Pascal's Christmas Tree

Blaise Pascal was a French mathematician in the 16th century, whose name has become attached to an "arithmetic triangle" that we now call Pascal's Triangle.

The start of the triangle looks like this:

The first and last number on each line are 1.  The other numbers are the sum of the two numbers in the line above, just to the left and just to the right.  This famous triangle has many interesting connections in mathematics.

But today, I want to share a Christmas activity for young mathematicians and artists:  Pascal's Christmas Tree.

Here's what to do:

1. Print a copy of  Pascal's Christmas Tree Template 
2. On a blank piece of paper, create the first 8 or 10 lines of Pascal's Triangle.  The first 5 lines are above, to get you started.



The first 5 lines

3. Assign a colour to the odd numbers (numbers ending in 1, 3, 5, 7, 9), and a different colour to the even numbers (numbers ending in 0, 2, 4, 6, 8).
4. On the printed tree, colour the squares that correspond to an odd number in the odd-number-colour, and the squares that correspond to an even number in the even-number-colour. See sample at right.
5. After a while you should see the pattern, so that instead of using the numbers from the triangle you created in Step 2, you can use the colours above a square to figure out the what colour is next.  For example, in the sample, when both squares above are green, the new square is red (ie. green + green = red.  Colour addition!).
6. Once you know the colour addition rules, continue to fill in the entire tree.  
7. How many different patterns can you find on your Christmas tree?  
8. Decorate your tree if you want.   Maybe a star at the top, gifts under the tree? 
9. Enjoy your mathematical Christmas creation!

Want to try a more complex version, with a somewhat different pattern?  This time choose 3 colours to use.  One colour will be for numbers that are multiples of 3 (eg. 3, 6, 9, 12 . . .), one will be for numbers that are one more than a multiple of 3 (eg. 1, 4, 7, 10 . . .)  and the third colour will be for numbers that are two more (or one less) that a multiple of 3 (eg. 2, 5, 8, 11 . . .).  You should be able to sort out the new "colour addition rules" and will see a more complicated pattern develop on the tree.  You can do the same with using 4 colours (using multiples of 4, one more, two more and 3 more than multiples of 4) or 5 colours.

Sneak preview of 3 colour tree

Here are two more templates:
A 27 line tree for 3 colours
A 64 line tree for 4 (or 2) colours)
Can you figure out why it is nice (but not necessary) to use the 27 line tree for 3 colours, the 32 or 64 line tree for 2 colours or the 64 line tree for 4 colours?

Have fun, and Merry Christmas and Happy Holidays!

12 Days of Christmas Gifts

A tradition, in our home, following in my mother's footsteps, is to have Pork Tourtiere for dinner on Christmas Eve.  For one of my sons this is the best dinner possible, and he has never understood why I don't make pork tourtiere throughout the rest of the year. (It wouldn't be as special then, would it?)

At 28, he has finally figured out a solution - he wants to make a minor modification to the 12 days of Christmas song:

On the first day of Christmas my mother gave to me, a very big delicious pork tourtiere.
On the second day of Christmas my mother gave to me, 2 pork tourtieres and a very big delicious pork tourtiere.
On the third day of Christmas my mother gave to me, 3 pork tourtieres, 2 pork tourtieres and a very big delicious pork tourtiere.

And so on.

So we decided to have some fun and do a bit of math to calculate how many gifts you would get (in his case, how many pork tourtieres) if you got all those gifts.  In my son's case, let's just say he would need a very large freezer and would have food for a year. Almost.

The sum is:
 1 +  (2 + 1) +  (3 + 2+ 1) + (4+ 3 + 2 + 1) + . . . + (12 + 11 + 10 + 9 +  . . . + 1)

I have shown the calculations on a separate document.  Take a minute to have a look!

Much to our surprise, the number of gifts you would receive, or the number of pork tourtieres my son would have, after the 12 days of Christmas, would be 364 - one less than the number of days in a year - one for every day EXCEPT Christmas day.  We liked that little coincidence.

And enjoyed a little recreational math along the way.

And needless to say, he's not getting all those dinners!

