Real Mathematics – Numbers #10

A game for kids who would like to get better at arithmetic operations, decimal system and numbers in overall:

European Championship

  • Only materials needed for this game are a twelve-faced dice, pen/pencil and a piece of paper.
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  • Game consists of encounters between two players.
  • In each encounter players roll the dice four times in order.
  • Outcome of a rolled twelve-faced dice is like the following:
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  • For every player only ambition is to write the biggest possible four-digit number.
  • Difference of players’ four-digit numbers decides the winner.

Scoring of the game

Player with the bigger number would get:

  • 4 points if the difference is a four-digit number.
  • 3 points if the difference is a three-digit number.
  • 2 points if the difference is a two-digit number.
  • 1 point if the difference is a single-digit number.

If the difference is zero; meaning that the numbers are equal to one another, then both players get no points.

Every encounter finishes when one of the players gets to 7 points.

League

In case there is enough number of students, it is possible to construct a league version of the game that finishes after playoffs. For instance if there were 20 students we could divide them into 4 groups with 5 teams. In each group every player would play 4 games and after the group stage group leaders would go onto the playoffs where the champion can be decided after semi-final and final games.

World Cup

In this version of the game players would roll the dice three times and write the biggest possible three-digit number. Although this time winner gets to be decided like following:

  • If the difference is odd, biggest number wins.
  • If the difference is even, smallest number wins.
  • Winner gets 3 points as loser gets nothing. Differences are kept as averages.
  • If numbers are the same, players get 1 point each.

M. Serkan Kalaycıoğlu

Real Mathematics: Pattern #4

Leonardo Pisano

Italian town Pisa was the home of an ingenious person named Leonardo Pisano, which means Leonardo from Pisa. He was not only essential to history of mathematics, but he was also influential for the birth of scientific revolution. It is not a surprised that Leonardo Pisano was from Italy as Italians were involved with Arabs through trading.

Arabs knew an amazing way of counting and calculating, which were done with a system called decimal system. I’ll talk about that story in another article.

Leonardo Pisano was the first known person who brought modern numbers Western Europe. Although this was an amazing accomplishment, his importance comes even more fascinating if you look at what he did for patterns.

The Rabbit Problem

If I wrote his name as Fibonacci, then majority of you would understand what problem I’ll mention in the following:

In a farm, there is one couple of baby rabbits. A rabbit couple can give birth to baby rabbits only after their 2nd month and they can continue giving birth each month after that. Leonardo Pisano tried to find out the number of rabbit couples after one year.

Solution

First month there is a baby couple. This couple will be adult in the second month and they will give birth to one couple baby rabbits in the third month.

In the fourth month first couple reproduces as the second couple becomes an adult.

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In the fifth month first and second couples have new babies as third couple becomes an adult.

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In the sixth month first, second and third couples have new babies as the fourth and fifth couples become an adult.

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IMG_5690At this point we can point out a pattern in the number of rabbit couples. After second month, total of previous two months gives the number of rabbits in the next month. For example number of rabbit couples in the third month becomes the summation of first and second months, which is 1+1=2.

Fourth month = Second month + Third month = 1 + 2 = 3… and so on.

 

Then number of rabbit couples after one year (twelve months) is:

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Beauty of Fibonacci

This number sequence is known as the Fibonacci sequence and it is visible to us in nature on so many occasions. I’ll be talking about the most popular examples of Fibonacci sequence in the following articles.

Real life examples of math subjects are crucial, especially the ones from nature itself. But most of the population lives in the cities and this force us math teachers to find out examples from modern life.

Stairs and Fibonacci

Imagine that you have to climb up from stairs inside your apartment.

  1. How many ways are there to climb 3 steps?
  2. How many ways are there for 5 steps, 6 steps, 8 steps and n steps?
  3. What is the relationship of this question and Fibonacci numbers?

M. Serkan Kalaycıoğlu

Real Mathematics: Numbers #8

Dotted Tic-Tac-Toe

Almost everyone knows how to play tic-tac-toe. Dotted tic-tac-toe is a multiplayer game to teach kids about number systems.

Rules:

  • Players throw a dice in turns.
  • Each square in the board has the capacity of 9 dots.
  • Players should put dots into one of the boxes as many as their dice shows.
  • While putting down dots, players must be careful not to exceed the capacity of the box. In such situations they have to put dots to another box that has capacity.
  • Whoever gets right to left, upside-down or diagonal three boxes will be the winner.

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Beauty of this game is although it is competitive, it requires help of your opponent.

An example:

In the first three rounds players get 5, 6 and 3 from the dice:

Fourth dice is 1 which gives second player a chance to complete a box.

In the next four rounds players get 5, 5, 6 and 4 which gives again second player to complete another box.

Game resolves in the next rounds as shown in the following:

Binary Tic-Tac-Toe

In this version of the game, use a coin instead of a dice.

Heads: 1

Tails: 0

  • Consider that a box can have either of the following: A head & tail (tail & head is the same) or a tail & tail.
  • A square can never had head & head, since 1+1=2 which is overload for the binary number system.
  • This means that every box is either 1+0=0+1=1 or 0+0=0.
  • So in this version of the game, x-o-x turns into 1-0-1. Right to left, upside-down or diagonal 1s or 0s wins the game.

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Once again, although the game is competitive, you’ll need your opponent’s help.

An Example:

First two coin tosses are both tails.

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Then four consecutive heads are tossed.

Game resolved after three consecutive tails.

M. Serkan Kalaycıoğlu

Real Mathematics: Game #1

Circle of Numbers

0, 1, 2, 3, 4, 5, 6, 7, 8, 9

Most of the countries in the world use these symbols for the remarkable decimal system. The system was designed so perfect, even though symbols might change in some regions, logic of the system is conserved all around the world. Decimal system is one of those things that are “universal”.

