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Fast-track promotion through a series of increasingly responsible administrative management positions in Constitution’s government affairs and lobbying division. Worked directly with the Director to provide executive-level administrative and operating management support.

Planned, staffed and directed all office management functions for the Department. Scope of responsibility was diverse and included A/P, A/R, budgeting, petty cash and corporate checking accounts. Recruited, trained and supervised administrative staff. Developed procedural and documentation manuals.

Managed confidential correspondence, appointments, meetings and schedule for the Director. Personally planned and coordinated industry and inter-company meetings involving the Corporate Affairs Department.

Independently researched, responded to and followed up on requests from Congress and company management.

Worked cooperatively with Corporate Secretary to direct the planning/execution of the 1986 and 1988 annual shareholders meetings.

Administered the Voluntary Contributors for a Better Government Program, nation- wide employee fundraising effort to increase support for corporate lobbying and legislative initiatives. Directed all Federal Election Commission filings and State Election Committee filings for the corporation’s Political Action Committee.

PREVIOUS PROFESSIONAL EXPERIENCE included several responsible administrative and customer service management positions. Completed a three-year position with Engravers of Europe, managing sales and service relationships with foreign embassy personnel, govern- ment officials and corporate VIPs.

EDUCATION
University College /University of Mawland, College Park, Maryland
Business Courses ( 1989 to Present)

I’M’ Business Institute, Bethesda, Maryland Secretarial & Administrative Training Program ( 1977)

Montgomerv College, Rockville, Maryland Music Major (1969 to 1970)

Taken From :100 Winning Resumes for $ 100,000 + Jobs

CAREER PROFILE
Over 15 years experience planning and directing executive-level administrative affairs and support to Chairmen, Boards of Directors and Senior Management. Combines strong planning, organiza- tional and communications skills with the ability to independently plan and direct high-level business affairs. Qualifications include:

Shareholder Meetings & Communications Executive Office Management
Regulatory Reporting & Communications Staff Training & Development
Corporate Political & Legislative Affairs
Customer Communications & Liaison Affairs Special Events Planning
Confidential Correspondence & Data Special Project Management
Budgeting & General Accounting

Proficient in the use of Wordperfect for Windows 5.2. Experienced with Lotus 1-2-3.

PROFESSIONAL EXPERIENCE
INTERSTATE SAVINGS BANK, Silver Spring, Maryland 1989 to Present
Corporate Secretary Executive Assistant to the Chairman and Board of Directors High-profile, executive-level administrative position supporting the Chairman, Board of Directors and other top management personnel throughout the organization. Scope of responsibility is diverse and includes Board affairs, customer and stockholder commu- nications, employee stock options, special events, regulatory reporting and executive administration.

Executive Liaison between Chairman and Senior Management Committee, Business Departments, and employees to plan, schedule and facilitate a broad range of corporate initiatives, company operations and large-scale business functions.

Handle confidential operating and financial information, maintain corporate records and minutes, and execute corporate contracts and agreements.

Project Supervisor for annual shareholders report. Coordinate annual shareholders meetings, manage liaison affairs with outside counsel regarding proxy statements, and facilitate print production of shareholder communications.

Maintain/update files for regulatory review, oversight and approval.
Provide training and supervision to Executive Department’s support staff.

Taken From :100 Winning Resumes for $ 100,000 + Jobs

Transitioned from manual operations to complete automation with the introduction of IBM mainframe technology. Significantly improved the ability to efficiently manage and catalog volumes of legal documentation while reducing staffing requirements by more than 10%.

Designed flexible administrative systems and processes to meet growth and expansion demands as the firm more than doubled its legal staff and client base.

Risk Manager
CITY OF ST. AUGUSTINE, St. Augustine, Florida 1978 to 1984
Directed a sophisticated internal risk management audit program of all City facilities and operations. Evaluated regulatory and safety compliance, coordinated on-site inspections with external agencies, and worked with City Attorney on pending lawsuits and litigation.

Earned a 1979 “Certificate of Appreciation” for service to the municipality.

EDUCATION.
MBA, University of South Florida, 1983
BS, Eastern Kentucky University, 1978
Completed numerous continuing professional education programs, workshops and seminars on topics including Communications, Negotiations, Information Systems Technology, LAN & Commu- nications Technology, Business Management and Profitability.

PROFESSIONAL AFFILIATIONS:
American Management Assn
American Bar Assn
American Marketing Assn
Medical Group Management Assn
American Academy of Procedural Coders
American Computer Systems & Technology Professionals
American Academy of Medical Administrators
American Institute of Professional Bookkeepers

Taken From :100 Winning Resumes for $ 100,000 + Jobs

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Redesigned and expanded capabilities of internal PC network through implementation of LAN client/server technology and a fully-integrated medical practice management program (ELCOMP). Resulted in tremendous improvements in operating efficiency and a $150,000 reduction in annual payroll costs.

Managed the selection, negotiation and acquisition of over $350,000 in capital technology investment.

Negotiated vendor contracts and maintained stable pricing for four consecutive years.

Designed and led in-house training programs on topics including PC technology, credit and collections, administration and OSHA regulations.

Authored business proposals to build partnerships and strategic alliances with health care provider organizations throughout the region.

Structured and negotiated contracts with providers, vendors and business partners.

Director of Administration
BROWN, GRANT, JACKSON & LEE, Lancaster, Pennsylvania 1990 to 1992
Planned, staffed, budgeted and directed all administrative functions for a law firm with three offices in Pennsylvania and a staff of 200+. Scope of responsibility was diverse and included human resources, employee relations, MIS and communications technology, accounting and financial reporting, purchasing, office services/ administration, marketing and client relations.

Managed a $1.1 million contract for the architectural design, layout and construction of a new law firm. Brought project in two weeks ahead of schedule and 10% under budget.

Directed the selection and implementation of a VINES network system on PC server and numerous other software, applications and technology development projects. Total capital investment exceeded $200,000.

Designed and produced marketing and direct mail communications in conjunction with several business/ client development programs.

Worked cooperatively with the firm’s partners to plan long-term operating, administrative, financial and marketing strategies.

Taken From :100 Winning Resumes for $ 100,000 + Jobs

QUALIFICATIONS PROFILE:
ADMINISTRATIVE DIRECTOR with over 10 years experience managing cross-functional business affairs for large service organizations. Expertise includes:

PC, MIS & Network Technology Human Resource Affairs
Administrative Policies & Procedures Finance, Budgeting & Banking
Front & Back Office Operations Client & Public Relations
Purchasing & Vendor Negotiations Joint Ventures & Partnerships
Marketing & New Business Development Facilities Management
Delivered strong operating results in productivity and efficiency improvement, technological advancement, cost reduction, and revenue and profit performance. Excellent training, supervisory and leadership skills. Organized and skillful project manager.

PROFESSIONAL EXPERIENCE:
Administrator
MEDICAL ASSOCIATES, INC., York, Pennsylvania 1992 to Present
Senior Administrative Director responsible for the strategic planning, staffing, budgeting and management of all administrative and business affairs for a specialized medical practice. Design and implement systems, processes and procedures for budgeting and financial affairs, office services, billing, records management, facilities management, purchasing, inventory control, and the complete human resource function (e.g., recruitment, training, scheduling, performance appraisal, benefits, compensation). Supervise a staff of 18.

Taken From :100 Winning Resumes for $ 100,000 + Jobs

Realigned key accounting positions, consolidated similar functions, avoided need for increased staffing, and improved information processing and reporting.

Designed and implemented a series of standards, policies and systems to more efficiently manage accounting/ financial data collection, analysis and reporting.

PUBLISHERS INC., New York, NY April 1984 to May 1986
Staff Accountant
Diversified accounting and financial reporting responsibilities with this nationwide book publisher and distributor. Analyzed and reconciled accounts, prepared journal entries, calculated monthly sales summary for 35 field sales representatives, and participated in month-end closing.
Computed and reported monthly commission, bonus and royalties. Conducted quarterly audit of consigned products with 20 book depositories nationwide.

CAPITAL JEWELERS, INC., New York, NY March 1983 to April 1984
Bookkeeper
Managed accounts payable, accounts receivable and billing for this NYC-based jewelry wholesaler. Prepared cash deposits, bank reconciliations and journal entries. Worked with Assistant Control- ler on month-end closings and month-end reporting.

EDUCATION:
B.S., Business / Major in Accounting, San Sebastian College, Manila, Philippines, 1982

Continuing ProfessionaZ Deve Zopment:
Graduate of numerous professional training programs, seminars and workshops on topics including management, organizational development, credit, collections and related legal affairs. Several courses were sponsored by Dun and Bradstreet.

Taken From :100 Winning Resumes for $ 100,000 + Jobs

Redesigned procedures to collect on past due accounts and increased cash flow by 37%.

Consistently surpassed all credit and collection targets. Maintained DSO of 35 days or less.

Captured a $250,000 annual cost savings through development/implementation of an Adver- tising Management System to integrate and consolidate billing, credit and collection informa- tion from three autonomous business units.

Designed a series of internal reporting mechanisms to measure sales, credit and collection performance.

SANFAX, Los Angeles, CA May 1986 to March 1990
Accounts Receivable Supervisor (1989 to 1990)
Lead Coordinator - MIS Systems (1988)
Lead Cash Processor (1987 to 1988)
Cash Processor (1 986)

Fast- track promotion through a series of increasingly responsible accounting, MIS and manage- ment positions with this $150 million fax machine manufacturer and wholesaler. Advanced to a final position training and supervising a staff of 10, where we provided accounting, credit and collections support for a 75-person customer service department.

Prepared monthly sales and cash receipts reports, account analyses and journal entries. Autho- rized customer credit adjustments and bad debt write-offs. Acquired extensive experience in customer communications and collection negotiations.

Taken From :100 Winning Resumes for $ 100,000 + Jobs

CAREER PROFILE
Accounting Professional / Accounting Department Supervisor with 1 3 years of progressively respon- sible experience with high-growth, turnaround and mature corporations. Qualifications include:

Accounts Receivable Credit & Collection Billing/ Invoicing
Accounts Payable Customer Service Financial Analysis/ Reporting
Account Reconciliation Cash Application C hargeback/ Adjustmen t s
General Ledger Staff Training/ Development Team Building/ Leadership

Delivered strong and sustainable operating, financial and service gains through expertise in systems design/implementation, standardization, workflow optimization and long-range planning. Excellent analytical, problem-solving and negotiating skills. PC proficient with IBM System 38, HP 9 18 Image Data, Lotus and Word.

PROFESSIONAL EXPERIENCE:
INVESTOR’S, INC., Los Angeles, CA March 1990 to December 1996
Accounts Receivable Supervisor
Recruited as Assistant Supervisor for the Accounts Receivable Department of this rapid growth national daily newspaper. Challenged to design and implement the systems, policies, procedures and technologies to regain control of the A/R function and establish a professional business unit. Promoted to Supervisor within six months. Assumed increased responsibilities for accounting, financial planning/reporting and MIS operations.

Scope of responsibility was diverse and included all billing, credit and collection activity for 1500 active accounts. Prepared monthly financial schedules, graphs, journal entries, sales commis- sions, and account analyses. Reviewed and approved credit applications, advertising agency dis- counts and advertising contracts. Monitored cash receipts application, contract adjustments and account reconciliations.

Worked in collaboration with Advertising Sales Department to provide data relevant to account history and credit status. Personally communicated with customers nationwide to resolve billing discrepancies and expedite collections. Consulted with executive management to establish corporate policy impacting A/R and credit operations.

Taken From :100 Winning Resumes for $ 100,000 + Jobs

Instrumental in the design and implementation of a computerized budget model.

INDUSTRIAL CREDIT CORPORATION, Stamford, Connecticut 1972 to 1982
Assistant Accounting Manager ( 1976 to 1982)
Personnel & Payroll Supervisor ( 1 974 to 1976)
General Ledger Supervisor ( 1972 to 1974)

Promoted rapidly through a series of increasingly responsible accounting and personnel management positions for this $550 million diversified leasing and finance company. Managed monthly and annual finan- cial statements, general ledger, loan reporting and management of a 15,000 account lease receivable system.

Administered pension, health insurance, stock purchase and credit union plans. Filed payroll tax returns in 21 states and to the Federal Government.

Designed and implemented daily accounting system to control $150 million of commercial paper and $9 1 million of long-term debt.

Championed implementation of on-line cash application system.

OLMER CORPORATION, New Haven, Connecticut
Promoted through several positions in Accounts Payable and Payroll.

EDUCATION:
B.A., Accounting
Quinnipiac College, Hamden, Connecticut, 1969
1969 to 1972

PERSONAL PROFILE:
Native of Cuba. U.S. Citizen since 1983. Fluent Spanish.

Taken From :100 Winning Resumes for $ 100,000 + Jobs

Created a complex system of internal and external partnership accounting responsive to the needs of the corporation and compliant with all SEC regulations.

Facilitated implementation of general accounting system and subsequent selection, development and imple- mentation of a state-of-the-art, PC-based lease accounting system. Managed parallel system opera- tions through transition to new technology.

Created the corporation’s human resource policies governing recruitment, retention, payroll, commis- sion, benefits, EEO, performance reviews, timekeeping, workers compensation and disability. Defined criteria for employment contracts and administration.
.
Negotiated policies and premiums with six different insurance carriers to optimize coverage and continue fixed costs to the corporation.

INVESTMENT GROUP, INC., New York, New York 1983 to 1987
Accounting Supervisor ( 1986)
Senior Staff Accountant (1983 to 1986)

Joined this investment management firm specializing in equipment leasing, limited partnerships, real estate investments and other tax shelters. Maintained accounting and financial records for Investment Group, 7 aMiliated companies, 38 limited partnerships and 3 real estate ventures. Managed staff of four.

Acquired expertise in inventory leased equipment, wrap leases, cash distributions, cash management, investor negotiations and letter of credit transactions for non-resource loans.

Taken From :100 Winning Resumes for $ 100,000 + Jobs

CORPORATE ACCOUNTING & FINANCE EXECUTIVE
Start-up, Turnaround & High-Growth Corporations
Dynamic professional career in strategic planning, development and leadership of complex accounting, budgeting, cash management, payroll, financial analysis and reporting. Successful in linking accounting and finance to business operations to drive forward organizational development, control costs and improve earnings. Strong project management, team building and leadership skills. Expert PC qualifications with proficiency in Real World, Excel, Lotus 1-2-3, Microsoft Word and Wordperfect. Extensive experience in systems selection and implementation.

PROFESSIONAL EXPERIENCE:
INVESTMENT CAPITAL CORPORATION, Greenwich, Connecticut 1987 to Present

Assistant Controller (1 995 to Present)
Manager - Accounting, Employee Benefits &Payroll ( I990 to 1995)
Accounting Manager ( 1 987 to 1990)

Recruited to affiliate company of Investment Group, Inc. (past employer) to direct the start-up of a complete corporate accounting function for this newly-formed syndicator. Assume additional accounting,
financial and human resource responsibilities as the company has grown from 5 to 87 employees and raised over $1 70 million in equity sales.

Serve as the senior accounting professional in the corporation reporting directly to CFO and Controller. Scope of responsibility is diverse and includes accounts payable, accounts receivable, general ledger, payroll,
monthly sales commission, and preparation of year-end financial documentation for external audit review, Coordinate accounting systems implementation and upgrades. Direct a staff of six.

Administer the entire cash management function for both company operations and its six limited partner- ships. Manage banking relationships to execute and administer escrow accounts. Coordinate fund transfers, cash receipts, disbursements and monthly reconciliations. Concurrent responsibility for the de- sign, administration and management of all employee benefit programs and related regulatory affairs. Con- tinue to provide cash management and accounting support to parent corporation on critical operating and long-term planning i sues.

Taken From :100 Winning Resumes for $ 100,000 + Jobs

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In CARiD you will find auto accessories from many major car manufacturers such as Acura, Audi, Cadillac, Chevy, and many more. The auto accessories are available for inside and outside of the car such as custom grilles, body kits, tail lights, wheels, floor mats, and many more.

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Restructured and tightened the company’s cash management policy to ensure prompt recovery of all receivables to meet cash requirements of payroll and vendor commitments.

Renegotiated payment schedule to recover $2 million from an Italian company. Personally collected first $500,000 payment and established framework for subsequent collection of all outstanding funds.

