An Evil King

There is an evil king, who takes six gnomes prisoner. However, deep down he is slightly merciful, so he will give them one chance to earn their freedom. He lines them all up so that the first in line can see straight ahead, the second can see the back of the first, the third the back of the second and first etc. etc. Taking out a bag of orange and pink hats, he tells them that he will place one on each of their heads and if they can guess the color of the hat on their head they can go free. (They can come up with a strategy beforehand.) How many of them can survive?

Coming up with a strategy:

The evil king lines them up:


He places the hats on their heads...:


...but on the first try, the gnomes fail.

The second group devises a touching strategy. When one gnome guesses the color on the top of their head, they will poke the back of the gnome in front of them if that is also their hat color.


And they all survive!


But what if no touching is allowed? An alternate strategy is to use the principle of odds and evens. Before starting, the group will determine whether "pink" or "orange" means an odd or even number of one color of hats. For example, they may decide that "pink" means an odd number of pink hats, and "orange" means an even number of pink hats. Then, the gnome in the back of the line will count how many pink hats there are, and say "pink" or "orange" depending on whether the number is odd or even. The second to last gnome will then be able to look ahead and determine what color of hat he/she has. While the gnome in the back will have a 50-50 chance of survival, everyone else will survive.

TGIF

Here are the class pictures from today!

Group 1:






Group 2:

Tomorrow is Picture Day!

Don't forget to wear your green t-shirt to class!

Today, Group 1 worked primarily with pictorial models, a method of problem solving where the word problem is translated to a visual representation. They then played a two-person game where one person picks a number between one and twenty-five and then takes turns with the other person, subtracting in units of 1, 2 and 3. The person stuck with 1 left (thus having to be the last to subtract) loses. They then took a 15-question Online Math League test to get a feel for the type of math problems used in competition.


Group 2 continued their work with exponents and basic geometry, focusing on solving area and circumference of circles and finding ("chasing") angles. They have been working everyday in a MathCounts workbook as well.

Throughout this past week, both groups have worked toward solving the Rubik's Cube. Step 1 is to form a white cross with a yellow center. Step 2 is to make a white cross while making sure all the white edge cubes are on the right side (ex. white and red edge cubes lined up with the red center). Step 3 is to fill in the corners to complete the white face. Over the next week, they will learn how to fill in the second layer and then the third. http://www.thedryeraseboard.com/ has a useful tutorial under "mechanical puzzles" if you'd like to learn outside of class.

Another game they have played is 24, a card game where four cards are dealt and you must put the four values in an arithmetic expression--using each only once--that evaluates to 24. One strategy is to manipulate the values to get 3 and 8, 5 and 4, or 2 and 12 which can then be multiplied to get 24. Others include trying to get 30-6, 29-5, 28-4, 18+6 etc. On the final day of class, 24 will be one of the competitions, so the groups will get more practice next week.

Learning to count and factorize

Group 1

Not learning to count 1, 2, 3, 4, 5, 6, ... learning to count figures!

Students in the morning class learned how to solve problems that involve counting the number of shapes in a given figure. For example, how many squares are in the following figures?

a.
┌─┐
└─┘

b.
┌─┬─┐
├─┼─┤
└─┴─┘

c.
┌─┬─┬─┐
├─┼─┼─┤
├─┼─┼─┤
└─┴─┴─┘

The answers are 1, 5, and 14. The students learned that these numbers can be broken down into 1, 1+4, and 1+4+9. They look like the sum of square numbers! And indeed it could be proven that this is the correct way to find the number of squares in squares of any size.

The students learned similar techniques for more types of figures.

Group 3

Today, the group learned advanced techniques for factoring, which is simplifying an expression into a product of smaller expressions. First, they went over some fundamental algebraic factoring techniques. Then, they learned a special technique called Simon's Favorite Factoring Trick (or SFFT for short).

Announcement! Let's Play Chess~

This Wednesday and Thursday, Michael Brown will be in the classroom during 11:30 and 12:30 pm to play simultaneous chess games with anyone who would like to challenge him!

