for D, 10.20

1/3+1/6=
1/3-1/6=
2/3+1/6=
1/2+1/4+1/8=
1/2+1/4+1/8+1/16=

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A, B and C are looking at a green breeble (a magical creature).

A: B says the breeble has three legs!
B: No, it has five thousand legs.
C: Just one of you is lying.

Indeed, just one of A, B and C is lying. Who?

%%%%%%%%%%%%%%%%%%%%%%%%

What is bigger:
9*9 or 99?
9*9*9 or 999?
5/2 or 7/3?
555/2 or 777/3?

%%%%%%%%%%%%%%%%%%%%%%%%
10x+2y=9
10x-y=0.

3x+y=0
2x-4y=0.

%%%%%%%%%%%%%%%%%%%%%%%%

September 25, 2016 - Math Circle 2.0

Warm-up

In a certain flower garden, each flower was either red, yellow, or blue.  All three colors were in the garden.  One person visited and said that whatever 3 flowers you picked, one was going to be yellow. Another person visited and said that whatever 3 flowers you picked, one was going to be red.  Is it true that for every 3 flowers you pick, one will be blue as well?  Why?

Fibonacci numbers 

We write all possible sequences of black and white dots, such that no two black next to each other. How many different sequences are there?

2  of length 1 of them; 

3 of length 2; 

5 of length 3... 

Writing them one under another - if longer sequence extends the shorter one - helps.

The students extended this picture and found that there were 8 sequences of length 3.

Battleship

The kids formed two teams of two and used coordinates to find each others' hidden battleships.

for Diza, 9.27

1. Find

3333 x 55=
33 x 5555=
333 x 555=

2. A: C is lying;
     B: C is lying;
     C: I do not!
In fact, exactly one of them is lying. Who?

3. Solve
a:
3x + 2y=1;
2x + 3y=4.

b:
 x - 3y=7;
2x - 8y=12.

4. Pete makes 3 pancakes every minute. At noon, when there was 12 pancakes on the plate, the Pancake Eating Monster appeared, and started to eat pancakes, 9 a minute. When he will find the plate empty and starts biting Pete?

Math 1.0 Sept 18 2016: Histograms, vector addition

How big are leaves? (introducing the histogram)

First we went outside and each person (4 students and 1 teacher) collected leaves. We collected two leaves from each of two trees.

One of the students four leaves,
two from each of two trees.

First, each student measured their leaves, from the tip of the petiole (stem) to the tip of the leaf. 

We recorded the length of each leaf as a table:

Length of all leaves as a table, organized by
collector. Checks were marking off leaves that
had been added to the histogram. 

Then we created a histogram. For each bin (0-1, 1-2, etc ... 13-14), we counted the number of leaves that fell in that range. We used open blocks to indicate a leaf from Tree 1 and closed blocks to indicate a leaf from Tree 2.

Then I asked the students:

Which tree has longer leaves?

How long would you expect a leaf from tree 1 to be? how about a leaf from tree 2?

We also looked at the outlier from tree 1 to confirm the recorded value and that it came from the same tree.

One of the student's measurements and class histogram.

Dragons and Unicorns at the cafe? (vectors, vector addition)

We re-visited the cafe that serves cabbages and apples. (Following from Yuliy's Feb 22 2015 lesson).

Recall that unicorns always eat 2 apples and a cabbage head, and dragons always take 2 cabbage heads and 1 apple each. How may dragons were there? The idea is draw a vector plane, with the horizontal steps representing cabbage heads, and vertical - apples. Each dragon eats a vector of (2,1) (two cabbages, one apple), and each unicorn eats (1,2). So, to get to (3,3) - three cabbages, three apples, shown as a purple point - one needs to make one "dragon step" and one "unicorn step": 

Conceptual drawing from Feb 22 2015

Homework

1. Drag Car Race 

Last week we watched a video about drag racing that showed the distance a car had gone at 5 second intervals.

We recorded this table:

time (s)	distance (ft)
0.0	0
0.5	21
1.5	75
2.0	125
2.5	231
3.0	420
3.5	783
3.9	1000

• Plot distance vs. time, with time on an x axis
• x axis = time in seconds from 0 to 4
• y axis = distance in feet from 0 to 1000
• connect the points

Questions: 

• At what time was the car going the fastest? What was the fastest speed?
• At what time was the car going the slowest?  What was the slowest speed?
• How far will the car have gone after 4 seconds?

