Combinations and Permutations with Children

When we talk about combinations in the English language, we have a problem.  People regularly use the word “combination” when that isn’t what they mean at all.  Combinations are a collection of things when the order DOES NOT matter.  Choosing which 3 stuff animals out of a collection of 10 to take with you on a trip is an example of a combination.  A combination lock is a great example of using the word incorrectly.  The order of the numbers matters in order to open the lock.  When the order matters, that is called a permutation.

Combinations and Permutations for Children

Another great way to look at this can be through sports.  If a basketball coach needs to pick his starting line-up from a team of 8 players, he can have any combination of 5 of the players.  Whatever order the players are selected, it is still the same group of 5 starting players for the line-up.  When a baseball coach is deciding the batting order, the order of the players matters.  The baseball example has to be a permutation.

Combination Activity with Dress-up Clothing:

I had my son select 3 dress-up hats and 2 weapons.  We then had a photo-shoot of him with the different combinations that he could have (assuming he has to have a hat and a weapon).  There are six different possibilities.

Combinations of 2 weapons and 3 hats

If we also can have no hat and no weapon OR just one of each, that adds another 6 combinations.

Combinations of no objects or just one of them



Permutation Activity with Play Animals in a Zoo:

In this example, we are going to use 3 animals to show the different permutations of the animals in cages at the zoo.  The order that we put the animals in their cages matters, so this a permutation.  (Other variations of the same idea would be having 3 children sit on 3 chairs or putting stuffed animals in order.)

With our 3 animals, we have a total of 6 different permutations.  If we have 4 animals, we would have 24 different permutations.  Permutations can become large very quickly!


Permutations of AnimalsI hope this helps you understand the difference between combinations and permutations and that you can have fun with these ideas with your children.  Please let me know if you have any questions about this!


Tumbling Dice: A Great Game That is Great Math Practice

Tumbling Dice is an exciting game of dexterity (and some luck), but it is also great for practicing addition and multiplication facts.

Tumbling Dice

It  is no longer in print, but it is easy to make your own.  I made a fairly nice one out of wood using these instructions, but you can make one out of a stack of books or any other flat objects.

Board Books

How to Play:

The first player places a die on the top level of the board and flicks it.  The goal is to get the die to stop on one of the other levels.  Players take turns flicking their dice.  A player can either try to score points or knock on of the opponent’s dice off of the board (or possibly both).  The game ends when all of the players have run out of dice.

Launching a Die

The scoring:

To find the value of a die, multiply the number on the die times the value of the level that it is on.  Starting from the top the values are 0, 1, 2, 3, and 4.   Then add up all of your dice to get your final score.  Highest score wins.

point values


The blue dice here give a good chance to apply the distributive property.

Distributive property


Dice graveyard



The original game came with a bunch of 6-sided dice.  This is nice, but it only lets you practice some of your times tables.   So we ordered some other dice. Here are some links to the dice we ordered: 60 Polyhedral Dice60 Polyhedral Dice.

We usually play with one 4-sided, two 6-sided, two 8-sided, one 10-sided, one 12-sided, and one 20-sided die.  This makes the game more fun and increases the values that the kids have to multiply and add.

If you want your kids to practice different times tables, you can always change the values of the levels.  This is especially great if there are sets of facts that your child struggles with.


Why we like it:

  • It is fun!
  • It is a great way to practice multiplication and addition.
  • You can use it to practice many different multiplication facts.
  • The kids are constantly multiplying trying to figure out what each die is scoring.
  • The randomness of the die rolls keeps kids interested even if they are not doing well.  Gideon won a game where he had no points until the very last roll.  Then he flicked the 20-sided die.  He got a 17 on the 3 level and scored 51 points.


I hope you try out this game because it is a lot of fun and is great math practice.  If you order the Polyhedral Dice that we use, you can use them for lots of different ways to practice math.  Becki likes to just have the kids roll 2 dice, multiply them, and roll again.




Math Picture Books

There are lots of great math picture books out there.  I wanted to share some of our favorites from our library.  The books get increasingly more difficult in concepts and reading level as you go through the list – starting with a preschool series and ending with geometry concepts.

Math Picture BooksPreschool/Early Math learning Series:

Great Source Mathstart books have three different levels.  These books are great for learning math concepts at a young age.  I used them more when my older children were preschool age and early elementary, but I want to get them all again for my youngest.  Our library has a large collection of these.  Here are just three titles in the series:

Beep Beep Vroom Vroom! is one of the books in level 1.  This one teaches patterns.

Super Sand Castle Saturday is from level 2 and teaches how to measure.  

Game Time! is from level 3 and focuses on days, weeks, hours, minutes, and seconds.

Math is Categorical Series:

We love Brian P. Cleary’s English books and were really excited to see he had math books as well.  These do a great job of introducing concepts and reinforcing ideas.  All of these are part of the series Math is Categorical.

The titles really clue you in to their subject matter so I didn’t describe each one.  We enjoy the fun cat illustrations and the sing-song rhyming throughout these books as well as the math content.

How Long or How Wide?: A Measuring Guide
On the Scale, a Weighty Tale
A Dollar, a Penny, How Much and How Many?
A Second, a Minute, a Week with Days in It: A Book about Time
The Mission of Addition
The Action of Subtraction
A Fraction’s Goal – Parts of a Whole


Charlesbridge Math Adventures:

The Charlesbridge Math Adventures are geared for a little older children.  The concepts go from upper elementary math into high school math concepts.  I shared about one of the single titles and a series within the Math Adventure series, but there are a number of others as well.  We have loved all of the books we have read in this series.

