Okay, so for our first module, ang i-discuss natin basically ay the nature of mathematics or similarly, yung parang math na may kita natin sa nature, sa kalikasan, or sa paligid natin. So, first I would like you to look around and look for patterns or designs na nakikita nyo sa paligid nyo. There are plenty, right?
So marami. From the pattern of your windows to your... tiles, yung mga spiderwebs natin, lahat yan ay mga patterns. And all those patterns are, can be described by mathematics.
Okay? So, according to this one, recognizing patterns feels natural like our brain is hardwired to recognize them. Okay? Kung familiar kayo nung bata kayo, naglalaro ba kayo ng 10, 20? So yung 10, 20, 30, 40. Okay, so isa rin yung na pattern with numbers with which nag-a-add tayo ng 10 para makuha yung kasunod na number.
Okay, so yun ang mga examples ng math na or patterns na makikita natin sa nature. Okay, sige. So let us proceed with patterns and numbers in nature and the world. Okay, so first, bago yun, kailangan muna natin i-define ko anong pattern. Okay, so patterns are regular, repeated, or recurring forms of design.
So here we have an example. So we have a smiley na walang ngipin, may ngipin, walang ngipin, may ngipin, walang ngipin. So intuition would tell us that yung susunod na term dito ay ano? Siyempre, smiley na...
Merong ngipin. Okay. So, yan yung susunod sa pattern natin. Okay.
And then, we have a next example. So, consider this one. So, based on the pattern, ano kaya yung susunod dito?
So, kung titignan natin yung shapes, parang wala syang pattern. So, hindi natin alam yung susunod. Kasi usually, square. So, naging triangle.
And then, naging... Circle. So ano kaya susunod dito?
Kung titignan natin yung gitna, makikita natin na naka-darkened yung middle. So since we only have two choices, so we have two stars, yung isa may dark, darkened yung center, yung isa naman ay hindi. Okay, so ang piliin natin ay yung hindi darkened. Kasi kung makikita natin, black, darkened, wala darkened. So yung susunod dito ay malamang C, letter B.
Okay, how about this one? here we have 2 or at least 3 patterns na makikita natin. So ano ba yung mga patterns na na-observe natin dito? So notice that, so parang nagro-rotate yung box natin. So for each step, yung box natin ay nagro-rotate ng 90 degrees counterclockwise.
So for the first one, meron tayong itong line. Then, iniikot natin sya ng 90 degrees kaya sya naging ganito. Okay?
Pero, nagdagdag tayo ng isang line na mas maikli sa kanya. Okay? And then, para mapunta dito, iniikot ulit natin tong box na ito ng 90 degrees.
Okay? So, naging ganito. Pero, again, nagdagdag tayo ng line na mas maliit.
So, basically, meron tayong tatlong patterns na nakikita. Iniikot natin yung box, nagdagdagdag. tayo ng line and then yung line na dinadagdag natin ay mas maikli kaysa dun sa susunod. So, intuition would tell us that the next image in this pattern would look like this. So, una, inikot natin yung box.
So, kopyahin natin yung mga lines and then magdagdag tayo ng isang line na mas maikli kaysa dun sa previous one. Okay? Sige. Let's proceed. Okay, so here we have examples of numbers with patterns.
So, in the first one, medyo madali lang naman. 1, 3, 5, 7. Ano kaya yung next? So, para lang tayo nag-i-skip ng plus 2, ano? So, para makuha yung 3, nagdagdag tayo ng 2 kay 1. And then, para makuha si 5, nagdagdag tayo ng 2 ulit plus 2. So, intuition will tell.
tell us that yung susunod na number dito ay 9. Am I right? The second one is kind of tricky. So, kailangan nyong pag-isipang mabuti to kung paano yung susunod.
So, to make things short, Yung pattern dito ay more like squared. So, eto ay 1 squared or 1 times 1. Kaya nakuha sa 1. Eto naman ay 2 times 2, 3 times 3. Si 16 ay 4 times 4 or 4 squared. Si 25 ay 5 squared.
And then, wala mang yung susunod ay 6 squared which is 6 by 6 or 36. So yung next term sa series natin ay 36. Okay. Sige. So a few other patterns na na-observe natin ay yung pag-change ng phase ng ating moon.
So as you can see here in the first image, so yung pattern natin ay nagugulat. From a half moon to this one. And then we have a full moon crescent.
And then next we have the crescent. So, this one is a repeating pattern, ano? Yung moon cycle natin. Next one is yung ating season. So, sa Pilipinas, hindi natin ito gaano na-observe.
Kasi nga, wala naman. Hindi tayo nag-i-snow. Pero usually, ganito yung nangyayari.
So, at a point closest to the sun, nag-o-occur yung ating summer. Okay? Tama.