Redesigned Bill of Materials for all products (in cooperation with Engineering Department) to facili- tate development/refinement of standard cost system.

Managed annual physical inventory in cooperation with Price Waterhouse auditors.

Director of Operations
COLORADO VIDEO, Mountain View, Colorado 1983 to 1991
Recruited as Controller for a start-up video production company servicing cable television, corporate training and expanding consumer video markets. Created all general accounting, cost accounting, budgeting, financial analysis/reporting, tax, payroll and long-range business planning systems.

Promoted to Director of Operations in 1985 with full P8.L responsibility for the entire business unit. Directed sales/marketing, studio and location production, post production, technology, human resources, purchasing, and all corporate finance and administrative affairs.

Built company from start-up to over $2 million in annual revenues.
Evaluated emerging technology, sourced vendors and directed over $200,000 in annual technol- ogy acquisitions to develop a state-of-the-art production facility.

Successfully marketed, negotiated and closed sales contracts with IBM, Apple, Pepsi, Department of Defense and numerous other corporate and government clients.

Previous Professional Experience:
Cost Accounting Manager, Katy Windows, Irvine, California 1987 to 1983
Cost Accountant, Anderson, Santa Monica, California 1980 to 1981
Cost S, Budget Administrator, Continental Group, Boston, Massachusetts
1977 to 1980

EDUCATION:
B.S., Business Administration Management, Honors Graduate, 1 979
NORTHEASTERN UNIVERSITY, Boston, Massachusetts Certified Tax Practitioner (CTP), State of California, 1992

Taken From :100 Winning Resumes for $ 100,000 + Jobs

NOTE: hstrumental in positioning division for sale at twice the acquisition cost afier only two years. Assisted Price Waterhouse in preparation of 8K filings for the SEC.

Accounting & Financial Systems Consultant
MOVERS 8 SHAKERS, Boston, Massachusetts 1992 to 1994
Recruited to design, implement and manage a PC-based accounting system for this sporting events management company and producer of the annual Boston Marathon. As the only accounting and finance professional in the organization, provided daily accounting, budgeting and cash management support in addition to long-range strategic, business, market and finance planning.

MOVERS 8, SHAKERS (Continued):
Created a comprehensive accounting and financial reporting system:

- Prepared financial statements for fiscal years 1992, 1993 and 1994.

- Redesigned chart of accounts to track sponsorship fees, entry fees and concession fees by

- Developed detailed job cost and profitability analysis reports as the foundation for budgeting multiple product classes and specific events.
and event planning to facilitate future growth and improved earnings.

Taken From :100 Winning Resumes for $ 100,000 + Jobs

Reengineered royalty and licensing agreements in cooperation with the Legal and Licensing Departments. Incorporated sophisticated financial analysis into intellectual property negotiations, designed financial models and saved over $800,000.

Designed an Excel relational database daily revenue flash report which graphically depicted gross shipments and returns by product line.
Provided senior executives with accurate data to evaluate actual sales performance versus plan in this rapidly changing commercial market.

Launched a process-by-process redesign of key accounting functions, consolidated workflow, stream- lined staffing requirements and captured over $190,000 in annual savings.

Accelerated internal automation and full use of existing system applications. Brought inventory control module on-line, resulting in annual physical adjustment of less than 1 %. Currently transi- tioning (via electronic transmission conversion) from AccPac to j .D. Edwards software.

Taken From :100 Winning Resumes for $ 100,000 + Jobs

Look carefully at each sample. There is a strategy in every instance for why a format and presentation style was selected, based upon each individual’s career history and current professional goals. Find a format that most closely matches your career track and then modify the text as appropriate.

Feel free to duplicate specific content, phrases, formats and styles as they relate to your experience. That’s why you bought this book!

JOHN P. SMITH
5555 North Avenue
Los Angeles, California 92009
(6 19) 222-3333

CORPORATE ACCOUNTING & FINANCE PROFESSIONAL
Start-Up, Turnaround & Hish-Growth Corporations Eighteen years experience in the design, development and management of comprehensive corporate accounting, budgeting, financial reporting, financial modeling, tax and MIS systems. Consistently successful in linking accounting with general operations to provide hands-on financial leadership for strategic planning, technology R&D, sales/marketing, purchasing, inventory, production and distribution.

Delivered strong and sustainable revenue and income gains. Equally effective in capturing cost reductions through process redesign and performance management.

Taken From :100 Winning Resumes for $ 100,000 + Jobs

Remember, people d o not meet you. They meet a piece of paper. Be sure that your resume reflects your style and quality. If it’s poorly prepared and presented, it doesn’t matter what it says - no one will read it.

Each and every resume is different. Your challenge is to develop a resume that will effectively market your qualifications and distinguish you from the crowd. There are no set rules for writing style, format, typing or printing. People are different, employment histories are different, career goals are different and resumes are different.

Each of the next 100 resume samples that follow are “real-life” examples (specifics have been changed to protect each executive’s confidentiality). What’s more important, each of these resumes has worked in opening doors, getting interviews and helping to close top- Level management opportunities. Use the sample words, formats, strategies and concepts as your solution and your foundation for a high-profile, high-performing resume and job search campaign.

Taken From :100 Winning Resumes for $ 100,000 + Jobs

1. Open a file in your PC.

2. Type your name, address and contact numbers.

3. Type in all major headings (e.g., Career Summary, Professional Experience, Education, Professional Profile).

4. Fill in Professional Profile. It’s easy. It’s only a list of information.

5. Fill in Education. It’s easy. It’s only a list of information.

6. Type in job titles, company names and dates.

7. Write your job descriptions. Start from your oldest job and work forward. The older jobs are easy to write. They’re short and to the point, and only highlight the most significant responsibilities and achievements. As you work forward, each position requires a bit more text and a bit more thought. Before you know it, you’ll be writing your most recent job. It takes the longest, but then you’re almost finished in no time at all.

8. Write your Career Summary. It will be easy now that you’ve just written all of your job descriptions and everything is fresh in your mind. Remember to highlight the skills, qualifications and achievements most supportive of yourcurrent career objectives.

NOTE: If you try to write the Summary first, it will do you in! How can you write the Summary when you haoe’t written the text. It can take hours instead of just a few minutes.

Taken From :100 Winning Resumes for $ 100,000 + Jobs

PERSONAL INFORMATION
I do not recommend the inclusion of personal information such as birth date, marital status, health, children, etc. Nor am I keen on hobbies and interests. I only recommend that you include this information if:

Required by the employer.

w Important because of a unique situation (e.g., you want someone to know that you are an American citizen; you want someone to realize that you are not as old or as young as they might think).

You have unusual interests that will grab someone’s attention. I know execu-
tives who were past Olympians, have ascended mountains on all seven conti- nents, have trekked through “untrekkable” places, are competitive triathletes and more. This type of information really sells.
Include it.

NOTE: Consolidate all of the “extras” - Affiliations, Publications, Languages, etc. is under one heading titled “Professional Profile. ” Try this format:

PROFESSIONAL PROFILE
Affiliations Chairman, National Industries Association Publications
Chairman, Industry Oversight National Association

“Database Management,” PC World, May 1992 “Cybernetics, ” Digital Design, January 199 1 “Netscape, ” Digital & Electronics Design, April 1987

Fluent in German, Dutch and French.

Microsoft Word, Excel, Lotus, PageMaker, Wordperfect Languages
PC Software

Taken From :100 Winning Resumes for $ 100,000 + Jobs

PUBLIC SPEAKING
List your public speaking engagements including title of presentation, audience, loca- tion and date. If the list is extensive, give a brief overview and only include the highlights (and, of course, those most related to your current career objectives).

TEACHING EXPERIENCE
If you are a teacher or trainer, this experience will be included in PROFESSIONAL EXPERIENCE. If, however, you are a business professional who has also taught, be sure to include this information. Just like publications, it immediately validates your qualifica- tions. List course, school, undergraduate/graduate and dates.

Taken From :100 Winning Resumes for $ 100,000 + Jobs

TECHNOLOGY SKILLS
Include your knowledge and proficiency with mainframe, PC, client/server, telecom- munications, internet and other technologies. If you are not in the Information Technol- ogy industry, a brief mention in your Career Summary may be appropriate. If you are in the industry, and depending upon your specific expertise, this section may be much more detailed and include your specific technical qualifications - hardware, software, operating systems, programming languages, network protocols, etc.

HONORS & AWARDS
Include your honors, awards, commendations and recognition. This information can be integrated into your job descriptions, included under Education (if appropriate) or put into a separate category at the end of your resume. If any are of significant note (e.g., national awards, advertising awards, #1 industry ranking), you may even want to include
them in your Career Summary. Use a subheading titled “Distinguished Credentials” or “Industry Honors & Awards” or “Professional Recognition”.

Taken From :100 Winning Resumes for $ 100,000 + Jobs

It is not necessary to include colleges you attended, but from which you did not receive a degree. However, if you do not have a degree, you would, of course, include atten- dance with major course of study, name of college/university and dates (as appropriate).

Be sure to include all relevant professional certifications and licenses. In relation to continuing professional education, give a good sampling of course titles, universities, sponsoring organizations and dates. It is not necessary to include them all - just the highlights (particularly those most relevant to your current career objectives).

Taken From :100 Winning Resumes for $ 100,000 + Jobs

Follow this same format all the way through your resume, becoming more and more brief as you get further back in time. Focus on achievements in earlier positions, not necessarily on your responsibilities unless than were unusual and/or vast.

How far back in time d o you go when listing your employment experience? You certainly d o not want to disqualify yourself because of your age. Unfortunately, there is no hard and fast rule. I like a “clean” picture. If you graduated from college anytime after 1968, include the date of graduation and all employment that follows. You don’t have to include much detail about the early positions (you can list them or summarize in a paragraph), but show your career path and progression.

If you graduated from college prior to 1968, it is recommended to “cut” your employ- ment history at some point (usually early to mid 1970’s) and then summarize key items from past experiences. These key items might include:

Well-known or well-respected companies.
Fast-track promotion.
Something you accomplished that was GREAT
rn Emerging products and technology experience.
rn International experience.

Taken From :100 Winning Resumes for $ 100,000 + Jobs

Architected the corporation’s 5-year strategic business plan that positioned XYZ for 25%t annual revenue and market growth.

Directed start-up of new manufacturing venture that grew to more than $22 million in first year sales.

Surpassed all turnaround objectives with 25% revenue and 37?6 profit gain.

Structured and negotiated 12 mergers and acquisitions as part of the
corporation’s aggressive growth and corporate development initiative.

Introduced quality circles, implemented TQM principles and led organization to IS0 9000 certification.

Personally negotiated $2 million capital financing transaction.

NOTE: These achievements are merely representative and given to you as “food for thought” in identifying and highlighting your own career achievements. Place your focus on the value you have delivered to your previous employers and the value you bring to a new organization.

Taken From :100 Winning Resumes for $ 100,000 + Jobs

Conceived and implemented a team-based project design and management process, increasing on-time project delivery and significantly enhancing customer retention.

Spearheaded acquisition and implementation of advanced robotics technolo- gies, virtually eliminated competition and won a $15 million, 5-year contract.

Honored as 1995 “Sales Director of the Year” (out of 25 candidates).

Won 6 sales recognition awards, 3 over-quota sales awards and the
corporation’s most prestigious leadership commendation.

Revitalized customer service organization, recruited qualified management team, and increased customer satisfaction ratings from 76% to 98%.

Identified opportunity and negotiated a strategic alliance with Uniroyal for development of on-site manufacturing and distribution operations in China.

Orchestrated the company’s successful and profitable expansion throughout emerging international markets (e.g., Africa, India, Middle East, Far East).

Taken From :100 Winning Resumes for $ 100,000 + Jobs

Increased sales by 48% across six major market segments and despite emerg- ing competition.

Accelerated annual profit contributions by more than 28% over previous year.

Led the development of emerging clientkerver technology, from concept
through R&D and manufacturing to nationwide market launch. Created what is now a $16 million revenue center for the XYZ Corporation.

Drove market share from 10% to 22% within first six months through transi- tion to customer-focused selling and account management strategies.

Reengineered all critical production planning, scheduling and manufacturing processes for a 12% reduction in annual operating costs.

Taken From :100 Winning Resumes for $ 100,000 + Jobs

Your challenge here is to briefly, yet aggressively describe your professional experience. For each position, you want to highlight four important things:

What did the companv do (A manufacturer? A distributor? A worldwide
technology leader? A multi-site service organization?)

Your challenqe (Was it a start-up venture, a turnaround or a high-growth organization? Why did they hire you? Was it to manage the “status-quo” or did you have a special mission?)

Your accountabilitv [Overall, what is it that you were responsible for in
terms of type and size of organization, number of employees, budgets, facili- ties, countries, regions, technologies, functions and operations?)

Your achievements [What did you produce and contribute?)

Taken From :100 Winning Resumes for $ 100,000 + Jobs

A Career Summary is a brief section that highlights your skills, qualifications, knowl- edge and achievements, and is designed to give a quick overview of your career history. The Career Summary can be written in a paragraph, as a bulleted list of items, or any combination thereof. In essence, you take your entire professional career and consolidate it into a 1-2 page resume. Now, take the resume and consolidate into an inch or two at the top - your Summary.

Career Summaries are the single most effective tool for establishing the reader’s perception of “who you are” based upon your current objectives. It is this section that allows you to “paint the picture” you want someone to see as they review your experience and your “fit” within their organization.

Taken From :100 Winning Resumes for $ 100,000 + Jobs

If you are certain of your career goal, and are not going to look at any opportunities that do not fall into that category, then an Objective is appropriate. However, if you want to be able to use your resume in a number of different situations, then d o not include an Objective, and certainly not one that is so broad that it tells your reader nothing.

Do not feel as though you must change your Objective each time to match the position to which you are applying. You want the resume process is to be easy and fast. In the vast majority of situations, you want to use the same resume over and over, modifying only your cover letter for each specific opportunity.

In situations where you do not use an Objective, let your cover letter speak of what your Objective is to each prospective employer. This gives you the opportunity to customize your Objective to that specific organization and their specific needs.

Taken From :100 Winning Resumes for $ 100,000 + Jobs

Here’s a quick example. Let’s say that from 1984 to 1994 you were General Managerof a large manufacturer. Then, from 1994 to the present, you were Vice President of Sales & Marketing. Your goal today is to get back into General Management. What types of achievements, responsibilities and skills will you highlight in your resume? General management of course, with sales and marketing as a secondary skill, even though it has been the most recent experience. If your goal is sales and marketing management, the strategy would be exactly the opposite.

I often refer to this as “painting the right picture”. In other words, you appropriately weight your skills and experience, shifting the focus to the areas most in line with your current objectives. Items on your resume do not have to be weighted according to actual responsibility, but rather in the manner that will alert your audience to what is most impor- tant in your job search today.

Taken From :100 Winning Resumes for $ 100,000 + Jobs

But, it can also be fun, challenging and tremendously worthwhile when you are finished. There is little else that is as personally and professionally rewarding than to view your own resume, your accomplishments, your contributions and your overall career success. It is energizing and a key component to your successful job search campaign!

Of course, you’ll want to include your work history and educational credentials in your resume. Other than that, what information you include, how and where you include it, and why you include it, are up to you. There no are limits; there are no restrictions.

Taken From :100 Winning Resumes for $ 100,000 + Jobs

Are there anv circumstances when a 3t page resume IS appropriate?
YES. Consider this. You’re the CEO of a Fortune 50 company applying for a position as CEO of another Fortune 50 company. Although your professional work experience is the most critical, so are your professional affiliations, civic affiliations, non-profit affiliations, public speaking engagements, Board of Director appointments, teaching experience and other professional activities. In a circumstance such as this, where the candidates are the top in the country, the search is quite selective and the stakes are high., often a longerresume is a more appropriate tool. The company is not just hiring the professional, they
are hiring the person, hidher network of contacts and hidher reputation. Longer resumes are quite acceptable in this situation.

For any of you in academia, scientific or medical research, or health care, you know that your curriculum vitae (not resume) can be longer than two pages. For those of you not familiar with CVs, they are vastly different than resumes. The purpose of a resume is to “sell”. The purpose of a CV is to highlight professional credentials, distinctions, research projects, technological skills, teaching and educational experiences, publications, sympo- sia, etc. CVs are often lengthy documents, and are appropriate tools for professionals in education, medicine, research and science.