If you are willing to participate in today’s modern world, you’d better know how to use numbers. You could go on and try to live a day without using any numbers: You’d see that it is impossible to finish even one single day. Actually, using numbers is not enough: you should be able to understand what a given number represents.

Understanding Numbers

What do I mean by “understanding numbers”?

If I show you 2125555555, can you associate it with anything?

How about 212 555 55 55?

Now most of you realize that 212 555 55 55 is a telephone number. Using three blank spaces inside the number 2125555555 changed the way you look at it.

Numbers take too much space in our daily lives. I think this is a great reason to work and develop a good understanding of them. Circle of Numbers is a game that helps children cultivate the ability of using numbers.

In order to play Circle of Numbers, all you need is a pencil and a pen. This game can have various numbers of versions.

Circle of Numbers 1.0.0

Draw a circle.

Place four boxes on the circle.

You have to place the numbers 0, 1, 2 and 3 inside the boxes such that difference of two adjacent boxes will be an odd number.

Example

Circle of Numbers 1.0.1

Again draw a circle with four boxes on it.

This time each of the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 should be used.

In order to do that, you will need ten boxes which means you will have to add six more boxes on the circle.

Difference of two adjacent boxes should be odd.

Example

Let’s assume 4 and 7 are placed as following.

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Player knows no matter which number is chosen, there will be number pairs that will not have odd difference. For instance if 5 is chosen, even though 5-4=1 is odd, 7-5=2 will be even. Player should add a new box to avoid this problem. Following will show you a working strategy step-by-step.

Circle of Numbers 2.0.0

Assume that there are five boxes on the circle and you are allowed to use the numbers 1, 3, 5, 8, 9. Can you construct a valid circle?

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Check it out

I just gave three different versions for the game Circle of Numbers with three different names: Circle of Numbers 1.0.0, Circle of Numbers 1.0.1 and Circle of Numbers 2.0.0. What do these numbers remind you? What would Circle of Numbers 2.0.1 look like?

M. Serkan Kalaycıoğlu

Real Mathematics: Numbers #6

Achtung! Following article contains advertisement for a chocolate company even though I won’t get paid.

How to get the most Milkinis?

Assume that

                – Class has 8 students.

                – All the students adore Milkinis and would do anything to have some.

                – There are 6 Milkinis bars in total.

Rules of the game:

                – 3 tables in the whole classroom named A, B and C.

                – Tables get 1,2 and 3 Milkinis bars in order.

                – On every table each student will get equal share of Milkinis bars. (eg. If there are 4 students on table A, those 4 students will get ¼=0,25 Milkinis bars each.)

                – Ambition is to select the table that has more Milkinis outcome for the student. (eg. While selecting, if student has a chance to get more chocolate from table A, he/she will choose that table.)

Q: Imagine you are one of the 8 students. In order to get the most Milkinis bars, what kind of strategy should you use? (On which turn you should select your table.)

Playing

As long as we don’t change the rules or anything, we will be starting the game in the same fashion every single time: First student will select table C, because it will give him/her 3 Milkinis bars which is greater than A’s 1 and B’s 2 bars.

Second student will choose B, so he/she will get 2 bars of Milkinis.

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Third one has to choose table C as he/she will get 3/2=1,5 bars of Milkinis. Now we have 1 student sitting in table B, 2 students sitting in table C. A is still empty.

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Forth student has 3 identical choices. In all three tables student will be getting same amount of Milkinis bar. (1 Milkinis bar.) Let’s assume forth student selects table B.

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Fifth one will get;

1/1=1 bar from table A,

2/3=0,66 from B,

3/3=1 from C. Let’s say the student selects table C.

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Sixth student will select table A as it will get him/her a full bar of Milkinis.

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Seventh will be selecting table C. Now table C has 4 students and 3 Milkinis bars. Each student here gets ¾=0,75 Milkinis.

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Eight student must select table B as each student on table B gets 2/3=0,66 Milkinis.

Result

Our game is finished with 1 student sitting at the table A, 3 at B and 4 at C. This would result that students get 1 bar of Milkinis from A, 0,66 from B and 0,75 from C. Student who goes to table A is the clear winners in the situation, who was the sixth choice.

One wonders…

  1. Is there a spot while selecting that guarantees the most chocolate?
  2. Why did I feel the need of using decimal point as using fractions would give me the same amount?
  3. One Milkinis bar has 4 little parts. Which student(s) would get the least Milkinis? How many parts of Milkinis would it be?

History of Decimal Point

In early math education teachers discuss decimal point right after teaching what fractions are. If they both mean the same thing, why do we teach both of them?

At the first sight it looks like a waste of time to show the same thing with two different notations. But in truth fractions and decimal point are both very useful and critical in math. Using decimal point might seem confusing, although when it comes to comparing two or more numbers, decimal point notation is easier to the eye than fractions are. (eg. Comparing 0,66 with 0,60 is easier and faster than comparing 2/3 and 3/5.) Also it takes less time when you write down huge numbers as decimals.

Fractions have a history of at least 4000 years. Decimal point notation is relatively a baby next to fractions. In his book History of Mathematics David E. Smith mentions a priest named Christopher Clavius (1537-1612). According to Smith, Clavius is the first known person who used decimal point systematically. In a book his book Clavius made a table called “Tabula Sinuum” where he wrote down his astronomical calculations in decimal point notation.

pellos

In 1492, Francesco Pellos wrote in his arithmetic book that 1/10th of 5836943 makes 583694.3 as shown. Although Pellos wrote the first known decimal point notation in his book, historians of mathematics claim that Clavius should be considered as the inventor of decimal point notation.

M. Serkan Kalaycıoğlu