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!

Math: How to Help our Children – a presentation to parents

Recently I was asked to speak to parents at my local high school about how to help our children with math.  After many years of helping students in one-on-one tutoring it was a new and interesting opportunity to put my thoughts together to present to parents.