And then, at the point farthest, ay nag-o-occur. Occur yung ating winter. Okay?
And then, naguulit na yung pattern natin every year. Spring, summer, winter. Ulit-ulit lang sya. Ano? So, it is one of the patterns na na-observe ng mga tao even noong simula pa lang.
Okay? So, even when it comes to animals. So, sabi nila, yung patterns daw ng mga tigers and then ang ating mga hainas ay governed by... math.
So, pwede nating malaman kung ano magiging itsura gamit yung mathematics. Bakit? Paano natin nasabi yun?
Kasi itong mga patterns na to ay nafo-form dahil sa chemical reactions, okay? And then diffusion ng diffusion processes sa mga cells. Okay?
Sige. So, ang galing ng math, no? Parang halos lahat ng bagay sa paligid natin ay governed by mathematics.
Okay? Sige. Another good example of application of mathematics na makikita natin sa natures ay packing problems. So dito, sige define muna natin.
Packing problems involve finding the optimum method of filling up a space such as a cubic or a spherical container. So pupunuin natin yung isang lugar na wala tayong ganong nasasayang na space. So, ang example daw nitong packing problem na ito ay si honeybees. Okay?
So, ito daw, yung kung familiar kayo, ito daw ay isang packing problem solution na nagawa ng mga bubuyo. So, as you can see, again, meron pa rin tayong pattern dyan. So, yung formation daw na yan, minamaximize niya yung storage while reducing the amount of wax that the bees use. Okay?
So, maximum storage. Pero minimum amount of wax. Okay, so optimized sya.
Okay, and the next pattern na may kita natin sa mga flowers ay usually yung flowers natin ay ganito. So meron tayong mga flowers with 3 petals and then 5 petals. So sabi nila, ang pinaka-common daw na bilang ng petals ay yung 5 petals.
Okay, so observe nyo yung mga bulaklak nyo sa paligid. Most likely, pag merong bulaklak dyan, lima yung petals nya. Okay, yun lang naman. Okay, and then a beautiful pattern can be seen in the sunflower. So as you can see, meron tayong mga spiral, okay, na nag-rotate from counterclockwise.
So as you can see, meron tayong makikita mga pattern na pag ganito, okay. Pero on the other hand, Makakakita ka rin ng pattern na reverse na counterclockwise arcs naman sya. Okay, so napakagaling nung structure ng sunflower.
Okay, so yung arrangement daw ng sunflower seeds na ito ay minamaximize nya yung access nya sa light at saka sa neutral. So pag ganito daw yung naging ganito yung form nya para masulit yun, para lahat sila maximize yung amount ng araw na tumatama sa kanila and then yung nutrients na nakaabsorb nila. Okay. The next one is the snail shell. Okay, so kung hihiwain niyo sa kalahati yung shell ng isang snail, ganito yung magiging itsura niya.
And then it follows an equiangular spiral. Okay, from the root word, equal angles. Sir, ano po yung angles na equal? So, as distance from the center increases, the angles formed by the radai To the point and the tangent to the point remain constant.
What do I mean by this? Okay. So, if we connect a point from the center and then i-connect natin sya sa labas.
And then we will form an angle here. Itong angle na to ay equal din sa angle na ito. Okay. Kaya sya tinawag na equiangular.
Habang lumalaki yung spiral natin, yung angle na ito na napoform ay equal lang rin. So, an equiangular spiral sometimes is referred to as logarithmic spiral. Okay?
Now, we proceed with next topic which is symmetry. So, I think na-encounter nyo naman na ito nung bata kayo. So, symmetry is indicated that you can draw an imaginary line across an object and the resulting parts are mirror images of each other. So, we have an example here. So, itong first image ay asymmetric kasi kaya natin maglagay ng guhit straight line sa gitna nya.
And then yung left at saka yung left side ay mirror images. So, with the second example, hindi natin pwedeng gawin yun kasi magkaiba yung tsura ng left at sa right side. So, we call objects that are not symmetric, asymmetric. Pag hindi sya symmetric, ang tawag natin sa kanya ay asymmetric.
Okay, so ang type... na symmetry daw to ay tatawagin natin na bilateral symmetry kasi nahahati natin sya into 2. So ang meaning kasi ng by 2, nahahati natin sya into 2 mirror images. Okay. Next, okay, so here we have Vitruvian man. So pag hinat, this one is an example of a bilateral symmetry.
Kasi pag hinati natin sya sa ginah, so yung man daw ay symmetric. Okay, so kasi yung left at saka yung right side ay mirror images ng isa't isa. Okay, maliban na lang kung sir paano kung may nunal sya dito. E di, hindi na sya simetrika no. Okay, so isang example lang yan.
