booklet 13 full - · A magic square is a square grid with numbers, such that all rows, columns and diagonals add to the same value. 8 18 4 6 10 14 16 2 12 Let us Verify that the

February 15 2017 @ UCI

Meeting 5 Student’s Booklet

Magical Patterns

1 Birds and Fish

2 Magic Squares

Contents

UC IRVINE MATH CEOhttp://www.math.uci.edu/mathceo/

STUDENT'S BOOKLET

We want to create several cards. Each card must have both birds and fish: between 1 and 3 birds, and

A Create all possible cards and draw them in the blank cards given.How many cards did you obtain?

UCI Math CEO • Meeting 5 (FEBRUARY 15 2017) 1 BIRDS AND FISH 3

1 BIRDS AND FISH B A card is called balanced if it has the same number of birds as fish. What fraction of the total number of cards are balanced?

How do you know that there are no more cards?

Who wants to read out loud?

How can you visualize your answer to the problem?

Give examples of balanced and unbalanced cards.

Why are these cards balanced or unbalanced?

between 1 and 3 fish. For example, the card in the picture has 2 birds and 3 fish (and so it has 5 animals).

Represent this fraction using a fraction bar:

1


C A card is called WET if it has more fish than birds.

Separate the pile of cards into WET and NON-WET.What fraction of the cards are WET? Represent your fraction using a bar (the bar is 1 unit)

Is the fraction that you obtain larger than 1/2? How do you know this?

D A card is called CROWDED if it has 2 or more fish, or 2 or more birds (or both!)

Separate the pile of cards into CROWDED and UNCROWDEDWhat fraction of the cards are CROWDED?


E Create a property of cards (similar to the “balanced” example), and divide the cards that you have into two categories: those who have the property and those who do not.

Each time you create a property, separate the cards according to it.

Modify the properties created by your friends!

1

2

3

1

2

3

1

2

3

1

2

3

Can you explain this diagram to someone else? How does it work?

Put a

Che

ck in

the

boxe

s th

at c

orre

spon

d to

thos

e ca

rds

that

obe

y yo

ur p

rope

rty

# of # ofCARD PROPERTIES

Use the template given to you to write your property and fraction bar. Give it to the volunteer, we will paste some of them on the wall!


Like in the 15 game, Teams take turns, this time choosing cards. A team wins if he fulfills one of the following missions:

M1 Have 3 cards all with the same number of birds

M2 Have 3 cards all with the same number of fish

M3 Have 3 cards all with the same total number of animals

M4 Have 3 balanced cards (cards that have the same number of birds as fish)

Play this game 4 times in teams!

THE BIRDS AND FISH GAME

Example: these three cards would make you win, as all have the same number of fish (Mission 2)

Each time you play, explain out loud why you chose that card.

Before playing, can we form some winning combinations?


F Color all sets of 3 cards that make you win the birds and fish game. Two examples are made for you!

BFFF

BBF

BBBF

BBBFF

BBFF

BF

BFF

BBFFF

BBBFFF

BFFF

BBF

BBBF

BBBFF

BBFF

BF

BFF

BBFFF

BBBFFF B

FFFBBF

BBBF

BBBFF

BBFF

BF

BFF

BBFFF

BBBFFF

BFFF

BBF

BBBF

BBBFF

BBFF

BF

BFF

BBFFF

BBBFFF

BFFF

BBF

BBBF

BBBFF

BBFF

BF

BFF

BBFFF

BBBFFF

BFFF

BBF

BBBF

BBBFF

BBFF

BF

BFF

BBFFF

BBBFFF

BFFF

BBF

BBBF

BBBFF

BBFF

BF

BFF

BBFFF

BBBFFF

BFFF

BBF

BBBF

BBBFF

BBFF

BF

BFF

BBFFF

BBBFFF

Same number of fish ( = 3) Same total number of animals ( = 5)

BBBFF

BF

BFF

BBFFF

BBBFFF

BFFF

BBF

BBBF

54321

BBFF

G How many times did you color each card?


H How would you arrange the 9 cards in a 3x3 grid so that when playing the game of Fish and Birds, it is equivalent to playing Tic Tac Toe? Go for it!

it’s just tic tac toe!

Conclusion: Tic Tac Toe, the 15 game and the game of Fish

and Birds are all mathematically the same!

What do you think the word “equivalent” means? Can you give some examples?

UCI Math CEO • Meeting 5 (FEBRUARY 15, 2017) 2 MAGIC SQUARES 10

A magic square is a square grid with numbers, such that all rows, columns and diagonals add to the same value.

8 18 4

6 10 14

16 2 12

Let us Verify that the square on the right is a magic square by showing that all lines add to the same value.

8 28 24

36 20 4

16 12 32

A Suppose that we know that the square above is a magic square. What is the sum of all the nine values of the square?

2 7 6

9 5 1

4 3 8

How did you solve it? Can you solve it in a different way?

Mental math teamwork: we add together!

Explain different ways to add

2 MAGIC SQUARES

Can you explain the concept in your own words?

2


B Complete the missing values to make these squares magic squares. 2 7 6

9 5 1

4 3 8

Did you remember to verify your answer?

8 6 1618 2

4 143 2

3 2

2 1

C If we want to create a magic square with the values from 1 through 9, what must be the sum of each line?