2. Dragons and Unicorns

• Draw a graph with # Cabbages on x axis and # Apples on y axis (each axis should go to 10)

Questions

• What does a Dragon vector look like? (Draw it)
• What does a Unicorn vector look like? (Draw it)
• If there is one dragon and one unicorn, how many apples and cabbages will be served?
• draw as addition of two vectors
• How many dragons and how many unicorns for the following orders:
• 5 cabbages, 4 apples?
• 5 cabbages, 7 apples?
• 0 cabbages, 2 apples?
• 8 cabbages, 7 apples?

  

Math 1.0 September 11 2016

Using plots to understand Velocity = distance / time

We are going to watch "How a top dragster works" by Car and Driver Magazine on YouTube:

It shows the distance at 0.5s intervals (also shows speed but we will focus on distance)

• watch video and record time and distance
• create a table with columns name time and distance

time (s)	distance (ft)
0.0	0
0.5	21
1.5	75
2.0	125
2.5	231
3.0	420
3.5	783
4.0	1000

• create a plot
• x axis = time in seconds from 0 to 4
• y axis = distance in feet from 0 to 1000
• ask: when was car going fastest? when was it going slower

Which way to the city of Lies?

Citizens of the City of Lies always lie. 
Citizens of the City of Truth always tell the truth. 
A citizen of one of those cities (you don't know which) is at the intersection. What question could you ask to them to find the way to the City of Truth?

You are at an unmarked intersection ... one way is the City of Lies and another way is the City of Truth. 

City of Lies or Truth Puzzle - Solution from mathisfun.com

Math 1.0 August 28: Two Dice Combinations and some Puzzles

Two dice combinations

Last week we had one dice, we figured out the possible outcomes, and then kids rolled dice, counted number of rolls for each case 1-6, and plotted them as histograms. 

This week we started by identifying possible outcomes of two die:

And then we counted the number of possible ways we could get each value between 1 and 12:

Left: possible outcomes of rolling two die
Middle: # of ways to get each outcome
Right: observed totals for class

So this week we saw that with two dice (unlike with one) we were more likely to roll numbers in the middle (7 is six times as likely as 2). The goal was not to begin computing probabilities though for those paying attention the probabiliity of rolling 7 (abbreviated p(7)) =1/6 and p(2) = 1/36.

  

While the students were busy rolling dice, Dr. Barishnikov found a website that simulated lots of dice. We looked at 1000 rolls of a single dice, and found the same as last week that there was an even probability of each outcome. Then we increased it to 2 dice. Below is what happens when we roll twenty dice (note: students thought rolling as many dice as many times as possible was very awesome).  

 Puzzles

Dr. Patel posed a puzzle: Who stole the diamond?

September 4, 2016 - Math Circle 2.0

Warm up

We reintroduced the number line and used it for addition and subtraction.

Coordinates

The robot cat and dog were visiting a new town and wanted to meet up.  Anya and Beans controlled the robot dog.  Willa and Ellie controlled the robot cat.  The students led the cat and dog to meet in the same square to have a party.

Die mapping

The students were give a single die and they then mapped it in two dimensions on a sheet of paper.  They then cut and folded the paper to reproduce the die.  They did a great job getting the sides of the die correct.

August 28, 2016 - Math Circle 2.0

Warm up

Beans has 6 cookies.  Ellie has 4 cookies.  How many more cookies does Beans have?

Coordinates

The robot dog was on the move in Urbana and she could move in all four directions.  The students successfully navigated the dog to the cat house party.  Next week.... The dog and cat will need to meet up by being programmed independently.

Patterns

We reviewed patterns from last week and then introduced a new pattern... A pattern that doubles.  Next week...  We will introduce a pattern that has two components - both shape and size.

Math 1.0 August 21: Length, Coordinates, Dice and Histograms

Semester introduction

The founder of the Urbana Math Circle Yuliy will be away for a semester, and so I slipped in to try my hand at teaching the older students for a few months.
I am framing the semester around two themes, and will add other bits and pieces.

The first theme is a fundamental law of physics: the harder you push something, the faster it goes. More specifically, that acceleration and speed are determined by how heavy something is and how hard it is pushed,  acceleration is proportional to force over mass a = F/m which is one arrangement of Newton's second law, F=ma. 

