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透過悅趣活動發展 數學解題能力. 张宝幼与您分享. Prof. Vincent J. Matsko. http:// staff.imsa.edu/~vmatsko /. 分配蛋糕. (a) 如何把七个蛋糕公平地分配给十位同学?每位同学将获得两片蛋糕。. 分配蛋糕. 提示 : 如何把上图所显示的七个蛋糕公平地分配给十位同学?每位同学将获得两片不同的蛋糕。. 分配蛋糕. 分配蛋糕. ( b ) 如何把七个蛋糕公平地分配给十二位同学?每位同学将获得两片蛋糕。. 分配蛋糕. (b).

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slide2

透過悅趣活動發展

數學解題能力

张宝幼与您分享

slide3

Prof. Vincent J. Matsko

http://staff.imsa.edu/~vmatsko/

slide4

分配蛋糕

(a) 如何把七个蛋糕公平地分配给十位同学?每位同学将获得两片蛋糕。

slide5

分配蛋糕

提示:

如何把上图所显示的七个蛋糕公平地分配给十位同学?每位同学将获得两片不同的蛋糕。

slide7

分配蛋糕

(b) 如何把七个蛋糕公平地分配给十二位同学?每位同学将获得两片蛋糕。

slide8

分配蛋糕

(b)

Or ------------------------------------------------------------

slide9

5

  • - 1,4
  • 2,3
  • LCM for 1, 4, 2, 3 = 12

2 answers

(c) How can you divide the 5 cakes among 12 students equally (fairly) where each student gets two pieces of cake?

slide10

(c) How can you divide the 5 cakes among 12 students equally (fairly) where each student gets two pieces of cake?

Or ------------------------------------------------------------

slide11

1 answer

  • 5
  • 2,3
  • LCM=6
  • 5 cakes divide among 6 students each one gets 2 pieces
  • 2x3
  • 3x2
slide12

No answer

  • 5
  • 2,3 (LCM=6)
  • 1,4 (LCM=4)
  • 1,2,3,4 (LCM=12)
  • 5 cakes divide among 7 students each one gets 2 pieces
slide13

跳出

思想框框

slide14

跳出

思想框框

Prof. HartwigMeissner, Germany

slide15

跳出

思想框框

A

B

Prof. Hartwig Meissner, Germany

slide16

Learning vs Unlearning

Learning makes us smart and proud,

Unlearning makes us wise and humble.

Charlie “Tremendous” Jones

slide17

“The illiterate of the 21st century will not be those who cannot read and write, but those who cannot learn, unlearn, and relearn.”

- Alvin Toffler -

slide18

学生的看法可能与我们不一样

Prof. HartwigMeissner, Germany

48x63
48x63=

48

x 63

144

2

4

288

1

1

3024

Show Your Working

slide20

Dice Construction

Top View

Side View

Front View

The diagram above shows a model constructed using 4 dice with its top view, front view and side view.

slide21

L2b

Dice Construction

Diagrams below show the front view and side view of a model constructed using some dice. Based on these views, can you construct this model using the least number of dice? Draw the top view of your answer.

Front view Side view

Reference: Prof. Hartwig

slide22

L2b answer

Dice Construction

slide23

L2b answer

2

2

2

2

2

2

2

2

Dice Construction

Answers:

slide24

Important Mathematical Thinking Skills

Reasoning from Evidence

Jim Ridgway

- organize unsorted data and draw conclusions

slide25

L9a

The Last Special Card

“Kruskal Count”

Special cards

1

2

First secret number = 3

3

1

2

3

4

1

The Last Special card

slide26

How to Increase

the Chances of Wining?

1-6

Average advance = (1+2+3+4+5+6)/6 = 21/6 = 3.5

Average Steps taken = 24/3.5 = 6.857, about 7 steps

The density of the victim’s cards is 1/3.5 = 0.286,

so there’s a 1 − 0.286 = 0.714 chance of a miss.

But you have to miss all 7 times,and this will occur about (0.714)7 = 0.0945

so you’ll win about 0.905 of the time.

slide27

How to Increase

the Chances of Wining?

1-5

Average advance = (1+2+3+4+5)/5 = 15/5 = 3

Average Steps taken = 24/3 = 8, 8 steps

The density of the victim’s cards is 1/3 = 0.333,

so there’s a 1 − 0.333 = 0.667 chance of a miss.

But you have to miss all 8 times,and this will occur about (0.667)8 = 0.0392

so you’ll win about 0.961 of the time.

slide28

How to Increase

the Chances of Wining?