Hint: First, find the total sum: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 =

Add using rainbow sum! (1+9) + (2+8) + ....

The 9 numbers are placed on the rows. Remember that rows have the same sum.

What is the sum of the values on each line?

1 2 34 5 67 8 9

2

8

D Complete placing values 1 through 9 to obtain a magic square:

2

8

Rotate90

If you ROTATE your magic square, is it still

a magic square? Why? Prove it!


Use the whiteboard. When you are done, Compare solutions

with peers.

Explain your reasoning visually, using pictures!

MULTIPLY ALL ENTRIES BY 2

2 9 4

7 5 3

6 1 8

Sum of each line: 15

Total sum: 3 x 5 = 45.

16

2 7 6

9 5 1

4 3 8

3

ADD 1 TO EACH ENTRY


x2+1

E Obtain new squares from magic squares as indicated:

Discuss:

Is this a magic square? Why?

Discuss:

Is this a magic square? Why?

1 F Obtain new squares from magic squares as indicated:

Share different ways to multiply

Sum of each line: 15

Total sum: 3 x 5 = 45.

18

14

2

6

10

H Create a magic square using all of the following numbers: 3, 4, 5, 6, 7, 8, 9, 10 and 11.


G Create a magic square using all of the following even numbers: 2, 4, 6, 8, 10, 12, 14, 16 and 18.

2

2 9 4

7 5 3

6 1 8

How do your answers relate to the 1-9 Magic square?

Which transformations turn one into the other?

10

14

18

16

2

4

6

8

12

3

5

7

10

11

6

4

98

The 15-game is played by two teams. The numbers 1 through 9 are available (face up cards). The teams, in turns, pick one number each time. The first team to have 3 numbers whose sum is equal to 15 wins the game. The game can end in a tie.

For example: In the game shown, team 1 has collected 5, 3 and 7 (and also 6). They win, because the numbers 5, 3 and 7 add to 15:

5 + 3 + 7 = 15.

The triplet 5,3,7 is called a winning triplet.

3 75

8

1

6

92

4

5 3 7


THE 15 GAMECan someone read the instructions out loud?

Can someone explain this table to everyone?

Team 1 Team 2

Move 1

Move 2

Move 3

Move 4

Move 5

Move 6

Move 7

5

4

35

7

35

35 6

6

4

4

8

8 2

Winning triplet

An example:

A Play this game 2 times. Record the winning triplets from all the games in which there was a winner. Note: Stop drawing numbers as soon as a team wins.

LET’S PLAY the 15 game! 5 7 31

86

92 4

Team 1 Team 2

Move 1

Move 2

Move 3

Move 4

Move 5

Move 6

Move 7

Move 8

Move 9

Team 1 Team 2

Move 1

Move 2

Move 3

Move 4

Move 5

Move 6

Move 7

Move 8

Move 9


B Play two more times. Now right after Team 1 starts, Team 2 writes all possible ways in which Team 1 can win.

Example: if Team 1 picked a 3, then Team 2 must calculate all ways in which Team 1 can win using 3:

3 + _____ + ______ = 15:

Two possibilities: 3+5+7=15 and 3+4+8=15.

5 73

1

8

6

92

4 Team 1 Team 2

Move 1

Move 2

Move 3

Move 4

Move 5

Move 6

Move 7

Move 8

Move 9


C Based on your experience, if you are the starting team, which number is the best to choose first?

Which numbers are NOTconvenient to choose first? Explain your answer

5 7

3 1

8 6

9

2

4

Team 1 Team 2

Move 1

Move 2

Move 3

Move 4

Move 5

Move 6

Move 7

Move 8

Move 9


D Color all winning triplets, one per board. Examples are given.FREQUENCIES

Choose a value from 1 to 9. The frequency of that value is equal to the number of winning triplets that contain that value.

Example: you can verify that 1+9+5 = 15 and 1+8+6 = 15, and there is no other sum using 1 that gives 15. So the value 1 has frequency 2.

Example: 2 is contained in three winning triplers: {2, 9, 4}, {2, 8, 5} and {2, 7, 6}, so it has frequency 3.

E Complete the graph of frequencies of the values 1-9:


1 2 3

4 5 6

7 8 9

54321

1 2 3

4 5 6

7 8 9

1 2 3

4 5 6

7 8 9

1 2 3

4 5 6

7 8 9

1 2 3

4 5 6

7 8 9

1 2 3

4 5 6

7 8 9

1 2 3

4 5 6

7 8 9

1 2 3

4 5 6

7 8 9

1 2 3 4 5 6 7 8 9

Freq

uenc

y

Value

To start the 15 game, should we choose a value of low or high frequency?

Justify your answer.

F Play the 15 Game again, by using the value 1 through 9 contained in the magic square. Any time you pick a number, you color the corresponding box of the magic square:

TEAM 1: REDTEAM 2: GREEN

What do you notice?

the 15 game in the magic square board 8 7

3 1

5 69

2

4

2 9 4

7 5 3

6 1 8


Documents

booklet 13 full - · A magic square is a square grid with numbers, such that all rows, columns and diagonals add to the same value. 8 18 4 6 10 14 16 2 12 Let us Verify that the