Taken From :100 Winning Resumes for $ 100,000 + Jobs

If you remember this, forget it! The times have changed and the market has changed. Today, resume writing is a competition among often very well-qualified professionals vying for a limited number of opportunities. There’s more competition than ever before. And the competition is more qualified than ever in the history of our job market. What strategies and tactics can you deploy that will differentiate you from the crowd?

Resume writing has evolved into an art. Your challenge is to consolidate the highlights of your professional career, creating a resume that is dynamic, distinctive, hard-hitting and competitive. You must determine what experiences, qualifications and accomplishments you have to “sell” that will distinguish you from the crowd and position you for job inter- views and offers. You must be aggressive and focused in conveying your message. And you must fit your resume onto 1-2 pages.

Taken From :100 Winning Resumes for $ 100,000 + Jobs

J A TALENTED MARKETING PROFESSIONAL with years of experi-
ence writing brochures, press releases, direct mail communications and more. You sit down to write your resume and nothing happens. Nothing. Hours, days and sometimes even weeks later, you finally have your resume complete. You hate it. It’s the same as everyone else’s. Nothing is sharp. Nothing is competitive.

J A TOP-PERFORMING SALES EXECUTIVE who can sell anything to anybody. No matter the challenge, the product or the market, you have always produced. You figure that writing a resume is nothing more than selling your qualifications. And, you’re right. You sit at your PC for the next two days tying to determine the best selling strategy. Finally you quit. You just don’t know how to sell the product when are the product!

J A SOPHISTICATED FINANCIAL EXECUTIVE with more than ZO years
experience as CFO of a Fortune 500 corporation. You’re known throughout the market for your expertise in deal making, negotiating, capital financing and corporate treasury. You are a pro. When you’ve finished your resume, you’re appalled. Your resume looks like that of every other senior finance executive. Nothing is different. Nothing is distinctive.

If any of these sound familiar, then this book is for you. If you don’t fall into any of these categories, but have struggled to prepare a resume, then this book is also for you. In it, you will find the SOLUTIONS for successful resume writing and job search management.

Taken From :100 Winning Resumes for $ 100,000 + Jobs

Because of the unconventional nature of the proof of this theorem, it is offered here.

Proof ? Since ?PYA is supplementary to ?PZA, quadrilateral PZAY is
cyclic.?? Draw PAPB, and PC.

Therefore,
m?PYZ = m?PAZ (I)

Similarly, since ?PYC is supplementary to ?PXC, quadrilateral PXCY is
cyclic, and
m?PYX = m?PCB (II)

However, quadrilateral PACB is also cyclic, since it is inscribed in the
given circumcircle, and therefore
m?PAZ = m?PCB (III)

? For other proofs of Simson’s theorem, see A. S. Posamentier and C. T. Salkind, Challenging Problems in Geometry (New York: Dover, 1996), pp. 43–45. ?? A quadrilateral whose opposite angles are supplementary is cyclic; that is, its vertices all lie on the same circle.

From (I)–(III), m?PYZ = m?PYX, and thus points X, Y , and Z are
collinear.

This invariant is beautifully demonstrated with the Geometer’s Sketchpad. There students would draw the figure and then, by moving the point on the circle to various positions, they can observe how the collinearity is preserved under all positions of the point P. Dynamic geometry of this kind can go a long way to impress your students and win them over toward a love for mathematics.

Taken From :Math Wonders to inspire teacher and student

bits and bytes
Your computer speaks a language made up of only two numerals:
0 and 1. This two-numeral form of communication is called machine
language; the numerals combine to form binary numbers. Machine
language uses binary numbers to form instructions for the chips and
microprocessing devices that drive computing devices—such as
computers, printers, hard disk drives, and so on.

You may have heard the terms bit (short for binary digit) and byte. A
bit is the smallest unit used by your computer and can be either a 1 or
a 0. A byte consists of 8 bits (more about bytes later). Because binary
numbers consist of only 1’s and 0’s, binary number values appear different from the decimal values we use in everyday life. For example,
in the decimal system when you see the number 1 followed by two 0’s
(100), you know it represents a value of one hundred. In binary, however, the same number combination—100—represents a value of four.

Going back to how machine language uses 1’s and 0’s, remember that
each digit in a binary number represents 1 bit. Machine language thinks of each bit as either “on” or “off.” A bit with a value of 1 is considered “on,” and a bit with a value of 0 is “off.” Therefore, determining the value of a binary number is simply a matter of adding up the columns that are “turned on,” in other words, wherever a 1 appears. (This “on/off” idea comes in handy later.)

In the decimal numbering system, each column (ones, tens, hundreds,
and so on) has ten times the value of the previous column. In the binary numbering system, however, each column has a value twice that
of the previous column (ones, twos, fours, eights, sixteens, and so on).

Because your computer understands binary values consisting of 1’s and
0’s, a binary value exists for every possible keyboard character. The
most widely accepted standardized system for numbering keyboard characters is called the ASCII system. (ASCII is pronounced “askee” and stands for American Standard Code for Information Interchange.)

We mentioned earlier that 8 bits—represented by 8 binary digits—
together are referred to as a byte. Almost all specifications of your
computer’s capabilities are represented in bytes. For example, memory
capacity, data-transfer rates, and data-storage capacity are all measured in bytes or multiples thereof (such as kilobytes or megabytes). The following table summarizes how to refer to quantities of bits and bytes.

Taken From: the ultimate memory guide

The beauty of this theorem lies in its simplicity. Begin by having your
students all draw a triangle with its vertices on a circle (something that is always possible, since any three noncollinear points determine a circle) and then they should select a point on the circle that is not at a vertex of the triangle. From that point, they should draw a perpendicular line to each of the three sides. The three points where these perpendiculars intersect the sides (points X, Y , and Z in Figure 5.24 on the next page) are always collinear (i.e., they lie on a straight line). Each accurate student drawing should reflect this fact. The line that these three points determine is often called the Simson line (further “injustice”!).

This can be more formally stated as follows.

Simson’s Theorem The feet of the perpendiculars drawn from any
point on the circumcircle of a triangle to the sides of the triangle are
collinear.

In Figure 5.24, point P is on the circumcircle of ABC. ?? PY ? ?? AC at Y , ?? PZ ? ?? AB at Z, and ?? PX ? ?? BC at X. According to Simson’s (i.e., Wallace’s) theorem, points X, Y , and Z are collinear. This line is usually referred to as the Simson line.

Taken From :Math Wonders to inspire teacher and student

While we can’t look at every possible memory configuration, we can
look at a system for depicting memory configuration requirements
called the bank schema. A bank schema is a diagram of rows and
columns that shows the number of memory sockets in a system. This
visual display is a theoretical bank layout and not an actual system
board layout; it is designed to help you quickly determine the configuration requirements when adding memory modules.

Kingston uses the bank schema system to help customers easily
understand the configuration rules of various computer systems. The
bank schema system appears in Kingston’s product documentation,
which includes the Kingston Upgrade Manual (a hard copy reference) and an electronic reference system called KEPLER”.

Taken From: the ultimate memory guide

? A cyclic quadrilateral is one whose four vertices lie on the same circle.

B’
C’
A’
F
C
A B
Figure 5.23b

With H as the orthocenter (the point of intersection of the altitudes), M is the midpoint of CH (see Figure 5.23d). Therefore, BM, a midline of ACH, is parallel to AH, or altitude AD. Since BC is a midline of ABC, BC BC. Therefore, since ?ADC is a right angle, ?MBC is also a right angle. Thus, quadrilateral MBCF is cyclic (opposite angles are supplementary). This places point M on the circle determined by points B, C, and F . We now have a seven-point circle.

We repeat this procedure with pointL, the midpoint ofBH (see Figure 5.23e). As before, ?BAL is a right angle, as is ?BEL. Therefore, points B, E, A, and L are concyclic (opposite angles are supplementary). We now have L as an additional point on our circle, making it an eight-point circle.

C’
B’ A’
C
D
A B
Figure 5.23c

M
H
F
E
D
C’
B’
C
A B
Figure 5.23d

To locate our final point on the circle, consider point K, the midpoint
of AH. As we did earlier, we find ?ABK to be a right angle, as is
?ADK. Therefore, quadrilateral ADKB is cyclic and point K is on the same circle as points B, A, and D. We have therefore proved that nine specific points lie on this circle. This is not to be taken lightly; it is quite spectacular!

K
A’
H
F
L
E
D
B’
C
A B
Figure 5.23e

Taken From :Math Wonders to inspire teacher and student

where memory comes from
As mentioned in the introduction, DRAM is the most common type
of memory chip. These chips are produced in very large, highly
specialized fabrication plants. From there, the chips go to memory
module manufacturers (like Kingston) where they’re used to make
different memory products. These products find their way through
various distribution channels to people who install them in computers.

where memory goes in the computer
For memory modules to do their job, they must be in direct communication with the computer’s CPU. It used to be that memory
was always soldered directly onto the computer’s system board (also
known as logic board or motherboard). But as memory requirements increased, it became infeasible to solder all the memory chips onto the system board.

This is how SIMMs and SIMM sockets became popular. The SIMM format offers a flexible method of upgrading memory while using less space on the system board.

Taken From: the ultimate memory guide

Perhaps you already know what it’s like to work on a system that doesn’t have quite enough memory. Things run a little more slowly at times, memory errors can occur more frequently, and sometimes you can’t launch an application or a file without first closing or quitting another. On a system with sufficient memory, however, you can do multiple tasks at once—such as printing one document while working on
another—and you can keep multiple applications open simultaneously.

what memory looks like
The Integrated Circuits, or ICs, that make up your computer’s
memory configuration are referred to as Dynamic Random Access
Memory, or DRAM. DRAM is by far the most common type of
memory chip. The quality of DRAM chips used in a memory module
is the most important component in determining the overall quality
and reliability of the module.

A common memory product is the Single In-line Memory Module,
or SIMM. As you can see from the illustration, a typical SIMM consists
of a number of DRAM chips on a small Printed Circuit Board, or
PCB, which fits into a SIMM socket on a computer’s system board
(more on this later).

SIMMs come in a variety of forms, including 30- and 72-pin formats.
These and other types of memory products are covered in greater
detail in the “Bits and Bytes” section on page 23.

Taken From: the ultimate memory guide

? Recherches sur la determination d’une hyperbole équilatèau moyen de quartes conditions données (Paris, 1820).

M
L
K
H
F
D
E
B’ A’
C’ B
C
A
Figure 5.23a

In Figure 5.23b, points A, B, and C are the midpoints of the three sides of ABC opposite the respective vertices. CF is an altitude of ABC. Since AB is a midline of ABC, AB  AB. Therefore, quadrilateral ABCF is a trapezoid. BC is also a midline of ABC, so that BC = 1 2BC. Since AF is the median to the hypotenuse of right BCF, AF = 1 2BC. Therefore, BC = AF and trapezoid ABCF is isosceles.

You will recall that when the opposite angles of a quadrilateral are supplementary, as in the case of an isosceles trapezoid, the quadrilateral is cyclic. Therefore, quadrilateral ABCF is cyclic.?
So far, we have four of the nine points on one circle.

To avoid any confusion, we redraw ABC (see Figure 5.23a) and include altitude AD. Using the same argument as before, we find that quadrilateral ABCD is an isosceles trapezoid and therefore cyclic. So we now have five of the nine points on one circle (i.e., points A, B, C, F , and D).

Taken From :Math Wonders to inspire teacher and student

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Satelliteinternetbroadband.com is one of the internet providers which is exist right now. It offers an internet connection that is hugh net. This internet bandwidth divides into several categories, such as home and professional. The bandwidth starts from 20 times faster than the dial-up. And the fastest can reach 50 times faster than the dial-up.

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To clarify this common mix-up, it helps to compare your computer to
an office that contains a work table and a set of file cabinets.

Another important difference between memory and storage is that
the information stored on a hard disk remains intact even when the
computer is turned off. However, any data held in memory is cleared
when the computer is turned off. (It’s like saying that any files left on
the work table at closing time will be thrown away.)

Taken From: the ultimate memory guide

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• The midpoints of the sides
• The feet of the altitudes
• The midpoints of the segments from the orthocenter to the vertices

Have your students do the necessary construction to locate each of these nine points. Careful construction will allow them to be on the same circle. This circle is called the nine-point circle of the triangle. Unmarked straightedge and compasses or the Geometer’s Sketchpad computer program would be fine for this activity.

In 1765, Leonhard Euler showed that six of these points, the midpoints of the sides and the feet of the altitudes, determine a unique circle. Yet not until 1820, when a paper? published by Brianchon and Poncelet appeared, were the remaining three points (the midpoints of the segments from the orthocenter to the vertices) found to be on this circle. The paper contains the first complete proof of the theorem and uses the name “the nine-point
circle” for the first time.

Theorem In any triangle, the midpoints of the sides, the feet of the
altitudes, and the midpoints of the segments from the orthocenter to the vertices lie on a circle.

Taken From :Math Wonders to inspire teacher and student

A good example of this is when the CPU loads an application program
—such as a word processor or page layout program—into memory,
thereby allowing the application program to run as quickly as possible.
In practical terms, this means you can get more work done with less
time spent waiting for the computer to perform tasks.

This “putting things the CPU needs in a place where it can get at
them more quickly” process is similar to placing various electronic
files and documents you’re using on the computer into a single file
folder or directory. By doing so, you keep them handy and avoid
searching in several places every time you need those documents.

Taken From: the ultimate memory guide

* State the most important idea or concept pertaining to your reading
purpose.
* List related key words, facts, and information in order of importance -
using as few words as possible - that pertain to your learning objectives.
* Finally, jot down important words or phrases in relation to the ideas
listed above. The most important things to jot down are key people,
important events, places, and dates. These will act as thought joggers or memory clues, which relate directly to the primary and secondary ideas listed.

For more about the inter-relationship of left and right brain functions, see Two Ways of Knowing in ‘Transforming the Mind’. Also the Bilateral Meter site contains information of profound significance in the field of psychotherapy. Words & Meanings, Semantic Development and The Semantic Differential are chapters in ‘Transforming the Mind’ that offer further insight into the power of words, as does Ken Ward’s Notes on Alfred Korzybski’s General Semantics.

This is taken further in The New Life Course - the complete personal
development program - where many exercises reveal how false thinking entraps us in conditioned thought.

Taken From: The Speed Reading Course

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To get an idea of how much things have changed over the last decade,
an excerpt from Inside the IBM PC, written by Peter Norton in 1983,
describes the merits of IBM’s new XT computer: “So IBM has equipped
all XTs with what it considers to be the minimum gear for a serious
personal computer. Now the 10-megabyte disk and the 128K [one
eighth of a megabyte] of memory are naturals for a serious machine.”

For some, the memory equation is simple: more is good; less is bad.
However, for those who want to know more, this reference guide
provides a general overview of what memory is and how it works.

Taken From: the ultimate memory guide

A very elegant (or somewhat sophisticated) method for verifying this is by taking an “extreme” point. By taking “any point” to be at a vertex, this can be easily established. Then the sum of the distances to two of the sides is 0, leaving the distance to the third side as the sum. This distance to the third side is simply the altitude.

We can show this in a number of more traditional ways.

We seek to prove: The sum of the distances from any point in the interior of an equilateral triangle to the sides of the triangle is constant (the length of the altitude of the triangle).

Here you can see an example of this with actual measurements (see
Figure 5.22a).

mPF = 338 cm
mPD = 088 cm
mPE = 139 cm
mPF + mPD + mPE = 565 cm
mBG = 565 cm

Taken From :Math Wonders to inspire teacher and student

Associate – Relate the information you’re trying to remember to something you already know. To help remember the three stages of memory (reception, retention, and recollection) you can associate the mind with a computer. By recalling the computer’s three processes (input, storage, and output) you will be able to remember the stages of
memory.

Visualize – Drawing out pictures and diagrams makes the information easier to recall by visualizing the drawing while taking the test. When memorizing the names of bones in the body, draw a human skeleton and label the bones. During the test, visualize the skeleton and you will be able to remember the names.

Flashcards – Write key words or terms that you need to know on one side of an index card. Write the explanation or definition on the other side of the card. Carry these cards with you and review them as often as possible.

Taken From: How to Improve your Memory

Lines ?? B2C2 and ?? B1C1 meet at A.
Lines ?? A2C2 and ?? A1C1 meet at B.
Lines ?? B2A2 and ?? B1A1 meet at C.