1-3

Average advance = (1+2+3)/3 = 6/3 = 2

Average Steps taken = 24/2 = 12, 12 steps

The density of the victim’s cards is 1/2 = 0.5,

so there’s a 1 − 0.5 = 0.5 chance of a miss.

But you have to miss all 12 times,and this will occur about (0.5)12 = 0.00024

so you’ll win about 0.9998 of the time.

slide29

L9a [P. 1-2]

Twinkle, Twinkle, Little Star

Twinkle, twinkle, little star,

How I wonder what you are.

Up above the world so high,

Like a diamond in the sky.

When the blazing sun is gone,

When he nothing shines upon,

Then you show your little light,

Twinkle, twinkle, all the night.

Then the traveller in the dark,

Thanks you for your tiny spark,

He could not see which way to go,

If you did not twinkle so.

slide30

Weighing Cards

This is a very impressive trick where the Magician can predict the number of cards selected by the spectator by just weighing the card in his palm.

slide31

L11a

Weighing Cards

slide32

Weighing Cards

  • How does the trick work?
  • If the Magician sees a red card on top of the faced up stack, he takes the number (x) multiply by 2 and then plus 2.
  • Number of cards = 2x + 2
  • If the Magician sees a blue card, he takes the number multiply by 2 and then subtract the result from 29.
  • Number of cards = 29 – 2x
  • Number of cards = 30 - (2x+2 – 1)
  • = 30 – (2x + 1)
  • = 29 – 2x
slide33

L11a [P1-2]

Weighing Cards

  • Variation for P1-2: No need to add the yellow cards.
  • Calculation for P1-2:
  • If the Magician sees a red card on top of the faced stack, he doubles the number (x)
  • Number of cards = x + x
  • If the Magician sees a blue card, he slides the top card down a bit to take a look at the next red card. He doubles the number on the red card (y) then plus one.
  • Number of cards = y+y+1
slide34

Ann's Tower

One of Ann’s Christmas presents was a box of building bricks. The bricks were all cubes with an edge of 5cm and completely filled the box which was also in the shape of a cube. Like many young children, Ann was fascinated with building towers. She started by building a large cube, then a smaller cube on top of that and a yet smaller cube on top of the latter. When the three cubes were complete she could still look over the top of her tower when standing, which disappointed her, but she had the satisfaction that she had used every single brick in her construction. How high was the tower?

[Bolt, (1989) p. 14]

slide35

a3 + b3 + c3 = d3

27 + 64 + 125 = 216

33 + 43 + 53 = 63

The height of the tower

= 5(3+4+5)cm

= 60cm

Make a table/list

slide36

马来西亚新加坡

通过数学提升创意的方法

  • 魔术
  • 趣题
  • 手工
  • 历史
  • 游戏
  • 项目
slide37

pp. 58-60

魔法卡片

slide42

0

1

0

1

4

2

1

8

slide43

1

1

1

0

4

2

1

8

slide45

?

0

1

1

1

4

2

1

8

slide46

20

要是有些卡片的号码更大呢?

slide47

1

0

1

1

0

4

2

1

8

16

slide48

1

2

0

Variation 2

1 x 9

2 x 1

0 x 3

slide49

马来西亚新加坡

通过数学提升创意的方法

  • 魔术
  • 趣题
  • 手工
  • 历史
  • 游戏
  • 项目
slide50

L5a

Calendar Game

The table below shows a list of 12 months in a year and 31 days in a month. Two players take turns to circle either a month or a date in ascending order. The first person to arrive at December 31st is the winner.

slide51

L5a

Calendar Game

MonthsDates

  • January 1 11 21
  • February 2 12 22
  • March 3 13 23
  • April 4 14 24
  • May 5 15 25
  • June 6 16 26
  • July 7 17 27
  • August 8 18 28
  • September 9 19 29
  • October 10 20 30
  • November 31
  • December
slide52

L5a answer

Calendar Game

The number of month is from 1 to 12; whereas the number of date can be 1 to 31. Let X be the number of month left to reach 12, and Y be the number of date left to reach 31. The winner needs to always match his opponent’s X or Y so that he can win the game.

slide53

L5a answer

Calendar Game

For example, if your opponent circles June, it is the 7th month from the end of the year. Therefore, you should circle 25 which is the 7th day counting backward from 31st.

Likewise, if your opponent circles 30, you should circle November.