Memory Plus: A Play and Practice Card Game

One of the things that is important (I think!) in being strong at what we call Mental Math, is being able to see, in your head, the "partner numbers" that add up to, say 10. 10 is particularly important, being the base for the number system we use.

This card game is based on the old standard, Memory. In the original game, a deck of cards is dealt face down on the table, and players take turns trying to turn over matching pairs. It might be numbers, or pictures, or some other representation on the cards.

In this game, we use a standard deck of cards, but only the numbers from 1 (Ace) to 9, so 36 of the 52 cards. Players try to turn over two cards whose values add up to 10. It's a great way to become comfortable with knowing that, for instance, "if I have a 3, I need 7 to get to 10".

Once working with 10 as your total becomes too easy, try taking out the nines, and turning over pairs that add to 9. Do the same for 8 or 7, by taking out all cards of a value equal to or higher than the total you are working with. Note that the number of cards on the table becomes a bit small for a good game as total gets lower.

 I have created an instruction sheet for Memory Plus if you need it.

 Enjoy!

Poetry in Math

My niece, Heather presented me with a math equation recently, which she had found and thought was rather fun.

Here it is:

[12 + 144 + 20 +  (3 x square root of 4)] / 7 + 5 x 11 = 9 squared + 0.

It does equate - both sides are 81.

You probably know that 12 is a dozen; But here are a couple of others:  144 is a gross; 20 is a score.

And so, if we put math into verse, we get:

A dozen, a gross and a score
Plus three times the square root of four
Divided by seven
Plus five times eleven
is nine squared and not a bit more!

Math in Limerick form.    Thank you Heather!

Puzzling Through Trig Identities

Proving trig identities is a bit like solving a puzzle:  if it doesn’t work this way, try that way . . . !

As was said of the Grinch:  “he puzzled and puzzled ‘till his puzzler was sore.” (Dr. Seuss)

When you are studying trigonometry in your math course (grade 11 or 12 ?) you likely have to  practice proving trigonometric identities.  And you don’t want to keep on puzzling and puzzling ‘till your puzzler is sore, so some strategies to aid you in your puzzling might not go amiss.

There is no one RIGHT way to prove a trig identity.  There are usually many ways to get from start to finish.  And there are many tips and tricks to help you along the way.  Try this Tips and Tricks sheet, and see if it helps at all.  Happy Puzzling!  

ADDEMUP! A Play and Practice Card Game

I often invent little games to play with students, to try to turn practice into play.  I find that many students need to work on their ability to add and subtract or multiply and divide numbers in their head.  "Mental math" it is often called.  The thing is, practicing once a week doesn't quite get us where we want to be (or so my piano teacher once told me!).  My hope is that if I can create a fun way to practice, maybe students can do it with their parents, siblings or friends at home.

Here is a game I created recently with a grade 5 student to practice adding and subtracting numbers up to 10, in your head.  We call it Addemup!

ADDEMUP!

Set-Up

Use a regular deck of 52 card deck of playing cards.  Remove the 12 face cards.
Pick a number between 1 and 20.  This number is the Game Number for the current game.
Shuffle the deck well.  Deal eight cards, face up, in a two rows by four cards.  This is the game grid.

Object of the Game

Each player tries to collect as many cards as possible, by picking up sets that total the Game Number, using addition and subtraction.

Game Play

This game can be played with two or more players.  Play is done by turn, until the entire deck is used.

On a turn a player attempts to make as many sets as possible, from the game grid, for his or her set collection.  A set is made up of a number of cards that total the Game Number, by adding or subtracting the value of each card.  The player must state what their calculation is as they pick up the cards. For example if the Game Number is 12, Player 1 may pick up 6, 9, 2, 1, in that order and say:

“6 plus 9 is 15, minus 2 is 13, minus 1 is 12”; or
“6 plus 9 is 15, 2 plus 1 is 3, 15 minus 3 is 12”.   

As a set is completed it is removed from the game grid and added to the player’s set collection.

At the end of a turn, cards from the deck are dealt to fill the spaces left in the grid by the removed sets.

