April 10, 2016 - Math Circle 2.0

1.  Ages with Legos.  If one 2x2 lego square represented 1 year, make a structure that represents the age of each student.  A 1x2 lego represents 1/2 year.   A tower representing David's age was also built.  It is much taller than the other students' towers.

2.  Three teams are in a basketball tournament:  team dinosaur, team flower, and team mushroom.  Each team played every other team exactly once. How many games were played in total? A fourth team - team square - then joined the league.  The students then counted the number of games in total that the four teams would play.  Homework:  A fifth team joined the league.  How many total games would need to be played?

3.  Review Lego Squares.   We can make squares with 1, 4, and 9 legos. The next number of legos to make a square is 16.  How do you differentiate between a square and a rectangle?

April 2, 2016 - Math Circle 2.0

Review previous lessons....

1.  Lego squares....  How many legos does it take to make a perfect square...  The students found that they can make a square with 1, 4, and 9 legos.   Can we make a square with more Legos?

2.  Number series.

0, 2, 4, 6, 8, .....  what comes next...  These are even numbers.


3. 5k Map - distance exercise.  Use a map of the Illinois 5k to learn directions of north, south, east, and west.

solving linear systems - with graph paper...

Let's explore a new way to solve systems of linear equations.

Assume you have something like

3x+2y=7
x+4y=9.

We draw the equations like this: 3x+2y is represented by 3 steps right, 2 up (and we mark the vector as 7, as we know that is the sum). Same thing for x+4y.

Now, we start doing jumps with the arrows. Say, doing two jumps along the "3x+2y" vector takes us to 14.

But the ultimate goal is to get to one of axes. We can do it, if we do the x+4y jump, but backward (shown as blue). As we go backward, we subtract 9.

Now, we landed on the x axis! And we ended up with 5. Where did we land? - it is 5 steps to the right yielding 5. So, each step is 1, that is x=1.

As getting to 7 takes 3 x steps (and 2 y steps), and each x step is 1, we conclude that 2 y steps are 7-3=4, and hence each y step is 2.

So the solution: x=1, y=2.

Easy.

Now, solve

4x+y=6
x+2y=5,

and

3x+4y=10
2x+2y=6.