1
2
2
1
2
1
A’ C’ B’
C
B
A
C
B
A
P
Figure 5.21

This is truly remarkable, but to make it even more astonishing, the converse is also true. Namely, if A1B1C1 and A2B2C2 are situated so that the pairs of corresponding sides intersect in three collinear points, then the lines joining the corresponding vertices, ?? A1A2, ?? B1B2, and ?? C1C2, are concurrent. For the teacher who wishes to pursue this theorem further, it is useful to know that it is a self-dual. That is, the dual? of the theorem is the converse.??

? See Unit 5.16 for a review of duality. ?? The proof of the theorem can be found in A. S. Posamentier, Advanced Euclidean Geometry: Excursions for Secondary Teachers and Students (Emeryville, CA: Key College Press, 2002).

Taken From :Math Wonders to inspire teacher and student

We are going to consider two triangles, whose “corresponding vertices” we will designate with the same letter and that will also determine their corresponding sides. This is important to keep in mind as we move along. The two triangles are going to be situated in a very specific manner, and their shape (or relative shape) is of no real concern to us. This is quite different from the kind of thinking used in the high school study of geometry.

We will be placing any two triangles in a position that will enable the
three lines joining corresponding vertices to be concurrent. Remarkably enough, when this is achieved, the pairs of corresponding sides meet in three collinear points. Let’s see how this looks in a more formal setting.

Desargues’s Theorem If A1B1C1 and A2B2C2 are situated so that
the lines joining the corresponding vertices, ?? A1A2, ?? B1B2, and ?? C1C2, are concurrent, then the pairs of corresponding sides intersect in three collinear points.

Taken From :Math Wonders to inspire teacher and student

Retention – Make a conscious effort to remember what is being said. If you set goals for your performance and motivate yourself this will give you the incentive to remember. Become an active reader by highlighting and marking your text. Review your notes frequently to increase your retention. Recite your notes aloud when possible. By using both your visual and auditory senses, you will increase your retention rate. Do all your homework when it is assigned. Using information in and out of the classroom will help you remember it better. See the list of Memory Aids for tips on improving your memory.

Recollection – Organize your material before the test. Group tests, summaries, and notes according to chapters and similar topics. Make a list of important topics and what you should know about them. The week before the test set up a block of time (2-3 hours) to thoroughly review the information. Remember to take breaks when studying! During the test visualize your diagrams and flashcards to help remember the
information. Use practice tests to study. Anticipate possible test questions and make up your own test or look at old tests if they are available from the professor.

Taken From: How to Improve your Memory

Types of Memory
Sensory Memory – We are constantly processing information gathered through our senses. Through selective attention, your mind determines what of the huge amount of incoming information is important and ignores the rest. When you concentrate on your professor’s lecture or the discussion that is going on you use selective attention to deem this information important. Although sensory information is only kept in your
mind a few seconds, by concentrating on a certain piece of information, you can transfer it to your short-term memory.

Short-Term Memory – Information in your short-term memory lasts only about a minute. When you meet someone and they tell you their name, chances are, an hour later, you won’t remember their name. By reciting and rehearsing information like names, lists or phone numbers, you can increase your retention of the information. Short-term memory is limited, however. The average number of items you can keep in
short-term memory is seven. To remember larger amounts of information you must group it into common themes, memorize “chunks” of information at once, or use other strategies to improve retention.

Long-Term Memory – Once information is moved to long-term memory, it is integrated with existing information. If this integration is not successfully done, the information may get “lost” and will be harder to recall. Long-term memory is like a giant warehouse full of file cabinets. You take information you know and you place it in existing “files.” If there is no existing file and you do not create one by integrating like information, the information may be more difficult to recall.

Taken From: How to Improve your Memory

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? A classic source for 370 proofs of the Pythagorean theorem is E. S. Loomis, The Pythagorean Proposition (Reston, VA: NCTM, 1968).

Begin by drawing a conveniently sized circle on a piece of cardboard.
Divide the circle into 16 equal arcs. This may be done by marking off
consecutive arcs of 22.5
or by consecutively dividing the circle into two parts, then four parts, then bisecting each of these quarter arcs, and so on.

These sectors, shown above, are then cut apart and placed in the manner shown in the figure below.

This placement suggests that we have a figure that approximates a parallelogram. That is, were the circle cut into more sectors, then the figure would look even more like a true parallelogram. Let us assume it is a parallelogram. In this case, the base would have length 1 2C, where C = 2r (r is the radius). The area of the parallelogram is equal to the product of its base and altitude (which here is r). Therefore, the area of the parallelogram is  1 2C r = 1 2 2r r = r2, which is the commonly known formula for the area of a circle. This should certainly impress your
students to the point where this area formula begins to have some intuitive meaning.

Taken From :Math Wonders to inspire teacher and student

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The same technique can be applied whenever you study materials that you intend to remember. It may be thought that with continued study of a subject, the reviews would accumulate and take over most of your study time. Actually, this is not the case. Supposing a person studied every day for one hour a day, and in addition set up a review programme for this study. On any one day he would need to review the work from the study session just finished (immediately after, a few hours after and before going to bed), and also material from one day, one week, one month and six months before.

Review of work done: Time taken:
Same day 5 minutes
1 week before 3 minutes
1 month before 1 minute
6 months before 1 minute
Maximum review time
on any one day: 10 minutes

Thus a person spending one hour a day on study would need to spend only a maximum total of 10 minutes a day to complete all the necessary reviewing, and improve his memory many times over. Thus a few minutes devoted to review makes the hours spent studying effective and worthwhile.

When you have acquired the discipline of organised review of previously studied materials, and received the benefits, the procedure will become automatic and easy.

Taken From: The Speed Reading Course

We now equate the two expressions for the area of the trapezoid
1
2
a2 + ab +
1
2
b2 = ab +
1
2
c2
1
2
a2 +
1
2
b2 =
1
2
c2

which is the familiar a2 + b2 = c2, the Pythagorean theorem.

There are more than 400 proofs? of the Pythagorean theorem available today; many are ingenious, yet some are a bit cumbersome. However, none will ever use trigonometry. Why is this? An astute student will tell you that there can be no proof of the Pythagorean theorem using trigonometry, since trigonometry depends (or is based) on the Pythagorean theorem.
Thus, using trigonometry to prove the very theorem on which it depends would be circular reasoning. Encourage your students to discover a new proof of this most famous theorem.

Taken From :Math Wonders to inspire teacher and student

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? In October 1851, he noted in his diary that “I have today commenced the study of geometry alone without class or teacher.”
?? This would appear to be a wrong reference, since we usually consider the proof that the base angles of an isosceles triangle are congruent as the pons asinorum, or “bridge of fools.”

shown in Figure 5.19, and that a trapezoid is formed. Notice also that
since m?AEB + m?CED = 90
, m?AED = 90
, making AED a right
triangle.

a
c
c
a
b
b
B
C
A
D
E
Figure 5.19

The area of the trapezoid =
1
2
(sum of bases)(altitude)
=
1
2
a + b a + b
=
1
2
a2 + ab +
1
2
b2

Taken From :Math Wonders to inspire teacher and student

About an hour later, have a second recall session, exactly as before, going through all the topics without undue strain, repeating them to yourself. New aspects and data will reappear by association. The third session should take place about three hours later, the next after six hours, preferably before going to sleep. This makes maximum use of the consolidation occurring during the dreaming process. Repeat the recall procedures three or four times on each of the second and third days, spacing the sessions out evenly through the day.

Matthew Erdelyi found that his subjects recalled information most easily if they were able to call up mental images associated with a particular topic. It seems that the mind handles images, especially vivid and unusual ones, far more effectively than it deals with words, numbers, or abstract concepts. You can make use of this fact by briefly forming a picture of each major topic when it is initially described and later as you review the topic; this will enhance retention and recall.

If you get stuck at any point make use of the picture association to jog your memory. Remain relaxed and think of the first thing that the previous item you were able to remember reminds you of. This should produce an association of some kind that can be used as a trigger, leading on to the next link in the chain.

After perhaps up to ten such links have been pulled out of your mind, one of the missing topics will reappear, like a rabbit out of the conjurer’s hat.

Taken From: The Speed Reading Course

After some moments of perplexity, you can relieve the class of their frustration by telling them that all three fellows proved the Pythagorean theorem. The first two bring no surprise, but President Garfield? He wasn’t a mathematician. He didn’t even study mathematics. As a matter of fact, his only study of geometry, some 25 years before he published his proof of the Pythagorean theorem, was informal and alone.?

While a member of the House of Representatives, Garfield, who enjoyed “playing” with elementary mathematics, came upon a cute proof of this famous theorem. It was subsequently published in the New England Journal of Education after being encouraged by two professors (Quimby and Parker) at Dartmouth College, where he went to give a lecture on March 7, 1876. The text begins with

In a personal interview with General James A. Garfield, Member of Congress from Ohio, we were shown the following demonstration of the pons asinorum,?? which he had hit upon in some mathematical amusements and discussions with other M.C.’s. We do not remember
to have seen it before, and we think it something on which the members of both houses can unite without distinction of party.

Taken From :Math Wonders to inspire teacher and student

Suppose instead one could turn this curve around and increase the amount of remembered facts with the passage of time. Studies have been carried out by Dr Matthew Erdelyi of New York University which showed that volunteers trying out his ideas, found themselves remembering twice as much information the day after the learning had taken place than five minutes after. From these studies practical techniques have been evolved which enable anyone to reverse the usual
forgetting curve and remember things better as time goes by.

The method is as follows. Suppose you have to attend a lecture or meeting where it is not possible to take notes or make a recording, yet it is vital to recall the salient points which were discussed. To ensure effective recall you must set up a programme in your mind which will act as a store for information. Therefore, as the session proceeds make a mental note of key points which are raised by repeating these subject headings to yourself in numerical order. Repeat this list from the beginning as each new heading is added. In this way you can keep a
running total of all the successive points that have been raised. This is possible because your inner thought-stream is much faster than the vocalised speech that you are listening to, so you can fill in the gaps with your review programming. It also helps to accompany each heading with a visual representation of the subject matter, particularly if that image is striking or humorous, i.e. memorable.

Taken From: The Speed Reading Course

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Visual Auditory Kinaesthetic
shapes volume pressure
colours pitch temperature
black/white pace of speech emotions
movement number of sounds speed of movement
size location of sound location of felt sensation
perspective rhythm texture

When reading a novel, many people fail to make adequate use of auditory imagery, even when they are good visualisers. If you use your auditory imagery to give all the ‘he said …’ and ’she said …’ dialogue, then your memory of the story will be vastly improved. When you read a book and use all the forms of imagery, you will experience the story as a three-dimensional movie in stereophonic sound, with imagery of emotion and movement, touch, taste and even temperature. You will be totally at one with the book and your subsequent recall will be nearly perfect. You will hardly be aware of reading the words, unless there is a gross printing error.

It may be difficult to construct concrete images when reading abstract material such as philosophy. A student who has both high right-brain and left-brain capacity will tend to form abstract patterns, rather like modern art, to hang the words and pictures upon. Modern physics has little that can be visualised as concrete imagery, however, when a psychologist asked Einstein about his thinking processes, Einstein replied, ‘I think in a combination of abstract visual patterns and muscular sensations; it is only later, when I wish to speak or write to another person, that I translate these thoughts into words.’

Taken From: The Speed Reading Course

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D
B
C
A
Figure 5.18a

Notice that ADC ? CDB ? ACB. In Figure 5.18a, ADC and
CDB are folded over ACB. So clearly AreaADC + AreaCDB = AreaACB. If we unfold the triangles (including the ACB itself), we
get the following (see Figure 5.18b) that shows that the relationship of the similar polygons (here right triangles) is the extension of the Pythagorean theorem:

The sum of the areas of similar right triangles on the legs of a right
triangle is equal to the area of the similar right triangle on the
hypotenuse of the triangle.

This essentially “proves” the Pythagorean theorem by paper folding!

P’
Q’
Q’
Q
P’
P
C’
D’
D” D
C
B
A
Figure 5.18b

Taken From :Math Wonders to inspire teacher and student

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1. The first step is to check that you have the ability to picture in your mind’s eye. Look at your desk and pretend that this desk is really your bedroom, and that you are on the ceiling, looking down at the four walls and everything contained inside. Mentally point to the wall where the bed is, the walls with windows, the door, the shelves, and so on. Do this exercise again with the layout of the whole house. This exercise will validate that you can make mental pictures of concrete objects, a right-brain skill.

2. Read a phrase or sentence out loud. The sentence is the easiest
grammatical unit to use for this particular method. A sentence should be chosen that uses nouns that are concrete and action verbs, rather than abstract nouns and the verb ‘to be’, as these will prevent the use of right-brain picturing abilities. As soon as you have stopped reading the sentence, close your eyes and picture in your mind what the sentence described. Notice the colour, size, shape, foreground, and distance of the picture in your mind. This will give you a further idea of your basic capacity to visualise. Used as a repetitive exercise, this will
improve your visualisation.

3. Once you can form a reasonably good mental picture from a sentence you have just read, the next goal is to find how many pictures you can hold on to. Read out between 3 and 9 visualisable sentences. If you go beyond your capacity, you will lose the first and second picture. This will tell you your capacity for a sequence of separate pictures. Practice will improve this ability. People who find it easy to create pictures and take in large amounts of information have the facility to take information spread out over several pictures and sequence this information into a movie. when you can do this well, you will have a seemingly infinite memory capacity, taking advantage of the right brain’s incredible powers (you will probably have noticed how much easier it is to remember peoples’ faces than their names).

Taken From: The Speed Reading Course

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Once they have drawn the above figure, students ought to be encouraged to draw quarter circular arcs, as shown in Figure 5.11c. The resulting figure approximates a logarithmic spiral.

P
S R
K
H L Q
T
N J
G
M F D
B
A
E
Figure 5.11c

We can locate the vanishing point of the spiral by drawing the diagonals of the two largest Golden Rectangles, as shown in Figure 5.11d.

P
S R
K
H L Q
T
N J
G
M F D
B
A
E
Figure 5.11d

Students who construct this figure accurately will see that BD and GF
contain the diagonals of the other Golden Rectangles as well. Moreover, BD?GF .

A similar spiral can be drawn by locating the centers of each of the squares in succession from largest to smallest and drawing (see Figure 5.11e).

S
R
L N M
K
J
G H
F
E
D
B C
A
Figure 5.11e
There is no end to the beauty of this rectangle!

Taken From :Math Wonders to inspire teacher and student

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Clearly, the Pythagorean theorem is the basis for much of geometry and all of trigonometry. For this reason, be careful about discovering a new proof to make sure that it is not based on a relationship established by the Pythagorean theorem itself. Such is the case with trigonometry. No proof of the Pythagorean theorem can use trigonometric relationships because they are based on the Pythagorean theorem—a clear case of circular reasoning. Students should be clear on what is meant by circular reasoning.

Now to the proof. It is very simple, but it is based on a theorem that states that when two chords intersect in the circle, the product of the segments of one chord is equal to the product of the segments of the other chord. In the figure below, this would mean that for the two intersecting chords:

p
q = r
s.
s
q
r
p

Now consider the circle with diameter AB perpendicular to chord CD:
c+b

a
c
b
c-b
a
c
D C
A
O
B
E

From the theorem stated above, c ? b c + b = a2. Then c2 ? b2 = a2 and, therefore, a2 + b2 = c2. The Pythagorean theorem is proved again. Although there have been many proofs after its publication, a nice collection of 370 proofs of the Pythagorean theorem is by Elisha S. Loomis.? This is a lovely resource for all mathematics teachers.

? A classic source for 370 proofs of the Pythagorean theorem is E. S. Loomis, The Pythagorean Proposition (Reston, VA: NCTM, 1968).

Taken From :Math Wonders to inspire teacher and student

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Ineffectual reading typically leaves out visually constructed imagery from the thought-stream. As a result the reader has a poor memory and poor contextual analysis skills. Without imagery to ‘reality test one’s comprehension, one may pass a totally anomalous word and fail to notice that it does not fit. Once the reader has a richly detailed internal picture, which includes colour, sound and movement, he will no longer be able to read past words and concepts that obviously do not make sense, because these will seem strange in the picture or movie that he has made. For example, a student reads: ‘The child was made to do
the maths problem in front of the class upon the skateboard.’ From his prior picture of a classroom, the student will realise immediately that the word should be ‘blackboard’, instead of skateboard, and will self-edit the word.

