infinity 2012

Volume 3, Issue 1 Official Publication of the Mathematics Teachers Association of the Philippines-Iloilo Chapter (MTAP-IC), Inc. November 2011- October 2012

Praeter Limites Eundum Est Going Beyond the Limits

www.mtapiloilo.blogspot.com MTAP-Iloilo Chapter [email protected]

∞

Themed “Coping with the Demands of the K-12 Curriculum,” the 4th Regional Convention of the Mathematics Teachers Association of the Philippines-Iloilo Chapter (MTAP-IC), Inc. has invited national experts to handle dialogues and discussions about the latest updates and trends that will equip teachers with the right tools for the new academic policy.

K-12, to note, is a flagship policy of the government that aims, among others, to have a

Liking MI (mathematical investigation) to the sharpening of a saw, Oquendo said, “we must not just routinely execute our task as teachers by just doing what is averagely required, but more importantly continue to look for better ways to make our strategies more pointed and sharper… and MI is one of the ways to do it.”

Teachers confab to zero in on K-12’s math curriculum

ISSN: 2244-3290

K-12 Quo vadis?

By Engr. Herman M. Lagon, PhD

MATHEMATICS TEACHERS and educators from all over Western Visayas will flock in Amigo Terrace Hotel on October 19-21, 2012 to talk about the new curriculum under the K-12 program of the P-Noy administration.

universal kindergarten and to add two more years in high school, apparently making students more skilled, competitive, and ready to face the challenges in the outside world.

Dr. Harold Cartagena, lead organizer of the event, said: “We will be focusing more on how should the teacher cope with the exigencies of the K-12 program. We will try to give our participants

Math investigation ‘every math teacher’s best tool’

NO LESS THAN the DepEd Regional Supervisor for High School Mathematics Jerry Oquendo appealed in the strongest possible way to the 86 Ilonggo participants of the MTAP-IC 2nd Regional Mathematical Investigation Seminar-Workshop to hone their skills “like a sharp knife” in mathematical investigation for it is “every mathematics teacher’s best tool in the classroom.”

“We need to be open and ready enough to master MI in order for us to use this in our classroom—so that students will see further how

elegant and exciting high school mathematics is. Hence, I congratulate you all, most especially the MTAP-IC (Mathematics Teachers Association of the Philippines-Iloilo Chapter) organizers, for this very successful seminar-workshop initiative to charge forward and face the educational frontier which is MI.”

Mathematical Investigation is an exploration of an open-ended mathematical situation. Through MI, students are free to choose an area of a situation for an in depth mathematical analysis. The claims or conjectures made are justified through a proof in written or oral form.

The three-day mathematical investigation seminar-workshop, held April 12, 13, and 21, 2012 at the Center for Teaching

Infinity, officially registered at last

ISSN 2244-3290For life, this is the

International Standard Serial Number (ISSN) of the official publication of the Mathematics Teachers Association of the Philippines-Iloilo Chapter (MTAP-IC), Inc.—The Infinity.

To note, a single ISSN uniquely identifies a title of a publication regardless of language or country in which published, without burden of a complex bibliographic description. As far as the National Library of the Philippines (NLP) is concerned, The Infinity is officially recognized and benchmarked among the formal papers in the country.

The Infinity editor in chief Dr. Herman Lagon said: “We are happy that this was realized. From now on, scholars, researchers, and librarians alike can now accurately cite

our official serial number, an integral component of the journal article citation used in investigations and even in regulating postal and copyright systems.”

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THINKING OUTSIDE THE BOX. Doing Mathematical Investigation is like maximizing all possible higher order thinking skills that a learner possesses—even thinking out of the box—in order to arrive at the most elegant proof of a conjecture. This sought-after MI expertise was not just learned, but earned by the participants in the 2nd Regional Mathematical Investigation Seminar-Workshop of MTAP-IC, Inc. held April 12-13, and 21, 2012 at the Center for Teaching Excellence building at West Visayas State University, La Paz, Iloilo City./The Infinity file photo

EARS-ON, EYES-ON, HANDS-ON. Participants in the MI seminar-workshop focus on the order of the day. Such kind of laser-like concentration is necessary in order to deliver what is expected from them—an excellent mathematical investigation masterpiece.

4Investigatingcracks on tableMath revolution

mth section of a line segment

Billiard ball triangles

Educational reform!?

Math for life!72

20What makes an international math competitor?

Dancing in the rhythm of mathematics5 9

Learningfrom the experts 13

Math teachers in MI action

The Infinity file photos

Triangles within a triangle

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NOVEMBER 2011 - OCTOBER 20122

CHRISTMAS OUTREACH. 30 teenagers in conflict with the law were given food, and school supplies last December 27, 2011 as part of the MTAP-IC, Inc. Christmas outreach program. The outgoing and incoming officers of the association distributed the said basic essentials personally to the beneficiaries of Balay Dalayunan at Zone II, Brgy. Bo. Obrero, Lapuz, Iloilo City. They also donated to the institution some water containers for their hygiene and sanitation concerns.

∞∞∞K-12 CONSULTATION. Last December 7, 2011, Dr. Herman Lagon, as deputized by the MTAP-IC, Inc. Board of Directors, participated in the K-12 Pre-Summit Conference in the Department of Education Training Center in Lahug, Cebu City. Representing the association, Lagon was able to cite in the said consultation concerns regarding mathmatics curriculum, most especially in the area of “progressive spiraling” syllabus of the new educational system. The said high-end dialogue was also participated in by school presidents, policy makers, deans, cabinet secretaries and other stakeholders coming from the Visayas Area and the Zamboanga Peninsula.

∞∞∞CAMP POSTPONED. Due to time constraints, the planned MTAP-IC, Inc. Math Camp for secondary teachers originally set August 18-19, 2012 in Iloilo National High School was shelved. Themed, “There’s more fun for mathematicians!” the supposed two-day, overnight affair is hoped to materialize next year.

∞∞∞NEW ACCOUNT. The organization has unanimously passed Resolution #7, series of 2012. Sponsored by Ms. Rosemarie Galvez and seconded by Prof. Alex Balsomo, the resolution changed the MTAP-IC, Inc. bank signatories—that is, the president (Dr. Alona M. Belarga) or the vice president (Dr. Herman Lagon), and the treasurer (Mrs. Rhodora A. Cartagena)—as the authorized signatories of MTAP-IC, Inc. savings account and savings account with automatic transfer facility at Banco de Oro-SM City Iloilo. It was processed and finalized June, 2012.

∞∞∞THEME CHOICE. The 4th Regional Convention of MTAP-IC, Inc. is themed “Coping with the demands of the K-12 mathematics curriculum.” In the July 8 meeting of the board at Grand Dame Hotel, the following are the other proposed themes: “Mathematics teachers sailing on the new Mathematics curriculum” and “Coping with the demands of the new Mathematics curriculum.”

∞∞∞MTAP REVIEWER. In an overnight session in Dao, Capiz early in December 2012, the MTAP-IC officers proposed and deliberated the different topics and questions (with answers) that will be included in the planned reviewer of mathematics problems for Grade 6 students. The organization hopes to publish this tailor-made instructional aid

“Since sad reality shows that Philippine mathematics education does not fare well as far as international and even national standards are concerned, there is a need for mathematics teachers to forge a sort of a ‘mathematics revolution’ in order to spark a pedagogical transformation necessary to regain the lost glory of mathematics education in the country.”

Sr. Coronel, MTAP founder and president since 1977, is just one of the many mathematics experts who graced the annual affair aptly themed “Revolutionizing Math Education.” The confab was also participated in by about 250 mathematics pre-service and in-service educators from about 60 schools in Western Visayas.

Also among the plenary speakers were Dr. Auxencia Limjap, chairperson of the De La Salle University Science Education Department and Dr. Diana Aure, Metrobank Most Outstanding Teacher Awardee from University of the Philippines-Visayas.

Sr. Coronel held a half-day workshop on problem solving and cooperative learning strategies. Dr. Limjap, on the other hand, facilitated a six-hour workshop on Understanding by Design while Dr. Aure discussed “what makes a mathematics teacher a winner?” respectively.

Ten more presenters also shared their educational research findings through parallel sessions. They were Dr. Harold Cartagena, Prof. Sybel Joy Labis, Prof. Michelle Callao,

next year.∞∞∞

STILL ONLINE. At present, the blogspot (www.mtapiloilo.blogspot.com), email ([email protected]) and facebook account (MTAP-Iloilo Chapter) of MTAP-IC, Inc. are all active online. In fact, the facebook group is now having 352 legitimate members. Questions and announcements are posted 24/7 in order to share, like, comment, or contact other mathematics enthusiasts and organization members in the most contemporary and fast way. It is maintained by MTAP-IC secretary Ms. Rosemarie Galvez.

∞∞∞BACK IN BAROTAC. West Visayas State University again held the Science and Math Enhancement Program last September 8-9, 2012 at Nueva Sevilla Elementary School in Barotac Viejo, Iloilo. Some MTAP-IC, Inc. officers and members also participated in the event.

∞∞∞FREE RESEARCH WORKSHOP. Three officers of MTAP-IC, Inc. who happens to be officers of the DOST-SEI scholars association in WVSU COE-GS (WDSA) initiated a Free Research Seminar-Workshop among basic education teachers in Guimaras. Last April 23-24, 2012. The two-day seminar, participated in by 60 teachers, was mainly organized by Ms. Rosemarie Galvez and assisted by Dr. Herman Lagon and Dr. Harold Cartagena, among others. Also some of the presenters are MTAP members and DOST-SEI scholars Edsel Llave and Dr. Myrna Libutaque. The training is themed, “Revolutionalizing mathematics and science education through action research and mathematical investigation.”

∞∞∞OUR SYMPATHIES. A valuable member of the mathematics teaching ranks in Iloilo passed away. A faculty member of University of San Agustin and an active member of MTAP-IC, Inc., Mr. Ananias Sustento, succumbed to the hands of the Lord last September 11, 2012. An organizational Death Aid amounting to PhP 2,000.00 was given to his family. Also, in a resolution made by the MTAP-IC BOD, another Death Aid was given to the family of Mrs. Rosalinda Sua-Palmani (sister of Board Member Engr. Ninfa Sotomil) who succumbed from chronic obstruction pulmonary disease (COPD). Mrs. Palmani passed away April 11, 2012 at the age of 73. The MTAP-IC BOD agreed to give another death aid to the family of the late Rosario Soriano Gutierrez, vda de Paspe, mother of former MTAP-IC vice president Sergio Paspe, Jr. last October 15, 2012.

∞∞∞SPONSORS. As of October15, MTAP-IC, Inc. has received sponsorships from media leader and philanthropist Rommel Ynion, Summerhouse (Midtown Hotel), Uygongco Foundation, Inc., Panorama Printing Press Inc., and the C & E Bookshop to defray the expenses for the 4th Regional Convention and the Inter-Tertiary Quiz Bee this October 19-21, 2011.

N E W S B L I T Z

Teachers’ org founder: Math Revolution a must

“THE MAIN purpose of mathematics is to enhance students’ reasoning and sense making abilities,” thus said by the bubbly 80-year-old mathematician Sister Iluminada Coronel, F.M.M., in the recently-concluded 3rd Regional Convention of the Mathematics Teachers Association of the Philippines-Iloilo Chapter (MTAP-IC), Inc. held October 28-29, 2011 at Iloilo Grand Hotel, Iloilo City.

Prof. Jessica Arsenal, Japanese researcher Tetsuhiro Takimoto, Prof. Josephine Lavilla, Dr. Wilhelm Cerbo, Prof. Alexander Balsomo, Dr. Helen Hofilena, and Prof. Amelia Navejas.

On the first day of the convention, the Second MTAP-IC Regional Inter-Tertiary Quiz Bee was also held with 59 participants coming from 22 schools representing all the provinces of Region VI. After solving 30 grueling mathematics problems, Ralp Joshua Sarrosa of University of the Philippines in the Visayas (UPV) ended up as champion. In close second was Keith Lester Mallorca of West Visayas State University (WVSU), narrowly followed by Vincent Gasataya of Central Philippine University (CPU). The rest of the top ten winners were: Arvin Escultero (4th, CPU), Abraham Porcal (5th, Western Visayas College of

Science and Technology), Mark Agustin (6th, WVSU), Michelle Olivares (7th, UPV), Allen Bibal (8th, UPV), Jevin Amago (9th, Colegio San Agustin-Bacolod), and Ryan Tercero (10th, Filamer Christian College).

The judges of the said competition were respected mathematicians Dr. Diana Aure, Dr. Sonia Formacion, and Dr. Pilar Arguelles.

Election of the seven new officers of the MTAP-IC Board of Directors to serve for the years 2012 and 2013 was also held. The following officers garnered the most votes: Dr. Heman Lagon of Ateneo de Iloilo, Ms. Rosemarie Galvez of USA, Prof. Balsomo of WVSU, Dr. Harold Cartagena of Iloilo Central Commercial High School, Dr. Alex Facinabao of USA, Prof. Rhodora Cartagena of USA, and Engr. Ramon

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NEWS

For her third year as head of one of the most active provincial math organizations in the country, Dr. Alona Belarga, the present Director of Instruction and Quality Assurance of West Visayas State University (WVSU) was re-elected as president while PhD in Science Education Major in Math graduate of WVSU Dr. Herman Lagon, who is presently teaching in Ateneo de Iloilo, was elected vice president of the organization that has about two thousand members all over Iloilo in its most recent three years of existence.

The other Taga-West who were also voted for office by its board of directors were Prof. Rosemarie Galvez (secretary) of University of San Agustin (USA) and Prof. Rhodora Cartagena (treasurer) of USA. Both are currently enrolled in the PhD in Science Education (Math) Program of the WVSU College of Education Graduate School. Engr. Ramon

Taga-west educators lead MTAP Iloilo orgIN A UNANIMOUS way, the new set of officers of the Mathematics Teachers Association of the Philippines Iloilo Chapter (MTAP-IC) for the fiscal year 2012 was elected January 29 at Summerhouse Hall, Iloilo City—mostly working in, enrolled at, or graduates of West Visayas State University (WVSU).

Jardiniano (auditor) of Western Institute of Technology (WIT) is the lone elected official who is not from the state university.

To complete the membership of the Executive Board are Dr. Wilhelm Cerbo of WVSU College of Arts and Science, Dean Alex Facinabao of USA, Engr. Ninfa Sotomil of WIT, Mrs. Ma. Aries Pastolero of Iloilo National High School Special Science Class (INHS-SSC), Ms. Catalina Reales of Maasin Central Elementary School, Engr. Roberto Neal Sobrejuanite of John B. Lacson Foundation Maritime University, Prof. Alfonso Maquelencia of USA, Prof. Alexander Balsomo of WVSU College of Arts and Science, Mr. Alex Jaruda of INHS, and Dr. Harold Cartagena of Iloilo Central Commercial High School.

Cerbo, Facinabao, Sotomil, Pastolero, Marquelencia, Balsomo, Jaruda, and Cartagena are all graduates of WVSU.

To note, the 15 members of the MTAP-IC board were elected in the MTAP-IC convention—eight of them last 2010, while the

other 7 in the 2011 convention October 29, 2011 held in Iloilo Grand Hotel.

For the past years, MTAP-IC has organized, among others, the Regional Math Camp in Maasin, Iloilo, the Math-Science Camp for elementary students in Barotac Viejo, Iloilo, the Regional Convention in Math Education at Iloilo Grand Hotel, the Regional Congress in Math Investigation (MI) at WVSU, and the launching of its first-ever Newsletter, the Infinity. It is currently working on its Reviewer in Math for Grade Six pupils.

“This year, we plan to continue our Infinity Newsletter, conduct MI Regional Congresses for Students and Teachers, Math Camp for Students, continue to support the MTAP DepEd Saturday classes, hold Math Quiz for Tertiary, publish a Math Reviewer, and hopefully launch our MTAP Journal, among others,” said Belarga./The Infinity

STILL IN COMMAND. MTAP founder and long-time president Sister Iluminada Coronel, FMM, shares her insights about mathematics education to the participants of the 3rd Regional Convention at Iloilo Grand Hotel last October 28, 2011. Her four decades of experience in teaching mathematics has given her the chutzpah to conclude that we must wage a “math revolution” in order to radically enhance students’ “reasoning and making sense” skills./Infinity file photo

NOVEMBER 2011 - OCTOBER 2012 3

linking Algebra and GeometryTriangles within a triangle:

A Winning (First Place; Best MI; Best Presenter; Best Poster) Mathematical Investigation presented April 21, 2012 during the 2nd Mathematical Investigation Congress, Center for Teaching Excellence Building, West Visayas State University, La Paz, Iloilo City

By Christina E. Carsula (Passi Montessori International School), Kim Jay C. Encio (PAREF – Westbridge School, Inc.), Nezel J. Francisco (Colegio de San Jose), Rutchell L. Gania (Passi Montessori International School ), Analie B. Guion (Buntatala National High School), Stephen Raymund T. Jinon (PAREF – Westbridge School, Inc.),

Jenever F. Nievares (PAREF – Westbridge School, Inc.), and Lowell N. Rublico (PAREF – Westbridge School, Inc.)

I. IntroductionThis mathematical investigation is

a collaborative effort and attempt of the investigators in relating Geometry and Algebra in the study of equilateral triangles. By means of this mathematical investigation, mathematical curiosity is satisfied, leading to learning and addition of existing mathematical literature which proved true of the characteristics of Mathematics – growing and dynamic.

