TOK TOK AND AND

MATHSMATHS

Туенок Ирина АнатольевнаТуенок Ирина Анатольевна,,Муниципальное образовательное Муниципальное образовательное

учреждение Красноярская учреждение Красноярская университетская гимназия «Универс» № университетская гимназия «Универс» №

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TOK AND MATHSTOK AND MATHS

How logical is mathematics? Consider the following:

0.1111….. = 1/9 0.2222.…. = 2/9

. .

0.8888…. = 8/9 0.9999…. = 9/9 = 1

Why can we say that 0.9999…. = 1?

TOK AND MATHSTOK AND MATHS

How important is it to be exact?How important is it to be exact? Discuss this with regard to Discuss this with regard to

different scenarios in different scenarios in mathematics and other areas of mathematics and other areas of knowledge such as science, knowledge such as science, medicine, architecture etc.medicine, architecture etc.

TOK AND MATHSTOK AND MATHS Is infinity a number? Is it ever correct to Is infinity a number? Is it ever correct to

write write xx = = ? ? Do some infinite sets have more elements Do some infinite sets have more elements

than others? than others? Give examples to back up your answer. Give examples to back up your answer. If infinite sets are not the same size then If infinite sets are not the same size then

what does "infinite" mean? what does "infinite" mean? Can you count the elements of some infinite Can you count the elements of some infinite

sets? Explain your answer as clearly as you sets? Explain your answer as clearly as you can. can.

Can you think of an infinite set where it is not Can you think of an infinite set where it is not possible to count all the elements?possible to count all the elements?

TOK AND MATHSTOK AND MATHS Do mathematical symbols such Do mathematical symbols such

as as , , , , etc. have meaning in etc. have meaning in the same sense as words have the same sense as words have meaning? meaning?

Can mathematical symbols be Can mathematical symbols be ambiguous or vague like certain ambiguous or vague like certain words, (for example: free, poor, words, (for example: free, poor, good etc)?good etc)?

TOK AND MATHSTOK AND MATHS What is the difference between What is the difference between

the empty set,the empty set,, the number , the number zero and nothing at all? zero and nothing at all?

Can there be such a thing as a Can there be such a thing as a complete vacuum? complete vacuum?

Do you think we can consider Do you think we can consider the universe to be the ultimate the universe to be the ultimate universal set U containing universal set U containing everythingeverything

TOK AND MATHSTOK AND MATHS

We must be very careful with We must be very careful with brackets in Mathematics. brackets in Mathematics.

Are brackets in Mathematics Are brackets in Mathematics more important than brackets more important than brackets used in other areas of used in other areas of knowledge? (e.g. History, knowledge? (e.g. History, Literature, Science etc?)Literature, Science etc?)

TOK AND MATHSTOK AND MATHS

Why is Why is orderorder important in important in some areas of mathematics some areas of mathematics and not in others?and not in others?

Do we really learn from our Do we really learn from our mistakes? Look at examples mistakes? Look at examples from Economics and History.from Economics and History.

TOK AND MATHSTOK AND MATHS Often there are several different ways to Often there are several different ways to

arrive at an answer. arrive at an answer. For each method there will be advantages For each method there will be advantages

and disadvantages. and disadvantages. Sometimes one method may be more Sometimes one method may be more

accurate than another. accurate than another. How do you decide what method is best? How do you decide what method is best? Do you think that everyone would agree Do you think that everyone would agree

with you?with you? Are there different ways to arrive at an Are there different ways to arrive at an

answer in other areas of knowledge such answer in other areas of knowledge such as the sciences, history, aesthetics?as the sciences, history, aesthetics?

TOK AND MATHSTOK AND MATHS It is very important for the timing to be as It is very important for the timing to be as

accurate as possible in the Olympic accurate as possible in the Olympic Games. However, the same degree of Games. However, the same degree of accuracy is not necessary when timing accuracy is not necessary when timing the cooking of a pot of rice. So, how the cooking of a pot of rice. So, how important is it to have “exact” values?important is it to have “exact” values?

What do you understand by an “exact What do you understand by an “exact value”?value”?

Can a measurement ever be exact? Can a measurement ever be exact? Is it more important to be “exact” in Is it more important to be “exact” in

Mathematics and Science than it is in Mathematics and Science than it is in other Areas of Knowledge such as other Areas of Knowledge such as Science, History, Aesthetics?Science, History, Aesthetics?

TOK AND MATHSTOK AND MATHS When conducting a survey how can When conducting a survey how can

you ever be sure that the replies you ever be sure that the replies given are true? given are true?

In a survey the people questioned In a survey the people questioned could lie about the information they could lie about the information they givegive

What type of questionnaire was used What type of questionnaire was used and to whom was it distributed?and to whom was it distributed?

What difference could this make to What difference could this make to the results?the results?

TOK AND MATHSTOK AND MATHS ““Stem and leaf diagram”, “box Stem and leaf diagram”, “box

and whisker plot”. and whisker plot”. Does the name given to a Does the name given to a

mathematical process make a mathematical process make a difference to the result?difference to the result?

How do titles influence us in How do titles influence us in other areas of knowledge?other areas of knowledge?

TOK AND MATHSTOK AND MATHS How do we know when two How do we know when two

things are related? things are related? Can two sets of data have a Can two sets of data have a

very strong correlation and yet very strong correlation and yet not be related? not be related?

Try to think of your own Try to think of your own examples. examples.

Can you find any examples in Can you find any examples in advertising?advertising?

TOK AND MATHSTOK AND MATHS Why do we have different Why do we have different

currencies? currencies? Would it not be easier to Would it not be easier to

have one universal have one universal currency? currency?

What would be the What would be the advantages and advantages and disadvantages?disadvantages?

TOK AND MATHSTOK AND MATHS

If mathematics is about If mathematics is about following rules and we know following rules and we know that 10/10 = 1 and 4/4 = 1 ,that 10/10 = 1 and 4/4 = 1 ,

then why does 0/0 then why does 0/0 1? 1?