One of the characteristics of visual storage is speed, so increasing the pace at which material is covered, with the assistance of speed-reading exercises, usually increases the powers of visualisation. Those students who can adapt to the visual mode of representation successfully are multi-sensory; however, there are some students who have difficulty. These are students who have failed to make the transition between an auditory mode of representation and a visual mode of representation. In normal development this transition occurs at about the age of ten. In the case of these students, retention can be so poor that one sentence later they are unable to remember what they have read. These students will attempt to retrieve the rote sound of words; they will try to store an auditory sequence of the word without transferring the words into pictures in their minds. A student with this problem will frequently state, ‘I don’t remember what it said.’

It is now known that reading involves both sides of the brain: the left side specialises in coding and decoding, the right side in synthesis of overall meaning. By using this as an operational definition, you can determine which side of a student’s brain is deficient when diagnosing his reading ability, and it can be used to formulate a prescriptive plan of how to improve his reading. For example, when a student is able to code and pronounce words disproportionately to his comprehension, his left brain is working in excess of his right brain.

Taken From: The Speed Reading Course

To make a Mind Map, one starts at the centre of a new sheet of paper, writing down the central theme very boldly, preferably in the form of a strong visual image, so that everything in the map is associated with it. Then work outwards in all directions, adding branches for each new concept, and further small branches and twigs for associated ideas as they occur. In this way one produces a growing and organised structure composed of key words and key images (see the previous article on ‘Key Words’).

Exercise
For more information about Mind Mapping and many examples, visit these web sites:

BrainLand | Peter Russell | Creative Mindmap
Shared Visions | Learn Mind-Mapping

When you’ve got a good reality on Mind Mapping, read through this Speed Reading Course and at the same time, based on your growing understanding, build up a Mind Map displaying the main ideas and how they connect.

Taken From: The Speed Reading Course

Pascal’s Theorem
The points of intersection of the opposite sides of a hexagon
inscribed in a conic section are collinear.
Brianchon’s Theorem
The lines joining the opposite vertices of a hexagon circumscribed about a conic section are concurrent.

In Figure 5.16a, the hexagon ABCDEF is circumscribed about the circle. As with Pascal’s theorem, we shall consider only the conic section that is a circle. According to Brianchon’s statement, the lines containing opposite vertices are concurrent. Your students can easily experiment with differentshaped circumscribed hexagons to verify that it is true. Again, we see that the simplicity of this figure and its result makes for its beauty.

F
E
D
C
B
A
Figure 5.16a

Right after stating his theorem, he suggested that if points A, F , and E were to be moved so that they would be collinear, with vertex F becoming a point of tangency, and thereby forming a pentagon, the same statement could be made. That is, since pentagon ABCDE is circumscribed about a circle, then CF AD, and BE are concurrent (see Figure 5.16b).

P
N
M
S
R
F
E
D C
B
A
Figure 5.16b

You are encouraged to demonstrate this wonderful relationship with the Geometer’s Sketchpad to get the full dramatic effect.

Taken From :Math Wonders to inspire teacher and student

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In 1806, at the age of 21, a student at the École Polytechnique, Charles Julien Brianchon (1785–1864), published an article in the Journal de L’École Polytechnique that was to become one of the fundamental contributions to the study of conic sections in projective geometry. His development led to a restatement of the somewhat forgotten theorem of Pascal and its extension, after which Brianchon stated a new theorem, which later bore his name. Brianchon’s theorem,? which states “In any hexagon circumscribed
about a conic section, the three diagonals cross each other in
the same point,”?? bears a curious resemblance to Pascal’s theorem, which we presented in the preceding unit on Pascal’s invariant.

To fully appreciate the relationship between Pascal’s theorem and what Brianchon discovered, it is best to first appreciate what the concept of duality in mathematics is. Two statements are duals of one another when all of the key words in the statements are replaced by their dual words. For example, point and line are dual words, collinearity and concurrency are duals, inscribed and circumscribed are duals, sides and vertices are duals, and so on. You might have your students practice with a few simple statements. Here is an example of the duality relationship. Notice how the terms “point” and “line” have been interchanged.

Two points determine a line. Two lines (intersecting, of course) determine a point.

? For a proof of Brianchon’s theorem, see A. S. Posamentier, Advanced Euclidean Geometry: Excursions for Secondary Teachers and Students (Emeryville, CA: Key College Press, 2002). ?? Source Book in Mathematics, edited by D. E. Smith (New York: McGraw–Hill, 1929), p. 336.

Taken From :Math Wonders to inspire teacher and student

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Such associative networks explain the incredible versatility and flexibility of human information processing. Memory is not like a container that gradually fills up, it is more like a tree growing hooks onto which the memories are hung. So the capacity of memory keeps growing - the more you know, the more you can know. There is no practical limit to this expansion because of the phenomenal capacity of the neuronal system of the brain, which in most people is largely untapped, even after a lifetime of mental processing.

Mind Maps
Because the brain naturally organises information in associative networks, it makes sense to record notes about information you want to remember in a similar way. Using the method of Mind Maps, all the various factors that enhance recall have been brought together, in order to produce a much more effective system of note taking. A mind map works organically in the same way as the brain itself, so it is therefore an excellent interface between the brain and the spoken or written word.

Taken From: The Speed Reading Course

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The more that notes consist of key words, the more useful they are and the better they are remembered. Ideally, notes should be based upon key words and accompanying key images, and incorporate summary diagrams and illustrative drawings. This concept is further expanded in the next article on ‘Mind Maps’.

Mind Maps
Meaning is an essential part of all thought processes, and it is meaning that gives order to experience. Indeed the process of perception is ultimately one of extracting meaning from the environment. If the mind is not attending, information will go ‘in one ear and out the other’; the trace it leaves may well be too weak to be recalled in normal circumstances. If concentration is applied, i.e. there is conscious involvement with the information, more meaning is extracted, more meaningful connections are made with existing understanding, the memory is stronger, and there will be more opportunity to make meaningful connections with new material in the future.

Taken From: The Speed Reading Course

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Here is a different-shaped hexagon inscribed in a circle (see Figure 5.15b). Again notice that, regardless of the shape, the points of intersection of the opposite sides of the hexagon meet at three points on a straight line (i.e., they are collinear).

G
I
H
F
E
D
A C
B
Figure 5.15b

If you do it on a computer, say, using the Geometer’s Sketchpad, you can actually see how, by changing the shape of the hexagon, the points G, H, and I always remain collinear. The amazing thing about this situation is that it is independent of the shape of the hexagon. You can even distort the hexagon so that it doesn’t look like a polygon anymore, and as long as you keep track of what were the opposite sides, the above collinearity will remain intact. Again, this is very easily and very dramatically demonstrated with the Geometer’s Sketchpad.

In Figure 5.15c, you can identify the original hexagon only by referring to the sides of the original one. The pairs of opposite sides and their intersections are

AB and DE intersect at point I .
BC and EF intersect at point H.
DC and FA intersect at point G.

These intersection points, G, H, and I , are still collinear.
G
I
H
F
E
D
A C
B
Figure 5.15c

This truly amazing relationship can give us motivation to look into why this behaves as it appears in the above illustrations.

Taken From :Math Wonders to inspire teacher and student

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In 1640, at the age of 16, the famous mathematician Blaise Pascal published a one-page paper titled Essay Pour les Coniques, which presents us with a most insightful theorem. What he called mysterium hexagrammicum states that the intersections of the opposite sides of a hexagon inscribed in a conic section are collinear.? We shall use the most common conic section, a circle.

Consider the hexagon ABCDEF inscribed in the circle (i.e., all its vertices are on the circle). You might have your class try this independently, either on paper or on a computer geometry program. The trick is to draw the hexagon shape that will allow you to get intersections of opposite sides— so don’t make the opposite sides parallel.

See Figure 5.15a to identify the pairs of opposite sides (extended) and their intersections:
AB and DE intersect at point I .
BC and EF intersect at point H.
DC and FA intersect at point G.

? For a proof of Pascal’s theorem, see A. S. Posamentier, Advanced Euclidean Geometry: Excursions for Secondary Teachers and Students (Emeryville, CA: Key College Press, 2002).

Taken From :Math Wonders to inspire teacher and student

Thanks to a clever experimental technique devised by Gordon Gallup for studies with primates, we have a way of ascertaining when the human infant first comes to view himself as a separate entity, an incipient person. It is possible, unbeknownst to the child, to place a tiny marker - for example, a daub of rouge - upon his nose and then to study his reactions as he peers at himself in the mirror. During the first year of life, the infant is amused by the rouge marking but apparently simply regards it as an interesting decoration on some other organism which he
happens to be examining in the mirror. But, during the second year of life, the child comes to react differently when he beholds the alien colouring. Children will touch their own noses and act silly or coy [embarrassed] when they encounter this unexpected redness on what they perceive to be their very own anatomy.

Awareness of physical separateness and identity are not, of course, the only components of beginning self-knowledge. The child also is starting to react to his own name, to refer to himself by name, to have definite programs and plans that he seeks to carry out, to feel efficacious when he is successful, to experience distress when he violates certain standards that others have set for him or that he has set for himself. All of these components of the initial sense of person make their initial appearance during the second year of life.

(From ‘Frames of Mind’ by Howard Gardner)

Looking at the marked key words separated from the text, the sense of the passage can be re-constituted:

infants
feelings
facial expressions
universal
specific experiences
intrapersonal knowledge
identity
other individuals
distinguishes
ten months
empathy
helpful
mirror
amused
second year
embarrassed
name
plans
standards

Taken From: The Speed Reading Course

4. Study in Depth
Keeping in mind what you want to learn, speculate on what the material will tell you. Begin to read with the satisfaction of your objectives in mind. Sometimes it is inappropriate to start at the beginning, so decide where to start reading. Your overall purpose for reading the material is your best guide. Note: the manner in which the author presents his ideas will demand that you constantly vary the rate of reading and the reading technique you are using, if you wish to be efficient. If you continue reading at the same rate for a prolonged period, it is a good
indication that you are not reading flexibly and that you are allowing yourself to become inefficient.

Make notes, jot down main ideas and Key Words and use Mind Maps (see later chapters). It also helps to mark or underline key words and concepts in the book itself, with a soft lead pencil that can easily be erased, to aid review. If it is your own book, do not be afraid to use different coloured pens; it helps memory and distinguishes different themes and topics.

Taken From: The Speed Reading Course

To be sure, these states are initially uninterpreted: the infant has no way of labelling to himself how he is feeling or why he is feeling this way. But the range of bodily states experienced by the infant - the fact that he feels, that he may feel differently on different occasions, and that he can come to correlate feelings with specific experiences - serves to introduce the child to the realm of intrapersonal knowledge.

Moreover, these discriminations also constitute the necessary point of departure for the eventual discovery that he is a distinct entity with his own experiences and his unique identity. Even as the infant is coming to know his own bodily reactions, and to differentiate them one from another, he is also coming to form preliminary distinctions among other individuals and even among the moods displayed by ‘familiar’ others. By two months of age, and perhaps even at birth, the child is already able to discriminate among, and imitate the facial expressions of, other individuals. This capacity suggests a degree of ‘pre-tunedness’ to the
feelings and behaviour of other individuals that is extraordinary.

The child soon distinguishes mother from father, parents from strangers, happy expressions from sad or angry ones. (Indeed, by the age of ten months, the infant’s ability to discriminate among different affective expressions already yields distinctive patterns of brain waves.)

Taken From: The Speed Reading Course

A key word or phrase is one which funnels into itself a range of ideas and images from the surrounding text, and which, when triggered, funnels back the same information. It will tend to be a strong noun or verb, on occasion accompanied by an additional key adjective or adverb. Nouns are the most useful as key words, but this does not mean you should exclude other words. Key words are simply the words that give you the most inclusive concept. They do not have to be actual words used in the text - you may have a better word that encapsulates and evokes the required associations, and a phrase may be necessary rather than just a word.

As an example, suggested key words have been indicated in bold type throughout the following text, starting on the next page. There may be words you do not understand, even when taking account of the context; in this case it is certainly necessary to look these up in a dictionary. Psychological terminology like ‘intrapersonal’ may not be in your dictionary, but the prefix ‘intra’ means within, so the meaning can be derived.

Taken From: The Speed Reading Course

You ought to try this with your class. Have students independently draw the figure described below and then have them compare their resulting diagrams.

Consider any two lines, each with three points located anywhere on the lines. Then connect the points of the first line to those on the second line, but do not connect the corresponding points. That is, don’t connect the rightmost point on one line to the rightmost point on the other line or don’t connect the two middle points.

Taken From :Math Wonders to inspire teacher and student

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* Imposing organisation upon the material.
* Allowing associations, inferences and ideas to be jotted down.
* Bringing attention to what is important.
* Enhancing later recall.

Since we do not remember complete sentences, it is a waste of time to write them down. The most effective note taking concentrates on the key words of the lecture or text. In selecting the key words, a person is brought into active contact with the information. The time which would have been spent making longwinded notes can be spent thinking around the concepts. He is not simply copying down in a semi-conscious manner but is becoming aware of the meaning and significance of the ideas, and forming images and associations between them. This increases comprehension and memory. Because the mind is active,
concentration is maintained, and review of the notes becomes quick and easy.

The ability to pick out the most appropriate word as a ‘key’ word is vital if you want to remember the most important information from any text. We mainly use the following parts of speech when we pick key words:

Nouns: identify the name of a person, place or object. They are the most essential information in a text. ‘Common nouns’ are whole classes of people or things, e.g. man, dog, table, sport, ball. ‘Proper nouns’ name a particular person or thing, e.g. Beethoven, the ‘Emperor’ Concerto, Vienna.

Verbs: indicate actions, things that happen, e.g. to bring, kiss, exist, drink, sing.

Adjectives: describe qualities of nouns (people and things) - how they appear or behave, e.g. old, tall, foolish, beautiful.

Taken From: The Speed Reading Course

Joining the points in order, we get the pentagon ABCDE in Figure 5.13b.
A B
N
Q
P
R
C
S
E
D
Figure 5.13b

Although the pentagon “looks” regular, the m?ABC is about 22
60 of a degree too large. That is, for ABCDE to be a regular pentagon, each angle must measure 108
; instead, we have that m?ABC ? 1083661202
. You might try to draw this with the Geometer’s Sketchpad or simply on the chalkboard. It ought to be easy to draw this artwork, following the instructions provided.

Taken From :Math Wonders to inspire teacher and student

Because of their greater meaningful content, key words tend to ‘lock up’ more information in memory and are the ‘keys’ to recalling the associated ideas. The images they generate are richer and have more associations. They are the words that are remembered, and when recalled, they ‘unlock’ the meaning again.

When a young child begins to speak, he starts with key words, especially concrete nouns, stringing them together directly - for example, ‘Peter ball’ or ‘Anne tired’. It is not until later that sentences include grammatical construction, to give expressions such as ‘Please would you throw me the ball’ or ‘I am feeling tired’.

Taken From: The Speed Reading Course

For years, engineers have been using a method for drawing what appears to be a regular pentagon; yet careful inspection will show that the construction is a tiny bit irregular.? This method, which we will provide below, was developed in 1525 by the famous German artist, Albrecht Dürer.

We refer to Fig 5.13a on page 162. Beginning with a segment AB, five
circles of radius AB are constructed as follows:

1. Circles with centers at A and B are drawn and intersect at Q and N.
2. Then the circle with center Q is drawn to intersect circles A and B at points R and S, respectively.
3. QN intersects circle Q at P.
4. ??SP and ??RP intersect circles A and B at points E and C, respectively.
5. Draw the circles with centers at E and C with radius AB to intersect at D.
6. The polygon ABCDE is (supposedly) a regular pentagon.

? For a discussion of where the error lies, see A. S. Posamentier and H. A. Hauptman, 101 Great Ideas for Introducing Key Concepts in Mathematics (Thousand Oaks, CA: Corwin Press, 2001), pp. 141–146.