SituationSuppose we will draw some equilateral

triangle with side n units partitioned evenly by n – 1 points per side and let us connect two points on any two sides such that the line connecting these points is parallel to the third side. Let us note the number of upright and inverted triangles using the legend/definition below:

Upright Triangle – triangle “pointing up”Inverted Triangle – triangle “pointing down”

II. Statement of the ProblemThe investigators would like to determine

the number of upright and inverted triangles with respect to the n units side of an equilateral triangle partitioned evenly by n – 1 points per side, such that two points on any two sides when connected by a line is parallel to the third side.

Specifically, it sought to answer the following questions:1. What relation or function describes the number of upright triangles, U(n), of an

FEATURE

TEAM WORK. Group leader Kim Jay Encio explains to his colleagues his proposed proof to one of their conjectures in their “Triangles within a triangle” mathematical investigation./The Infinity file photo

equilateral triangle of side n units partitioned evenly by n – 1 points per side, when two points of any two sides are connected by a line parallel to the third side?2. What relation or function describes the number of inverted triangles, I(n), of an equilateral triangle of side n units partitioned evenly by n – 1 points per side, when two points of any two sides are connected by a line parallel to the third side?

III. Data – Gathering and ConjecturesThe table on the next page summarizes the number of upright triangles of all sizes in each of the following equilateral triangles of side n units partitioned evenly by n – 1 points per side, provided that the line connecting any two points is parallel to the third side:

Taking the differences in n’s and U(n)’s, we have

It is distinctively clear from the diagram that equal first differences in n’s resulted to equal third differences in U(n)’s. Therefore, the relation or function describing the number of upright triangles of an equilateral triangle of side n units where each side is partitioned by n – 1 points, provided that the line connecting any two points is parallel to the third side, must be a function of the third degree/ cubic function which takes the form

U(n) = An3 + Bn2 + Cn +D,where A, B, C and D ℜ∈ . Since there are four arbitrary constants in the said form (i.e. A, B, C, and D), in this case we will arbitrarily pick (1, 1), (2, 4), (3, 10) and (4, 20). Substituting each of these points to the form

U(n) = An3 + Bn2 + Cn +D,we will have,

1 = A (1)3 + B (1)2 + C (1) +DOr

1 = A + B + C +D (equation1)4 = A (2)3 + B (2)2 + C (2) +D

Or 4 = 8A + 4B + 2C +D (equation 2)

10 = A (3)3 + B (3)2 + C (3) +DOr

10 = 27A + 9B + 3C +D (equation 3)20 = A (4)3 + B (4)2 + C (4) +D

Or 20 = 64A + 16B + 4C +D (equation 4)

Exp ress ing t he f ou r equa t i ons in its augmented matrix form, we have

P e r f o r m i n g p e r t i n e n t elementary row operations, we have

Thus D = 0.

Substituting D = 0 to 6C + 11D = 2, we have 6C + 11(0) = 2 6C = 2 C = 1/3Substituting C = 1/3 and D = 0 to 4B + 6C + 7D = 4, we have 4B + 6 (1/3) + 7(0) = 4 4B + 2 = 4 4B = 2 B = 1/2Substituting B = 1/2, C = 1/3 and D = 0 to A + B + C + D = 1, we have A + 1/2 + 1/3 + 0 = 1 A + 5/6 = 1 A = 1/6Substituting these values to the form U(n) = An3 + Bn2 + Cn +D, we have

U(n) = 1/6 n3 + 1/2 n2 + 1/3 n

or in complete factored form, we haveU(n) = 1/6 n (n + 1) (n + 2).

Thus, we arrived to our first conjecture as follows:

Conjecture 1The number of upright triangles in an equilateral triangle of side n units partitioned evenly by n – 1 points per side, provided that the line connecting any two points is parallel to the third side, is defined by

U(n) = 1/6 n (n + 1) (n + 2), .

Testing the formula for some known cases…For n = 1 U(n) = 1/6 n (n + 1) (n + 2) U(1) = 1/6 (1) (1 + 1) (1 + 2) U(1) = 1/6 (1) (2) (3) U(1) = 1For n = 2 U(n) = 1/6 n (n + 1) (n + 2) U(2) = 1/6 (2) (2 + 1) (2 + 2) U(2) = 1/6 (2) (3) (4) U(2) = 4 For n = 3 U(n) = 1/6 n (n + 1) (n + 2) U(3) = 1/6 (3) (3 + 1) (3 + 2) U(3) = 1/6 (3) (4) (5) U(3) = 10

The results of actual substitution to the formula match with known existing data for n = 1, 2, 3. Thus, the formula could possibly model the number of upright triangles in an equilateral triangle of side n units partitioned by n – 1 points per side, provided that the line connecting two points is parallel to the third side.

The following table summarizes the number of inverted triangles of all sizes in each of the following equilateral triangles of side n units partitioned evenly by n – 1 points per side, provided that the line connecting two points is parallel to the third side:

Taking the differences in n’s and I(n)’s, we have…

As what we could notice, equal first differences in n’s did not resulted to equal [third] differences in I(n)’s. But one can notice alternating 1, 0, 1, 0. In this case, we can still find a relation or function by partitioning the odd and even-numbered dimensions of an equilateral triangle partitioned by n – 1 points per side, provided that the line connecting two points is parallel to the third side.

For an equilateral triangle of side n units, where n , the table of values is as follows:

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I. STATEMENT OF THE PROBLEM While cooking in the kitchen, I found an

evidence of poor construction materials – cracked tiles. My father would surely worry about that problem. I, on the other hand, worried about the mathematical side of the problem, the heart of this investigation.

Consider a rectangular table covered with square tiles, which have cracked along a diagonal of the rectangle. Investigate (Bastow, Hughes, Kissane& Mortlock, 1984).

Investigating cracks on tableBy Ms. Rosemarie G. Galvez

University of San Agustin

II. CONJECTURESConjectures were formed through

preliminary skirmishing and systematic exploration. The construction of the diagrams should also be done systematically to see the

patterns that will beautifully come out of the illustrations.

I observed that the diagonal intersects the segments inside the rectangle. Furthermore, there are cases when the diagonal contains the points of intersection of the segments. But, what conditions should be met for the diagonal to intersect a particular number of segments or to contain the points of intersection? With this question as the focus of this investigation, I defined the following variables.

Let r =number of rowsc =number of columns s =the number of segments in the interior

of rectangle intersected by the diagonal (Note that whenever the diagonal passes through an intersection of segments, then it intersects two segments.)

v=the number of vertices or lattice points (may be viewed as the intersections of segments, the corners of tiles, or the vertices of the small squares) intersected by the diagonal in the interior of the rectangle. The vertices of the rectangular

table intersected by the diagonals are no longer counted. By counting the number of intersections in the figures, I came up with the following tables summarizing my observations. Table 1

I set r as constant. So, I just looked at the relationship of c and s . It is quite obvious that the relationship is linear.

For s=1 and c = 2,

Hence, the linear equation is s = c - 1Table2

The slope is

For s=2 and c = 2,

Hence, the linear equation is s = c+0

Table 3

The slope is

For s=2 and c=1, 2=(1)(1)+b3 b3=1

Hence, the linear equation is s=c+1

Table 4

r 4 4 4 4 4c 1 2 3 4 5s 3 4 5 6 7

The slope is

For s=3 and c=1,


Table 5

r 5 5 5 5 5c 1 2 3 4 5s 4 5 6 7 8

The slope is

For s=5 and c=2,


Observe that the linear equations have all slopes of 1, but the intercepts are increasing. Again, the linear equations are: s=c-1 f or r = 1 s=c+0 f or r = 2 s=c+1 f or r = 3 s=c+2 f or r = 4 s=c+3 f or r = 5

We could form relationship for the s-intercepts, bi, and the number of rows, r.

bi -1 0 1 2 3 r 1 2 3 4 5

Clearly, bi = r – 2.

We know that the slope is 1 and bi = r – 2. By substitution in the linear equation s=m1c+b1, we obtain

s=c+r–2

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GALVEZ. Reflecting on reflections/The Infinity file photo

Hence, the first conjecture is: On rectangular table covered by square

tiles, the diagonal would intersect segments determined by the tiles. The number segments intersected by the diagonal in the interior of the rectangle is determined by s=c+r–2, where r=number of rows and c=number of columns.

I also investigated the number of lattice points in the interior of the rectangle, contained by the diagonal.

r 2 2 3 4 4 5

c 2 4 3 2 4 5

v 1 1 2 1 3 4Here I observed that the number of

lattice points will be greater than zero if r is a multiple of c or vice versa. Of course the number of vertices intersected is greater than 0 also when r = c . It was observed that

v = r – 1I decided to have another illustration, say

for r = 6 . Six has more factors, so patterns here could be seen. The following is the table for r=6.

r 6 6 6 6 6 6

c 1 2 3 4 5 6

v 0 1 2 1 0 5

From this table, I observed that the equation v = r–1 does not hold anymore. Instead, the following equation holds.

v=gcf(r,c)-1

Here are the two conjectures I made based on the patterns that I observed. 1.) For r x c rectangular table covered with square tiles, the diagonal intersects

s = c+r–2segments in the interior of the rectangle, where:

r = number of rowsc =number of columnss =number of segments intersected

by the diagonal in the interior of the rectangle

2.) For r x c rectangular table covered with square tiles, the diagonal intersects v=gcf(r,c)-1

vertices, where: v =number of vertices or lattice

points intersected by the diagonal in the interior of the rectangle

gcf =greatest common factor.

III. VERIFYING CONJECTURESA. Using rectangular table By substitution, s=c+r–2 s=7+7–2 s=12

The diagonal intersects 12 segments. By counting in the figure, this is true.

By using a similar diagram, we could verify conjecture 2. If we substitute in the formula, we get v=gcf(r,c)-1 v = gcf (7,7) – 1 v = 7 – 1=6

Verifying by using the figure tells us the formula is true for this case.

B. Using 8 x 10 rectangular table

By substitution, s = c + r – 2 s = 10 + 8 – 2 s = 16

Indeed, the figure shows that the diagonal intersects 16 segments.

Verifying the number of vertices intersected of the same figure, we have

v = gcf (r,c) – 1v = gcf (8,10) – 1

v = 2 –1 = 1Correct again!

C. Using rectangular table

The number of segments intersected is s = c + r – 2 s = 7 + 6 – 2 s = 11

Wow! I got it right again. Try counting in the figure. The number of vertices intersected is v = gcf (r,c) – 1 v = gcf (7,6) – 1 v = 1 –1 = 0

Obviously, the figure shows that the diagonal intersects no vertex in the interior of the rectangle.

I verified the two conjectures by using three cases. Observe that in the first case, the dimensions are all odd. The second case has both dimensions even. The third case, on the other hand, has one even dimension while the other is odd. The cases, I believe, are well-chosen to somehow represent the infinite possible rectangular tables that could be constructed.

Nevertheless, the verified conjectures still need to be proven.

V. JUSTIFICATIONA rectangular figure covered with

square tiles may be represented by a rectangle drawn on a Cartesian plane with one side on the x-axis and another side on the y-axis. With c representing the number of columns of squares and r representing the number of rows of squares, the vertices of the rectangle

FEATURE


Oftentimes, teachers worry what con-cept to teach to their students. They do not only know that the answer lies within the students. So, it is just a matter of looking what the students know and use whatever concepts they have in their store knowledge. The kinds of experience teachers provide clearly play a major role in determining the extent and quality of students’ learning.

Students’ understanding of mathemati-cal ideas can be built throughout their years if they actively engage in tasks and experi-ences designed to deepen and connect their knowledge. In doing this, the teacher makes use of the students’ own concept about the subject and thus provides the learners with meaningful learning. Therefore, the teach-er’s proper use of diagnosis is very helpful to determine what students have and what stu-dents need to learn. Moreover, looking into the students’ minds will help teachers design materials appropriate for bringing out essen-tial ideas or concepts from the students.

Students exhibit different talents, abili-ties, achievements, needs and interests in mathematics, teaching, therefore, should give students the opportunities to learn im-portant mathematics under the guidance of competent and committed teacher. With the teacher’s guidance, students can thus begin to develop the ability to articulate using his prior knowledge. Students’ skills in visualiz-ing and reasoning about mathematical con-cepts are likewise developed.

At the beginning of the 21st Century, mathematics for both the elementary school and high school has been undergoing two major changes. The first one is in teaching where one moves from routine exercises and memorized algorithms toward creative solutions to conventional problems. The second one consists in spreading problem solving: culture throughout the world. Most international math competition that Filipino kids participated in for the past seventeen years reflects both trends. It ranges from the essay-type to non-routine and open-ended problems.

On the other hand, it is important for us to know the difference between an exer-cise and a problem so that we can have a better understanding of their functions. Our definition of an exercise is you know imme-diately how to complete the given item just by looking at it. It is just a question of doing the work. Whereas a problem, we mean a more intricate question for which at first one has probably no clue on how to approach it. But by perseverance and inspired effort, one can transform it into a sequence of exercis-es. Those are important elements that really matter in any international competitions. And in my presentation today, we chose mainly the latter because they are beautiful, inter-esting, fun to solve, and they best reflect mathematical ingenuity and elegant argu-ments.

Mathematics Competitions are well es-tablished and popular in our country. Proof of this is the growing number of academic contests in the region, such as Philippine Invitational Mathematics Examination of the MTG, the Math Challenge of the Metro-bank–Department of Education–Mathemat-ics Teachers Association of the Philippines, and the Philippine Mathematics Olympiad. Math competitions aim to (a) enhance stu-dents’ competence and interest in solving mathematics problems with varying levels of difficulty and (b) challenge their mathemat-ics skills and ingenuity. Math tournaments set a high standard of qualification for aspir-ing competitors. For local competitions, such as those mentioned above, participants go through a process of elimination, usually in two or more stages, which increases in dif-ficulty at every stage. Those who excel in all the elimination rounds are eligible for na-tional contests and become contenders for international events.

To give students an edge in math learn-

What makes an international math competitor? A Philippine Setting

Dr. Simon L. ChuaPresident, Mathematics Trainers’ Guild (MTG), Philippines

ing, especially in high pressure competitions, they must be provided with extensive train-ing to develop their mathematical abilities.

Mathematical AbilitiesV. A. Krutetskii, a Russian psycholo-

gist, was the first to define the parameters of mathematical abilities. These parameters were then revised by the famous Russian mathematician, A. N. Kolmogorov and finally by the Mathematical Educational Committee and the Mathematics Committee of the US Research Association. Together they identi-fied the components of mathematical abili-ties as follow:

1. Ability to perform appropriate mental calculation and mathematical operations as well as effective prediction by using numbers clues and other signs

2. Ability to practice logical reasoning 3. Ability to shorten the process of

reasoning 4. Ability to record mathematical

generalization, figuration and logical mode to memory

5. Ability to form spatial conceptsA report from the Special Committee of

World Federation of National Mathematics Competition states that high school students who are being considered as participants in mathematical contests must possess the fol-lowing abilities:

1. Observative Ability – the ability to recognize quickly the “number” or “figure” represented by an object and connects this to a mathematical figure and relation. In most international mathematics competition, this ability is demonstrated by individuals who are able to:(a) find out the structural

features and interrelations of mathematical relations

(b) recognize special figures and relations from a geometric figure.

2. Associative Ability – based on the mathematical concept of association as the process of forming connections between relevant ideas and/or knowledge.

3. Computational Ability – the ability to (a) memorize the definitions, formu-las, and rules of operation; (b) simpli-fy an operational process; (c) reverse a computational process and be able to check it; (d) predict and estimate values; and (e) recognize recurrence and induction.

4. Abstractive Summary Ability – This math ability requires students to summarize a particular problem, generate an abstract conclusion through analysis and synthesis, and then apply the conclusion to the specific problem.5. Ability of logical reasoning – This ability is the core of mathematical abilities and includes the following:

(a) Understanding and mastery of the relationship among formulas, principles, theorems, and axioms in a conceptual system.

(b) Mastery of relevant logical knowledge such as sufficient condition and necessary condition, inductive reasoning, deductive reasoning, and analogical reasoning.

(c) Mastery of commonly used mathematical methods such as analytical, synthetic, inductive, and reductive

methods.(d) Capacity to think in a concise

manner by simplifying the reasoning process.

6. The ability of writing and expressing oneself – the ability to express ideas clearly and accurately such as in the presentation of solutions to problems.

With these important inputs, we can have a better perception of our priority with regard to our focus in uplifting mathematics education. As always, I make a special emphasis that mathematics contents are worth the time and attention of the students. Mathematics topics can be considered important for different reasons, such as their utility in developing other mathematical ideas, in linking different areas in mathematics, or in deepening students’ appreciation of mathematics as a discipline and as human creation. Ideas may also merit curricular focus because they are useful in representing and solving problems within or outside mathematics.

Careful analysis and consideration allow us to arrive at a conclusion that the classroom, the lesson, and the teacher really matter at making a student an international math competitor. With this thought in mind, every teacher must realize that chances of our students to do better in any mathematics competition largely depend in his/her hands. This is why foundational mathematical concepts and contents should have a prominent place in the teaching of mathematics. The improvement of mathematics education for all students requires effective mathematics teaching in all classrooms.

In closing, it is a wonderful privilege to be in the classroom of dedicated mathematics teachers where enthusiasm, curiosity and strategies of young children are valued and built upon, with lasting effects upon their understanding, their attitudes, their love of mathematics, and their confident views of themselves as learners of mathematics. Indeed, it is still the effective mathematics teachers that make a difference. Compilation of International Math Contest Problems Problem 1 Refer to the diagram below. The hypotenuse of the given right triangle is 6 cm. Determine area of this right triangle without Using Trigonometry.

Problem 2 Given three squares with sides equal to 2 units, 3 units, and 6 units, perform only two cuts and reassemble the resulting 5 pieces into a square whose side is equal to 7 units. Note: By a cut we understand a polygonal

line that decomposes a polygon into two connected pieces.