 A Place for Zero  is a fun story about the number zero trying to find his place.  It teaches about the Identity Property of Addition, the Zero Property of Multiplication, a x 0 = 0, and place value all during an adventure.  The book mentions words like factors and products as well.

Sir Cumference and the First Round Table is the first Sir Cumference book in a series of fun math adventures in the Middle Ages.  The knights, castles, and dragons add a lot of fun to the series while teaching math concepts.  This book teaches radius, diameter, and circumference.

Other books in this series include:

Sir Cumference and the Dragon of Pi which teaches an estimation of Pi and where the number comes from using the circumference formula through the book.

Sir Cumference and All the King’s Tens teaches place value up through the thousands place through a story planning a party for the king and having tents to hold the guests.

Sir Cumference and the Great Knight of Angleland involves measuring angles with a protractor.

Sir Cumference and the Isle of Immeter works with perimeter and area of a circle.

Sir Cumference and the Sword in the Cone deals with 3-dimensional objects and is 6th in the series.  (Our library doesn’t have this one so I haven’t had a chance to read it.)

There seem to be new books still being added to this series.  Here’s the whole list.

It seems like every time I search for math picture books, I find new ones to try.  These three series are our favorites so far.  Do you have any math picture books you like?

The Mathematics Of A Reading Chair

We recently bought a very large bean bag chair that was a lightning deal on Amazon.   Instead of the traditional bean bag filling, it has little chunks of memory foam in it.  The kids love to curl up in it to read, which was the main reason that we purchased it.  I thought that one child at a time would sit in it to read, but they have enjoyed piling 3 or 4 of them in it.

Math of a reading chair

Since it was memory foam, it came compressed in a box.  The directions said that it would take 7 to 10 days for it to reach its full size, but I fluffed it vigorously and it reached full size by the next day.

One of the kids commented that the chair was “twice as big as the box that it came in.”  That sounded like a math problem to me, so we got to work.  I explained that volume is measured in cubic inches and it is really saying how many 1 inch cubes would fit inside the object.  Then the boys measured the box and the chair.  Ruth recorded the numbers on the whiteboard.  I added some diagrams and gave them the formulas that they would need.  Then Elijah and Ruth completed all of the calculations on the board.  I helped out a few times when they forgot what to do or made a small mistake, but they really did the work themselves.



We found that the box had a volume of 8,209 cubic inches, and the chair had a volume of 45,216 cubic inches.  The chair has a volume five and a half times bigger than the box!

Shortly after we finished this activity, we were eating dinner and discussion the size difference in the chair and box.  Ruth glanced up at the U.S. map that hangs by our table and said, “It is like Pennsylvania and New Jersey.”  So I looked it up.  Keep in mind that the volume of the box and chair were 8,209 cubic inches and 45,216 cubic inches.  The areas of New Jersey and Pennsylvania are 8,729 sq miles and 46,055 sq miles.  So PA is 5.28 times bigger than NJ.  Wow that is excellent estimating!

I love when learning happens like this.  Volume was not just some idea on a math worksheet, it was the means to answer a real life question.  Yes, Elijah practices 2 digit multiplication in his math curriculum, but this was an opportunity to use his skills for a purpose.  Isaiah had just been learning about measurement in his math book, and we had a reason to actually measure something.

20141212_172823  Curling up with a good book


The kids tend to just pile on so that the chair squishes flat, but if you sit in the middle, it conforms around you in more of a chair-like shape.

Look for those opportunities in life to respond to your children’s questions in a way that they can learn something new or practice the concepts that they have already learned.

If you are interested in one of these cool Fuf chairs, here are some links to different sizes (ours is the king):


4-Foot Large


6-Foot XL

7-Foot XXL


P.S.  I am sitting in the chair right now as I write this post.  It is very comfortable. (I only link to products that I like.)

 This post contains affiliate links.

Circles That Roll (But Not Like You Think)

Circles that Roll

I have listed a few of my favorite YouTube channels in previous posts, but I neglected to mention Numberphile.  Numberphile posted a new video a few days ago, where Matt Parker showed how linked circles can roll.  He demonstrates how it works and how far the circles must overlap.  They must be overlapped the correct amount so that the center of mass of the two circles does not move up and down.   The video goes through the proof that this is the correct way to link the circles, but it made me think, “What if we overlapped the circles too much or too little?”

So I took some old CD’s and cut the slots 1.76 cm (ummm… well it was somewhere close to that).  We tried it out by rolling it on the floor and it worked well, but it was kind of hard to figure out how to push this strange shape.  So we rolled it down a very slight incline.



Then I cut smaller slots in another pair of CD’s.  They clearly did not roll as well. (Sorry for the sideways video)


Then we tried larger slots.  Once again, it did not roll as well.


Isaiah wanted to race the good one and the one that had small grooves.


Taking someone else’s idea and creating variations is a great way to invent your own science experiments.  Next time you see a science activity, try to think of some variations (or have your kids think of them) and turn it into a real experiment.

If you get a chance to make these, it is very fun and interesting to watch them roll.  I wonder if there is or ever will be a useful application of this idea?