Taken From :Math Wonders to inspire teacher and student

Association plays a dominant role in nearly every mental function, and words themselves are no exception. The brain associates divergently as well as linearly, carrying on thousands of different actions at the same time, searching, sorting and selecting, relating and making syntheses as it goes along, using left and right brain faculties. Thus a person often finds that in conversation, his mind is not just behaving linearly, but racing on in different directions, exploring to create new ideas and evaluating the ramifications of what is being said. Although a single
line of words is coming out, a continuing and enormously complex process is taking place in the mind throughout the conversation. At the same time subtle changes in intonation, body position, facial expression, eye language, and so on, are integrated into the overall process.

Similarly the listener or reader is not simply observing a long list of words; he is receiving each word in the context of the ideas and concepts that surround it, and interpreting it in his own unique way, making evaluations and criticisms based upon his prior knowledge, experience and beliefs. You only have to consider a simple word and start recognising the associations that come into your mind, to see that this is true.

Taken From: The Speed Reading Course

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Let’s take a closer look at this pentagon (Figure 5.12a). Point D divides AC into the Golden Section, since

DC
AD =
AD
AC

We can say that the segment of length AD is the mean proportional between the lengths of the shorter segment (DC) and the entire segment (AC).

A
B C
D
G H
Figure 5.12a

For some student audiences, it might be useful to show what the value of the Golden Section is. To do this, begin with the isosceles triangle ABC, whose vertex angle has measure 36
. Then consider the bisector BD of ?ABC (Figure 5.12b).

72°
36°
36°
36°
x
x
x
1
B C
A
D
Figure 5.12b

We find that m?DBC = 36
. Therefore, ABC ?BCD. Let AD = x and AB = 1. However, since ADB and DBC are isosceles, BC = BD = AD = x.

From the similarity above,
1 ? x
x =
x
1

This gives us
x2 + x ? 1 = 0 and x =
?5 ? 1
2
(The negative root cannot be used for the length of AD.)

We recall that
?5 ? 1
2 =
1


The ratio for ABC of
side
base =
1
x = 
We therefore call this a Golden Triangle.

Taken From :Math Wonders to inspire teacher and student

Two proofs of this interesting property are provided here. The first compares the length of each perpendicular segment to a portion of the altitude, and the second involves area comparisons.

Proof I In equilateral ABC, PR?AC, PQ?BC, PS ?AB, and AD?BC. Draw a line through P parallel to BC, meeting AD, AB, and AC at G, E, and F , respectively (see Figure 5.22b).

T
R
S
Q
N
P
H
G
E F
B D C
A
Figure 5.22b

Since PGDQ is a rectangle, PQ = GD. Draw ET ?AC. Since AEF
is equilateral, AG  ET (all the altitudes of an equilateral triangle are
congruent). Draw PH  AC, meeting ET at N. NT  PR. Since EHP

is equilateral, altitudes PS and EN are congruent. Therefore, we have
shown that PS + PR = ET = AG. Since PQ = GD, PS + PR + PQ = AG + GD = AD, a constant for the given triangle.

Proof II In equilateral ABC, PR?AC, PQ?BC, PS ?AB, and
AD?BC. Draw PA, PB, and PC (see Figure 5.22c).

R
S
B Q D C
A
Figure 5.22c

AreaABC = AreaAPB + AreaBPC + AreaCPA
=
1
2
AB PS +
1
2
BC PQ +
1
2
AC PR

Since AB = BC = AC, the area of ABC = 1
2BC %PS + PQ + PR&.
However, the area of ABC = 1
2BC AD . Therefore, PS +PQ+PR = AD, a constant for the given triangle.

Your students now have ample justification for this very interesting phenomenon.

Taken From :Math Wonders to inspire teacher and student

B P A

Simply, for the segment AB, the point P partitions (or divides) it into two segments, AP and PB, such that

AP
PB =
PB
AB

This proportion, apparently already known to the Egyptians and the
Greeks, was probably first named the “Golden Section” or “sectio aurea” by Leonardo da Vinci, who drew geometric diagrams for Fra Luca Pacioli’s book, De Divina Proportione (1509), which dealt with this topic.

There are probably endless beauties involving this Golden Section. One of these is the relative ease with which one can construct the ratio by merely folding a strip of paper.

Simply have your students take a strip of paper, say about 1–2 inches wide, and make a knot. Then very carefully flatten the knot as shown in the next figure. Notice the resulting shape appears to be a regular pentagon, that is, a pentagon with all angles congruent and all sides the same length.

Taken From :Math Wonders to inspire teacher and student

Continually ask WHO, WHAT, WHY, HOW, WHERE and WHEN questions, as an interactive dialogue between yourself and the study material, in order to extract the important facts.

The Who question helps you to hold in mind any significant people. Why
classifies purposes. How classifies cause and effect sequences, time sequences, procedure or process instructions or where the new information fits into your life. The Where question points to where the action is taking place or where the new information can be used. The When question can both denote when a subject takes place and when you can use the information. Finally, the What question allows you to take a quick survey of your current knowledge.

Take regular breaks every thirty or forty minutes. After each short rest break, take a minute to review the previous work: this consolidates the retention.

Taken From: The Speed Reading Course

We can extend from the statement of the Pythagorean theorem:

The sum of the squares on the legs of a right triangle is equal to
the square on the hypotenuse of the triangle.

By replacing the word “squares” with “areas of similar polygons,” to read:

The sum of the areas of similar polygons on the legs of a right triangle is equal to the area of the similar polygon on the hypotenuse
of the triangle.

This replacement can be shown to be correct and holds true for any similar polygons appropriately (correspondingly) placed on the right triangle’s sides.

Taken From :Math Wonders to inspire teacher and student

Have students try to find some Golden Rectangles in their environment.
To construct a Golden Rectangle, begin with a square (see Figure 5.11a). Locate the midpoint M of one side and make a circular arc with center at M and radius length ME. Call the point D where the arc intersects ?? AF . Then erect a perpendicular to AD at D to meet ?? BE at C. Rectangle ABCD is a Golden Rectangle. This can be done with straightedge and compasses, or students may be adept at using the Geometer’s Sketchpad and then ought to be encouraged to use it.

C
M F D
B
A
E
Figure 5.11a

Here is a nice extension of the Golden Rectangle. If we continuously
construct squares in the Golden Rectangle, as shown below (Figure 5.11b), each resulting rectangle is a Golden Rectangle; that is, it is similar to the original rectangle, since all Golden Rectangles are similar.

P
S R
K
H L Q
T
N J
G
M F D
B
A
E
Figure 5.11b

Remove square ABEF from Golden Rectangle ABGD to get Golden
Rectangle EGDF.
Remove square EGJN from Golden Rectangle EGDF to get Golden
Rectangle JDFN.
Remove square PJDT from Golden Rectangle JDFN to get Golden
Rectangle TFNP.
Remove square TFHQ from Golden Rectangle TFNP to get Golden
Rectangle HNPQ.
Remove square HNKL from Golden Rectangle HNPQ to get Golden
Rectangle KPQL.
Remove square KPRS from Golden Rectangle KPQL to get Golden
Rectangle SRQL.
And so on.

Taken From :Math Wonders to inspire teacher and student

H I g
G
A
B
C
F
E
D
Figure 5.14

In Figure 5.14 we did just that. We marked the three points of intersection G, H, and I . Now here is the amazing part: Regardless of how you drew the original lines or where you located the points on the lines, the points G, H, and I are always collinear (i.e., they lie on the same straight line)!??

You might have your students either do their drawings on an overhead transparency or on a computer drawing program so that the rest of the class can see each student’s drawing.

?? For a proof of Pappus’s theorem, see A. S. Posamentier, Advanced Euclidean Geometry: Excursions for Secondary Teachers and Students (Emeryville, CA: Key College Press, 2002).

Taken From :Math Wonders to inspire teacher and student

? For a proof of this theorem and its extensions, see A. S. Posamentier, Advanced Euclidean Geometry: Excursions for Secondary Teachers and Students (Emeryville, CA: Key College Press,
2002), Chapter 4.

K
L
M
D
F
E
C
B
A
Figure 5.10d

With a computer geometry program, such as the Geometer’s Sketchpad, you can see that, regardless of the shape of the original triangle, the above relationships all hold true. The question you might ask your students is what would they expect to happen if point C were to be on AB, thereby collapsing the original triangle. See Figure 5.10e.

K
L
M
D
F
E
C
B
A
Figure 5.10e

Lo and behold, our equilateral triangle is preserved. Perhaps even more astonishing (if anything could be) is the generalization of this theorem. That is, suppose we constructed similar triangles, appropriately placed on the sides of our randomly drawn triangle, and joined their centers (this time we must be consistent as to which “centers” we choose to use— centroid, orthocenter, incenter, etc.). The resulting figure will be similar to the three similar triangles.

With the aid of a computer drawing program, students can see that all that we said above about triangles drawn externally on the sides of our randomly selected triangle can be extended to triangles drawn internally as well.

Taken From :Math Wonders to inspire teacher and student

Read the Table of Contents. Note the sequence and check for Chapter
summaries. Chapter summaries are an abstract of the Chapter contents.
They will frequently inform you whether or not a particular publication is
suitable for your purposes.

The next step is to look at the visual material. Read the maps, graphs,
illustrations, charts, and bold headings.

Get a close feel for the actual contents of the book by looking at
beginnings and ends of chapters, subsection headings and anything else
which catches the eye - bold print, italicised sections, etc. Read any
summaries the author may have provided. If there are study questions at the end of each chapter, you should look at these also. This will give you an indication of the level of the book in relation to your present
knowledge.

Now you have completed these steps, then decide to use the book or not.

Taken From: The Speed Reading Course

O
D
F
E
C
B
A
Figure 5.10b

There are quite a few most unusual properties in this figure. For example, your students probably paid little attention to the notion that the three segments AE, BD, and CF are also concurrent. This concept is not exploredGeometric much in the typical high school geometry course. Yet, they ought not take this for granted. It must be proved, but for our purposes we shall accept it without proof.?

Not only is point O a common point for the three segments, but it is also the only point in the triangle where the sum of the distances to the vertices of the original triangle is a minimum. This is often called the minimum distance point of the triangle ABC.

As if this weren’t enough, this point, O, is the only point in the triangle where the sides subtend equal angles (Figure 5.10c). That is, m?AOC = m?COB = m?BOA = 120
.

120 °
120 ° 120 °
A
B
C
O
Figure 5.10c

Taken From :Math Wonders to inspire teacher and student

Your answer to this question is your purpose for reading. It may help at this stage to review your current knowledge of the subject. This increases expectancy of what is to come, and exposes gaps in one’s knowledge and a corresponding desire to fill the vacuum.

2. Survey
A book or publication should be surveyed as follows:

Read the title, any subtitles, jacket summaries (in the case of a book), and identify the source of the publication, i.e. the author and publisher.

Read the date of publication or copyright. The book may well have gone
beyond its sell-by-date, e.g. a book on electric motors written in 1950
would be irrelevant, unless perhaps you were trying to mend Grandma’s
lawnmower.

Analyse the Index. The particular concepts listed and the way in which
they are organised will tell you a particular author’s bias and whether or
not the book will cover the ideas that you are trying to get wise on.

Frequently, the Index is a better guide for these purposes than the
Contents page.

Taken From: The Speed Reading Course

The holistic approach to study parallels the normal activity of shopping: one prepares a list of what is required, goes only down the relevant streets (noticing other shop windows on the way in case they contain unexpected items of interest), and visits only those stores that contain all that one needs, with time and energy to spare.

In-Depth Reading or ’study’ is the most complicated and slowest of the reading processes. After an initial survey or pre-reading (scanning), gathering the context and main concepts, the in-depth reading involves critical and analytical thinking to interpret, evaluate, judge, and reflect on information and ideas. There are four main aspects to in-depth reading:

1. Gathering facts and ideas.
2. Sorting facts and ideas for relative importance and their relationship to one another.
3. Measuring these ideas against one’s existing knowledge base.
4. A process of selection, separating the ideas into those that you wish to remember or act upon, and ideas that you wish to reject.

In-depth reading techniques are a form of Self-Questioning. As we read we try to answer questions of HOW and WHY together with the implied suggestions: explain, describe, evaluate, interpret, illustrate, and define. When reading nonfiction and other serious material, the full procedure is as follows:

Taken From: The Speed Reading Course

The theorem bears the name of Napoleon, but today’s historians instead credit one of Napoleon’s military engineers.

The theorem states that the segments joining each vertex of a given triangle (of any shape) with the remote vertex of the equilateral triangle (drawn externally on the opposite side of the given triangle) are congruent.

O
F
E
D
C B
A

Figure 5.10a
That is, when ADCBCE, and ABF are equilateral, AE, BD, and
CF are congruent to one another (Figure 5.10a). Your students should take note of the unusual nature of this situation, since we started with any triangle and still this relationship holds true. If each of your students were to draw an original triangle, each would come up with the same conclusion. Either straight edge and compasses or the Geometer’s Sketchpad would be fine for this; however, the latter would be better.

Taken From :Math Wonders to inspire teacher and student

The most common approach to the study of a new text is the ’start and slog’ approach. The reader opens the book at page 1 and reads through to the end. This might seem the most obvious approach, but it is in fact an inefficient use of the reader’s knowledge and time and has a number of disadvantages:

1. Time may be wasted going over material that is already familiar, or that is irrelevant to the study in question, or which may be more conveniently summarised later.
2. The reader has no overall perspective until he finishes the text, and possibly not even then.
3. Any information that is retained is usually disorganised; it is seldom well integrated with the rest of the book nor with the reader’s whole body of knowledge.
4. Motivation is low and the reader tends to become bored, dull and tired, leading to poor reading efficiency.

Taken From: The Speed Reading Course

The theorem states the following:

The adjacent angle trisectors of any triangle intersect at three
points determining an equilateral triangle.

Let us look at the following figure.

F
E
D
A
B
C

Notice how the points D, E, and F are the intersection points of the adjacent trisectors of the angles of ABC and DEF is an equilateral triangle. Wow! This equilateral triangle evolved by beginning with any shaped triangle. Drawing on the Geometer’s Sketchpad allows you to change the shape of the original ABC and observe that each time DEF remains equilateral, although of different size.

The following figures demonstrate a few variations that you can create on the Geometer’s Sketchpad to witness this amazing relationship. This is truly one of the most dramatic (i.e., surprising) relationships in geometry and should be presented that way. Be cautioned, the proof is one of the most difficult in Euclidean geometry.?

F
E
D
A
B
C

? Several proofs can be found in A. S. Posamentier and C. T. Salkind, Challenging Problems in Geometry (New York: Dover, 1996), pp. 158–161.

F
E
D
A
B
C
F
E
D
A
B

Taken From :Math Wonders to inspire teacher and student

1. Make a rapid scan of a light novel. Start at a rate of 15 seconds per page. Later, with practice, this time can be reduced to 12 or 10 seconds per page or even less.
2. You are scanning for significant people, events and conflicts. At the end of each chapter stop to review what you have just read. Then try and speculate about the contents of the next chapter.
3. When you have scanned several chapters, no more than five, then you will probably need to ask yourself some questions relating to missed events and information, in order to be able to follow the development of the story. Speculate on these answers, then go back and re-read these chapters normally, to see if you were correct.
4. When you have reached the end of the book in the above manner, take some time to summarise the story mentally. Form and answer any unanswered questions about the story and evaluate what you gained from this book.

By using the above exercises you will soon find that you have much greater concentration and retention. Through these procedures you will have developed a lasting and very useful skill.

Taken From: The Speed Reading Course

The sum of the areas of the semicircles on the legs of a right triangle
is equal to the area of the semicircle on the hypotenuse.

Thus, for the figure below, we can say that the areas of the semicircles relate as follows:

Area P = AreaQ + AreaR
R
Q
P
T
A
C
B

Suppose we now flip semicircle P over the rest of the figure (using AB
as its axis). We would get a figure as shown below.

J2
J1
T
L2
L1
C
A
B

Let us now focus on the lunes formed by the two semicircles. We mark them L1 and L2.
J2
J1 T
L2
L1
C
A
B

Earlier we established that
Area P = AreaQ + AreaR

In the figure above, that same relationship can be written as follows:

Area J1+Area J2+Area T =Area L1+Area J1+Area L2+Area J2

If we subtract Area J1 + Area J2 from both sides, we get the astonishing result:

Area T = Area L1 + Area L2

That is, we have a rectilinear figure (the triangle) equal to some nonrectilinear figures (the lunes).