Problem 3Can you work out which three of the shapes at the bottom can be joined together to make this pyramid shape? None of the pieces can be rotated or turned over.

Problem 4Can you work out which of the shapes at the right can be joined together to make the larger shape at the left? None of the pieces can be rotated or turned over, and no shape can be used more than once.

Problem 5

2013 x 2012 2012 2012 – 2012 x 2013 2013 2013Problem 6The number 142857 is known as a 6-digit Cyclic Number because the first 6 multiples of this number will still be the same 6-digit number in different sequential order.The results are:

142857 × 2 = 285714

142857 × 3 = 428571

142857 × 4 = 571428

142857 × 5 = 714285

142857 × 6 = 857142We can illustrate this concept with the diagram above.Using the 6-digit number above as an example, find a 16-digit Cyclic Number.Problem 7The vertices of a square are connected to the midpoint of another side, as shown in the figure, in the process forming a smaller square. The area of the smaller square is what fractional part of the area of the given square?

Problem 8Given an equilateral triangle with its inscribed and circumscribed circles, what is the ratio of the area of the larger circle to the area of the smaller circle?

Problem 9

Determine all of the digits represented by X in the long division and also determine the remaining four digits of the five-digit answer of which 8 is the third digit, as shown in figure above./The Infinity

∞∞∞Dr. Chua is the president and co-founder of the Mathematics Trainers Guild (MTG), Philippines. This article is part of his talk on “Upgrading mathematics standards for K-12: building teachers skills in training students for local, national, and international mathematics competitions” in the 4th Regional Convention of MTAP-IC, Inc. on October 19, 2012 at Amigo Terrace Hotel.—The Infinity Editor

ACCORDING TO Gelman and Gallistel (1978), children learn many mathematical ideas quite naturally even before they enter school. If we only hold on to this principle, we would relate our lessons to what students actually have and what they know.

FEATURE


mth section of a line segment cut into n equal divisions

By Mr. Matthew T. LasapAteneo de Iloilo - Santa Maria Catholic School

to page 15

SNAKE STYLE, “How can I find for the coordinates of the mth section of a line segment with n equal divisions?” Thus was asked by Mr. Lasap when he was preparing for his Analytical Geometry class in Ateneo de Iloilo. His “serpentile” proof says it all.

Introduction:As I was preparing for my lesson on

Analytic Geometry, I saw the topic “Division of a Line Segment”. I know that midpoints will be a part of the topic, but what made me worry are the points of trisection of a line segment. I have never gone through the formula in finding for this. Thus, I tried to derive it. After coming up with such formula, I was so amazed with it that I began asking myself, “How can I find for the coordinates of the mth section of a line segment with n equal divisions?” and thus, this investigation.

For the benefit of the readers, sections as used in this investigation refer to points dividing the segment into equal parts.

Problem:What are the coordinates of the mth

section of a line segment cut into n equal divisions?

Investigation/Derivation:

Let the point ( ),m mM h k be the mth section of a line segment, whose endpoints

are at ( )1 1,A x y and ( )2 2,B x y , with n divisions.

To express ( ),m mM h k in terms of the coordinates of the endpoints of the segment, it should be noted that the horizontal

distance of M from A is ( )2 1m x xn

− and its

vertical distance is ( )2 1m y yn

− . [Sides of

similar triangles are proportional.]

Hence, ( ),m mh k is the same as

[ ] [ ]2 1 2 11 1,

m x x m y yx y

n n − −

+ +

2 1 2 11 1,mx mx my myx y

n n− − + +

2 1 1 2 1 1,mx mx nx my my nyn n

− + − +

[ ] [ ]2 1 2 1,mx n m x my n m y

n n + − + −

Conjecture:A line segment, whose endpoints are

at ( )1 1,x y and ( )2 2,x y , cut into n equal divisions has an mth section at

[ ] [ ]2 1 2 1,mx n m x my n m y

n n + − + −

where n m> where .

Justification:

Let AB be line segment joining

arbitrary points ( )1 1,A x y and

( )2 2,B x y and with length ABd in an

coordinate plane. Suppose ( ),m mh k or

[ ] [ ]2 1 2 1,mx n m x my n m y

n n + − + −

is the

mth section when the line segment is cut into n equal divisions. Clearly, n m> where

.The conjecture can be proved by

showing that, a) The distance from point A to point

M is mn

of the distance between A and B,

i.e. . This will be the aim of Part 1.

b) M is a point on AB . Part 2 and Part 3 focuses on this.

This is outlined by the following illustration,

Figure 2. Paradigm of the Proof

Part 1. AM ABmd dn

=

Clearly, the length AB of is

( ) ( )2 22 1 2 1ABd x x y y= − + − .

The distance from A to M should be mn

of this distance. So, using the distance

formula,

[ ] [ ]2 22 1 2 1

1 1AMmx n m x my n m y

d x yn n=

+ − + −− + −

2 22 1 1 2 1 1

1 1AMmx nx mx my ny myd x y

n n=+ − + − − + −

2 22 1 1 1 2 1 1 1

AMmx mx nx nx my my ny nyd

n n=− + − − + − +

( ) ( )2 22 1 2 12

2

1 1AMd mx mx my my

n n= − + −

( ) ( )2 2

2 22 1 2 12 2AM

m md x x y yn n= − + −

( ) ( )2 22 1 2 1AM

md x x y yn= − + −

Hence, AM ABmd dn

=

It has been shown that M has the required distance from A. However, it is not the only point satisfying this distance from A. All the points on the circle with A as its center has this distance from A. Thus, it has to be shown that M is on the segment. This

would require that the line containing AB contains the point M.

Part 2. ( ),m mM h k is a Point on The equation of the line containing the

segment is

2 1 1

2 1 1

y y y yx x x x− −

=− −

. ( ),m mM h k should

satisfy this equation. So,[ ]

[ ]

2 11

2 1

2 12 11

my n m yyy y n

mx n m xx xx

n

+ −−−

=+ −−

−

2 1 1 1

2 1

2 1 1 12 1

my ny my nyy y n

mx nx mx nxx xn

+ − −−

=+ − −−

2 1

2 1

2 12 1

my myy y n

mx mxx xn

−−

=−−

2 1 2 1

2 1 2 1

y y y yx x x x− −

=− −

Hence, ( ),m mx y is on the line containing the segment.

Having shown that M is on the line is not sufficient to prove that M is on the segment. This is because there are two points on this line with the specified distance from A. Hence, it is imperative to show that M is on

AM , i.e. by proving that BM BAd d< . This will, finally, identify a unique point (M) on the segment with the necessary distance from A.

Part 3. ( ),m mM h k is a Point ABThis can be shown by establishing that

BM BAd d< . So, using the distance formula we have,

[ ] [ ]2 22 1 2 1

2 2BMmx n m x my n m y

d x yn n

+ − + −= − + −

2 22 1 1 2 1 1

2 2BMmx nx mx my ny myd x y

n n+ − + − = − + −

( ) ( )2 22 1 1 2 2 1 1 22 2

1 1BMd mx nx mx nx my ny my ny

n n= + − − + + − −

[ ] [ ]( ) [ ] [ ]( )2 21 2 1 2

1BMd n m x n m x n m y n m y

n= − − − + − − −

[ ] ( ) ( )2 2 21 2 1 2

1BMd n m x x y y

n = − − + −

( ) ( )2 21 2 1 2BM

n md x x y yn−

= − + −

Since 0n mn−

< , it can be concluded

that the alleged section is on the line

segment.

It was shown that ( ),m mx y or

[ ] [ ]2 1 2 1,mx n m x my n m y

n n + − + −

has a

distance mn

of the length of the segment

and that it is on the line segment.

∴ [ ] [ ]2 1 2 1,

mx n m x my n m yn n

+ − + −

is the mth section of the segment, whose

endpoints are at ( )1 1,x y and ( )2 2,x y , cut into n divisions.

Application:This investigation aimed primarily in

finding for the coordinates of any section of a segment cut in any number of equal divisions.

It should be noted that the above conjecture is a generalized formula. The midpoint formula can be derived by setting

2n = (since the segment is cut into two equal parts) and 1m = (the midpoint is the first and only section when a segment is bisected). Hence,

( ) [ ] ( ) [ ]2 1 2 11 2 1 1 2 1,

2 2x x y y + − + −

2 1 2 1,2 2

x x y y+ +

and, thus, the midpoint formula.

A sample problem on point of trisection is given below.

1. What are the coordinates of the first and the second points of trisection of the

segment with endpoints at ( )15, 4− and

( )6,8 ? Solution:

Let, 1 1 2 215, 4, 6, 8x y x y= = − = =Trisecting a segment means to divide it

into 3 equal parts, thus, 3n = .

To solve for the first trisection, we let 1m = . Substituting the values,

[ ] [ ]2 1 2 1,mx n m x my n m y

n n + − + −

( )( ) [ ]( ) ( )( ) [ ]( )1 6 3 1 15 1 8 3 1 4,

3 3 + − + − −

6 30 8 8,3 3+ −

( )12,0

FEATURE


IntroductionNowadays, the billiard world is beginning

to emerge as a favorite sport of many. Many enthusiasts come to witness every billiard tournament happening in the world. Billiard in fact has nourished the hunger for leisure but what many are unaware of is that playing billiards requires skill. There are several types of equipment used in billiards. One mathematical aspect of this sport has been highlighted in the study. This research will be dealing with the technical side of the game. The triangle has been given importance in this investigation. The sport billiards has been adopted in this study, however, the scope extends all the way to situations including significantly larger and more balls as opposed to the norms of the billiard world.

Billiard Ball Triangle (Rack)A rack is the name given to a frame (usually

wood, plastic, or metal) used to organize billiard balls at the beginning of a game. Rack may also be used as a verb to describe the act of setting billiard balls in starting position in billiards games that make use of racks (usually, but not always, using a physical rack), as well as a word to describe the balls in that starting position.

The most common shape of a physical rack is that of a triangle, with the ball pattern of 5-4-3-2-1. Racks are sometimes called simply “triangles” (most often by amateur shooters) based on the predominance of this form. Triangular-shaped racks are used for eight-ball, straight ball, one-pocket, bank pool, snooker and many other games. (wikipedia.org)

Several possible mathematical situations arise from the concept of billiard triangles. One such situation would be finding the perimeter of the triangle when a certain number of layers of billiard ball triangles are given. One should also consider increasing the radius of each ball and its relationship to the triangle’s perimeter. Another common situation that arises from this concept would be the area of the triangle with respect to the number of layers of billiard balls and the radius of each ball. In the above said situations comes the heart of this investigation – the generation of a general formula for each situation to fit the stated conditions (number of layers of billiard balls and the radius of each ball).

This study focused mainly on generating formulas to determine the two major geometric data that can be calculated from the billiard ball triangle with respect to the number of layers of billiard balls, and the radius of each ball.

Situation

Make a billiard ball triangle of varying layers and billiard ball radii. Investigate its geometric properties.

Statement of the ProblemThis section describes the purpose in

conducting the study, and enumerates the specific objectives of the research.

Generally, this study aimed to determine the new formula of determining the area, perimeter, and total number of billiard balls in a billiard ball triangle setup given the number of layers of the billiard ball set and the radius of each ball.

Specifically, this study sought to answer the following: 1. What is the formula to determine the

perimeter of a billiard ball triangle to be constructed to enclose billiard balls given a specific number of layers of billiard balls and the radius of each billiard ball?

2. What is the formula to determine the total number of billiard balls in the billiard ball triangle from the first layer to a specific layer given the specified layer of billiard balls?

3. What is the formula to determine the area of a billiard ball triangle to be constructed to enclose billiard balls given a specific number of layers of billiard balls and the radius of each billiard ball?

ConjecturesThis section shows the preceding data where

the formula was derived. It also shows, step by step, how it was formulated, and states the conjecture from it.

Billiard ball trianglesWinning entry in the 2011-2012, Regional Science and Technology Fair, Mathematical Investigation, Individual Student Category

Researcher: Mr. Rey Philip J. Gallos, Advisers: Mrs. Portia J. Estorque and Mr. Julio J. VillalonIloilo National High School – Special Science Class, Luna St., La Paz, Iloilo City

The following is a table of the perimeter of the billiard ball triangles given x as the number of layers of billiard balls and r as the radius of each ball.

Based on the given table, conjecture 1 was formulated.

Conjecture 1The following formula is used to determine

the perimeter of the billiard ball triangle given x as the number of layers of billiard balls, and r as the radius of each ball: P = 2r[3(x-1)+π]

The following table shows the total number of billiard balls denoted by n from the first layer to the specified layer denoted by x.


Conjecture 2The following formula determines the total

number of billiard balls denoted by n, given x as the number of billiard balls, from the first layer to the specified layer:

The following table shows the area of the billiard ball triangles in square units given x as the number of layers of billiard balls and r as the radius of each ball.


Conjecture 3The following formula determines the area

of the billiard ball triangle in square units given x

as the number of layers of billiard balls, and r as the radius of each ball:

Verifying ConjecturesThis section tests the conjectures against

existing cases, extreme cases. Conjectures are used to make predictions and predictions are tested. The data may support or provide a counter-example indicating the need to revise or reject the conjectures.

Testing Conjecture # 1:If x = 1, r = 1, P = 2π P = 2r[3(x-1)+π] P = 2(1)[3(1-1)+π] P = 2[3(0)+π] P = 2(0+π) P = 2π 2π = 2πIf x = 2, r = 2, P = 12+4π P = 2r[3(x-1)+π] P = 2(2)[3(2-1)+π] P = 4[3(1)+π] P = 4(3+π) P = 12+4π 12+4π = 12+4πIf x = 3, r = 3, P = 36+6π P = 2r[3(x-1)+π] P = 2(3)[3(3-1)+π] P = 6[3(2)+π] P = 6(6+π) P = 36+6π 36+6π = 36+6πIf x = 4, r = 4, P = 72+8π P = 2r[3(x-1)+π] P = 2(4)[3(4-1)+π] P = 8[3(3)+π] P = 8(9+π) P = 72+8π 72+8π = 72+8πIf x = 5, r = 5, P = 120+10π P = 2r[3(x-1)+π] P = 2(5)[3(5-1)+π] P = 10[3(4)+π] P = 10(12+π) P = 120+10π 120+10π = 120+10πIf x = 9, r = 10, P = 480+20π P = 2r[3(x-1)+π] P = 2(10)[3(9-1)+π] P = 20[3(8)+π] P = 20(24+π) P = 480+20π 480+20π = 480+20πIf x = 98, r = 99, P = 57618+198π P = 2r[3(x-1)+π] P = 2(10)[3(9-1)+π] P = 20[3(8)+π] P = 20(24+π) P = 57618+198π 57618+198π = 57618+198πIf x = 101, r = 202, P = 121200+404π P = 2r[3(x-1)+π] P = 2(99)[3(98-1)+π] P = 198[3(97)+π] P = 198(291+π) P = 121200+404π 121200+404π = 121200+404πIf x = 1234, r = 4321, P = 31966758+18642π P = 2r[3(x-1)+π] P = 2(4321)[3(1234-1)+π] P = 8642[3(1233)+π] P = 8642(3699+π) P = 31966758+18642π

31966758+18642π = 31966758+18642π

Testing Conjecture # 2:If x = 1, n=1

n = 1 1 = 1If x = 2, n=3

n = 3 3 = 3If x = 3, n=6

n = 6 6 = 6If x = 4, n=10

n = 10 10 = 10If x = 5, n=15

n = 15 15 = 15 If x = 25, n=315

n = 315 315 = 31 If x = 203, n=20706

n = 20706 20706 = 20706

to page 12

GALLOS: Math whiz from Iloilo National High School-Special Science Class.

FEATURE


And yes it was.As a deputized representative

to the said historic event, I was able to share insights with lawmakers, school presidents, deans, cabinet secretaries and other stakeholders covering the whole Visayas Area and the Zamboanga Peninsula. And I realized that a significant number, if not majority of us, were groping in the dark as to where is K-12 coming from and where is it going.

And so it was. In one whole day of dialogue, debate, and sharing, I was able to catch some vital points that may be relevant to the present dispensation. In the most straightforward way possible, let me enumerate them to you in

To some, the new curriculum makes education more expensive and burdensome; to others, it is the gateway to having more competitive and competent graduates. The debate goes on even in the ranks of teachers who are currently implementing the newly-fangled government policy.

However, we find it more productive to instead focus on how to cope with what is already served on the table.

Of course, the main concern, at least to most educators, is the curriculum. How will it be implemented from Kinder to Grade 12? How will it be designed in such a way that lessons will be decongesting but far-reaching? What are the subjects and concepts to be taught per quarter? When, how, and in what way will the teachers be prepared to face this new kind of pedagogical prospectus?

The questions seem endless. Hence, it is not surprising that MTAP-IC, Inc. thought it would be worth it to focus more

on the K-12 issues in its 4th Regional Convention this October. There are many patches that need to be fitted in this K-12 tapestry and so the association believes that it has a social responsibility to respond to this challenge by giving its membership the right venue to learn from the experts, and weave things in the process for the sake of mathematics literacy.

In other quarters of the country, many of these similar dialogues go on. Such is necessary for the government needs an impetus that must propel the program up to its full circle in 2018. And so we enjoin all Filipinos to cross elbows and embrace the K-12 challenge with an open heart and mind.

Thus, we call all mathematics teachers to be more receptive to change and to adapt to the revolutionary program with more progressive pair of eyes. For in the long run, it will be our students who will be affected by the K-12 vision. The more we understand what it is, where it is coming from, and where it is going, the better we see how our future will come into sight./The Infinity

The strong team spirit among the officers and Board of Directors with the unwavering support of the advisers is instrumental for leap frogging the attainment of the organization’s mission, vision and objectives. Extensive collaborative efforts with partner institutions and agencies were made possible to achieve synergistic impacts towards the attainment of math literacy in this part of the country.