On Wednesday, he will play with students from the morning group, and on Thursday, he will play with the afternoon group. Students from the morning group will have to stay after class, and those in the afternoon group will have to come one hour early.

Michael is a two-time Western States Champ for 3rd and 4th grade. He placed third in the 4th Grade National Championships 2006, tied for second in the 5th Grade National Championships 2007, and tied for third in the 5th Grade Bert Lerner K-5 Championships. He has also won the following tournaments: Agoura Hills 2007, La Palma Spring 2008, and Pacific Southwest 2008.

Please bring your own chess set.

Problem Solving and Math Magic

Today, Group 1 started the class with some interesting tidbits about the Fibonacci Numbers and the Golden Ratio in real life. Fibonacci sequences are found in nature: in the arrangement of leaves and branches or the sections of a pineapple. The Golden Ratio can be found in art and architecture. Da Vinchi, in particular, is known for applying the ratio, also known as the divine proportion, to his masterpieces, dividing the canvas into golden sections. It can even be found in music composition! The Golden Ratio is also claimed to be the parameters for a perfect face.

For further reading on the Fibonacci Numbers and the Golden Ratio: http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/

The rest of the lesson included learning about Polya's problem solving model, an efficient alternate to the "brute force" method. They also learned about the Gauss method for calculating sums when dealing with sequences. Finally, the class learned some math, birthday, and card magic!

The second group covered exponents, simplifying expressions, and mixture problems, before learning math magic as well.

Happy Birthday, Tyler!

Friday

Today the first group learned the pigeon hole principle, which states that if x number of objects are distributed among y pigeonholes, and x > y, then at least one pigeonhole will have more than one object.

With relevant application as always, here is an example: Li is the largest surname in the world. More than 80 million people in the world have the last name Li. Is it possible to find at least two Mr. Li's with the same number on their heads? Note: Each head has a maximum of 2 million hairs. So, if we filled each "pigeonhole", or number of hairs, with one person, there would be 78 million people left to distribute...thus it is possible to find at least two Mr. Li's with the same number of hairs.

They also learned how to solve proof by contradiction.

The second group worked on inequalities and coin problems. They learned how to make a table as an efficient way to problem solve.

Class Pictures!

Rubiks Cube Craziness

On Wednesday, the students started to learn how to solve the rubiks cube.

Step 1: make a white cross with a yellow center


Step 2: make a white cross

Step 3: make an entire white face and complete the first layer making sure the colors bordering the white face match up with the colors of the center cube on each respective side.





They will continue to learn how to solve it for the remainder of the session, and hopefully by the end be able to complete the whole thing!





Sites to check out:
http://www.rubiks.com
http://www.thedryeraseboard.com/mechpuz/333/

Day 2

Group 1



After reviewing the problem of the day, the kids took a speed mental math quiz to practice multiplying by 11 and squaring numbers ending with "5". Mental math practice included looking for pairs adding to ten and grouping terms. Lesson 2 covered sequences: what is a sequence, how to identify arithmetic sequences, calculating the number of terms in a sequence etc.



A favorite example was when Dr. Li asked if the class would rather get $100 for every A received in school, or to receive one cent for the first A, then double that for the second A, and so on. While most of the class chose the first option, it turns out it would be much more lucrative to choose the second option in the long run! If one made As in 36 classes, one would accumulate a total of $687,194,767.35, as opposed to $3600!

The class ended with the card game, 24, where one is shown four cards and must apply mathematical operations to obtain the number "24".

Group 2

The second group did work with fractions, decimals, and percentages. They also practiced translating word problems into algebra.

To apply this, we had a problem solving relay competition! With four people per team, each received a different word problem and had to translate it into a variable expression. The first team player was given a number, which he/she subsequently plugged into their expression and then passed the answer to the next person in their team, who plugged that number into their expression etc. After a few tries, one team emerged the winner.






The class also ended with 24.

The First Day in Pictures

Group 1 - at play...





and at work.
Group 2




working in partners.


break time.