Okay, so we have other types of symmetry. So yung kanina bilateral, tatrong symmetry sya. Here we have rotational symmetry.
Okay, ano, sige. Consider natin itong flower na ito. So, pag hinati natin sya dito, symmetric yung left at right side. Pero notice, pag hinati natin sya dito, symmetric pa rin ito, saka ito.
Tama? And then, pag hinati natin sya dito, symmetric ito, tapos itong side na ito. Okay? Again, with the starfish, ganun din.
Marami syang lines of symmetry. Actually, five. Ito.
Okay? So, ang tawag natin sa symmetry na yan ay rotational symmetry. Ano? Kasi, rotation, meaning iikutin natin. Pag inikot natin itong sunflower ng some degrees, makukuha pa rin natin yung same na itsura.
Same goes with the starfish. Okay? So, rotational symmetry is the property a shape has when it looks the same after some rotation by a partial term. So, ang tanong na next ay, so ilang degrees po ba namin iikot yung object?
Para makuha namin yung angle of rotation natin. So here we have the formula, 360 divided by n, where the n is the order of rotation. Sir, paano po namin malalaman yung order of rotation?
As we go back here, malalaman natin yung order of rotation kung ilan yung lines na pwede natin ilagay. For example dito, meron tayong 1, 2, 3, 4, and 5. So meaning, Number of, ang n natin ay equal sa 5. Dito naman, ang n natin ay equal kay 3 kasi 1, 2, 3. So kung kukumpitin natin ang angle of rotation, etong unang image, ang angle of rotation niya ay 360 divided by 3 which is 120 degrees. Okay?
And then dito naman sa pangalawa, 360 divided by 5 is? 72 equal to 72 degrees So, yan yung tinatawag natin na angle of rotation, okay? Sige, so hinanap na natin itong unang dalawa Itong pangatlo, pwede nyo itry na hapin yung angle of rotation nya. Okay? Next, we have definition.
So, ito naman yung patterns na may kita natin sa numbers. Okay? So, a sequence is an ordered list of numbers called terms that may have repeated values.
The arrangement of these terms is set by a definite rule. So, sequence sya meaning in order. So, pag iniba mo yung order, hindi na siya sequence kasi magugulo yung rule na nag-govern sa kanya. For example, this one.
So, we have 1 and the next term is 10 and then 100 then 1000. So, what happens? Sir, nagdadagdag lang naman po ng isang 0. You can be right, pero since math yung subject natin, gawin naman natin siya mathematical. Sir, hindi naman tayo pwede magdagdag ng 0. Pag nagdagdag tayo ng 0. 1 plus 0 is still 0. So, what we're actually doing is multiply the first number by 10. And then, yung susunod by 10, by 10, by 10. Okay? So, by doing this, 1000 times 10, we get the next term to be 10,000.
How about this one? We have 2, 5, 9, 14, 20. Okay? So, another pattern that we can notice is ito. So, 2 naging 5, anong ginawa natin? Nag-plus tayo ng 3. 5 naging 9, nag-plus tayo ng 4. 9 naging 14, nag-plus tayo ng 5. 14 naging 20, nagdagdag tayo ng 6. So, para makuha yung next term, kailangan natin magdagdag ng 7. So, yung next term natin ay 27. Okay?
So, inotice nyo yun na hindi lang addition yung pattern natin. Meron din tayong subtraction, multiplication. It's up to you.
To figure things out. Okay. So, we have another example.
So, I think you can try this naman, no? So, try 16, 32, 64, 128. I think medyo obvious naman to. Nag-multiply lang tayo ng 2. Okay.
And then, yung second term, medyo complicated sya. Yung 1, 1, 2, 3, 5, 8. Ano kaya yung susunod? Okay. So, actually, yung pattern dito, sige, sabihin ko na. We have 1 and 1. pag inadd mo sya makukuha mo si 2 1 plus 2 makukuha mo si 3 2 plus 3 makukuha mo si 5 3 plus 5 8 5 times 8 5 plus 8 is 13 so our next term is 13 actually meron tayong tawag sa sequence na to etong sequence na to ay yung tinatawag natin na Fibonacci sequence ok so yung Fibonacci sequence daw ay nagmula sa pag nonotice ng pattern nito ok, so dito daw sya nang galing, so first sa unang buwan meron tayong pares ng rabbit itong pares ng rabbit na to, hindi muna sya manganak, tatanda muna sya so sa second month, meron pa rin tayong dalawang number dalawang rabbit, so isang pares so isang pares, isang pares sa pangatlong buwan, malaki na yung rabbit natin, so pwede na sya mag-anak Okay, so meron na tayong dalawang pares.