Taken From :Math Wonders to inspire teacher and student

To master the Pacing Technique you must understand the training you are going to give your mind. You are being asked to look at words so fast that you cannot possibly pronounce them, and so fast that you cannot understand them either.

Every time you do the above exercises you will comprehend a few words. As you continue with these exercises, you will begin to grasp thoughts and eventually, you will read at a much higher speed. When performing this type of exercise, you should always go back and re-read the passage at a comfortable rate, i.e. at a rate at which you can obtain understanding. Every time you do a speed-exercise and then return to what appears to be your normal speed, you will find that your normal speed has become faster.

Since written English is often highly redundant, i.e. much of the material can be omitted without any loss of meaning, a large proportion of information in a text can be absorbed through peripheral vision. Words that are highly likely to occur in a given context do not have to be checked by looking directly at them - peripheral vision can check that they are what is expected even while the eye is fixating elsewhere. The Pacing Technique helps prepare you to read in this expanded way, reading not along each line, but from side to side of the centre of the page, taking in most of a line in one glance, and also peripherally absorbing several further lines beneath it.

Making fuller use of peripheral vision, the skilled reader is able to get a better idea of the general sense of what is to follow, and this helps to speed up reading as well as to understand and integrate the material. This is why many students find that as soon as they become adept at speed reading, their comprehension actually increases. They have a broader perspective of what they are reading, and since they are reading faster, the short-term memory for what has just been read goes back several sentences further and the words currently being read are
understood within a larger context.

High-speed training has two further advantages: It encourages you to see the key words in the text; and it brings the right hemisphere (which controls peripheral vision) into the reading process, increasing integration and thereby facilitating the right-brain’s ability to synthesise relationships within the material.

Taken From: The Speed Reading Course

Let’s first recall the Pythagorean theorem. It stated the following:

The sum of the squares of the legs of a right triangle is equal to
the square of the hypotenuse.

This can be stated a bit differently with the same effect.

The sum of the squares on the legs of a right triangle is equal to
the square on the hypotenuse.

We can take this a step further.

The sum of the areas of the squares on the legs of a right triangle
is equal to the area of the square on the hypotenuse.

As a matter of fact, we can easily show that the square can be replaced by any similar figures drawn on the sides of a right triangle:

The sum of the areas of the similar polygons on the legs of a right
triangle is equal to the area of the similar polygon on the hypotenuse.

Taken From :Math Wonders to inspire teacher and student

Pacing Exercise
1. Place the book you intend to read in front of you and place the plastic ruler or strip as above.

2. Use your right index finger or a pen as a pacer, moving it smoothly down the centre of the page, over the transparent strip. This may be helpful until you have disciplined your eyes to ‘pace the page’. You may find that moving a 3 x 5 cm card down the plastic strip will be less distracting. The reason to use either the card, a pen, or your fingers in this way is to keep your eyes moving down.

3. When you reach the bottom of the page, jot down any words you
remember. If you do not remember any words at all, don’t let this upset you - you will improve with practice. Eventually you will remember thoughts and groups of words. By pausing frequently to mentally summarise what you have read, you will organise your thoughts and improve retention.

Taken From: The Speed Reading Course

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Now for an even more elegant solution. This unit lends itself to a very
powerful problem-solving strategy that may be called considering extreme cases.

Have students consider the original problem mentioned above. They
should realize that the solution was independent of the circumference of the earth, since the end result did not include the circumference in the calculation. It only required calculating 1
2 .

Here is a really nifty solution using an extreme case.

Suppose the inner circle (on the previous page) is very small, so small
that it has a zero-length radius (that means it is actually just a point). We were required to find the difference between the radii, R?r = R?0 = R.

So all we need to find is the length of the radius of the larger circle and our problem will be solved. With the circumference of the smaller sircle now 0, we apply the formula for the circumference of the larger circle:

C + 1 = 0 + 1 = 2R then R =
1
2

This unit has two lovely little treasures. First, it reveals an astonishing result, clearly not to be anticipated at the start, and, second, it provides your students with a nice problem-solving strategy that can serve as a useful model for future use.

Taken From :Math Wonders to inspire teacher and student

Pacing
A plastic ruler or strip of transparent plastic 5 cm wide, is placed vertically down the page, as shown below, to delineate the section of the page where your Pacing Technique will be used.

By fixating only the words in the pacing zone, you reduce your reading time by about one half. But you don’t reduce your comprehension by one half because you are forced to think beyond the words your eyes are seeing. When your thoughts are on the same subject as the material you are reading, the addition of your personal experience to the reading increases your understanding and memory.

If you read within the pacing zone by sliding back and forth in a Z or S-type pattern to the bottom of the page, you will find that you have read about 200 words with no more than 50 or 60 fixations. All the time you are reading in this way, your eyes are seeing and picking-up the odd word from peripheral vision and you are thinking all the time and putting together ideas, because the mind abhors a vacuum.

Using a 5cm transparent plastic ruler:

Taken From: The Speed Reading Course

Consider the globe of the earth with a rope wrapped tightly around the equator. The rope will be about 24,900 miles long. We now lengthen the rope by exactly 1 yard. We position this (now loose) rope around the equator so that it is uniformly spaced off the globe. Will a mouse fit under the rope?

The traditional way to determine the distance between the circumferences is to find the difference between the radii. Let R be the length of the radius of the circle formed by the rope (circumference C + 1) and r the length of the radius of the circle formed by the earth (circumference C).

Earth
Rope
r R

The familiar circumference formulas give us
C = 2r or r =
C
2
and
C + 1 = 2R or R =
C + 1
2

We need to find the difference of the radii, which is
R ? r =
C + 1
2 ?
C
2 =
1
2 ? 0159 yards ? 57 inches

Wow! There is a space of more than 51 2 inches for a mouse to crawl
under.

Taken From :Math Wonders to inspire teacher and student

7. Practice reading (with comprehension) for one minute at approximately 100 w.p.m. faster than your highest normal speed.

8. When you can do that, continue increasing your speed in approximately 100 w.p.m. increments. If you calculate how many words there are on an average line, then it is easy to convert w.p.m. into lines per minute. E.g. if a line has 10 words and you are reading at one line per second, then you are reading at 600 w.p.m.

9. Start from the beginning of a chapter and practice reading three lines at a time, with a visual aid (such as a card) and at a fast reading speed, for 5 minutes.

10. Read on from this point, aiming for comprehension at the highest speed possible. Do this for five minutes, then calculate and record your reading speed in w.p.m.

11. Take an easy book and start of the beginning of a chapter. Skim for one minute using a visual guide at 4 seconds per page.

12. Return to the beginning of the chapter and practice reading at your
minimum speed for five minutes.

Taken From: The Speed Reading Course

As librarian of Alexandria, Eratosthenes had access to records of calendar events. He discovered that at noon on a certain day of the year, in a town on the Nile called Syene (now called Aswan), the sun was directly overhead. As a result, the bottom of a deep well was entirely lit and a vertical pole, being parallel to the rays hitting it, cast no shadow.

At the same time, however, a vertical pole in the city of Alexandria did cast a shadow. When that day arrived again, Eratosthenes measured the angle (?1 in the figure on page 140) formed by such a pole and the ray of light from the sun going past the top of the pole to the far end of the shadow. He found it to be about 7
12, or 1
50 of 360
.

Sun’s rays Sun
Syene
(Aswan)
center
Alexandria
1
1
Earth

Assuming the rays of the sun to be parallel, he knew that the angle
at the center of the earth must be congruent to ?1, and hence must also measure approximately 1 50 of 360
. Since Syene and Alexandria were almost on the same meridian, Syene must be located on the radius of the circle, which was parallel to the rays of the sun. Eratosthenes thus deduced that the distance between Syene and Alexandria was 1 50 of the circumference of the earth. The distance from Syene to Alexandria was believed to be about 5,000 Greek stadia. A stadium was a unit of measurement equal to the length of an Olympic or Egyptian stadium. Therefore, Eratosthenes concluded that the circumference of the earth was about 250,000 Greek stadia, or about 24,660 miles. This is very close to modern calculations. So how’s that for some real geometry! Your students should be able to appreciate this ancient use of geometry.

Taken From :Math Wonders to inspire teacher and student

2. As you move your finger along the line that you are reading, try and take in more than one word at a time.

3. When you have reached the limits of the previous exercise, then take some light reading material and try to read more than one line at the same time. Magazine articles are good for this purpose because many magazines have narrow columns of about 5 or 6 words, and often the material is light reading.

4. Various patterns of visual guiding should be experimented with. These
include diagonal, curving, and straight-down-the-page movements. Exercise your eye movements over the page, moving your eyes on horizontal and vertical planes and diagonally from the upper left of the page to the lower right and finally, from the upper right to the lower left. Try to speed-up gradually day by day. The purpose of this exercise is to train your eyes to function more accurately and independently.

5. Practice reading as fast as you can for one minute, without worrying about comprehension. Don’t worry about your comprehension this is an exercise of perceptual speed.

Taken From: The Speed Reading Course

Percentage
of area
of circle P
represented
by the sum
Sum of Each Sum of of the areas
Each circle’s the circles’ circle’s the circles’ of the smaller
Circle Diameter circumference circumferences area areas circles
P 12 12 12 36 36 100
R 6 6 12 9 18 50
Q 4 4 12 4 12 331
3
S 3 3 12 2.25 9 25

An inspection of the above chart shows that the sum of the circumferences for each group of circles is the same yet the sum of the areas is quite different. The more circles we formed with the same total length of string, the smaller the total area of the circles. Just what you would not expect to happen!

That is, when two equal circles were formed, the total area of the two circles was one-half that of the large circle. Similarly, when four equal circles were formed, the total area of the four circles was one-fourth of the area of the large circle.

This seems to go against one’s intuition. Yet if we consider a more extreme case, with say 100 smaller equal circles, we would see that the area of each circle becomes extremely small and the sum of the areas of these 100 circles is one-hundredth of the area of the larger circle.

Have students explain this rather disconcerting concept. It ought to give them an interesting perspective on comparison of areas.

Taken From :Math Wonders to inspire teacher and student

In contrast, by turning pages as fast as possible and attempting to see as many words per page as one can, perception and the will are conditioned into much more rapid and efficient reading practices. This high speed conditioning can be compared to driving along a motorway at 100 miles per hour. Imagine that you have been driving for an hour at this speed. Suddenly you come to a road sign saying ‘Slow down to 30 m.p.h.’ Now imagine that your speedometer is not working; what speed would you actually slow down to? The answer would probably be 50-60 m.p.h.

The reason for this is that your perceptions have become conditioned to a much higher speed, which becomes ‘normal’. There is a ratchet effect by which previous ‘normals’ are more or less forgotten as the result of the perceptual conditioning. The same principle applies to reading; after high speed practice, you will often find yourself reading at twice the speed, without even feeling the difference.

Taken From: The Speed Reading Course

6
4
9
1
5
5
A B
D C
H I
K J
P Q
S R

With a perimeter of 20, the area of rectangle ABCD is 24.
With a perimeter of 20, the area of rectangle HIJK is 9.
With a perimeter of 20, the area of rectangle PQRS is 25.

It can be shown that the maximum area of a rectangle with a fixed perimeter is the one with equal length and width, that is, a square.

It is interesting to compare areas of similar figures. We will consider
circles.

Suppose you have four equal pieces of string. With the first piece of
string, one circle is formed. The second piece of string is cut into two
equal parts and two congruent circles are formed. The third piece of
string is cut into three equal pieces and three congruent circles are formed. In a similar way, four congruent circles are formed from the fourth piece of string. Note that the sum of the circumferences of each group of congruent circles is the same (see p. 138 for illustration).

Taken From :Math Wonders to inspire teacher and student

The following diagram illustrates the visual processing of a line of text. When reading left to right, the material yet to be read is taken in with peripheral vision and analysed for content by the linguistic left hemisphere. This helps the brain decide the best next point of fixation and increases the efficiency of reading.

Book-worms are nearly always using the right hand visual field (connected to the left hemisphere), and dyslexics use the left visual field (connected to the right hemisphere). Both these extreme cases tend to have maladaptive scanning patterns, because they are nearly always using one side of the nervous system exclusively. Maladaptive patterns will include back-skipping, missing lines, and reading the same line twice. Practicing the speed reading techniques as presented
in this course should help to correct these patterns.

Taken From: The Speed Reading Course

In ADB, the midpoints of sides AD and AB are F and G, respectively.

Therefore, FG  DB and FG = 1 2BD, and EH  DB and EH = 1 2BD.

Therefore, FG  EH and FG = EH. This establishes that FGHE is a
parallelogram (see p. 136).

E
F
G
H
A
D
C
B

Furthermore, if the diagonals, DB and AC, are congruent (as above), then the sides of the parallelogram must also be congruent, since they are each one-half the length of the diagonals of the original quadrilateral. This results in a rhombus.

Similarly, if the diagonals of the original quadrilateral are perpendicular
and congruent, then since the sides of the parallelogram are, in pairs,
parallel to the diagonals and half their length, the adjacent sides of the parallelogram must be perpendicular and congruent to each other, making it a square.

Taken From :Math Wonders to inspire teacher and student

Step 4. Once you can read silently whilst counting silently, begin to increase your reading speed. When your reading speed exceeds 360 w.p.m., the two types of subjective reading will become more differentiated. (Test). By using thoughtstream you can read much faster, whereas subvocalisation is limited by the speed of motoric response.

Step 5. Now that you can easily read with thought-stream, leaving behind any subvocalisation, it is time to add more character to the inner speech, so that it is not just a silent stream of thought but is also a stream of visualisation. Image the dialogue of the novel, adopting different voices in your inner speech to suit the characters. This should further differentiate your thought-stream from subvocalisation, which would always tend to be a reflection of your own voice. At the same time, visualise the scenarios of the story, hear the environmental sounds, smell all the various scents, and feel the emotions portrayed.

Continue with the above exercises until you have a reality about the two types of reading (subvocalised and thought-stream) and can choose between them. This approach is better than trying to suppress subvocalisation altogether.

By suppressing both types of subjective reading, one can learn to skim at more than 2000 w.p.m., however, there will be very little retention of what has been read. This type of reading is valuable only when one is searching for a particular datum, or when one is doing this as a perceptual exercise.

Taken From: The Speed Reading Course

Here are a few possible results:

A question that ought to be asked at this point is how might the original quadrilateral have been shaped for the parallelogram to be a rectangle, rhombus, or square?

Either through guess and check or by an analysis of the situation, students should discover the following: When the diagonals of the original quadrilateral are perpendicular, the parallelogram is a rectangle.

When the diagonals of the original quadrilateral are congruent, then the parallelogram is a rhombus.

When the diagonals of the original quadrilateral are congruent and perpendicular, then the parallelogram is a square.

For the teacher who wishes to demonstrate this for the class, the Geometer’s Sketchpad software is highly recommended. For the teacher who wishes to prove that all of the above is “really true,” a short proof outline is provided, one that should be within easy reach for a high school geometry student.

Taken From :Math Wonders to inspire teacher and student

Step 1. Choose a page from a light novel. Easily understood material is required because even when a good reader is reading something that he finds difficult to comprehend, there will be a tendency to revert to subvocalisation, when a phrase or sentence containing unfamiliar or foreign words is presented. Unfamiliar words can only be held in mind either by having extremely good powers of auditory visualisation or by rehearsing them subvocally.

Note: a reader using thought-stream, rather than subvocalisation, will find he is able to detect misunderstood words more easily, because he will revert to subvocalisation as he strives to give meaning to the unfamiliar. If you find yourself suddenly subvocalising when you would otherwise use thought-stream, this is a strong indication that you have just gone past a word that is misunderstood, or a group of words forming a concept that does not make sense. Misunderstood words should each be defined and then the concept re-evaluated.

Step 2. Count out loud from one to ten repeatedly, whilst reading the page silently using thought-stream. Counting out loud will occupy the motor-vocal system, so that the mind is unable to subvocalise.

Step 3. When you are able to read silently whilst counting out loud, then begin to read silently using thought-stream and to count silently at the same time using sub-vocalisation.

Taken From: The Speed Reading Course

This is obviously a very primitive approximation of . A late-18th-century rabbi, Elijah of Vilna (Poland), one of the great modern biblical scholars, who earned the title “Gaon of Vilna” (meaning brilliance of Vilna), came up with a remarkable discovery, one that could make most history of mathematics books faulty if they say that the Bible approximated the value of  as 3. Elijah of Vilna noticed that the Hebrew word for “line measure” was written differently in each of the two biblical passages mentioned above.

