The sequence of exercises through which the child is introduced to the group operations with the Golden Beads Math operations include addition, multiplication, subtraction, and division. All these operations require full understanding of quantities on a concrete level before moving to the abstract level of performing these operations mentally. So the first exercise introduced to the child to prepare him for these operations; is the “Number Rods”. The ten number rods are graduated in length from 4 inches length for the shortest one, to 40 inches length for the longest one. The different four inch sections are alternately colored red and blue and can be counted on each rod. So , each rod consists of distinct countable units, united together to represent a number. This overcomes the difficulty of adding one unit after another in a sum total. “The fact that a group is enlarged through the addition of a unit, and that this increasing must be considered; constitutes the chief obstacle for the children of three and a half to four, in learning how to count.” Maria Montessori, The Discovery of the Child, page 263 Several activities are done with the number rods. All of them are activities of moving and joining. One time the child builds them next to each other starting with the shortest and ending with the longest, and another time, he builds them on top of each other. All this joining and moving is an introduction to arithmetic. Then the “Sand Paper Numerals” are introduced to the child, to teach him their names and how to write them. Later on the child learns to match the card to the corresponding rod, which “forms the basis for a lengthy task which a child can continue by himself. The sums of the rods can be written so that they correspond to the numbers.” Maria Montessori, The Discovery of the Child, page 265 Two other objects are used to help the child begin arithmetic; The “Spindle Box”, and “The Cards and Counters”. ”The Spindle Box” attains three main objectives. It gives a child a concept of numerical groups , and at the same time; it fixes before his eyes the succession of numerals, and finally he is introduced to the “Zero”. In this activity; the child reads a number on the wall of a compartment, and then he groups the corresponding number of spindles in his hand, then he places them in the compartment. “Cards and Counters” accomplish another two objectives. It shows that the child has understood the order of the numbers, and that he recognizes the figures representing the numbers. Moreover; it is by this activity that the child realizes the difference between odd and even numbers. By then; the foundation for counting and arithmetical operations is laid.
Till now, the child is working with numbers not more than ten. “Number Games” are then played to make sure that the child deeply understands the quantities and the written symbols of numbers from 1 to 10, and to make sure that he really understands the concept of “Zero”. When this understanding is shown; the child is introduced to the “Decimal System”. First the child is taught “The Names of The Power of Ten” through the bead material. By the power of ten, we mean the unit, ten, hundred and thousand. For this lesson; the teacher uses one single bead to represent the “Unit”, ten beads wired together in the form of a bar to represent the “ Ten”, ten of the ten bars wired beside each other forming a square of hundred beads to represent the “Hundred”, and finally, ten hundreds wired on top of each others forming a cube of thousand beads to represent the “Thousand”. The teacher introduces these names with a three period name lesson, and to make him understand this on a more concrete level; “The Counting Through” activity is introduced. He would see how repeating a digit 10 times makes it become another digit. He will see hoe ten units make a ten, ten tens make a hundred, and ten hundreds make a thousand. This is...
...SEQUENCE
* In mathematics, informally speaking, a sequence is an ordered list of objects (or events). Like a set, it contains members (also called elements, or terms). The number of ordered elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Most precisely, a sequence can be defined as a function whose domain is a countable totally ordered set, such as the natural numbers.
* For example, {M, A, R, Y} is a sequence of letters with the letter 'M' first and 'Y' last. This sequence differs from {A, R, M, Y}. Also, the sequence {1, 1, 2, 3, 5, 8}, which contains the number 1 at two different positions, is a valid sequence. Sequences can be finite, as in this example, or infinite, such as the sequence of all even positive integers {2, 4, 6,...}. Finite sequences are sometimes known as strings or words and infinite sequences as streams. The empty sequence { } is included in most notions of sequence, but may be excluded depending on the context.
ARITHMETIC SEQUENCE
* A sequence such as 1, 5, 9, 13, 17 or 12, 7, 2, –3, –8, –13, –18 which has...