I would like to take this opportunity to express my deepest gratitude to all MTAP-IC Officers & Board of Directors, Advisers, Members, Trainers, Speakers, Judges, Student Assistants, Sponsors, the Department of Education, the Commission on Higher Education, MTAP National and Local organizations, and the institutions where the Board of

1. It’s not about you; It’s about them. - Some teachers think they are the only experts whose role is to impart their knowledge to students. 2. Study your students. - Knowing the content of the lesson is not sufficient.3. Students take risks when teachers create safe environment. - For students to learn, they need to let themselves be vulnerable. 4. Great teachers exude passion as well as purpose.- The difference between a good and great teacher is not really expertise. It is true passion - for the students, for the content,for the art of teaching itself. 5. Students learn when teachers show them how much they need to learn. 6. Keep it clear even if you

Editor in Chief: Dr. Herman LagonAssociate Editor: Ms. Rosemarie Galvez

Staff: Dr. Harold Cartagena Mr. Matthew Lasap ∞ Mr. Keith Malorca

Layout Artist: Mrs. Lennie Yunque (Panorama Printing, Inc.)Blogspot: www.mtapiloilo.blogspot.comFacebook Account: MTAP-Iloilo Chapter

E-Mail: [email protected]

Official Publication of the Mathematics Teachers Association of the Philippines-Iloilo Chapter (MTAP-IC), Inc.Volume 3, Issue 1, November 2011 - October 2012

Praeter Limites Eundum Est (Going Beyond the Limits)

Educational reform!?

IMPULSES

Engr. Herman M. Lagon, Ph.D.Ateneo de Iloilo

MTAP-IC, INC. was privileged enough to be officially invited in the gathering of minds dubbed as the K-12 Pre-Summit Conference held December 7, 2011 at the DepEd Training Center in Lahug, Cebu City. Then, the K-12 was considered in most schools to be an emerging pedagogical species that everybody talks about but nobody is so sure of.

THE COUNTRY is now in its educational milestone with the entry of the K-12 initiative. The two added years in high school and the compulsory kindergarten system make it both challenging and controversial to say the least.

Principles of Good Teachingcan’t keep it simple. - A good teacher can make a complex idea understandable. An essential ingredient of teaching and learning is good communication. 7. Practice vulnerability without sacrificing credibility. - A good teacher is not afraid of saying, “I do not know”. 8. Teach from the heart. - The best teaching isn’t formulaic; its personal. 9. Repeat the important points. - The first time you say something, it is heard. The second time, it is recognized, the third time, it is learned. 10. Good teachers ask good questions. - Learning is exploring unknown territory, and what better way to explore than to have the courage to ask questions.

bullet form. What are the very important

facts about the state of education in the country vis-à-vis K-12 program?

• The Philippines must catch up with the rest of the world (5th in Quality of Education and last in the Quality of Science and Math Education and Capacity for Innovation in the World Economic Forum Global Competitiveness Report.

• We are the last country in Asia and one of only three countries in the world with a 10-year pre-university program. What does the K-12 program

aspire?

Dr. Alona M. BelargaMTAP-IC President (2008-2012)

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THE MATHEMATICS Teachers Association of the Philippines-Iloilo Chapter (MTAP-IC) Incorporated is blessed to have hard-working and committed bunch of wonderful people whose commendable work ethics made an indelible mark in the lives of mathematics teachers it has served in the past four years of its revival as a professional organization.

Directors are affiliated with for all the help extended in various forms to help MTAP-IC trail blaze a sound mathematics instruction within and outside the realm of our respective working spheres in the region.

Indeed, Andrew Carnegie’s words were re-lived among the working staff of MTAP-IC and I quote:

“Teamwork is the ability to work together toward a common vision. It is the ability to direct individual accomplishments toward organizational objectives. It is the fuel that allows common people to attain uncommon results.”

May the MTAP-IC Team continue to work hand-in-hand to make a difference in the lives of the people it touches.

To one and all, I greet you in advance: “A grace-filled and joyous Christmas!”/The Infinity

Teamwork

FROM THE PRESIDENT’S DESK

Ms. Rosemarie Galvez, MTAP-IC secretary, replied through the MTAP-IC official facebook account, “Thank you ma’am for asking a question that could motivate any mathematics enthusiast to open a book, think critically, and argue logically.

We say x is a multiple of y if xn=y, for some integer n. For instance, 6 is a multiple of 3 since 3(2)=6. Also, 12 is a multiple of 3 because 3(4)=12. Now, is zero a multiple of 3? Could we find an integer n such that 3(n)=0?”

Join our online mathematics discussion. You may like MTAP-Iloilo page on facebook, post on MTAP-IC’s facebook account, or leave a comment at www.mtapiloilo.blogspot.com.

Ms. Analiza Opoan Ojerio asked,

“Is zero a multiple of every number?”

• To produce holistically developed learners who have 21st century skills and are prepared for higher education, middle-level skills development, employment, and entrepreneurship

•Enhanced Basic Education curriculum

• Acquire mastery of basic competencies, be emotionally mature, be socially aware, pro-active, involved in public and civic affairs, be adequately prepared for the world of work or entrepreneurship or higher education, be legally employable with potential for better earnings, be globally competitive.Is there a K-12 law in the

offing?• House bill 4219, by Rep.

Feliciano Belmonte, Jr.-Pending• Senate Bill 2700, by Sen.

Ralph G. Recto-Pending• Kindergarten Education Act-

Passed Already• K-12 may still be implemented

through Executive OrderWhen will the K-12 be

implemented?• Universal Kindergarten

started SY 2011-2012.• The new curriculum for

Grade 1 and Grade 7 (High

EDITO

RIAL

OPINION

NOVEMBER 2011 - OCTOBER 2012 9OPINION

Algebra alone would make us cringe in our seats for symbols that do not seem to exist and numbers that kept flying in the air. Well, to start with, the name algebra itself even sounds like it belongs in the lingerie section and not in the mathematics section; however, some authors would attribute algebra to the word al jabr from the title of the book written by Al-Khwārīzmī entitled Hisâb al-jabr w’al muqâbalah (The science of reunion and reduction), so it is too far from the lingerie section after all.

Aside from the weird name, algebra is also abundant with invisible. If dealing with the supernaturals would mean unscientific, well, then surprise surprise! We have one of those in mathematics. Have you ever experienced feeling something that could not be seen? I mean a ghost or perhaps a spirit? Well, try algebra. We have these instances when we have to worry why 3x-x is 2x and people would argue that x is actually 1x. In the same manner that x is actually x1. Where did 1 come from? Is it a ghost that momentarily appeared beside x? Boohoo! That is why mathematics is a waterloo of many.

Mathematicians decided not to write numbers explicitly. Did

My warm afternoon greetings to all of you present on this gathering of minds. Let me extend by deep appreciation giving me once again this rare opportunity to be with you math enthusiasts of Western Visayas.

First of all, I warmly extend my felicitations to the organizers of this activity—The Second Regional Mathematics Investigation Training-Workshop—spearheaded by the competent and committed Chairperson, Dr. Herman M. Lagon, and the dedicated president of MTAP-Iloilo Chapter, Dr. Alona M. Belarga. To our dear teachers who have spent your time, effort and resources just to participate in this affair, my heartfelt congratulations to all of you.

Today’s mathematics teachers are experiencing major changes not only in the mathematics content they teach, but also in the way they teach. Nearly half of these teachers came through school when mathematics consisted of a collection of facts and skills to be memorized or mastered by a relatively homogenous group of students taught using lecture approach. Now teachers are called in to teach new, more challenging mathematics to very diverse audience using active learning approach designed to develop understanding.

As we all know, math is one of the most difficult subjects to teach. While other subjects are typically easy to master and so can be taught with little effort, math requires a dedication of time and energy on the mart of students that teachers often find difficult to command. In addition, some students are troubled by the need to think logically and perform complex calculations. These and other problems create challenges for math teachers, which, if properly addressed, can be overcome.

One of these problems is the lack of commitment. Some students don’t even try to learn math. They complain that the subject is beyond their mental abilities. This attitude of capitulation cannot produce any result but failure. To encourage a student with such a negative attitude to apply maximum effort to learning mathematics, you as a teacher need to work closely with his parents to monitor how the child does his daily assignments. The time the student spends learning math is important and needs to be closely monitored. Still, to prevent the student from just staring at textbooks or chalkboards, he needs to be genuinely interested in his success at math. This interest can be fostered by parental promises of incentives or encouragements on condition of improvement in his math grades.

Another problem is the students’ computational weakness. There are students who despite an understanding of math concepts, make basic errors and so don’t do well at math. Such students misread signs and carry numbers incorrectly. These students need to be told their weaknesses so they can spend more time doing basic calculation at home.

Making connections of the students to the real world is also one of the problems. Some students

RANDOM BRUSHSTROKES

Of mathematical ghosts and gender issues

By Ms. Rosemarie GalvezUniversity of San Agustin

MATHEMATICS has a language of its own. In fact, the language of mathematics is an enigmatic one. Well, just like English. Sometimes we ask, “Why is oxen the plural of ox but chicken is not the plural of chick?” Mathematics has its fair share of language stumbling blocks.

they decide to hide this from us so that mathematics will just be for the sophisticated? Well, perhaps the ghost-like characteristic of 1 is a divine secret that the brotherhood or perhaps sisterhood of mathematicians would keep under lock and key. But, it is not cipher text or some difficult codes. Could you just imagine how the world will be if we decide to write 1 as exponent every time? Yes, I understand that x expressed as x1 is indeed helpful in calculations. But is it not true that 3 is the same as 31? What if you have to get my cellphone number expressed in this explicit manner of exponentiation? You’ll start getting my number by recording 01, 91, 11, 61, but now I have to stop because I am running out of breath. See, explicitly stating the exponent 1 every time will only lead to knowing if a person has globe or smart number. Hence, it is reasonable indeed to just simply leave the exponent 1 as a ghost. Well, in mathematical parlance, implicitly stated.

While you’re thinking of other stumbling blocks in algebra, let me shift the gear to face another issue, the gender issue. History tells us that male mathematicians abound. Pythagoras, Thales, Pascal, Euclid, Fermat, Descartes, the list is almost

endless. Would that mean there is indeed a superiority of the y chromosome? Should I go back to my fourth paragraph and delete the word sisterhood? Bah! I could sense some male lips smiling. Well, I just want to make it clear that I am a woman and that this column would not pay tribute to testosterone laden individuals, though I have to admit that men have dominated history, sad to say, in almost all fields.

The thick mathematics history book of David Burton (who by the way is male) has set aside some lines for the first woman mathematician, Hypatia, whose rise in the intellectual ladder was interpreted to be a threat against the Christians. So, instead of being respected for her scholarly contributions, she ended up being ambushed to death by a mob of angry religious fanatics who believe that her lectures centered on paganism. Well, this is not shocking in history. Women, have struggled much to be respected for their contribution. But we should be glad that today, women have been recognized for what they can do. We don’t need to look far to find an epitome of women empowerment. Our beloved MTAP-IC president, Dr. Belarga, is a good example. Without her impeccable leadership and expertise in mathematics, I am sure we won’t be able to have a strong and active organization.

These days, gender does not always come in black and white; there are those who would rather prefer to be in the gray areas. As mathematics teachers, we have to be aware that these learners, no matter what shade of gray they choose, have the right to learn just like those who are quite sure with their whiteness

Let me postulate that mathematics appears in places that we sometimes least expect it to be – just like in dance. Let us see how both, dancing and mathematics, equate to each other.

For about 3 1/10 of a decade of existence, I have tried different dances. Looking into some of dance-moves, one can notice some mathematical symbols. Just like stretching your arms on the sides to form a ‘plus sign’ or holding hands together in front of your chest and make a wave just like the ‘sine graph,’ or manipulating your arms forming’ perpendicular’, ’parallel’ and ‘intersecting lines’. These are just a few of the many unbounded possible dance-moves you can create from mathematical concepts using your body.

TWISTS AND TURNS

Dancing in the rhythm of mathematics

By Dr. Harold CartagenaIloilo City Community College

LOOKING into old photos I realized that as a child, my happiest moment, aside from playing, is when I am dancing. Dancing was my first love then. Being a weakling in mathematics I usually uttered words “I will rather choose to dance in front of people than deal mathematics,” This might sound odd but it is true. Ironically, at present, I am a holder of PhD in Science Education major in Mathematics. See the twist?

To add variations, fun and energy, you can divide the usual whole 8 counts in different patterns and sequences which will depend of course with the type of music you are using. You can have 1 move in every count or beat, or if you like it faster you can have two to three moves in a beat. If it is a group dance, dancers can take fraction of the music as they alternately take turns in the dance floor. This is math right?

To make the dance number more dynamic, dancers can make formations just like square or circles, or they can form sets based on colors, sizes, costumes or movements. Likewise, geometrical, topological and abstract properties may be observed as you look into the formation, transformation, translation and reflection of each

dancer. See how fun it is?Realistically, dance and

math can be integrated. Imagine a class where students exhibit math concepts through a dance number. For sure it will be more enjoyable than having the usual board games.

So what brings out the great ‘shift’ in my career? It is simple as doing the ‘negation’of my negative attitude in mathematics. Logically, it will turn out positive. Correct?

Being in the world of math is no miracle. I, myself, did have struggles. Learning the beat and dance steps or figures necessary and persistently finish the dance is the algorithm I used to survive.

Indeed embracing once weaknesses and connecting ‘coplanar points’ lead to prove that you can make ‘variances’ and ‘significantly’ make a better change.

Evidently, mathematics and dance are independent yet interrelated. No wonder I became a math specialist.

As word of advice from a dancer-turned-math educator: “Learn to dance with the rhythm of mathematics.”

∞∞∞Dr. Cartagena is still into dancing while teaching the scholars of Iloilo City at ICCC math and its implication to tourism. He can be reached at [email protected] Infinity Editor

INSPIRATIONS

Math for life!

By Mr. Jerry A. OquendoEPS, DepEd RO 6

have trouble applying mathematics to real-life situations. They find it challenging to comprehend what numbers represent in the physical world. For example, a child can easily divide 15 by 5, but she might struggle to divide 15 apples among five people. Students with such problems can do better by practicing their math skills in the real world. For example, urge their parents to challenge the students with mathematical problems on a daily basis. A mother can ask her daughter how much sugar she needs to put in a cake to preserve a certain proportion.

Difficulty with math language is also identified as a factor affecting students’ poor performance in math. Particularly challenging are verbal instructions. There is a list of English math vocabulary encountered in textbooks in which all students must know. So, before a teacher gives any meaningful math instruction, he must make sure his students understand and can use specialized math vocabulary. These children can improve their math scores by writing the problem at hand in symbols, for example, by assigning the letters “x” and ‘y” to variables instead of using variables’ names directly in their calculations.

Another difficulty is the visual and spatial aspects. Visual and spatial aspects of math are challenging to many. To improve their skills at solving geometric problems, these students need to make models of their problems. For example, in addition to drawing a pyramid on paper, they might need to see and touch a model of a pyramid to understand the relations of its sides. Accommodate such students with models of space figures—cubes, spheres, cones, prisms, and pyramids.

Another barrier is the language used in instruction. When math concepts are not properly understood because students do not understand the term used by the teacher, learning will never take place. In some schools, problems are being translated first in their mother tongue in order for these children to understand what the problem is all about.

I fully believe that your stay in this venue is proof that you are trying to be equipped with all the knowledge, skills and right attitude to arrest some if not all these problems. This Math Investigation would be an efficient and effective learning and teaching tool in the delivery of more in-depth, relevant, and lasting math awareness and literacy to our young mathematicians back at your respective stations.

Can we look forward when our students are soaring high in their achievement and performance in mathematics—something soon?

Thank you so much./The Infinity

∞∞∞

(This article is an excerpt from the speech delivered by Mr. Oquendo during the opening rites of the 2nd Regional Math Investigation Seminar-Workshop on April 11, 2012 at the CTE Building at West Visayas State University, La Paz, Iloilo City.—Infinity Editor)

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LIKE THE famous Simeon Poisson who once said “Life is good for two things, discovering mathematics and teaching mathematics,” I, too, share the same sentiment. And I should say Mathematics is, for life.

NOVEMBER 2011 - OCTOBER 201210 SnapshotsHERE IS the photographic smorgasbord of activities of the Mathematics Teachers Association of the Philippines-Iloilo Chapter (MTAP-IC), Inc. in year 2012. For the past months, the organization has been trying its best to be faithful to its thrust in developing math literacy in the region through seminars, workshops, camps, trainings, information dissemination, resource sharing, and researches catering not just regular teachers and administrators but also students, parents, out of-school youth, and pre-service math majors. The pictures here, though incomplete, somewhat represent what MTAP-IC, Inc. and its active members have done so far.(Photos taken by Dr. Herman Lagon, Dr. Rosemarie Galvez, Dr. Harold Cartagena, Mr. Neal Sobrejuanite, Ms. Marjie Pineda, Ms. Ma. Eva Claire Sayson, and the Black Box-The Infinity Editor)

NOVEMBER 2011 - OCTOBER 2012 11SnapshotsHERE IS the photographic smorgasbord of activities of the Mathematics Teachers Association of the Philippines-Iloilo Chapter (MTAP-IC), Inc. in year 2012. For the past months, the organization has been trying its best to be faithful to its thrust in developing math literacy in the region through seminars, workshops, camps, trainings, information dissemination, resource sharing, and researches catering not just regular teachers and administrators but also students, parents, out of-school youth, and pre-service math majors. The pictures here, though incomplete, somewhat represent what MTAP-IC, Inc. and its active members have done so far.(Photos taken by Dr. Herman Lagon, Dr. Rosemarie Galvez, Dr. Harold Cartagena, Mr. Neal Sobrejuanite, Ms. Marjie Pineda, Ms. Ma. Eva Claire Sayson, and the Black Box-The Infinity Editor)


If x = 1010, n=510555

n = 510555 510555 = 510555 If x = 9999, n=49995000

n = 49995000 49995000 = 49995000

Testing Conjecture # 3:If x = 1, r = 1, A = units2

If x = 2, r = 2, A = units2

If x = 3, r = 3, A = units2

If x = 4, r = 4, A =units2

If x = 5, r = 4, A =units2

If x = 101, r = 10, A =units2

If x = 121, r = 20, A =units2

If x = 171, r = 5, A =units2

JustificationThis section shows the proof to show that the formula is true for all circumstances.

