A Plea for Pure Mathematics |
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byJosef Böhm | |||||||||||||||||||||||||||||||||
BHAK St.Pölten, DERIVE User Group and T3 AUSTRIA | |||||||||||||||||||||||||||||||||
I
am a teacher at a business college (high school level, 14-19 years) and
I have been working with CAS in teaching mathematics for a long time.
For many reasons I prefer DERIVE and I have prepared a lot of papers for
various purposes. In one of them we repeated and extended our knowledge
about GCD and LCM. I´ll present the worksheet and add some comments (all in Italics). It was interesting and fascinating to observe the students´ attitude towards that kind of mathematics and I did not find any real differencies in attitudes comparing more or less gifted students |
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TI 92 Worksheet 3 | |||||||||||||||||||||||||||||||||
Greatest Common Divisor and Least Common Multiple (GCD and LCM) | |||||||||||||||||||||||||||||||||
Using
the factor(844074000)
= ................................................ |
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(Store
the GCD as ): .......................... ![]() |
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Try once
more but use another name because gcd is obviously in use. So take: ![]() |
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If your answer is no then check your result ! |
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(2) Can we prove the statement given in the box above?: | |||||||||||||||||||||||||||||||||
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We want to show: If gcd(a,b) = T then gcd(a/T, b/T) = 1 We show that gcd(a/T, b/T) ¹ 1 implies T is not the gcd of a and b
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Find and check in a similar way the GCDs of:
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(Comment: it seems that it is not possible to factorize general expressions and the coefficients in one operation. Try to find an alternative!) | |||||||||||||||||||||||||||||||||
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The TI-92 knows the function gcd(number1,number2) for calculating the GCD of number1 and number2. Check that using two randomly chosen pairs of numbers: |
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(6) Calculate GCD(844074000, 45585540000, 4765246200) and compare with (1)! | |||||||||||||||||||||||||||||||||
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Have you ever tried gcd for non-integer numbers? Do that now and report your results:
I assume that you have tried only rational numbers, take now at least one irrational number:
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(8) I also assume that you have worked with decimal numbers. It is "more mathematical" to work with fractions:
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Find the LCMs for the expressions from (3), (4) and (5). Check the result for at least one of them: |
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(11) LCM(problem 3) = ............................................... | |||||||||||||||||||||||||||||||||
(12) LCM(problem 4) = ............................................... | |||||||||||||||||||||||||||||||||
(13) LCM(problem 5) = ............................................... | |||||||||||||||||||||||||||||||||
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Work systematically! Start your investigation using smaller numbers and try to find a rule.
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When you are sure you know the relation then you should check your conjecture using two "larger" numbers or expressions from the 1st sheet: | |||||||||||||||||||||||||||||||||
You should be able to express the relation as a formula: | |||||||||||||||||||||||||||||||||
GCD(a,b) = |
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Express this formula
in words: Until now that conjecture has only been an unproved conjecture and not a theorem. The proof is still lacking. Try to follow carefully the exact proof: |
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a = 8400, b = 1440 a = 24*3*52*7 and b = 25*32*5 (*70) gcd(8400, 1440) = 24*31*51*70 and lcm(8400, 1440) = 25*32*52*71 gcd * lcm = 24+5 * 31+2 * 51+2 * 70+1 = a*b Let us generalize: a = p1x1 * p2x2 *............* pnxn and b = p1y1 * p2y2 *............* pnyn (with n = greatest prime factor occuring and if pk does not occur in one factorization then xk or yk = 0).
gcd(a,b)*lcm(a,b) = p1min(x1, y1)+max(x1, y1) * p2min(x2, y2)+max(x2, y2) * .................. = = p1x1+y1 * p2x2+y2 * .................. * pnxn+yn = a * b |
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As ggt
and kgv are related very closely:
.............................................................., it will be sufficient to find a function for one of them. Do you know any application of GCD and/or LCM? |
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We will stick to the LCM. | |||||||||||||||||||||||||||||||||
The TI-function
getdenom(fraction) gives back the denominator of a fraction. What are the denominators
of the fractions: Use ............................................. Check it: ................................................................ getdenom will support our task to find the function kgv! Are you ready? Do you have a formula, which returns the LCM of two expressions t1 and t2? Then define your function in the following way: |
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define kgv(t1,t2) = ............................................................................................. | |||||||||||||||||||||||||||||||||
and check its validity by applying it to the two expressions on the TI-screen - is it correct? Congratulations! But to make sure: just one more little test: Doublecheck your function for two numbers: kgv(3,6) = .................... Everything ok? Why not?? What has happened?? Find an explanation and improve the function. |
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Improved version: define kgv(t1,t2) = ............................................................................................. |
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which leads immediately to define ggT(t1,t2) = ............................................................................................. |
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From the TI 92 Worksheet 3 | |||||||||||||||||||||||||||||||||
Train your skills in calculating and manipulating with fractions | |||||||||||||||||||||||||||||||||
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Calculate without the TI. Write down your result: ................................. | |||||||||||||||||||||||||||||||||
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Here it
is.
Can you find a reason why this calculation could cause a very special mistake? |
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Compare
the common denominator and the factors used to bring all the numerators
over a common denominator.
Explain the last part in the edit line: -4 * gn: |
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How to
work with an expanding factor ew( )? -4 * ew(....). The remaining work is very easy: Explain the relation between left and right hand side in the last row of the history area: |
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I want to close with a short summary. I like applied mathematics but I am sure that we can not do without pure mathematics. I found that the students liked investigating the problems presented and I spent a lot of time during the breaks to answer their questions (what if?, why not?, am I right?, do you think?,....). I never heard the question: What is the use of all that stuff? And I am teacher at a vocational school!!! So I think we can tempt pupils to have their "Adventures in Their Heads". And it is necessary to explain sometimes what they have done, like: "Now you found an important rule by your own!", "This is your first selfmade function!",... The students are really proud and their selfconfidence is growing. There were some very emotional moments when I felt that the sparkles were spreading and when I then left the classroom I felt something like: "that is it, I´ve got them". This are the precious moments when we learn why we have become teachers. I know that the use of a CAS in whatever form is not a necessary condition, but it is helpful in providing a new motivation for students and ............ not to forget, for teachers, too. One example of this species is just now standing in front of you. What I have shown here using the TI-92 is platform - independent. I am sure that many chapters of our traditional curriculums can be changed and that it is not necessary to add many new topics as a consequence of the use of modern technologies. On the contrary, I must say, that I would now need more time in the classroom than before, because for me mathematics teaching has a much wider scope than ever before. And that without new subject matter. Let me repeat in some keyterms what we have done besides revising the chapter CLM and GCD from Lower Secondary School (in order of appearence): |
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Working
with larger numbers and expressions
Indirect Proof Exactness in Editing the Data Concept of Nested Functions Generalization 1 Interpretation and Reinterpretation Reasoning, Defining, Checking Using the Manual Pattern Recognizing 1 Defining and Verbalising Proof Generalization 2 Try and Error, Exploring Principle of Duality Giving Reasons Pattern Recognizing 2 Compiling the Results in an Appropriate Form Defining Functions Using Functions for other Functions Checking functions even for trivial cases Improving functions Working with the functions |