...Anatolia College 
Mathematics HL investigation

The Fibonacci sequence 
Christos Vassos

Introduction
In this investigation we are going to examine the Fibonacci sequence and investigate some of its aspects by forming conjectures and trying to prove them. Finally, we are going to reach a conclusion about the conjectures we have previously established.
Segment 1: The Fibonacci sequence
The Fibonaccisequence can be defined as the following recursive function:
Fn=un1+ un2
Where F0=0 and F1=1
Using the above we can find the first eight terms of the sequence. An example of calculations is given below:
F2=F1F0F2=1+0=1
We are able to calculate the rest of the terms the same way:
F0  F1  F2  F3  F4  F5  F6  F7 
0  1  1  2  3  5  8  13 
Segment 2: The Golden ratio
In order to define the golden ratio we need to examine the following sketch:
The line above is divided into two segments in such a way that ABAP=APPB
The ratio described above is called the golden ratio.
If we assume that AP=x units and PB=1 units we can derive the following expression:
x+1x=x1
By solving the equation x2x1=0 we find that: x=1+52
Segment 3: Conjecture of φn
In this segment we examine the following geometric sequence:
φ,φ2,φ3…
Since x=1+52 can simplify φ by replacing the value of x to the...
...Sequence of Operations (Gas Furnace)
Natural Draft Furnace
24 volts is always present to the thermostat R terminal.
1. When the thermostat closes its switch to call for heat, 24 volts is sent out of the thermostat on the W terminal.
2. This 24 volts goes back into the furnace, then typically through 1 or 2 safety devices to the gas valve.
3. If the standing pilot is lit. The gas valve opens, and the gas to the burners is then ignited by the pilot light.
4. The heat exchanger is heated until a device tells the blower to come on, usually a fan and limit control.
5. When the thermostat is satisfied, the switch in the thermostat opens, the gas valve closes, stopping fuel and the fan continues to blow until the fan and limit cools, turning off the blower.
High Efficiency Furnace
24 volts is always present to the thermostat R terminal.
1. Thermostat sends 24 volts through white wire to furnace.
2. Furnace checks pressure switch, and safety switches for proper positions (open/closed).
3. Draft inducer blower motor, starts to run.
4. If vent and vacuum hoses are clear the pressure switch, closes. (Only a few pressure switch's open to prove venting)
5. After inducer power purges the heat exchanger, either a pilot lights, a hot surface igniter (HSI), or a spark begins.
6. If the furnace is an intermittent pilot model, once the pilot lights, current is sent through the flame by a...
...Exercise 1 page 224
a. Using Maximax, the worst payoffs for the alternatives are as follws:
Do nothing: $60 thousands
Expand: $80 thousands
Subcontract: $70 thousands
Hence, since $80 thousands is the best, choose to expand the firms using the maximax strategy
b. Using Maximin
Do nothing: $50 thousands
Expand: $20 thousands
Subcontract: $40 thousands
Hence, since $50 thousands is the best, choose to do nothing using the maximin strategy
c. Using Laplace
For the Laplace criterion, first find the row totals, and then divide each of those amounts by number of states of nature. Thus, we have
Alternative  Next year’s demand  Row total  Row average 
 Low  High   
Do nothing  50  60  110  55 
Expand  20  80  100  50 
Subcontract  40  70  110  55 
Because the “Do nothing” facility or “Subcontract” facility has the Highest average, they would be chosen under the Laplace.
d. Using Minnimax regret
Alternative  Regrets  Worst 
 Low  High  
Do nothing  0  20  20 
Expand  30  0  30 
Subcontract  10  10  10 
The best of these worst regrets would be chosen using minimax regret. The lowest regret is 10, which is for a subcontract facility. Hence, that alternative would be chosen.
Exercise 2 page 224
a. Determine the expected profit of each alternative.
EP Do nothing = 0.3x50+0.7x60 =$57
EP Expand = 0.3x20+0.7x80 =$62
EP Subcontract =...