Sa pangatlong ban, itong matatanda na rabbit na ito, pwede ulit silang mag-anak. And then, meron pa rin tayo ng original na rabbit. Itong rabbit na ito, hindi pa manganak kasi bata pa lang.
So meron tayong three patterns. Continuing this one, makakakaroon tayo ng five pairs. And then, sa next one is eight, okay?
So dyan daw nang galing yung idea nung Fibonacci sequence, okay? Sige, so the Fibonacci sequence is a sequence named after Leonard of Pisa who has a nickname Fibonacci. Okay, so tatawagin natin na Fib N daw yung nth term ng ating Fibonacci sequence.
So Fib of N. So yung first one, Fib 1 equal to 1, Fib 2 ay 2, Fib 3 ay 2, Fib 4 is 3. Okay? So geometrically yung Fibonacci sequence pwede natin siyang isolat as a spiral.
So paano pong naging spiral yung Fibonacci sequence natin? Mag-start tayo sa 1 by 1 na square and then 1 by 1. Sa tabi nya magdagdag tayo ng 2 by 2 kasi yun yung susunod na term. And then we add a 3 by 3 square.
Wait lang, ibahin ko na lang yung color. So ito yung 1 by 1 na square, 1 by 1. Then 2 by 2, 3 by 3, 4 by 4. And then 5 by 5 na square, 8 by 8. 13 by 13 and then 21 by 21 So, pag nakakabuo tayo ng tinatawag natin na golden rectangle pwede naman natin iyan ituloy pero hindi nakasya sa screen natin and then notice that pag kinonnect natin yung mga intersections natin makakakuha tayo ng isang spiral yung spiral na yan ay yung tinatawag natin na golden spiral Okay. Okay. Okay.
Proceed tayo. So, yung Fibonacci sequence has interesting properties. Kung pinag-divide daw natin yung dalawang magkasunod na Fibonacci number, for example, 1 divided by 1, 1. 2 divided by 1, 2. 3 divided by 2, 1.5.
5 divided by 3, 1.667, etc. Titigil tayo sa 1.1618. Maski ituloy nyo yan, yung pattern na yan. Ang kakalabasan ay 1.618.
And then, meron tayong tawag sa number na yan. Ang tawag natin sa number na yan ay yung ating golden ratio. Okay. So, ito na yung last part na ating discussion. Yung mathematics para sa ating mundo or mathematics for our world.
Okay. So, saan ba natin pwedeng gamitin yung math? So, one obvious part of application of mathematics is pwede natin siyang gamitin sa ... pag-organize. So, sa iba't-ibang aspeto ng ating bagay.
Example nito ay, pwede tayong mag-gather ng data kung ano yung mga mabenta na produkto sa ating sari-sari store. Siyempre, kung alam natin na mabenta siya, adi yun yung bibiliin natin, yun yung i-restock natin. Pag hindi naman mabenta yung produkto na yun, babawasan natin yung bilang niya. So, math din ang gamit doon.
Another example is, ginagamit ng mga scientist yung mathematics para i-plot. yung daanan ng or migration patterns ng mga ibon. Okay?
Para makonserve yung populations nila. Next one is mathematics for prediction. So ginagamit natin sya para manghula. One very special application of mathematics for prediction is yung prediction ng ating mga bagyo. Okay?
So gamit ang math na pre-predict natin kung magiging gano'ng kalakas ang isang bagyo. And then kung ano yung daan na, tatahakin niya, okay? And then next one is yung mathematics for control, ano?
So, yung the observations of nature as well as their interactions and relationship could be more elegantly described by means of mathematics. So, actually everything around us can be described by mathematics, okay? So yung population, yung pag-grow ng population natin, pwede natin i-compute gamit yung math.
So yung pagdami ng COVID cases, pwede rin kinocompute din yan gamit yung mathematics. So very, napakaganda talaga ng, napakadami talaga ng applications ng mathematics. And then last but not the least is that the fact that mathematics is indispensable. What do we mean by indispensable? Hindi natin siya pwedeng alisin sa buhay natin.
Maski ano pang major mo, kung gusto mong maging chef, forester, maging engineer, kung gusto mong maging doktor, kung ano man ang pangarap mo sa buhay, kailangan mo ng mathematics. Maski pag gusto mong maging language teacher or English teacher, kailangan mo pa rin ng math. Siyempre, meron ka pa rin expenses, nagko-compute ka pa rin ng utang, sahot, salary deduction, and all of these things use mathematics. So, what? I'm trying to say is mathematics is an important part of our life in everything we do.
Okay? So, I hope you learned how to appreciate mathematics in this lecture. So, yung module natin ay i-upload ko na lang rin sa ating group. And then, at the end of it, you will find an assessment. Madali lang naman yung module assessment and then yung quiz natin.
More on patterns lang naman siya. Okay? So, thank you for listening.