The game ends when no cards are left to be dealt and the player whose turn it is can make no more sets.

The object is to collect as many cards (not sets) as possible.  A set with four cards is therefore more valuable than a set with two cards. 

Variations

·         Instead of a 4 x 2 layout of cards try 3 x 3 cards, 3 x 2 cards or 4 x 3 cards instead. 

·         The red cards are negative integers and the black cards are positive integers, with addition and subtraction allowed, but if you have a red 4, you must subtract -4 or add -4 to get your final result.  For this variation, your Game Number may be a negative integer.

·         Instead of choosing a Game Number prior to the start of the game, for each turn, after filling the grid or face-up cards, flip the top card of the deck over and place it on top of the deck – this becomes the Game Number for this player’s turn.  For the next player’s turn, this card becomes the first one placed in the grid to replace the cards removed as sets.

The Joy of Discovery

Recently we had relatives visiting from Germany.  A family of four, with a daughter 4 1/2 (not 4, I was told!) years old.  While mother and father were busy with the baby, I had the opportunity to play with the 4 1/2 year old.  I looked through my stack of games for something suitable for her age, preferably something that didn't involve a lot of struggling between English and German.  After a game of Memory (which youngsters are far too good at!) I pulled out Bee Lines - a math related game that uses addition and subtraction of numbers up to 10.

The idea is to be the first to create a continuous line from one side of the board to the other, or from top to bottom, with your bees (red or blue background).  To do so, on your turn you spin two spinners, each with the numbers 1 through 10 on them.  If you spin a 5 and a 9, you can place a bee on a 4 (9 minus 5) or a 14 (9 plus 5) to help build your line.

That's the game.  Our visitor was happy to work at adding and subtracting the necessary numbers, but after a few turns, we decided to try a number line instead of counting on our fingers. We drew a nice large number line, and she would start with the bigger number (she could figure that out) and count, first up, and then down, from there to get to the two numbers she could choose from to place her bee on.

So when she spun a 10 and a 5, the excitement, the sheer joy and the laughter when she counted up to 15, then down to 5, and realized that her spin (the 5) was the same as her answer was incredible!  And not only the 5, but the 15 ended in the same number!  Then when she spun an 8 and a 4, and one of her answers was 4.  More joy and laughter.  And when 8 and 5 gave her 13 and 3.  Wow!  Her joy of discovery was a joy to watch.  Her excitement and laughter were infectious.

A young girl was discovering math for herself and loving it.  Who says math is hard?  Math is fun!

By the way, Bee Lines is a great math game to play with young people.  We bought it to play with our kids many years ago.  I now use it with my young tutoring students when we need a break from pencil and paper.

A Fun Factoring Trick

Factoring – fun?  Is that a bit of a stretch?  Maybe, but let’s pretend. The more geeky – or mathy? -  among us might find it so.  I do!
Ever had to factor quadratics?  You need to find factors to be able to do it.  Ask any grade 10 student!
Or how about reducing fractions (or ratios)?  How do we find out if we can reduce the fraction 123/78 ? We need to do that sort of thing in math class!
There are a few tricks.
Let’s try checking if 3 is a factor.  It’s a trick people are less likely to know, and it’s easy to do:   Take a really big number.  Add up all the digits. Can you divide that sum by 3?  If so, the really big number can also be divided by 3!
So 123/78.  Can we reduce it? 
First let’s check 123:  1+2+3=6.  3 is a factor of 6.  That means that 3 is a factor of 123 also.
Now let’s check 78:  7+8 = 15. 3 is a factor of 15 so 3 is a factor of 78.
So, we can reduce 123/78 by dividing top and bottom by 3.  Now we need to do the division by hand, in our head, or if really necessary, on a calculator! 
123/3=41;
78/3=26;
Therefore123/78 = 41/26
Shall we try a really big number?  354296505:  3+5+4+2+9+6+5+0+5= 39;  Hmm still big-ish; We can do the same trick on 39 now.  3+9=12 – now THAT’s divisible by 3. So 39 is too, and 354296505 is too!
But we can make it easier on ourselves.  We can ignore all the 3’s, 6’s and 9’s (and 0’s) in the big number. [Why?  If we add a multiple of 3 (like 3, or 6, or 9) to a multiple of 3, it’s still a multiple of 3.] So let’s try 354296505 again, ignoring 3’s, 6’s, and 9’s:  5+4+2+5+5 = 21 – and 21 is divisible by 3 [and 21 plus the 3’s, 6’s and 9’s must also be a multiple of 3] and therefore the big number, 354296505 is divisible by 3.
The same trick works for checking if a number is divisible by 9.  Add up the digits and see if that sum can be divided by 9.  If it can, the big number is also divisible by 9.
One note of clarification, if needed:  I am using “divisible by”, a “multiple of” and “can be divided by” interchangeably.  They all mean the same thing. And reversing the numbers, “is a factor of”:  20 is divisible by 5; or 5 is a factor of 20.
Here are a few other tricks, some of which you may already know.