...Through the Eyes of a Child
Why are some people judgmental towards others? Whether we judge on race, gender, or something as simple as age, judging not only causes anguish but can leave emotional scars people never recover from. Racism is one of the most common of judgmental forms still seen today in our society. Although today we do not find racism as prominent as back in the 1920s and 1930s. When reading the story “The Angel of the Candy Counter” written by Maya Angelou, we can see the damage of racism from a child’s view, and how she dealt with the experience in order to successfully allow us to understand the negative power of racism. As a result of Angelou incorporating strategies such as ethos, pathos, shocking words, and even fantasies to bring out intense emotions she is able to connect with her audience.
Clearly Angelou writes as if she is speaking to a wide range of audience. She immediately opens telling a story of a child feeling as though she is being punished with a toothache: “The Angel of the Candy Counter had found me at last, and was exacting excruciating penance for all the stolen Milky Ways, Mounds, Mr. Goodbars and Hersheys with Almonds” (146). Angelou later states, “It seems terribly unfair to have a toothache and a headache and have to bear at the same time the heavy burden of Blackness” (147). With these two statements we can conclude she is focusing on children and people of color, for both are able to...
...Exercise on group behavior
1. Based on knowledge from your company group relate to and explain the 5stage group development model and the punctuated equilibrium model (both very important for exam.
Group analysis based on 5stage model:
* Forming. Firs task was to find out people with the same level of motivation, expectations and similar point of view through the vision of business idea. After finally forming the company group another task was to find out the purpose of the project and understood why this experience could be useful for us (It took almost all day). Furthermore we had to take decisions about how group is going to look like and what tasks we have to do personally: we find out which persons are going to be leaders who are generating the idea and who are going to work with developing and presenting it.
* Storming. Secondly we as a very fresh and inexperienced group had to come up with real and innovative business plan. It was the hardest part of all because we didn’t knew each other very good. We had many conflicts about different ideas while finally after many disagreements we find out the best decision of our idea.
* Norming. During the working time our relationships grown up significantly. It became much easier to work with each other. We began to understand each other easily and that let us...
...Unit 16 – Exercise for Specific Groups
Task One
Fitness suite has been open for a Year now and wants to help all of the people in local area get involved in exercise especially specific groups such as obese. You are a trainee personal trainer and you are considering setting up exercise courses for special populations in an identified area. It is your role to research the range of provision currently on offer in a range of local facilities and benefits that it can bring.
Specific groups are not always aware of where they can go and the benefits of engaging in exercise. Product a leaflet which describes (P1) the provision of exercise for 3 of the target groups you have studied: disabled people, antenatal, postnatal, older adults, children etc.
The three gym sectors are Public local government which are ran by the local government an example of these gyms would be Blidworth leisure centre Gym, Dukeries Gym, Southwell Gym etc. Another sector is the private provision which includes nationwide gyms such as DW Sports Gym, Pure Gym, Oasis and Bannatynes Gym. The final sector in the voluntary sector which may just be a local Zumba class this class will have very little cost involved and is not there to gain masses of profit. Before I research what each provision has to...
...Which social groups are marginalized, excluded or silenced within the text?
Social groups are significant in Golding’s novel “Lord of the Flies”, as they exhibit and accompany the development of a group of British schoolboys, which socially deteriorates into savagery, splitting into certain social subgroups.
In a context shaped by the world wars and the resulting communal imbalance, perhaps creating or already foreshadowing a sense of rivalry and social disharmony, Golding employs several characters that differ according to age, physical capability, political approaches and have different positions in hierarchy. In my opinion, Golding creates social groups, which lead to marginalization, exclusion and silencing.
Physically weaker boys such as Piggy and the younger “littlunes” seem to be increasingly marginalized and perhaps silenced by the “biguns”, as community shifts to animality and chaos. In a meeting, Jack vigorously accuses the physically inferior boys for the bad “fear talk” about the beast:
“You littluns started all this (…) fear talk. Beast! Where from? (…) What does that mean but nightmares? Anyway, you don’t hunt or build or help – you are a lot of cry babies and sissies”.
Jack, whose masculine and dominant character very much represents
the power of the “biguns” and particularly the choir boys, reveals his oppressing and antagonistic nature...
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