Conjecture # 1: Perimeter = 2r[3(x-1)+π]

The billiard ball triangle (Figure 1) can be sliced to look like Figures 2 and 3. The first part of the triangle is Figure 2. Looking at Figure 2, it can be observed that there are 3 straight sides. Each side is equal twice the radius times the number of layers of billiard balls minus 1. This should then be multiplied by 3 since there are 3 sides. Therefore, the perimeter of the highlighted areas in Figure 2 is 3[2r(x-1) ]. By simplifying this, the value 6r(x-1) is obtained. On the other hand, Figure 3 is composed of 3 arcs. You’ll notice that when the 3 arcs are puzzled together, a circle is formed. Thus, the formula for getting the circumference of a circle is used to get the perimeter of the 3 arc corners of the billiard ball triangle. Knowing this, the perimeter of the 3 arc corners is 2πr. Therefore, the total perimeter of the billiard ball triangle is 6r(x-1)+2πr,, however, this can still be simplified to get the formula: 2r[3(x-1)+π]. Thus, it can be concluded that the formula for getting the perimeter of a billiard ball triangle, given x as the number of layers of billiard balls and r as the radius of each ball, is 2r[3(x-1)+π] .

Conjecture # 2: Total number of balls =

The number of billiard balls in each layer is equal to 1 more than the previous layer, thus, there is a common difference of 1, and the trend appears to be in an arithmetic sequence. Therefore, the formula for determining the arithmetic sum of a series will be used to establish the total number of billiard balls from the first to the specified layer. For any arithmetic series, the formula for finding the sum is . In this case, x was used for the number of layers, so in order to be consistent, the formula for finding the arithmetic sum is . The value for a1 in this sequence is 1, and the last term (ax) is equal to x. Thus, the total number of balls is . This can be simplified further to obtain the formula:

Therefore, it can be concluded that the total number of billiard balls in x layers can be solved using the formula:

Conjecture # 3: Area = r^2 {[6+√3 (x-1) ](x-1)+π}

To find the area of the billiard ball triangle, one way would be to slice the triangle into a set of rectangles (Figure 4), a smaller equilateral

triangle (Figure 5), and a set of sectors (Figure 6). Starting with Figure 4, the area would be equal to thrice the area of a rectangle. The area of a rectangle is equal to its length times its width. The width of one rectangle is equal to the radius of one circle. Its length, on the other hand, is equal to twice the radius times the number of layers minus 1. This, of course, has to be multiplied by 3 because there are 3 rectangles. Thus, the formula for getting the area of Figure 4 is 3{[(x-1)2r]r}. This can then be simplified to get 6r2 (x-1). For Figure 5, an equilateral triangle is the figure to be solved. Based on Heron’s Formula, the area of an equilateral triangle, given

the length of the side, is Each side of the

equilateral triangle in Figure 5 is equal to twice the radius times the number of layers minus 1. By substituting this to the previous formula, the area

of the equilateral triangle is . This

can still be simplified to get . The 4

in the numerator and denominator cancel each other out, thus, simplifying the formula further to get r2 (x-1)^2 √3.. It is important to note that r2 was not multiplied to (x-1)2 in order to use it as a common factor to be brought out of the other elements of the formula. Lastly, the areas of the sectors in Figure 6 make up the remaining areas to be solved in order to get the total area of the billiard ball triangle. When the 3 sectors in Figure 6 are pieced together, a circle is formed. Thus, the combined area of the 3 sectors is equal to the area of one circle. The area is, therefore, equal to πr2. By combining the 3 formulas, the formula:6r2(x-1)+r2 (x-1)2 √3 + πr2 is obtained. Removing r2 from the equation, and placing it outside grouping symbols as a common factor to simplify the formula yields the formula: r2 [6(x-1)+(x-1)2 √3+π]. This can still be simplified by doing some factoring inside the grouping symbols. By doing this, the formula: r2{[6+√3(x-1)](x-1)+π} is obtained. Therefore, it can be concluded that the formula for determining the area of a billiard ball triangle with x as the number of layers of billiard balls and r as the radius of each ball is r2 {[6+√3 (x-1) ](x-1)+π}.

SummaryThis study made use of Mathematical

Investigation, which aimed to determine the formula of determining the area, perimeter, and total number of billiard balls in a billiard ball triangle setup given the number of layers of the billiard ball set and the radius of each ball. Specifically, it sought to answer the following questions: 1. If given a specific number of layers of billiard balls and the radius of each billiard ball, can a general formula be created to determine the perimeter of the billiard ball triangle to be constructed to house the billiard balls? 2. If given the number of layers of billiard balls, can a formula be generated to determine the total number of billiard balls from the first layer to the specified layer? 3. If given a specific number of layers of billiard balls and the radius of each billiard ball, can a general formula be created to determine the area of the billiard ball triangle to be constructed to house the billiard balls?

Billiard ball triangles are triangular in form and have round edges used to house billiard balls in a game of billiards. Like any closed figure, the basic geometric properties are an important part of its mathematical makeup. These would be area and perimeter, and since it houses billiard balls, it’s also important to know the total number of balls it houses given the number of layers it has. This has been the foundation of the establishment of this study.

Three conjectures have been tested and proven with regard to the billiard ball triangle. These are for its perimeter, area, and the number

of balls it houses. The formula for the perimeter of a billiard ball triangle is 2r[3(x-1)+π], where x is the number of layers of billiard balls and r is the radius of each billiard ball. The formula for the number of balls in a billiard ball triangle is , where x is the number of layers of billiard balls. The formula for the area of a billiard ball triangle is r2{[6+√3(x-1)](x-1)+π}, where x is the number of layers of billiard balls and r is the radius of each ball. The formulas were based on a table of values and from geometric principles both basic and advanced.

The formulas were tested and verified by substituting the smallest and the largest and most extreme cases possible. Each formula was tested using this method. This was done to test the formula for possible counter-examples because of the extreme deviation of values.

The first and third formulas were proven by dissecting the billiard ball triangle into common geometric parts in order to solve for the needed values. These values were then put together in order to get the required data of the whole figure itself. Geometric concepts such as Heron’s formula, the area and circumference formula, and many more were utilized in the derivation and the proving of the formulas. The second formula implied the use of the formula for arithmetic sums. The total number of balls in each layer appeared in an arithmetic sequence. Thus, the arithmetic sum formula was used to generate the second formula – the formula to determine the total number of balls from the first to the specified layer in the billiard ball triangle.

Possible ExtensionsThis section may serve as further

investigations for later use and may arise from student’s investigation. These are clearly stated from the data.

Based on the findings and the justification, the following possible extensions were made:

1.) Since the 3 formulas were tested and proven to be true, students may use them for questions related to billiard ball triangles and similar figures or for entirely different questions but with the same basic geometric principles.

2.) It is recommended that Mathematics teachers use and introduce these formulas to their students as an easier of way of finding solutions and as an introductory concept to deeper geometric problems with similar concepts and figures.

3.) The researcher should explore new formulas related to the billiard ball triangle concept and other concepts and scenarios related to this main concept. Such scenarios could be if the billiard ball triangle had sharp instead of round corners. Also, the researcher could visualize concepts on other scenarios having the same concepts such as if the billiard balls were enclosed by a square container with round edges. Or a scenario where the square container had sharp edges instead and myriads of other related scenarios that branch out from the billiard ball triangle concept.

4.) Another possible extension to investigate on would be the number of balls with a unit radius that can be contained in billiard ball triangle, square, and/or other polygons.

5.) It is suggested that one investigates on concepts analogous to this study in 3-dimensional figures such as the number of balls considering another variable – height.

6.) It is also suggested that one investigates on taking the concept of thickness of the material to be used in the construction of the triangle as an additional determinant in the surface area and/or volume of the billiard ball triangle to be constructed./The Infinity

Billiard..... from page 7

Getting the writing job donewriting tips based on experience

By Ms. Rosemarie G. GalvezUniverity of San Agustin

WITH MY limited experience in writing, I felt like I was wrung out with writing juice several times when I was writing my dissertation as a graduate school requirement. Three months before the looming end of my scholarship, I realized that I have to overcome slacking moments and very low points of depression, and I decided to dish out list of advice that I could give myself. Somewhat like a list fitted for a self-help book, my advice could also help any researcher, mathematics investigator, or anybody who is in the dilemma of finishing a writing job.

I started whipping up some words of advice for myself based on the popular

theories of learning. I did not bother much to recall the complexities of each theory, but I bore in mind the basics that each espoused. Well, to make sure that writing is done without mishap, I made sure that the stimuli are right for writing. I observed that I fell asleep when I wrote on my bed, I ended up eating when I worked in the kitchen, and I preferred to pull the weeds when I decided to do the paper works outdoors. Hence, I believe that it is important to have a writing nook that would serve as good stimulus for conditioning the mind for writing. Also, it would be better to reprogram the brain such that the writing job is placed in the long term memory. Make the writing task a strong pedestal in your mind that nobody could ever topple.

Writing is also a learning process, and writers may be considered as learners who construct knowledge through the help of a facilitator. Usually a friend or a colleague could help. In my case, I had my research advisers, Dr. Belarga and Dr. Loriega. My dear friend, Dr. H said, “Pester your adviser.” I bet advisers would love it when their advisees do that.

Aside from existing theories and numerous tidbits of advice I got from friends, I made a list of simplified three tips that I borrowed from Plato, cervical cancer vaccine ad, and Nike.

1. Know thyself. What did they do that they have finished on time and what have I done wrong that I have been slacking? I, myself, should know that. I have been with myself every single moment of the day, but

I sometimes don’t see myself. I see other people. A writing job could be done best when the writer develop a relationship with himself or herself.

Knowing oneself entails knowing what works and what does not work. Do I write well when I am alone or when I am in a crowd? Do I want to have variety in my tasks or do I want to focus on one task only? Do I work well with pressure or positive motivation? These are some of the questions that could be used for reflection. The self should be the coach and the friend of the writer. Sad to say, however, that the self could also be the primary source of factors that could put the writer down. So,

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FEATURE


Learning from the experts!Nearly 100 math educators from all corners of

Western Visayas learned from the national-caliber experts in the WVSU Center for

Excellence main function hall for the MTAP-IC 2nd Regional Math Investigation Seminar-Workshop held April 12, 13, and 21, 2012. Themed “Conquering the Mystery of Math Investigation (MI),” the math workout was facilitated in by Dr. Emellie Palomo, Dr. Myrna Libutaque, and Dr. Elvira Arellano. This photo feature shows how the whole lecture-discussion-presentation went through.


Taking the differences in n’s and I(n)’s, we have…

In this case, since equal first differences in n’s produced equal third differences in I(n)’s, thus the relation or function that possibly describes it is of degree 3/ cubic function that has the form

I(n) = An3 + Bn2 + Cn +D,

where A, B, C and D . Since there are four constants in the said form (i.e. A, B, C, and D), just like in our first case, we will arbitrarily pick (2, 1), (4, 7), (6, 22) and (8, 50). Substituting each of these points to the form

I(n) = An3 + Bn2 + Cn +D,we will have,

1 = A (2)3 + B (2)2 + C (2) +DOr

1 = 8A + 4B + 2C +D (equation5)

7 = A (4)3 + B (4)2 + C (4) +DOr

7 = 64A + 16B + 4C +D (equation6)

22 = A (6)3 + B (6)2 + C (6) +DOr

22 = 216A + 36B + 6C +D (equation7)

50 = A (8)3 + B (8)2 + C (8) +DOr

50 = 512A + 64B + 8C +D (equation8)

Expressing the four equations in its augmented matrix form, we have

Performing pertinent elementary row operations, we have…

Thus D = 0.Substituting D = 0 to 12C + 11D = -1, we have 12C + 11(0) = -1 12C = -1 C = - 1/12Substituting C = - 1/12 and D = 0 to 16B + 12C + 7D = 1, we have16B + 12 (- 1/12) + 7 (0) = 1 16B – 1 = 1 16B = 2 B = 1/8Substituting B = 1/8, C = - 1/12 and D = 0 to 8A + 4B + 2C + D = 1, we have8A + 4 (1/8)+ 2 (- 1/12) + 0 = 1 8A + 1/2 - 1/6 = 1 8A + 1/3 = 1 8A = 2/3 A = 1/12Substituting these values to the form U(n) = An3 + Bn2 + Cn +D, we have

I(n) = 1/12 n3 + 1/8 n2 – 1/12 n.Or in complete factored form, we have

I(n) = 1/24 n (n + 2) (2n – 1).

Testing the formula,

For n = 2 I(n) = 1/24 n (n + 2) (2n – 1) I(2) = 1/24 (2) (2 + 2) (2(2) – 1) I(2) = 1/24 (2) (4) (3) I(2) = 1

The results of actual substitution to the formula match with existing, known data for n = 2, 4, 6.Thus, the formula can possibly describe the number of inverted triangles in an equilateral triangle partitioned evenly by n – 1 points per side, provided that the line connecting two points is parallel to the third

side, ∈∀n 2N.

For an equilateral triangle of side n units, where ∈n 2N – 1, the table of values is shown below. Taking the differences in n’s and I(n)’s, we have

In this case, since equal first differences in n’s produced equal third differences in I(n)’s, thus the relation or function that possibly describes it, still, is of degree 3/ cubic function that has the form

I(n) = An3 + Bn2 + Cn +D,where A, B, C and D ℜ∈ . Since there are four constants in the said form (i.e. A, B, C, and D), just like in our first case, we will arbitrarily pick (1, 0), (3, 3), (5, 13) and (7, 34). Substituting each of these points to the form

I(n) = An3 + Bn2 + Cn +D,We will have,

0 = A (1)3 + B (1)2 + C (1) +DOr

0 = A + B + C +D (equation9)3 = A (3)3 + B (3)2 + C (3) +D

Or3 = 27A + 9B + 3C +D (equation10)

13 = A (5)3 + B (5)2 + C (5) +DOr

13 = 125A + 25B + 5C +D (equation11)

34 = A (7)3 + B (7)2 + C (7) +DOr

34 = 343A + 49B + 7C +D (equation12)

Expressing the four equations in its augmented matrix form and solving for the constants by performing elementary row operations, we have…

Thus, 48D = -6 D =-1/8Substituting D = -1/8 to 120C + 184D = -33, we have 120C + 184(-1/8) = - 33 120C – 23 = -33 120C = -10 C = -1/12 Substituting C = - 1/12 and

D = -1/8 to 18B + 24C + 26D = -3, we have18B + 24 (- 1/12) + 26 (-1/8) = - 3 18B -21/4 = - 3 18B = 9/4 B = 1/8Substituting B = 1/8, C = - 1/12 and D = -1/8 to A + B + C + D = 0, we have A + 1/8 - 1/12 -1/8 = 0 A - 1/12 = 0 A = 1/12Substituting these values to the form U(n) = An3 + Bn2 + Cn +D, we haveI(n) = 1/12 n3 + 1/8 n2 - 1/12 n -1/8,

Or in complete factored form, we haveI(n) = 1/24 (n – 1) (n + 1) (2n + 3).

Testing the formula,For n = 1 I(n) = 1/24 (n – 1) (n + 1) (2n + 3) I(1) = 1/24 (1 – 1) (1 + 1) (2(1) + 3) I(1) = 1/24 (0) (2) (5) I(1) = 0

Based on the procedure, it is noted that to find the number of inverted triangles in an equilateral triangle of side n units partitioned evenly by n – 1 points per side, provided that the line connecting two points is parallel to the third side, we use two separate functions depending whether n is even or odd. Thus, we define our next conjecture by the following Piecewise Function:

Conjecture 2The number of inverted triangles in an equilateral triangle of side n units partitioned by n – 1 points per side, provided that any two points connected from two sides is parallel to the third side, is defined by

IV. Proof/ Justification of Conjectures 1 and 2Before we proceed with the actual proof of Conjecture 1, let us revisit the following table of values:

As what we could notice, each succeeding term is derived by adding successive numbers of the form For example…U(1) = 1 = 1U(2) = 4 = 1 + 3U(3) = 10 = 1 + 3 + 6U(4) = 20 = 1 + 3 + 6 + 10

Now we will proceed with the proof of Conjecture 1.

Proving Conjecture 1 by Using the Principle of Mathematical Induction

Conjecture 1The number of upright triangles in an equilateral triangle of side n units partitioned evenly by n – 1 points per side per side, provided that the line connecting two points of any two sides is parallel to the third side, is defined by

i. Verification. Verify true for n = 4, 5

For n = 4 U(4) = 1/6 (4) (4 + 1) (4 + 2) = 1 + 3 + 6 + 10 1/6 (4) (5) (6) = 20 20 = 20 For n = 5U(5) = 1/6 (5) (5 + 1) (5 + 2) = 1 + 3 + 6 + 10 + 15 1/6 (5) (6) (7) = 35 35 = 35 The formula was verified true for n = 4, 5.

ii. Assumption. Since the formula was verified true for n = 4, 5, assume true for n = k, i.e.

iii. Proof by Induction. Prove true for n = k + 1.

iv. Conclusion. Since the proposition was verified true for n = 4, 5 and was proven true for n = k + 1, thus the proposition is true and valid for all N.