We start today reviewing a few basics - understanding length, understanding units, how to draw a graph with x and y axes. Later we will explore position, time, and the relationships between these. We will also learn about functions.

Another topic will be 'expectation'. What do we expect will happen if we roll a dice, flip a coin, pull a card, etc. This will provide some foundations for probability some day when students are ready.

Finally student choice: which of Yuliy's lessons were their favorites? 

Length and Measurement

First, students used a ruler to draw a number line. Twenty squares long. Then they measured this in centimeters. They came up with 12 or 13 cm until we looked into the notches and learned to read the millimeter marks. Then we translated this to inches (20 squares is 5" on a 1/4" grid), a non-standard grid that elicited an ode to 1/2 cm grids (and where to find them) from the professor in the back of the room.

Next, we enumerated the number line. every other square was one unit. This then became the x-axis, and we built a y axis.

For good measure, we plotted a sequence of numbers, starting at x=0 and y = 2:

From here, we thought about what number would go at x=-1 or x=3 ...

Dice (or expectation)

This week we discussed what could happen if you throw a single dice. Students realized that possible outcomes were 1, 2, 3, 4, 5, or 6.  

I asked them to write them down and start filling in a square corresponding to the number of dots each time they rolled, thus:

After we did this for a while I asked each student to say what value they got the most of. I enumerated those. Different students got different answers. Then we set up a second 'trial' and began rolling, to see if the same value won both times:

One student came to what seems to be a very unexpected outcome on the second trial:

He got one one, two twos and so on. Perfectly. We were all amazed. Did this fit in with what we expected?

Student choice: favorite topics

At the end of the meeting I asked students to list their favorite activities from Yuliy's math circle. 

I was amazed that the students came up with 'Roots', 'Vectors', 'Roman Numerals' and 'Algebraic notation', then there were some logic puzzles (remembered as 'truth or mad' ... which of the two twins is lying? homework, below).

Hopscotch came up. An app that lets kids build their own games. Emil built a variant of geometry dash.

Homework:

Bab and Bob are twins. One always lies and one always tells the truth. You meet one, but you can't tell them apart. How do you find out if you are talking to Bab or Bob?

Math 2.0 March 14: Piaget games

1. Length 

Two strings of same length. Lined up, which is longer?

Shift one, which is longer? 

Make one curved, which is longer?

Travis added dots at each inch. Line them up and then 

2. Volume 

Play dough. Asked children to make two of equal amount 

Students tried rolling and flattening them.

Then they used cookie cutter to measure  equal amounts.

Rolled them into a ball and all agreed these are equal amounts 

Then flattened one. All agreed flat one was bigger.

Then rolled them back again and they look the same.

Repeat squishing experiment 

Still all agree flat one is bigger. 

3. Area 

Travis has a hungry horse and three patches of grass.

Arrange patches touching and one town has them separate. 

Are these the same?

Now if the farmer splits up the patches

Which farmer has more grass for the horse?

Now count the patches.

3. Counting paths

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

4. Distance - maximal and minimal
The distance from my home to the nearest ice cream store is 8 blocks. The distance from my home to school is 6 blocks. What could be a maximal distance from my school to the ice cream store? What could be a minimal distance?

5. 3 and 5 cent stamps
Suppose you have an unlimited number of 3 and 5 cent stamps.
Prove that you can make any amount of postage 8 cents or more.

6. Experiment - oil and water

August 21, 2016 - Math Circle 2.0

Welcome back 2.0.  It was great to see everyone after the summer.  There was a full class in Altgeld Hall!

Warm up

Addition errors....  The students identified addition equations on the chalk board that were wrong and corrected them.

Coordinates

The robot cat was back in Urbana.  It could only travel two directions (left and up).  Using their programming skills, the students gave the robot instructions to navigate the cat through Urbana to the dog party.  They sucessfully negotiated through road construction!  Next week...  keep with idea but add more directions.



Patterns

The students were able to repeat a pattern developed by Max.   They also were able to reconstruct missing parts of the pattern.  Next week...  add additional pattern symbols.

Book stacking problem

Introduced the book stacking problem that the Math O 1.0 explored earlier.  The students experimented with playing cards and noticed that the card when placed on their desk would not fall if over half the card was on the desk.

http://mathworld.wolfram.com/BookStackingProblem.html

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.

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