Number	How to know if it is a factor	Easy ways to do the division (for a few of them)	Examples
2	If it’s even, it’s divisible by 2 (The even numbers end in 0, 2, 4, 6, or 8)		95376:  ends in 6, which is an even number, so is divisible by 2.
3	add up the digits of the number; if the sum is divisible by 3, the original number is also divisible by 3		76431:  7+6+4+3+1 = 21; 21 is divisible by 3 therefore 3 is a factor of 76431. 76430:  7+6+4+3+0 = 20 3 is NOT a factor of 20, therefore it is not a factor of 76430
4	Ignore all but the final 2 digits; subtract 20, 40, 60 or 80 to bring it to a lower number; if this number is divisible by 4, the complete number is also divisible by 4;		238594: look at the last 2 digits: 94. Subtract 80; 94 - 80 = 14;  14 is NOT divisible by 4 so 238594 is not either 7608372:  72 – 60 = 12;  12 is divisible by 4 so 7608372 is also.
5	If it ends in a “5” or a “0” it is divisible by 5	Double the number (or add it to itself); then get rid of the last 0.	115 / 5 = (115 + 115) / 10 = 230 / 10 = 23
6	if the number if both even (divisible by 2) and divisible by 3 (see above) then it is divisible by 6		54270:  54270 ends in “0” therefore is even; 5+4+2+7+0= 18; 18 is divisible by 3 therefore 3 is a factor of 54270; even AND divisible by 3 = divisible by 6
9	add up the digits of the number; if the sum is divisible by 9, the original number is also divisible by 9		54270:  5+4+2+7+0= 18; 18 is divisible by 9 therefore 9 is a factor of 54270; 76431:  7+6+4+3+1 = 21; 21 is NOT divisible by 9 therefore 9 is a factor of 76431;
10	If it ends in a “0”  it is divisible by 10	Get rid of the last “0”	120 / 10 = 12 23400 / 10 = 2340

Have fun impressing your friends with your wizardry!

Exploring Transformations - With the Help of Blokus

 In working with a grade 5 student, for whom the current topic in class is transformations, we pulled out our game of Blokus to do some exploring and practicing.

Transformations being studied included translations (left, right, up, down), rotations (spinning the image about a fixed point) and reflections (mirror images).    Blokus was a great way to look at reflections and rotations, the more difficult of the three transformations to grasp.

For the reflections, we first chose one piece and then placed the same piece of each of the four colours, one in each of the four corners, all being reflections of each other.  We then took turns placing another piece on the board, after which the student gets to place the three like-shaped pieces of different colours in the appropriate place to match the left-right (horizontal) and up-down (vertical) reflections.  Eventually we had all the pieces in place, every piece being reflected appropriately.

Our second project was to do a similar thing, but this time, our transformation was a 90 degree rotation.  For this, we started at the centre of the board, again choosing one piece but this time rotating it by 90 degrees for each of its like-shaped pieces of different colours.  We placed the pieces so as to create a spiral when we were done.

Challenging at times, but it definitely got easier as we progressed and got the hang of it.  Good fun, good learning, picturesque result!