Let us extend the formula to other extreme cases, say n = 11…

U(n) = 1/6 n (n + 1) (n + 2)U(11) = 1/6 (11) (11 + 1) (11 + 2)U(11) = 1/6 (11) (12) (13)U(11) = 286

In this case, we have to expect 286 upright triangles for an equilateral triangle of side 11 units whose sides are partitioned evenly by 10 points.

Let us take a glimpse on our tables relating the measure of side n of our equilateral triangle with the number of inverted triangles formed and taking the first differences in n’s and I(n)’s…

A. n∈ 2N

B. n ∈2N – 1

Notice that regardless whether n is even or odd, the difference of I(n) and I(n-1) is always a number of the form (n (n- 1))/2.

For n∈2N, For n∈2N - 1,1 = 1 0 = 07 = 1 + 6 3 = 0 + 322 = 1 + 6 + 15 13 = 0 + 3 + 1050 = 1 + 6 + 15 + 28 34 = 0 + 3 + 10 + 2195 = 1 + 6 + 15 + 28 + 45 70 = 0 + 3 + 10 + 21 +36

Now we are ready to prove our second conjecture.

Proving Conjecture 2 by Using the Principle of Mathematical Induction

Conjecture 2The number of inverted triangles, I(n), in an equilateral triangle of side n units partitioned by n – 1 points per side, provided that the line connecting two points is parallel to the third side, is defined by

This one is a special case. Since this Piecewise Function is comprised of two different functions defined on certain, restricted domains, thus two mathematical inductions are necessary to prove each case.

A. The number of inverted triangles in an equilateral triangle of side n units partitioned by n – 1 points per side, provided that the line connecting two points is parallel to the third side ∈∀n 2N, is defined by

Triangles ..... from page 3


i. Verification. Prove true for n = 8,10.

For n = 81 + 6 + 15 + 28 = 1/24 (8) (8 + 2) (2(8) – 1) 50 = 1/24 (8) (10) (15) 50 = 50For n = 101 + 6 + 15 + 28 + 45 = 1/24 (10) (10 + 2) (2(10) – 1) 95 = 1/24 (10) (12) (19) 95 = 95The conjecture is verified true for n = 8, 10.ii. Assumption. Since the conjecture was verified true for n = 8, 10, assume true for n = k. That is,

iii. Proof of Induction. Prove for the next even integer n = k + 2.

iv. Conclusion. Since the proposition was verified true for n = 8, 10 and was proved true for n = k + 2, thus the proposition is valid

∈∀n 2N.

B. The number of inverted triangles, I(n), in an equilateral triangle of side n units partitioned by n – 1 points per side, provided that the line connecting two points is parallel to the third side is defined by

i. Verification. Verify the proposition to be true for n = 7, 9.

For n = 70 + 3 + 10 + 21 = 1/24 (7 – 1) (7 + 1) (2(7) + 3) 34 = 1/24 (6) (8) (17) 34 = 34For n = 90 + 3 + 10 + 21 + 36 = 1/24 (9 – 1) (9 + 1) (2(9) + 3) 70 = 1/24 (8) (10) (21) 70 = 70The conjecture is verified true for n = 7, 9.

ii. Assumption. Since the conjecture was verified true for n = 7, 9, assume true for n = k. That is,

iii. Proof of Induction. Prove for the next odd integer n = k + 2.

iv. Conclusion. Since the proposition was verified true for n = 7, 9 and was proved true for the next odd integer n = k + 2, thus the proposition is valid ∈∀n 2N – 1.

V. SummaryThis study focused on determining the

number of upright and inverted triangles with respect to the n units side of an equilateral triangle partitioned evenly by n – 1 points per side, provided that the line connecting two points is parallel to the third side.

Conjecture 1 The number of upright triangles in an

equilateral triangle of side n units partitioned evenly by n – 1 points per side, provided that any two points connected from two sides is parallel to the third side, is defined by

U(n) = 1/6 n (n + 1) (n + 2), Nn∈∀ . In attempt to describe the number of

inverted triangles, I(n), based on the table, the researchers took the differences in n’s and I(n)’s. Equal first differences in n’s did not result to equal [third] differences in I(n)’s but alternating 1’s and 0’s are noticed.

In this case, the researchers partitioned the odd and even-numbered measures of sides n of an equilateral triangle partitioned by n – 1 points, provided that any two points connected from two sides is parallel to the third side. By doing so, and taking differences in n’s and I(n)’s of each table, it is noted that equal first differences in n’s equaled the third differences of each table. Thus, the relation or function describing the number of inverted triangles, I(n), in an equilateral triangle of side n units must be a function of the third degree/ cubic unction. Using the same procedures as to conjecture 1, we arrived at our second conjecture as follows:

Conjecture 2The number of inverted triangles, I(n),

in an equilateral triangle of side n units partitioned by n – 1 points per side, provided that any two points connected from two sides is parallel to the third side, is defined by

The formulas are tested against known existing cases and results match with the data. The principle of Mathematical Induction was used to prove the conjectures true and valid under their respective domains. The formulas are also used to predict the number of upright

and inverted triangles an equilateral triangle of side n units partitioned by n – 1 points per side.

VI. Possible ExtensionsIn light of the aforementioned results of

this study, the following recommendations are advanced:

1. Further invest igat ions may be conducted on determining the relationship, if there exists any, between the number of points in an equilateral triangle of side n partitioned evenly by n – 1 points on each side and the total number of triangles.

2. It is highly recommended to conduct similar investigations to find a function rule, if there exists any, relating the side n of an equilateral triangle partitioned by n – 1 points per side given that all of these points are connected, to the total number of intersections at the interior of the equilateral triangle.

3. It is proposed to carry out a similar study to other kinds of triangles (scalene, isosceles, right, obtuse, etc.).

4. Using the same conditions in this study, it is suggested to conduct investigations to find formulas, if there exists any, to determine the number of quadrilaterals, pentagons, n – gons at the interior of the triangle./The Infinity

∞∞∞

The lead proponent of this MI is Kim Jay Encio. He graduated summa cum laude in BS Secondary Education and managed to clinch the Top 3 spot in the Licensure Exam for Teachers in 2010. Corollary to this, The Infinity would like to correct the inadvertence it had in its 2011 publication more specifically in the article entitled “An Oblique Problem: A Mathematical Investigation” in page 8 and page 12 that was also authored by Encio during his BSEd years in WVSU College of Education as part of the requirements for the course Mathematical Investigation and Modeling under the tutelage of Dr. Helen Hofileña. Also part of his team were his school mates Louie Jee Labrador, Rimbrant Padernal, Roel Rocero, and Bryan Tacayon. The said article was inadvertently named after PSHS-WV. -The Infinity Editor

are (0, 0), (0, r), (c, 0), and (c, r).

The diagonal intersects vertical segments between 0 and c. So, the number of vertical segments intersected by the diagonal in the interior of the rectangle is c-1.

A similar argument may be used to obtain the number of horizontal segments intersected by the diagonal in the interior of the rectangle. The diagonal intersects r-1 horizontal segments. The total number of segments intersected by the diagonal in the interior of the rectangle is

s = number of vertical segments + number of horizontal segmentss = (c – 1) + (r – 1)

Simplifying, we prove the conjecture that s = c + r–2.

The diagonal of the rectangle passes through the points (c, 0) and (0, r). Hence, the equation of the line that contains the diagonal is rx+cy=rc . The lattice points intersected by the diagonal would be the points intersected by the diagonal containing integral coordinates.

In the linear Diophantine equation, if one solution of the linear equation ax + by = d is (x0, y0 ), then the other solutions are found by

In a linear equation rx+cy=rc , we have

We know that the line intersects the point (0, r). So, we could have xo=0 and yo=r.

The values of x should be less than c and should be greater than 0.

The values of y should be less than r and should be greater than 0.

The same inequality is obtained for both cases. The allowable value for t should

be less than the greatest common factor of r and c and should be greater than zero. The number of values that are allowed for t is the number of integral coordinates in a given set of restrictions. That is, t is also the number of lattice points intersected by the diagonal in the interior of the rectangle represented by v.

Hence, v is the number of integers in the set {1,2,…,[gcf(r,c)-2 ],[gcf(r,c)-1 ]}.

Therefore, v=gcf(r,c)-1.

VI. SUMMARY AND RECOMMENDATIONThis investigation focused on the

rectangular table covered by square tiles and its diagonal. With the use of analytic geometry, geometry, and elementary number theory, two conjectures were constructed and proven. The first one is on the number of segments intersected by the diagonal in the interior of the rectangle, s=c+r-2. . The second one is on the number of lattice points (or vertices of the tiles) contained by the diagonal in the interior of the rectangle, v=gcf(r,c)-1, where s=number of segments intersected, c=number of tiles in a column, r=number of tiles in a row, v=number of vertices intersected, gcf= greatest common factor. The values of v and s could be further investigated when the other variables are odd or even. An extension of this investigation could also focus on the diagonal of a rectangular prism. This investigation is indeed fun. Math learners will never look at tiles the same way again – especially the ones with cracks!/The Infinity

ReferenceBastow, B., Hughes, J., Kissane, B.,

& Mortlock, R. (1984). 40 mathematical investigations. The Mathematical Association of Australia.

∞∞∞Ms. Galvez teaches Mathematical Investigation at the University of San Agustin. Reactions and comments may be sent to [email protected].–The Infinity Editor

To solve for the first trisection, we let 2m = . Substituting the values,

[ ] [ ]2 1 2 1,mx n m x my n m y

n n + − + −

( )( ) [ ]( ) ( )( ) [ ]( )2 6 3 2 15 2 8 3 2 4,

3 3 + − + − −

12 15 16 4,3 3+ −

( )9,4Thus, the points of trisection is at

( )12,0 and ( )9,4 ./The Infinity

∞∞∞

After graduating summa cum laude in Bachelor of Secondary Education (Math) course at WVSU under a DOST-SEI scholarship, he was immediately absorbed by Ateneo de Iloilo-Santa Maria Catholic School to teach Intermediate Algebra, Statistics, Trigonometry, and Analytic Geometry. He is presently finishing his Masters degree in Mathematics at University of the Philippines in the Visayas. This MI of his was also proofread by his mentor, Dr. Lourdes Zamora of UPV-CAS Division of Professional Education.-The Infinity Editor

mth section ..... from page 6Investigating... from page 4


some updates about the new policy, tips on how to implement the K-12 mathematics curriculum, and in-depth reflections as to the implication and intentions of the program.”

The first day of the convention, October 19, is intended for the Inter-Tertiary Quiz Bee in the morning. The formal opening program shall be held in the afternoon, followed by the talk on “Upgrading mathematics standards for k-12: building teachers skills in traning students for local, national and international mathematics competitions” to be facilitated in by Dr. Simon Chua, President of the Mathematics Teachers Guild (MTG)-Philippines.

The next day, October 20, will be greeted by a session on the “Basics of Math Curriculum,” followed by parallel sessions which will feature about 10 research papers that concern

K-12...from page 1

Infinity...from page 1

mathematics instruction. The afternoon assembly shall be used for the discussion on “Coping with K-12 Mathematics Education” to be led by Dr. Ian June Carces, a renowned mathematics professor of the Ateneo de Manila University. Business meeting which includes reports and election of new set of members of the Board of Directors will follow suit.

The last day, October 21, shall be graced by Dr. Purita Bilbao, director of the Center for Teaching Excellence of West Visayas State University to talk about the implication of the K-12 program in the tertiary level. The whole three-day affair shall be capped with a closing program thereafter.

To note, the convention is officially endorsed by the Department of Education and the Commission on Higher Education. /The Infinity

Excellence building at West Visayas State University, was participated in by mathematics teachers from all provinces of Western Visayas, reaching as far from Caluya Island, Antique, Boracay Island in Aklan, and Kabankalan City in Negros Occidental.

“We are so blessed with the immense help of our DepEd officials, most especially to Regional Director III Dr. Corazon Brown who helped us encourage teachers to join our seminar,” Dr. Herman Lagon, overall organizer of the event, said, adding, “in order to maintain a manageable size of participants, we had to regrettably decline the application of scores of teachers who were late in making the reservation, but we promise them to hold the same set of sessions next year.”

Themed “Conquering the Mystery of Mathematics Investigation (MI),” the mathematics workout is facilitated in by three main resource speakers who were trained by the University of the Philippines National Institute for Science and Mathematics Education (UPNISMED). They are Dr. Emellie Palomo, director of the Integrated Laboratory School of WVSU, Dr. Myrna Libutaque, mathematics teacher of Philippine Science High School-Western Visayas Campus, and Dr. Elvira Arellano, associate dean of the College of Education and director of the Center for Research in Science and Mathematics Education of WVSU. They are being assisted by about 10 more mathematics doctors, professors, and experts of the field coming from the MTAP-IC.

The first day of the training, April 12, focused more on inputs given by the resource speakers. Dr. Palomo covered the “Nature and Purposes of Mathematics Investigation,” “Types of Mathematics Investigation,” and “Stages in the Conduct of Mathematicas Investigation.” Dr. Libutaque followed suit and talked about “Conducting Mathematics Investigation,” and “Writing Proofs in Mathematics Investigation.” She also shared the results and findings of her dissertation on Mathematics Investigation entitled “MI Approach in Teaching Algebra in the Development of Problem Solving and Proof Writing and Mathematical Habits

Math...from page 1

In an e-mailed letter addressed to Lagon, the chief of the Bibliographic Services Division of the NLP Elizabeth Arevalo said, “We would like to inform you that we assigned ISSN 2244-3290 for the printed copy to your publication entitled ‘The Infinity.’ In this regard, it is necessary that the ISSN mentioned above should be printed on the front page of each

issue (top right hand corner) for easy identification and retrieval.”

With this, articles published in The Infinity will be considered formally logged to the NLP, hence, will have greater weight as far as authenticity of the written output is concerned.

“I humbly hope that more mathematics educators will send their articles to The Infinity for future publication,” said Dr. Lagon./The Infinity

Jardiniano of Western Institute of Technology, in that order. They joined the eight other BOD members who are to comprise the 15-strong MTAP-IC BOD for 2012.

“We are so pleased with the turn-out of the convention,” Dr. Alona Belarga, MTAP-IC president said, adding, “it was packed not just with updates on innovations and trends in teaching mathematics but

Teachers’...from page 2

also of teachers showing their commitment to indeed create a culture shift—a revolution of sort—in their own classrooms—making mathematics fun, positively challenging, appealing and most of all sensible, to students.”

“We just hope we have inspired mathematics teachers to wage their own little ‘mathematics revolutions’ in their classrooms as they deliver the beauty of

or blackness. We may not (and we should not) come to the point of mobbing someone to death, but our words could sometimes hurl some stunting effects to the learner’s intellectual growth.

I hope my random thoughts would make you, the

Of mathematical...from page 8

mathematics teacher, consider the importance of unlocking the mathematics language for the learners, and the significance of dealing with the gender issue. May mathematics be available to everyone no matter what language we speak or gender

of the Mind.”The last to give inputs

was Dr Arellano who discussed “Mathematical Habits,” “Using Investigation in the Classroom,” “Assessing and Formulating Rubrics,” “Format in Writing Mathematics Investigation,” and Writing Mathematics Investigation.” The first day was capped by preliminaries in MI writing workshop. Here, participants were grouped by six or seven to form a team that will eventually design, work on, and create their very own Mathematics Investigation.

The second day, April 13, was used entirely by the MI groups for the skirmishing, formulating of problem statements, making conjectures, and writting for proofs, among others. Four main facilitator-trainers guided them in the whole duration of the workshop.

After seven days (April 14-April 20) of independent group work, the teams went back to the CTE building Saturday, April 21, and formally presented their finished MI masterpieces before panel of experts namely Dr. Palomo, Dr. Arellano, Dr. Alona Belarga, and Mr. Mario Bañavo. These outputs were entitled “Over-Laughing Triangles,” “Oblong, Oblong, Where Do We Belong? (Oblong Numbers),” “Amazing Cubes,” Behind Reflections: Angles and Images,” “Total Perimeter of a Square and the Inscribed Squares Formed by Joining Endpoints,” Triangles from a Chorded Circle,” “(10n-1) Times ‘P’ in Seconds,” “Angles in a Fan,” “The Ladderized Cubes,” Rectangles: How Many Are You?,” “Twin in Grid Squares,” and “The Ultimate 8.” Part of the reporting was the power point presentation and an MI poster that showed the summary of their investigation.

Group 1 (Over-Laughing Triangles) was declared the landslide winner bagging the Best Presenter, Best MI and Best Poster awards. They were composed of Rowell Rublico, Stephen Jinon, Kim Jay Encio, and Jenever Nievares of PAREF-Westbridge School, Inc., Analie Guion of Buntatala National High School, Christina Carsula of Passi National High School, Christina Carsula and Rutchell Gania of Passi Montessori

International School, and Nezel Francisco of Colegio de San Jose. Their MI paper will be printed in The Infinity, the official publication of MTAP-IC.

The last day of the seminar was also capped with the oath-taking rites of the new set of MTAP-IC officers. Inducted were president: Dr. Alona Belarga, Director of Instruction and Quality Assurance of WVSU; vice president, Dr. Herman Lagon, subject area coordinator of Ateneo de Iloilo Santa Maria Catholic School (AdI); secretary, Prof. Rosemarie Galvez, college mathematics teacher of University of San Agustin (USA), treasurer, Prof. Rhodora Cartagena, head of the Mathematics and Physics Department of USA, and auditor, Engr. Ramon Jardiniano, mathematics professor of Western Institute of Technology (WIT).

