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.

March 6, 2016 Math 2.0 Shape sorting, Symmetry, Odd/Even, Gravity + Water tension

Venn diagrams

We have shapes. Some are triangles and others are squares. Some are blue and some red.
    ('blue' was used to refer to both the purple and light blue shapes until confusion struck, at which point purples were removed)  

First, students sorted objects by shape (triangles inside, squares outside). Then they were asked to wrap a string around all of the triangles, and all of the squares.

Then the students were asked to sort objects by color and put a string around red, and another around the blue 

Finally, students were asked to wrap the string around the blue triangles. The goal was to create a venn diagram. This was not completed - but left for pondering (how can this be done?)

Symmetry

We use a string to define the line of symmetry 

Then I made an asymmetric pattern and asked students to add pieces to create symmetry

This one was tricky because students had to figure out that they needed to make a square from two triangles:

Review odd/even

Students were given cards with the numbers 1-10 in random order 

Then they were asked to put them in order

Some interesting questions came up, such as

'What does 0 mean?' 
'Where does it go?'

And some answers:
'It doesn't make sense, it's like a snake'

Which numbers are odd and which are even?

Is there a pattern?

• Separate odd and even
• Point to a number and decide if it is odd or even 
• Identify if the following numbers are odd or even:
• 1
• 7
• 13
• 1,010,003 is odd or even, etc.

Gravity free water experiment

Darrin demonstrated gravity by pouring water out of a glass
then put a plate on the glass, tipped it over, and it didn't spill out

Feb 28, 2016 - Math Circle 2.0

Odd and even number pattern

We sorted numbers 0 to 20 on the table and then examined the odd and even pattern of numbers.  We continued the pattern even with 21, 22, and beyond.

Grouping of objects

We borrowed a bunch of Ellie's toy animals and grouped them.  We grouped them many ways:  Anya's likes and dislikes, humans and animals, water creatures and land creatures.

Probability

The students did an experiment with coins to determine how many flips it takes to get a "heads."  Sometimes it took 5 tries, other times it took only one try.

Doubling numbers

Cookie Monster starts a cookie eating marathon.  He eats one on Sunday, two on Monday, four on Tuesday.  He then eats twice as many cookies as the previous day.  The students made the pattern to Thursday and figured out that it would be a challenge - even for Cookie Monster.

Raw egg and hard boiled egg experiment

Darrin made a hard boiled egg and got it mixed up with the raw eggs.  We made an experiment to determine how to find the hard boiled egg.

For next time....

If a b and c are cities, if 4 roads go from a-> b and 3 roads from b->c, how many roads go from a->c

Constancy, Odd ones, and Symmetry Feb 14, 2016 Math Circle 2.0

What are there more of: rectangle or triangle blocks?

Based on experiments by Piaget, used by Zvonkin in his math circle.

First we lined up 8 rectangle and 7 triangle blocks. Which are there more of? It was clear that there were more rectangle blocks. Then lined up 7 of each in rows of the same length. Then, it was clear that there were the same number of each. But when I spread the triangles apart (see below) and asked "are there more triangles or rectangles" most thought there were more triangles. But then they thought about it 

Then I pushed all of the triangle blocks together. Which were there more of?

Which is longer?

After working on the 'what are there more of for a while, we went into "which line is longer". And we did different permutations of this: rectangles on their long or short edge, triangles on the long or short side. Below, I thought that the rectangles on their long side and triangles on their short sides would make lines of the same length. But, this was not correct. Each triangle is just a bit shorter so when lined up the row of triangles is shorter.

Odd one out

from Zvonkin 

Put together sets of cards. First three, then more. We started easy, like the one below (giraffe, lion, snake):

Later we moved onto more challenging sets with no clear answer:

These weren't as easy as counting the number of legs. For [elephant rhino lamb], the lamb was seen as the odd one because it was furry! For [chicken, eagle, flamingo] all were stuck until one child chose the eagle as the odd one out because it had long claws.

Symmetry

Last, we moved to drawing symmetry. I started with a connect four, trying to have students repeat patterns I made on one side, but it was difficult for them to understand the symmetry. So I went back to some exercises we did last year: I folded a post-it note, and drew on one side, and then students drew across the line of symmetry.

After they had the 'symmetry' down, I tried a more difficult challenge: the original drawing crossed the line of symmetry. This was confusing even though I tried a few permutations.

Admittedly, this one was pretty tough:

Reference: Zvotkin Ch 1 http://www.msri.org/people/staff/levy/files/MCL/Zvonkin.pdf

Feb 7, 2016 - Math Circle 2.0

Third dimension

To review coordinates with three axes (x, y, z) we placed Legos on a grid and asked a series of 'what are the coordinates of the pink Lego?

This was the third time we presented the idea, and now all of the kids more or less get it, but I presented this more as a fact than an open ended puzzle. Will try more puzzle next time, and hope the students recover!

Dividing stones

2x + 3 = 9

3x + 1 = 4

2x + 3 = 15

Splitting coins

Split in 2s and 3s

experiment

Do oranges float in water?

Why does the orange float but the pieces don't?

Jan 24, 2016 - Math Circle 2.0

Stone distribution

Anya brought 12 beautiful stones.  If we were going to distribute the stones to 3 friends, how many stones would each friend receive. How many stones would each friend receive if we were going to distribute to only 2 friends.

Third dimension

We played battleship on a grid.  How would we track an airplane by adapting the grid system?

Making shapes with coins

We tried to make shapes with a defined number of coins on the grid.  We could make squares and rectangles with a certain number of coins.