(\phi = \frac{1 + \sqrt{5}}{2}\).  $$\phi = \frac{1}{1 + \phi} = \frac{1}{1 + \frac{1}{1 + \phi}} = \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \phi}}} = \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \ddots}}}}.$$  The reason that \(\phi\) has such a nice continued fraction representation is that \(1 + \frac{1}{\phi} = \phi\).  Can you figure out why that's true? This is another great math circle topic idea, and is probably most appropriate for high school students.

Generalizations of the Integers

Long division is fundamental to number theory, the area of math that I work in. We've already seen a little of this in modular arithmetic and continued fractions, but it goes even further.  One of the main things we study in number theory is generalizations of the integers and rational numbers.  In order to do this, we must look closely at the properties that the integers and rational numbers satisfy.  One important property is unique prime factorization, and another is the existence of a division algorithm that produces a quotient and remainder.  This is what is meant when the first problem set of PROMYS and ROSS state that "long division has unexpectedly deep consequences and can be generalized in far-reaching ways".

Thursday, May 19, 2016

Math You Can Play: Math games using everyday materials

This week, I took at look at the first two books in the Math You Can Play series by Denise Gaskins.  The first book (Amazon Affiliate link here) focuses on counting and number bonds, and is aimed at pre-K through 2nd grade, and the second book (Amazon Affiliate link here) focuses on addition and subtraction, and is aimed at Kindergarten through 4th grade.  

Both books contain math games that only use standard materials in addition to printable game boards: a few decks of playing cards, dice, dominos, graph paper, at least two kinds of tokens (beans, coins, poker chips, etc).  For each game, she starts by listing the relevant math concepts, and also includes some variations.

Why play these games? From the introduction:
Math games push students to develop a creatively logical approach to solving problems. When children play games, they build reasoning skills that will help them throughout their lives. In the stress-free struggle of a game, players learn to think things through. They must consider their options, change their plans in reaction to new situations, and look for the less obvious solutions in order to outwit their opponents.
 Even more important, games help children learn to enjoy the challenge of thinking hard. In the context of a game, children willingly practice far more arithmetic than they would suffer through on a workbook page, and their vocabulary grows as they discuss options and strategies with their fellow players. Because their attention is focused on their next move, they don’t notice how much they are learning. 
And games are good medicine for math anxiety. Everyone knows it takes time to master the fine points of a game, so children feel free to make mistakes or “get stuck” without losing face. 
If your child feels discouraged or has an “I can’t do it” attitude toward math, take him off the textbooks for a while and feed him a strict diet of games. It will not be long before his eyes regain their sparkle. Beating a parent at a math game will give any child confidence. And if you’re like me, your kids will beat you more often than you might want to admit.
A good math game is still fun and challenging when the math concepts involved have been mastered, and so older siblings and parents can still enjoy joining in.  For example, how can you make 24 using each of the numbers 1, 8, 4, and 3 exactly once, combining them using addition, subtraction, multiplication, and division? Even for me, there's a moment of excitement on figuring out the answer.  (You can buy the 24 Game, or make your own version with a deck of cards.)

In addition to the games, there are chapters about "Workbook Syndrome", mastering the math facts, and lots of resources ranging from books to board games (including more than a few that I regularly play with friends).  

If you want to try out a few games without going all in and buying the books, you can find many of them on her blog.  If, like me, you're cost conscious, you can also buy the combined kindle edition for a significant discount (Amazon Affiliate link here).  She also has a book entitled Let's Play Math: How Families Can Learn Math Together and Enjoy It (Amazon Affiliate link here).  I haven't read it, but it looks good!

Friday, May 6, 2016

Tiny Polka Dot: A deck of math cards for ages 3 - 8

I already posted about the first Math For Love game, Prime Climb.  They recently started a new Kickstarter campaign for a specialized deck of math cards, called Tiny Polka Dot, which is aimed at kids in pre-K to 2nd grade.  There are lots of mathematical games and activities that use a standard deck of cards (here are some examples), but these have 6 different suits each of which represent the numbers from 0 to 10 in a different way - ten frames, big and little dots, circles, doubles plus one, numerals, and dice patterns.

Tiny Polka Cards
Photo of Tiny Polka Cards from their Kickstarter page

They include 12 different games which teach conservation of number, one-to-one correspondence, subitizing, addition, and subtraction. They also encourage parents and kids to make up their own games or modify standard card games for this deck.  This is a great way to make mathematics playful and fun! I just backed this project and highly recommend that you do too!

Thursday, May 5, 2016

Blokus: A strategic game of spatial thinking

Last night, I played Blokus (Amazon affiliate link here) with a few friends (none of which consider themselves to be math people).  In Blokus, four players take turns placing one of their colored tiles on the board, starting at their corner. Each piece played must touch another piece of the same color, but only on the corners.  As the board fills up, players must block their opponents, protect their territory, and plan out a strategy in order to fit as many of their remaining pieces as possible.  The game requires a ton of spatial thinking, and can be enjoyed by everyone from early elementary schoolers to adults.

Clicking this image leads to an Amazon affiliate link
My roommate asked an awesome question after we had finished our game: If we worked together, could we fit all the pieces on the board and still follow the rules of starting from a corner and having same-color pieces touch by corners? I immediately started unpacking the pieces so that we could give it a try.  Before getting started, we did a few things:

1. We counted how many squares there were on the board and on all of our tiles, so that we could get a sense of how tightly we would need to pack them in.  The board was 20 by 20, and hence had 400 squares.  Each color had 1*1 + 2*1 + 3*2 + 4*5 + 5*12 = 89 squares, so there were 89*4 = 90*4 - 4 = 360 - 4 = 356 squares on the tiles.  This means that we can leave 44 squares uncovered.  Note that all of this was talked through out loud without needing paper or a calculator.

2. We decided that we would allow ourselves to move around pieces after placing them.  We just wanted our final product to be a valid Blokus arrangement, and so didn't care about following the usual rules during the placement process.

3. My roommate suggested that we start by mirroring each other, as we had often done that for the first few moves while playing the game.

My neighbor took the lead most of the time in placing pieces, and we quickly realized that copying someone else's placement really challenged our spatial thinking skills, especially as the colors began to mingle.  We liked the symmetric arrangement, and decided to try and keep that going the entire time.  We also realized that it was easier to fill in the corners as much as possible with two colors, and then worry about connecting each color later:

A recreation of our board after filling in the corners

Here was our final result, which we were all very happy with:


Although there were no children involved, this is a great example of how to encourage more mathematical thinking at home! If my roommate's question had been shot down, we would have just put away the game and been done for the night, but we followed the principal of "Yes, and..." and created something we were really proud of (as evidenced by the fact that a picture was immediately posted to Facebook).  Mathematics is about asking and exploring interesting questions, and can happen naturally in the right sort of environment!

Although this didn't come up last night, I wanted to mention another mathematical aspect of Blokus:

A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. They are classified by how many square cells they contain. A monomino has one cell, a domino has two cells, a tromino has three cells, a tetromino has four cells, etc. How many different tetrominos are there? Well, it depends on how you define different.

How many different tetrominos are pictured above?  
Since Blokus pieces can be rotated and flipped, it makes sense in our situation to consider all of the ones pictured above to be the same.  It might be fun to take out a piece of paper and try to draw and count all of the polyominos with one, two, three, four, and five cells.  You will find 21 shapes which correspond exactly to the Blokus pieces!

21 Blokus pieces in red
Counting polyonimos quickly becomes a very hard problem - the number of polyonimos will n cells is only currently known up to n = 28.  They are great ways to explore area, perimeter, symmetry, and counting in the elementary school classroom and beyond!