Also took their oath were the other members of the Board of Directors, namely: Dr. Alex Facinabao, dean of the USA College of Education, Dr. Wilhelm Cerbo, associate dean of the WVSU College of Arts and Science, Engr. Ninfa Sotomil, mathematics professor of WIT, Mrs. Ma. Aries Pastolero, mathematics teacher of Iloilo National High School Special Science Class (INHS-SSC), Ms. Catalina Reales, elementary mathematics teacher of Maasin Central Elementary School, Engr. Roberto Neal Sobrejuanite, mathematics and computer professor of John B. Lacson Foundation Maritime University, Prof. Alfonso Maquelencia, mathematics teacher of USA, Prof. Alexander Balsomo, head of the mathematics department of the WVSU College of Arts and Science, Mr. Alex Jaruda, mathmetics teacher of INHS, and Dr. Harold Cartagena of the Iloilo City Community College (ICCC).

“It is just apt for us to be inducted before Sir Oquendo and the active participants of this MI seminar-workshop; we really hope that with you as witnesses, we will be inspired more to initiate worthy projects for the betterment of our mathematics education in Iloilo and in Region VI,” Dr. Belarga told the participants after the oath-taking rites./The Infinity

MTAP-IC BOD Meetings1st Regular Meeting: January 29, 2012 at Summerhouse, Midtown Hotel, Yulo Street, Iloilo City

Present: Dr. Alona Belarga (presiding officer), Dr. Herman Lagon, Mrs. Maria Aries Pastolero, Mr. Alex Jaruda, Engr. Ninfa Sotomil, Dr. Harold Cartagena, Ms. Rosemarie Galvez, Ms. Catalina Reales, Dr. Wilhelm Cerbo, Prof. Alexander Balsomo, Prof. Rhodora Cartagena, and Engr. Ramon Jardiniano

2nd Regular Meeting: April 9, 2012 at Summerhouse, Midtown Hotel, Yulo Street, Iloilo City

Present: Dr. Alona Belarga (presiding officer), Dr. Herman Lagon, Prof. Rhodora Cartagena, Dr. Wilhelm Cerbo, Prof. Alfonso Marquelencia, Prof. Alexander Balsomo, Ms. Rosemarie Galvez, Engr. Ramon Jardiniano

1ST Special Meeting: April 21, 2012 at Center for Teaching Excellence Building, WVSU, La Paz, Iloilo City

Present: Dr. Alona Belarga (presiding officer), Dr. Herman Lagon, Prof. Rhodora Cartagena, Mrs. Maria Aries Pastolero, Prof. Alfonso Marquelencia, Prof. Alexander Balsomo, Ms. Rosemarie Galvez, Engr. Ramon Jardiniano, Mr. Alex Jaruda, Dr. Harold Cartagena, Dr. Alex Facinabao, Engr. Ninfa Sotomil, Ms. Catalina Reales

3rd Regular Meeting: June 10, 2012 at Summerhouse, Midtown Hotel, Yulo Street, Iloilo City

Present: Dr. Alona Belarga (presiding officer), Dr. Herman Lagon, Dr. Wilhelm Cerbo, Prof. Alexander Balsomo, Prof. Alfonso Marquelencia, Prof. Rhodora Cartagena, Ms. Rosemarie Galvez, Ms. Catalina Reales, Mrs. Maria Aries Pastolero, Engr. Ninfa Sotomil, Mr. Alex Jaruda, Dr. Harold Cartagena

4th Regular Meeting: July 8, 2012 at Grand Dame Hotel, La Paz, Iloilo City

Present: Dr. Alona Belarga (presiding officer), Dr. Herman Lagon, Mrs. Maria Aries Pastolero, Mr. Alex Jaruda, Engr. Ninfa Sotomil, Dr. Harold Cartagena, Ms. Rosemarie Galvez, Ms. Catalina Reales, Dr. Wilhelm Cerbo, Prof. Alexander Balsomo , Prof. Rhodora Cartagena, Engr. Ramon Jardiniano, Prof. Alfonso Marquelencia

5th Regular Meeting: September 16, 2012 at Summerhouse, Midtown Hotel, Yulo Street, Iloilo City

Present: Dr. Herman Lagon (presiding officer), Prof. Alfonso Marquelencia , Ms. Rosemarie Galvez, Engr. Ninfa Sotomil, Dr. Harold Cartagena, Dr. Wilhelm Cerbo, Ms. Catalina Reales, Engr. Ramon Jardiniano, Prof. Rhodora Cartagena

2nd Special Meeting: October 7, 2012 at Summerhouse, Midtown Hotel, Yulo Street, Iloilo CityPresent: Dr. Alona Belarga (presiding officer), Dr. Herman Lagon, Engr. Ninfa Sotomil, Dr. Harold Cartagena, Dr. Wilhelm Cerbo, Ms. Rosemarie Galvez, Ms. Catalina Reales, Engr. Ramon Jardiniano, Prof. Rhodora Cartagena, Mr. Alex Jaruda, Prof. Alexander Balsomo, Mrs. Maria Aries Pastolero

3rd Special Meeting: October 14, 2012 at Summerhouse, Midtown Hotel, Yulo Street, Iloilo CityPresent:Dr. Alona Belarga (presiding officer), Dr. Herman Lagon, Engr. Nimfa Sotomil, Prof. Alfonso Marquelencia, Dr. Rosemarie Galvez, Engr. Ramon Jardiniano, Mrs. Ma. Aries Pastolero, Mrs. Rhodora Cartagena, Dr. Harold Cartagena, Ms. Catalina Reales.

Note: Meetings were done in three to four hours that start with the reading, revision, and approval of the minutes of the previous meeting, followed by the regular order of business./The Infinity

mathematics to their students—that will eventually cause students to love mathematics and use it to serve others.”

To note, the whole two-day affair was officially endorsed by the Department of Education (DepEd) and the Commission in Higher Education (CHEd) through a regional memorandum. The next convention is set October 19, 20, and 21, 2012 in Amigo Terrace Hotel, Iloilo City./The Infinity

we belong.∞∞∞

Rosemarie Galvez is interested to discuss more about the language barriers in the mathematics classroom. Email reactions to [email protected]. –The Infinity Editor


Parvane Mae Lagon

School Year 1) will be implemented in SY 2012-2013 and will progress in the succeeding school years.

• Grade 11 (HS Year 5) will be introduced in SY 2016-2017, Grade 12 (HS Year 6) in 2017-2018.

• The first batch of students to go through K-12 will graduate in 2018.Where will the additional

two years be added?• The two years will be

added after the existing four-year high school program. This will be called Senior High School.Why is the K-12 program

better than the current program?• K-12 offers a more

balanced approach to learning that will enable children to acquire and master lifelong learning skills (as against a congested curriculum).

• It will help in freeing parents of the burden of having to spend for college just to make their children employable.Will this address the drop-

out problem?• The decongested

curriculum will allow mastery of competencies and enable students to better cope with the lessons. This should partly address those who drop out because they cannot cope with schoolwork.

• The curriculum will be learner-centered, enriched, and responsive to local needs. It will also allow students to choose what suits their interest. This should partly address those who drop out because of lack of personal interest.

• DepEd will also continue to offer programs such as home schooling for elementary students and the dropout reduction program for high schools. These programs address the learning needs of marginalized students and learners at risk of dropping out. How will K-12 help in

ensuring employment for our graduates?

• The K-12 basic education curriculum will be sufficient to prepare students for work.

• The curriculum will enable students to acquire Certificate of Competency (COCs) and National Certifications (NCs). This will be in accordance to TESDA training regulations. This will allow graduates to have

Educational reform!?...from page 8

middle level skills and will offer them better opportunities to be gainfully employed.

• There will be school-industry partnership for techvoc tracks to allow students to gain work experience while studying, and offer the opportunity to be absorbed by the companies.How will the K-12 program

help working students (college level)?

• DepEd is in collaboration with CHEd to provide more opportunities for working students to attend classes.

• DepEd is working with the DOLE to ensure that jobs will be available to K-12 graduates and that consideration will be given to working students.How will the K-12 program

help students intending to pursue higher education?

•The K-12 basic education curriculum will be in accordance with the college readiness standards from CHEd which sets the skills and competencies needed of K-12 graduates who wish to pursue higher education.

• CHEd will download its general education subjects to Grades 11 and 12 (HS years 5 and 6) of K-12 ensuring mastery of core competencies for K-12 graduates. This may lead to a reduction in the number of years of college courses resulting to a decrease in educational expenses of households. How close is DepEd in

addressing the resource gaps (i.e. classroom, teachers)?

• DepEd has targeted to close the resource gaps in the next two years.

• Aside from increasing the budget of DepEd, it is also enjoying support from local governments, private partners, and donor agencies. How about the additional

cost to parents?• Grades 11 and 12 (HS

Years 5 and 6) will be offered for free in public schools.

• K-12 graduates will have higher earning potential since they will be more competent and skilled compared to graduates of the previous 10-year system.

• DepEd is in discussion with CHEd on the possibility of decreasing the number of years of certain courses in college.

• K-12 graduates will have national certification from TESDA, which will enable them to have higher employment opportunities.What will happen to the

college and universities during the 2-year transition period (SY 2016-2017 and SY 2017-2018)?

• DepEd is in the process of formulating a transition management plan which includes working in collaboration with other educational institutions during the two-year gap. The arrangements may include using private school facilities and teachers for senior high school during the transition period.

• DepEd is working closely with private educational institutions to address these transition management issues.Will senior high school be

implemented in existing high schools or will new schools be built?

• Existing schools will be used for the additional 2-year program. DepEd is likewise in discussions with CHEd, TESDA, and private schools to use their existing facilities during the transition period and beyond.Is K-12 required for private

schools as well? Will the same implementation timeline apply to private schools?

• Since private schools follow the DepEd curriculum, they will also be implementing the 12-year basic education program but the implementation plan will differ. This will be discussed with the representatives of the private schools.

• Private schools are active participants in developing the K-12 program.

• Note that a number of private schools offer at least 12 years of basic education: 2 years kindergarten, 6 to 7 years of elementary, and 4 years of high schools. How will the college and

technical-vocational courses be adjusted due to the K-12 curriculum? Will adjustments be made in time for the first graduates of K-12?

• TESDA will download some of its basic technical competencies while CHEd ill transfer the general education subjects to basic education.

• CHEd will be releasing its updated college readiness standards which will be the basis for the competencies in Grades 11 and 12 (HS Years 5 and 6).

• These activities will be completed before SY 2016-2017.What will happen to the

curriculum? What subjects will

be added and removed? • There will be continuum

from Kinder to Grade 12; and to technical and higher education.

• The current curriculum will be decongested to allow for mastery of learning.

• In Grades 11 and 12 (HS Years 5 and 6), core subjects like Math, Science, and English will be strengthened. Specializations in the students’ areas of interest will also be offered.

• Right now, a technical working group has formulated the new curriculum framework, standards, and competencies for K-12. Experts from CHEd, TESDA, and other stakeholders are part of this working group. After this, the changes in terms of subjects added, removed, and enhanced will be clearer.What specializations will be

offered in senior high school?•Among the specializations

offered will be on academics, middle-level skills development, sports, and arts.

• Specializations will also be guided by local needs and conditions. How will students choose

their specializations?• Students will undergo

several assessments to determine their interests and strengths. These will include an aptitude test, a career assessment exam, and an occupational interest inventory for high schools, and should help students decide on their specialization.For senior high school,

what will happen if majority of our students want to specialize in agriculture and only one is interested to take math or academics? How will this be accommodated?

• This is an extreme situation.

• The areas of specialization will be offered according to the resources available in a locality and the

needs of students.Will teachers be burdened

by additional teaching load due to the K-12 program?

• There will be no additional workload due to the K-12 program. The Magna Carta for Public School Teachers provides that teachers should only teach up to six hours a day.

• The decongested K-12 curriculum will allow students and will enable them to focus on their areas of expertise.How will teachers be

prepared for the K-12 program?• Teachers will be given

sufficient in-service training to implement this program. The pre-service training for aspiring teachers will also be modified to conform with the requirements of the program.I am sure that there are still

lots of questions unanswered in the issue of K-12. I am also certain that after months of fine-tuning, there might be things enumerated above that may not be germane to the demands of the present anymore. Perhaps, there are already changes in the policy, approach, or requirement in the new curricular design that make the K-12 of December 2011 outdated or inaccurate.

But that exactly is my point.The K-12 is a living organism,

still adapting, still evolving. Hence, the demand for us teachers to be at pace with the new educational policy is towering. To just wait and see is neither a convenience nor an option anymore.

Essentially, each teacher must take all possible initiatives and options to be informed about the new policies of the mutating K-12 in order not to be left biting the dust of change.

∞∞∞Dr. Lagon is a subject area coordinator and physics teacher of Ateneo de Iloilo-SMCS High School Department. He also trains gifted high school mathematics students in the Mathematics Trainers’ Guild (MTG) Iloilo. He may be reached through hermanlagon1@gmai l .com.-The Infinity Editor

“know thyself.” Thank you, Plato for that tip.

2. No more excuses. Kris Aquino said this to encourage women to get the cervical cancer vaccine. This applies also to those who wish to start writing. Yes, excuses abound.

Fac ing the laptop and ignoring the noise and the fun possibilities that the world has to offer, I asked myself several times if writing is indeed worth my time. Should I type while my garden is in need of weeding? Should I prioritize this over the pile of papers to check? Why should I be sitting when my tummy fats need trimming, my laundry is piling up and the furniture needs dusting? But if you want to have the writing job done, then prioritize. No more excuses.

3. Just do it. Start now. As the old adage goes, a journey of a thousand miles begins with a single step. A single letter, a single word, a phrase, a sentence, or a paragraph could go a long way. But there are just some moments when I had no motivation to write or I was too tired to think. To condition myself, I did tasks that did not require much creativity and thinking. For instance, I opted to search for related literature or organize my writing files. When the idea bulb lit up, I tapped the keyboard. If I forgot the right words, I used blanks

initially. Filling in the blanks was easier when I had written several sentences.

When the Pandora’s Box was opened there was no stopping of the escape of its contents. Similarly, when I opened my brains for ideas, I was surprised and overwhelmed wi th a l l the switches that turned on. I could feel my neurons making connections and sometimes my hand could not cope with the speed of my thoughts.

Write it now. Write it right. Write it wrong. Just write it.

Not numbered but very important is to recognize the power of health and prayers. Surv ive the wr i t ing s t ress unscathed. It took me a perforated ear drum to realize that health is precedence for any successful endeavor. Also, prayers could also bring in miracles. We might believe in different gods and we might doubt the existence of God but the healing process of prayer worked every time./The Infinity

∞∞∞

It turned out having a dose of one’s medicine worked out. Ms. Rosemarie Galvez successfully f i n i shed her d isser ta t ion , Quantifying artworks: an analysis of the mathematics principles used by Ilonggo artists.-The Infinity Editor

Getting...from page 12

NOVEMBER 2011 - OCTOBER 201218ARTICLE I – Name and DomicileSection 1 – The Association shall be known as the

Mathematics Teachers Association of the Philippines, Inc. – Iloilo Chapter (MTAP-IC) or referred to as MTAP-IC.

Section 2 – The office of the Association shall be located at the place/office/institution of the incumbent president or at such other places that the Board of Directors may designate.

ARTICLE II – MembershipSection 1 – Admission of Members. Any mathematics

teacher seeking admission into the Association must be proposed/vouched by any member in good standing and this proposal should appear in his/her application form to be forwarded to the office of the Association.The President has the power to accept new members subject to the approval of the Board of Directors. Membership will be effective after paying a membership fee to the Treasurer which will be dated back to January 1 of that year.

Section 2 – Duties and Responsibilities of the Members. The member shall have the following duties and responsibilities:a. To comply with the by-laws, rules and regulations

that may be promulgated by the Association from time to time;

b. To attend all the meetings of the Association that require their attendance; and

c. To pay membership dues and other assessment of the Association.

Section 3 Members. The board of directors shall consist of fifteen (15) members to be elected at large by a secret ballot by the members of the Association during the General Assembly. Upon assumption to office, the Directors shall elect from among themselves by secret ballot a President, a Vice President, a Secretary (preferably that he/she belongs to the same institution with the President), a Treasurer, and an Auditor.Any MTAP-IC member of good standing who already served for two consecutive terms are not eligible to run for re-election. However, he/she is qualified to run for the following year’s MTAP-IC BOD election.

Section 4 – Membership Dues. The annual dues of he members of the Association shall be determined and fixed from time to time by the Board of Directors with the approval of the majority of the members.

Section 5 – Resignations. Any member may resign from the Association by notifying the Secretary. There shall be no reimbursement of dues under these circumstances.

Article III – Board of DirectorsSection 1 – Members. The board of directors shall consist

of fifteen (15) members to be elected at large by a secret ballot by the members of the Association during the General Assembly. Upon assumption to office, the Directors shall elect from among themselves by secret ballot a President, a Vice President, a Secretary (preferably that he/she belongs to the same institution with the President), a Treasurer, and an Auditor. Any MTAP-IC member of good standing who already served for two consecutive terms are not eligible to run for re-election. However, he/she is qualified to run for the following year’s MTAP-IC BOD election.

Section 2 – Powers and Duties. The Board of Directors shall serve as the Governing Body of the Association. It shall have the power to:a. Establish policies and actions which promote the

welfare of the Association;b. Formulate necessary rules and procedures;c. Prepare plans and programs for the Association;d. Fix the time and place of meetings of the

Association; ande. Discharge such other responsibilities as it may

deem necessary.Section 3 – Term of Office. Members of the Board of

Directors shall serve a term of 2 years for those who got a top 8 rank during the election at large and one (1) year for those who got the ranks of 9-15. Any member of the Board may be reelected for another term.

Section 4 – Vacancies. If any vacancy shall occur among the Directors by reason of death, resignation or for any other reason, such vacancy shall be filled by a majority vote of the members of the Board of Directors constituting a quorum. The Director chosen shall serve only for the unexpired term of the position to which he/she is elected.Any member of the Board of Directors who incurred four (4) successive absences both in regular and special meetings of the Association without justifiable reasons will be subjected for expulsion. The remaining members of the Board of Directors are authorized to appoint or designate from among the members to replace the expelled member of the Board.

Section 5 – Quorum and Meetings. The Board of Directors shall meet at least once every two months. Meeting shall be held at the call of the President or on a written petition signed by at least three members of the Board.

By-Laws

Eight members shall constitute a quorum.Special meetings of the Board may be called by the President. Notices of any such meeting must be given at least five (5) days in advance of the date of the meeting.Minutes of all meetings of Board shall be kept and

carefully preserved as a record of the business transacted at such meetings and must contain such entries as are required by law.

Section 6 – Powers of the Board of Directors. The Board of Directors shall exercise the powers of the Association, conducts its management, administer its properties, and exercise such power as are herein conferred by this by-laws. At the annual meeting, the Board shall approve the new members accepted by the President.

ARTICLE IV – Executive OfficersSection 1 – Executive Officers. The Executive Officers of

the Association shall be President, Vice President, Secretary and Treasurer, and an Auditor, all of whom are members of the Board of Directors.

Section 2 – Election and Vacancy. The executive officer shall be elected by the Board of Directors at their first meeting after their election. Every officer including the President shall be subjected to removal at any time by majority of all the Board of Directors, but all officers, unless removed, shall hold office until their successors are appointed or elected and qualified. If vacancy shall occur among the officers of the Association, such vacancy shall be filled by the Board of Directors constituting a quorum at a special meeting called for the purpose, provided, such officers elected shall serve only the unexpired term of their predecessors.

Section 3 – The President. The President shall be the Chief Executive of the Association. He/she shall:a. Preside at all meetings of the Board of Directors and

of the members, and enforce the by-laws;b. Exercise general supervision of the business affairs

and property of he Association;c. Appoint and dissolve all committees for the

Association and be a member ex-officio of all committees;

d. Countersign all orders drawn and signed by the Treasurer for the payment of money, and countersign all checks drawn and signed by the Treasurer;

e. Call special meetings of the Board of Directors and members when necessary;

f. Make an annual report to the Board of Directors and to the members; and

g. Perform such duties as may be imposed on him/her by the Board of Directors.

Section 4 – The Vice President. The Vice President shall preside in the absence or inability of the President and shall execute all the duties of the President whenever the latter may delegate that authority. He/she shall perform such duties as may be imposed on him/her by the Board of Directors.

Section 5 – The Secretary. The Secretary shall:a. Keep a record of all proceedings, such as minutes of

meetings and activities of the Association/ Board of Directors in a book or books kept for the purpose. This record shall be open at all times to the Board of Directors;

b. Keep a true and accurate record of the name, addresses, and offices of the members;

c. Shall take charge of the reports, records, documents and the papers as the Board may direct;

d. Shall perform such other duties as are incidental to his/her office or are properly required by the Board of Directors and by the President;

e. Shall attend to the giving and serving of all notices; and

f. Shall assist the President in the execution of his/

her duties.Section 6 – The Treasurer. In addition to whatever duties

the Board of Directors may impose, the Treasurer shall:a. Collect and receive all moneys due the Association

from any source whatever and accurately the same;

b. Keep and record of the indebtedness of each member of the Association and the amount receive from each on forms provided for that purposes;

c. Deposit in the name of the Association, in an approved bank, all moneys received and report at each meeting the moneys received and spent since the last meeting. All bank deposits withdrawals should be properly certified and presented to the President.

d. Keep an account record of all financial business transacted by the Association. These records shall be open at all times to the inspection of the Board of Directors. The Treasurer shall pay by check all bills and obligations of the Association on order of the President.

Section7 – The Auditor. The Auditor shall:a. Audit all receipts of disbursements and expenditures

of the Treasurer;b. Inspect the book of accounts of the Treasurer;c. Countersign financial reports and records of the

Treasurer; andd. Perform such other duties assigned to him/her by

the President.ARTICLE V – AdviserSection 1 – There shall be advisers of the governing body.

The adviser shall be composed of the incumbent math supervisors of the Divisions of Iloilo, Iloilo City, and Passi City, the immediate MTAP-IC past President, or whoever the Board of Directors may designate.

ARTICLE VI – The Election CommitteeSection 1 – There shall be a Committee on Election to

conduct the Election of Officers. The members of the Board of Directors are authorized to appoint three (3) members to compose the Committee. The duties and functions of the Committee will be prescribed by the Board of Directors.

ARTICLE VII – Members’ MeetingSection 1 – There shall be annual meeting for all members

of the Association called by the President on the day and at the time approved by the Board of Directors. At this meeting, the annual report of the President should be made. This report should cover in general the entire work of the Association for the past year.

Section 2 – At the annual meeting, a program should be presented in keeping with the purpose of the Association.

Section 3 – Special meetings may be called by the President according to need.

Section 4 – Notice of the time and place of the annual or special meetings of the members, shall be given in the same manner as provided with regard to the Board of Directors.

Section 5 – A quorum at any meeting of the members shall consist of a majority of the members and a majority of such quorum shall decide to any question at the meetings, except in those matters when the law requires the affirmative vote of a greater proportion.

Section 6 – At every meeting of the members of the Association, every member is entitled to one vote only.

Section 7 – Minutes of all meetings of the members shall be kept and carefully preserved as a record of the business transacted.

ARTICLE VIII – Investments and AccountsSection 1 – No investment of any character shall be made

without the approval of the Board of Directors and/or the members as the case may be.

Section 2 – All books, accounts, and records of the Association shall be open to inspection by any member of the Board of Directors at all times and to the members at the reasonable time during office hours.

Section 3 – No director or member shall be entitled to or benefited by any of the profits of the Association except as a reasonable compensation for services rendered.

ARTICLE IX – Fiscal YearSection 1 – The fiscal year of the Association shall begin

on the first day of January in each year and shall end on the last day of December of that year.

ARTICLE X – Annual FeeSection 1 – The annual fee of the members shall be two

hundred pesos (P200.00) subject to the changes determined by the Board of Directors. Failure to pay for the annual fee for the year shall result in being dropped from membership in the Association.

ARTICLE XI – Amendment of By-LawsSection 1 – These By-Laws or any part thereof may be

amended or repealed by a majority of all the members at any meeting called for that purpose. The power to amend, repeal or adopt new by-laws may be delegated to the Board of Directors for consideration and deliberations./The Infinity

Mathematics Teachers Association of the Philippines, ILOILO CHAPTER (MTAP-IC), Inc.

Please refer to the official blogsite of the organization, http://www.mtapiloilo.blogspot.com/, for the

financial report. Other real-time updates are also posted in this blog for every member’s perusal and appreciation.

CONSTITUTIONand


PROF. ALFONSO S. MARQUELENCIA, Board Member, is an Assistant Professor 1 at the University of San Agustin. He is a graduate of Bachelor of Secondary Education major in Mathematics at West Visayas State University in 1992. In 2005, he finished the degree

DR. HERMAN M. L AGON, Vice President, is a graduate of Bachelor of Science in Civil Engineering (Most Outstanding Engineering Graduate) in University of Iloilo. He finished his Master of Arts in Science Education in the same school in 2008 and his PhD in Science Education Major

ENGR. NINFA S. SOTOMIL, Board Member, is a Professor at Western Institute of Technology. She finished the the degree of Bachelor of Science in Chemical Engineering, With Highest Distinction in 1980, Bachelor of Science in Mathematics, With Highest Distinction, in 1985

MRS. MARIA ARIES A. PASTOLERO, Board Member, is a Master Teacher 1 at Iloilo National High School. She earned her Bachelor of Secondary Education Major in Mathematics at West Visayas State University, with units in Master of Arts in Mathematics at

MS. ROSEMARIE GALVEZ, Secretary, is a graduate of Bachelor of Secondary Education Major in Mathematics (Magna Cum Laude & Rotary Awardee for Outstanding Graduate) at West Visayas State University. She clinched the top seven spot in the Professional

DR. ALONA M. BELARGA, President, holds the academic rank of Professor 1 at West Visayas State University College of Education. She finished the degree of Bachelor of Science in Education (BSED) major in Mathematics at West Visayas State University in 1984

DR. ALEX B. FACINABAO, Board Member, is the Dean of the College of Education of the University of San Agustin since 2009. He was chairperson of mathematics and physics in the same school prior to his present assignment. He is a graduate of BS in Math Education

MS. CATALINA M. REALES, Board Member, is a Master Teacher 2 at Maasin Central Elementary School and a District Mathematics Coordinator of the District of Maasin, Iloilo. She earned her Bachelor of Elementary Education Integrated Teacher Education

MR. ALEX A . JARUDA , Board Member, is a Master Teacher 1 at Iloilo National High School. He was awarded by the Japan Ministry of Education a MONBUSHO scholarship for the Teacher Training Course in Mathematics. He earned the degree of Bachelor

E N G R . R O B E R T O N E A L S . SOBREJUANITE, Board Member, is faculty member of the Natural Sciences Department and the Quality Assurance Manager of the John B. Lacson Foundation Maritime University (Arevalo), Inc..

MS. HERMOSISIMA L. ALTILLERO, Adviser, is an Education Supervisor (Mathematics-Secondary) of the Department of Education, Division of Iloilo. Prior to her appointment as Education Supervisor, she once served as Principal of Major Manuel A. Aaron Memorial National High School in Janiauy and Head Teacher of Cadagmayan National High School in Sta. Barbara.

MRS. LIGAYA H. MONTELIJAO, Adviser, is an Education Program Supervisor in Mathematics in the Division of Iloilo City. She finished the degree of BSED major in Math at West Visayas State University.

DR. KIM S. ARCEÑA, Adviser, is the Education Supervisor (Mathematics-Elementary) and Research Coordinator of the Department of Education in the Divison of Iloilo. He is a graduate of Bachelor of Elementary Education (BEED) with specialization in Mathematics, Master of Education (MA Ed) in Administration and Supervision, and Doctor of Education (Ed. D.) major in Educational Management at West Visayas State University. Dr. Arcena has attended a JICA Training Program for Young Leaders (Education Category) in Kochi, Japan in 2007, the Regional Training of the TIP Teams on the Mass Implementation of the Teachers Induction Program, and the DedpEd-MTAP National Conference on Mathematics Education in 2008. He is a writer of the Division and Regional Mathematics Module, and the Module on Local Taxation in the Division of Iloilo, and has constructed the Division test Reviewer in Mathematics for the National Achievement Test (NAT)./The Infinity

ENGR. RAMON S. JARDINIANO, Auditor, is a graduate of Western Institute of Technology (WIT) with distinction and became a faculty of the Department of Mathematics of the same school since 1982. A mechanical engineer by profession, he also completed his academic

PROF. RHODORA A. CARTAGENA, Treasurer, is the chairperson of Math and Physics Department of the University of San Agustin and the in-charge of the USA Statistical Analysis Center. She is a graduate of Bachelor of Science in Applied Mathematics and Master

PROF. ALEXANDER J. BALSOMO, Board Member chairperson of the Department of Mathematics of West Visayas State University, is a graduate of Bachelor of Science in Applied Mathematics and Master of Education (Mathematics) in University of the Philippines in the Visayas. He is engaged in pure

BOARD OF DIRECTORS

2012

DR. HAROLD F. CARTAGENA, Board Member, is the department head and faculty member of Iloilo City Community College. He graduated cum laude in Bachelor of Secondary Education (Math) at West Visayas State University under the SM Foundation and COEXSTEP

(Magna Cum Laude) as WVSU-Teacher Education scholar. In 1995, she finished her Master of Arts in Teaching Mathematics at the University of the Philippines, Diliman, Quezon City as Academic Excellence Awardee under the Department of Science and Technology Science Education Institute (DOST-SEI) Scholarship. In AY 1996-1998, she was awarded a MONBUSHO scholarship by the government of Japan for her post graduate in teacher training program at Ehime University, Matsuyama City, Ehime, Japan. She earned her Doctor of Philosophy in Mathematics Education (PhD Mathematics Education) in 2001 at the University of the Philippines in Diliman, Quezon City as DOST-SEI scholar and as Academic Excellence Awardee. Dr. Alona M. Belarga served as Associate Dean of the WVSU College of Education Graduate School in June 2007-June 2010, and as chair of the Mathematics Education Division from June 2010 to Ocotber 2011. She serves as the Director of Instruction and Quality Assurance of the WVSU from November 2011 to the present.

in Mathematics in 2011 at WVSU under the DOST-SEI scholarship. Dr. Lagon is formerly the president of WDSA, an association of post-grad DOST-SEI scholars in WVSU, from 2008-2010, and was awarded Most Outstanding Teacher in Region 6 (PRISSAAP) and in the Philippines (PERAA), and Most Outstanding School Paper Adviser in the region (PIA) and in the country (DepEd). He worked as a professional journalist for six years before teaching physics (and supervising the science program) in Ateneo de Iloilo High School (2000-present). He is presently the editor in chief of MTAP-IC’s Infinity, and is about to finish his Master in Business Administration (MBA) course.

Licensure Examination for Teachers (PLET) in 2004 and earned her Master of Education Major in Mathematics at the University of the Philippines in the Visayas. She is a former high school teacher of Sun Yat Sen High School and Ateneo de Iloilo, repectively. She served as the President of WDSA, an association of post-grad DOST-SEI scholars in WVSU from November 2011 to June 2012. Currently, she teaches college math in University of San Agustin. She has finished defending her dissertation for the degree PhD in Science Education-Mathematics under a DOST-SEI scholarship in WVSU.

of Education (Mathematics) in University of the Philippines in the Visayas. She is currently finishing her PhD in Science Education (Mathematics) at WVSU.

requirements for Master of Engineering in WIT last year.

math research and sought-after lecturer in statistics and actuarial math. Sir Alex is a candidate for the PhD Math program in Ateneo de Manila University under the Commission on Higher Education scholarship. He is also currently an active officer of the Mathematics Society of the Philippines (MSP) and Philippine Statistics Association (PSA).

Scholarships. He is a graduate of Master of Education (Mathematics) of University of the Visayas and PhD in Science Education (Mathematics) in WVSU under the DOST-SEI ASTHDP-SECC Scholarship.

at Western Institute of Technology and Master of Arts in Education at West Visayas State University. Amid his Doctor of Education degree for Curriculum, Instruction, and Evaluation taken in WVSU, he now finishes his PhD in Science Education Major in Mathematics course in the same university. He is currently the Business Manager of Association of Deans of Teacher Education Institutions, Inc. (ADTEI) Region VI.

of Secondary Education Major in Mathematics and is presently enrolled at West Visayas State University for his Master of Arts in Education Major in Mathematics.

of Master of Arts in Education major in Mathematics at the University of San Agustin. In 1998, he finished the Bachelor of Laws (LL.B) at the University of Iloilo. Presently, he is finishing the degree PhD in Science Education major in Mathematics at West Visayas State University.

the University of the Philippines in the Visayas. She was a recipient of the MONBUSHO Scholarship of the Japanese Ministry of Education from October 2000-March 2002 in Kyushu University (Japanese Language Course) and Fukuoka University of Education (Teacher Training Course in Mathematics). She worked as an Assistant Language Teacher in Tokyo, Japan from 2003-2005. Presently she is the Subject Chairperson in Math of the Special Science Class-Iloilo National High School.

Program (BEEd-ITEP) at the West Visayas State College and her Post Graduate at West Visayas State University. She is a Scholarship grantee of the University of the Philippine Institute of Science and Mathematics Education Department (UP-ISMED) in 1993.

He is a graduate of BS Civil Engineering at Western Institute of Technology and Master of Science in Maritime Education at the John B. Lacson Foundation Maritime University. He has completed the academic requirements for the degree of Doctor of Rural Development at Iloilo State College of Fisheries.

and Bachelor of Science in Computer Education in 2002 at Western Institute of Technology (WIT). She holds the degree of Master in Engineering Education and has completed the requirements for the degree the PhD in Science Education major in Mathematics at West Visayas State University. She is presently an associate professor of WIT.

DR. WILHELM P. CERBO, Board Member, holds the academic rank of Professor 3 and presently the Associate Dean of West Visayas State University College of Arts and Sciences. He is a graduate of Bachelor of Secondary Education Major in Mathematics in 1993 as a University Scholar, Master of Arts in Education in 2003 and Doctor of Education Major in Curriculum Instruction and Evaluation in 2008 at the West Visayas State University. As Training Affiliate of the Statistical Research and Training Center

(SRTC), he spearheaded several trainings for the improvement of Statistics and Statistics Education in the region such as Statistics for Gender Responsive Local Development Planning and Statistics for Poverty Analysis among the Local Government Units. He also serves as Chair of the Philippine Statistical Association (PSA) Region 6 Chapter and sits as member of the PSA National Board representing the different PSA Chapters in the Philippines since 2009.


T h i s p h o t o c o l l a g e presents the participants in the MTAP-IC’s three-

day Mathematical Investigation Seminar-Workshop held April 12,

13, and 21, 2012 at the Center for Education Excellence building at West Visayas State University. Working their heart out to design the most elegant and authentic Mathematical

Investigation that they can imagine, the part ic ipants

found the workshop—as the final evaluation reveals—to

be humbling, challenging, enlightening, exciting, and lasting.

The Infinity file photos

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