In 1 Kings 7:23, it was written as , whereas in 2 Chronicles 4:2 it
was written as . Elijah applied the biblical analysis technique (still used today) called gematria, where the Hebrew letters are given their appropriatenumerical values according to their sequence in the Hebrew alphabet, to the two spellings of the word for “line measure” and found the following.

? Alfred S. Posamentier and Noam Gordon, “An Astounding Revelation on the History of ,” Mathematics Teacher, Vol. 77, No. 1, Jan. 1984, p. 52.

The letter values are = 100, = 6, and = 5. Therefore, the spelling
for “line measure” in 1 Kings 7:23 is = 5 + 6 + 100 = 111, while
in 2 Chronicles 4:2 the spelling = 6 + 100 = 106. He then took the
ratio of these two values: 111 106 = 10472 (to four decimal places), which he considered the necessary correction factor, for when it is multiplied by 3, which is believed to be the value of  stated in the Bible, one gets 3.1416, which is  correct to four decimal places! “Wow!” is a usual reaction. Such accuracy is quite astonishing for ancient times. To support this notion, have students take string to measure the circumference and diameter of several circular objects and find their quotient. They will most likely not get near this four-place accuracy. Moreover, to really push the point of the high degree of accuracy of four decimal places, chances are if you took the average of all the students’  measurements, you still
probably wouldn’t get to four-place accuracy.

Taken From :Math Wonders to inspire teacher and student

Generally speaking, subvocalisation is unnecessary to the adult reader, except perhaps when reading poetry (in which case rhythm, rhyme, and alliteration are an important component, and so subvocalisation may be more enjoyable when poetry is read silently). However, subvocalisation limits the maximum reading speed to about 300 w.p.m. +/- 20%. In contrast, a trained reader may read at more than 1000 w.p.m. with a pure thought-stream.

A thought-stream is essential for full understanding. Although it may be possible to read light material such as a novel without using a thought-stream at all, memory will be impaired. The thought-stream is particularly important when reading abstract material that cannot be easily visualised, and when long and complicated sentence constructions are used. When this type of material is read and the thought-stream is suppressed, it is nearly impossible to preserve word
order and syntax. When the material is difficult to visualise, syntax and word order may be the only guides to meaning and understanding.

Before a student can learn to let go of subvocalisation without at the same time suppressing inner speech altogether, he has to learn to differentiate between subvocalisation and the thought-stream. This first step can be done by a process of localisation. Most people will experience subvocalisation as being connected with the mouth or the throat, and also the breath. When asked to attend to it fully, a person will tend to look down.

Taken From: The Speed Reading Course

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Take a tall and narrow cylindrical drinking glass. Ask a student if the
circumference is greater than or less than the height. The glass should be chosen so that it would “appear” to have a longer height than its circumference. (The typical tall narrow drinking glass fits this requirement.) Now ask the student how he/she might test his/her conjecture (aside from using a piece of string). Recall for him/her that the formula for the circumference of a circle is C = d ( times the diameter). He/she may recall that  = 314 is the usual approximation, but we’ll be even more crude and use  = 3. Thus, the circumference will be 3 times the diameter, which can be easily “measured” with a stick or a pencil and then marked off 3
times along the height of the glass. Usually, you will find that the circumference is longer than the tall glass, even though it does not “appear” to be so. This little optical trick is useful to demonstrate the value of .

Now for a real “mindblower”! To appreciate the next revelation on , you need to know that virtually all the books on the history of mathematics state that in its earliest manifestation in history, namely the Bible (Old Testament), its value is given as 3. Yet recent “detective work” shows otherwise.?

Students always relish the notion that a hidden code can reveal long
lost secrets. Such is the case with the common interpretation of the value of  in the Bible. There are two places in the Bible where the same sentence appears, identical in every way except for one word, spelled differently in the two citations. The description of a pool or fountain in King Solomon’s temple is referred to in the passages that may be found in 1 Kings 7:23 and 2 Chronicles 4:2 and that read as follows:

And he made the molten sea of ten cubits from brim to brim, round in compass, and the height thereof was five cubits; and a line of thirty cubits did compass it round about.

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A
D
B
E
C
A
D
B
E
C
A
D
B
E
C

A
D
B
E
C
A
D
B
E
C
Again, notice how, through the sequence of angle moves, the pencil’s
direction changed by 180
.

For those who feel more comfortable with a geometric “proof,” the following demonstration is provided. Note that we are accepting the notion that the angle sum of the “corners” of a pentagram is the same for all pentagrams. Since the type of pentagram was not specified, we can assume the pentagram to be regular, or that it is one, which is inscribable in a circle (i.e., all of its vertices lie on the circle). In either case, we notice that each of the angles is now an inscribed angle of the circle, and so has half the measure of the intercepted arc (see below).

A
C
E
B
D

Consequently, we obtain the following:
m?A =
1
2
m

CD (this reads “the measure of angle A
equals one-half the measure of arc CD”)
m?B =
1
2
m

ED
m?C =
1
2
m

AE
m?D =
1
2
m

AB
m?E =
1
2
m

BC

If we add these equalities, we obtain
m?A + m?B + m?C + m?D + m?E
=
1
2
m

CD +

ED +

AE +

AB +

BC
=
1
2
360
= 180

That is, the sum (of the measures) of the angles of a pentagram is one-half the degree measure of a circle, or 180
. Again, note that there was no loss of generality by allowing the nonspecified pentagram to assume a more useful configuration.

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Most students know that the sum of the angles of a triangle is 180
and a quadrilateral’s angles have a sum of 360
. But what is the sum of the angles of a pentagram? Although easily provable, we shall assume that all pentagrams have the same angle sum. This implies that we ought to be able to get the answer by finding the angle sum of a regular pentagram and then simply generalize it to all pentagrams. Students ought to be able to “stumble” on this angle sum once they have been able to find the measure of one vertex angle, not very difficult since the angles are all congruent and there is lovely symmetry throughout.

However, suppose we didn’t make this connection and were simply trying to get the angle sum of an “ugly” pentagram, such as the one below.

A
D
B
E
C

We could determine this by placing a pencil on AC in the direction pointing at A and rotating it through ?A, so that it is now on AD pointing at A. Then rotate it through ?D so that it is now on BD pointing at B. Then rotate it through ?B so that it is now on BE pointing at B. Then rotate it through ?E so that it is now on EC pointing at C. Lastly, rotate it through ?C so that it is now on AC pointing at C, which is the opposite direction of its starting position. Therefore, the pencil reversed its direction, which is the same thing as a rotation of 180
, implying that the angle sum of the pentagram (through which the pencil was rotated, angle by angle) is 180
.

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However, it is also nice to show why this folding procedure has the vertices meet at a point on the side of the triangle. Establishing this phenomenon is tantamount to proving the theorem of the angle sum of a triangle.

The proof of this theorem follows directly from the paper-folding exercise. By folding the top vertex along a parallel crease (i.e., DE  BC), AF ?ED at M. Since MF  AM, or M is the midpoint of AF , D and E are midpoints of AB and AC, respectively, since a line parallel to one side of a triangle (either BAF or CAF ) and bisecting a second side AF of the triangle also bisects the third side. It is then easy to show that since AD  DF , DB  DF and similarly EF  EC, so that the folding over of vertices B and C would fit at F , forming a straight line along ?? BFC (Figure 5.4, p. 125).

The most important part of this unit is to convince your students that a paper-folding exercise can be quite valid in demonstrating a property. Mention the difference between a paper-folding demonstration and a proof.

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So now how does the angle sum of a triangle relate to this? The simplest and perhaps the most convincing way to demonstrate this angle sum is to tear the three vertices from a paper triangle and place them together to form a straight line. This straight line represents one-half of a complete revolution, hence, the 180
.

It is perhaps more elegant to use a folding procedure. Students should be told to cut a conveniently large scalene triangle from a piece of paper. They should then fold one vertex so that it touches the opposite side and so that the crease is parallel to that side. (See Figure 5.1).

Figure 5.1

Figure 5.2

Figure 5.3

A
B
C
D
E
F
M

Figure 5.4

Taken From :Math Wonders to inspire teacher and student

There are a number of units that demonstrate the beautiful concept of an invariant in geometry. What that means is that, in some situations, critical aspects of a figure remain constant even when other parts are changed. These invariants can be nicely demonstrated on the computer with the help of the Geometer’s Sketchpad program. For example, the perpendiculars drawn to the three sides of a triangle from any point on its circumcircle intersect the sides in three collinear points (Simson’s invariant). These points will always be on a straight line. This invariant is just one of several
shown in this chapter.

There are several very entertaining proofs of the Pythagorean theorem, one by paper folding, one extraordinarily simple, and one done by a former U.S. president. There are units that will require the students to do some activities, “moving” along a figure, paper folding, and exposing some extraordinary properties/phenomena.

The chapter is full of unusual geometric properties, all pointing to the
beauty of the subject matter. It is for you to present the material in
the most interesting way possible. This will, of course, depend in large measure on your personality and your taste for the delights of the geometric menu offered here.

Taken From :Math Wonders to inspire teacher and student

We will now show that (a b c) is a primitive Pythagorean triple where a = m2?n2, b = 2mn, and c = m2+n2. It is simple to verify algebraically that (m2 ? n22 + 2mn2 = m2 + n22, thereby making it a Pythagorean triple. What remains is to prove that (a b c) is a primitive Pythagorean triple.

Suppose a and b have a common factor h > 1. Since a is odd, h must
also be odd. Because a2 +b2 = c2 h would also be a factor of c. We also have h a factor of m2 ? n2 and m2 + n2 as well as of their sum, 2m2, and their difference, 2n2.

Since h is odd, it is a common factor of m2 and n2. However, m and n
(and as a result, m2 and n2) are relatively prime. Therefore, h cannot be a common factor of m and n. This contradiction establishes that a and b are relatively prime.

Having finally established a method for generating primitive Pythagorean triples, students should be eager to put it to use. The table below gives some of the smaller primitive Pythagorean triples.

Pythagorean Triples
m n a b c
2 1 3 4 5
3 2 5 12 13
4 1 15 8 17
4 3 7 24 25
5 2 21 20 29
5 4 9 40 41
6 1 35 12 37
6 5 11 60 61
7 2 45 28 53
7 4 33 56 65
7 6 13 84 85

A fast inspection of the above table indicates that certain primitive
Pythagorean triples (a b c) have c = b + 1. Have students discover the relationship between m and n for these triples.

They should notice that for these triples m = n + 1. To prove this will be true for other primitive Pythagorean triples (not in the table), let m = n+1 and generate the Pythagorean triples.

a = m2 ? n2 = n + 12 ? n2 = 2n + 1
b = 2mn = 2nn + 1 = 2n2 + 2n
c = m2 + n2 = n + 12 + n2 = 2n2 + 2n + 1

Clearly, c = b + 1, which was to be shown!

A natural question to ask your students is to find all primitive Pythagorean triples that are consecutive natural numbers. In a method similar to that used above, they ought to find that the only triple satisfying that condition is (3 4 5).

Students should have a far better appreciation for Pythagorean triples and elementary number theory after completing this unit. Other investigations that students may wish to explore are presented below.

1. Find six primitive Pythagorean triples that are not included in the table.
2. Prove that every primitive Pythagorean triple has one member that is divisible by 3.
3. Prove that every primitive Pythagorean triple has one member that is divisible by 5.
4. Prove that for every primitive Pythagorean triple the product of its
members is a multiple of 60.
5. Find a Pythagorean triple (a b c), where b2 = a + 2.

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Since b is even, we may represent b as
b = 2r

But
b2 = c + ac ? a

Therefore,
b2 = 2p2q = 4r2 or pq = r2

If the product of two relatively prime natural numbers (p and q) is the
square of a natural number (r), then each of them must be the square of a natural number.

Therefore, we let p = m2 and q = n2, where m and n are natural numbers. Since they are factors of relatively prime numbers (p and q), they (m and n) are also relatively prime.

Since a = p ? q and c = p + q, it follows that a = m2 ? n2 and c = m2 + n2.

Also, since b = 2r and b2 = 4r2 = 4pq = 4m2n2, b = 2mn.

To summarize, we now have formulas for generating Pythagorean triples:
a = m2 ? n2 b = 2mn c = m2 + n2

The numbers m and n cannot both be even, since they are relatively prime. They cannot both be odd, for this would make c = m2 + n2 an even number, which we established earlier as impossible. Since this indicates that one must be even and the other odd, b = 2mn must be divisible by 4. Therefore, no Pythagorean triple can be composed of three prime numbers. This does not mean that the other members of the Pythagorean triple may not be prime

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2k + 12 = 4k2 + 4k + 1 = 4kk + 1 + 1

Since k and k + 1 are consecutive, one of them must be even. Therefore, 4kk + 1 must be divisible by 8. Thus, 2k + 12, when divided by 8, leaves a remainder of 1.

The next lemmas follow directly.

Lemma 2 When the sum of two odd square numbers is divided by 8, the remainder is 2.

Lemma 3 The sum of two odd square numbers cannot be a square
number.

Proof Since the sum of two odd square numbers, when divided by 8,
leaves a remainder of 2, the sum is even, but not divisible by 4. It therefore cannot be a square number.

We are now ready to begin our development of formulas for Pythagorean triples. Let us assume that a b c is a primitive Pythagorean triple. This implies that a and b are relatively prime.? Therefore, they cannot both be even. Can they both be odd?

If a and b are both odd, then, by Lemma 3, a2 +b2 = c2. This contradicts our assumption that a b c is a Pythagorean triple; therefore, a and b cannot both be odd. Therefore, one must be odd and one even.

? Relatively prime means that they do not have any common factors aside from 1.

Let us suppose that a is odd and b is even. This implies that c is also
odd. We can rewrite a2 + b2 = c2 as

b2 = c2 ? a2
b2 = c + ac ? a

Since the sum and the difference of two odd numbers are even, c+a = 2p
and c ? a = 2q (p and q are natural numbers).

By solving for a and c, we get
c = p + q and a = p ? q

Taken From :Math Wonders to inspire teacher and student

Ask your students to supply the number(s) that will make each a
Pythagorean triple:

1. (3 4 __)
2. (7 __ 25)
3. (11 __ __)

The first two triples can be easily determined using the Pythagorean theorem. However, this method will not work with the third triple. At this point, your students will be quite receptive to learning about a method to discover the missing triple. So, with properly motivated students as your audience, you can embark on the adventure of developing a method for establishing Pythagorean triples.

However, before beginning to develop formulas, we must consider a few simple “lemmas” (these are “helper” theorems).

Lemma 1 When the square of an odd number is divided by 8, the
remainder is 1.

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Consider the following series for which we are asked to find the sum:
1
?1 + ?2 +
1
?2 + ?3 +
1
?3 + ?4
+· · ·+
1
?2001 + ?2002 +
1
?2002 + ?2003

Students are taught that they cannot do much with a fraction where the denominator is irrational and so must seek to change it to an equivalent fraction with a rational denominator. To do this, they know to multiply the fraction by 1 so as not to change its value. Yet, the form that 1 should take on is the conjugate of the current denominator in both the numerator and the denominator.

The general term of this series may be written as 1 ?k+?k+1 .

We shall now rationalize the denominator of this fraction by multiplying it by 1 in the form of
?k??k+1 ?k??k+1 to get
1
?k + ?k + 1 
?k ? ?k + 1
?k ? ?k + 1 =
?k ? ?k + 1
?1

That is, we have found
1
?k + ?k + 1 = ?k + 1 ? ?k
We can then rewrite the series as

?2 ? ?1 + ?3 ? ?2 + ?4 ? ?3
+· · ·+?2002 ? ?2001 + ?2003 ? ?2002

which then becomes simply

?2003 ? 1 ? 44
754888 ? 1 = 43
754888

Students can readily see how rationalizing the denominator is not just
an exercise without purpose. It is useful, and here we have had a prime example.

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Find four consecutive numbers whose product is 120.

Give your students a bit of time to begin to tackle this question. Most will probably write an algebraic equation to depict the situation. It may look like this:

xx + 1x + 2x + 3 = 120

Ridding the parentheses here leaves us with a fourth-degree equation
(a quartic equation) in one variable. Rather than to try to solve this quartic equation, a nonalgebraic solution might be preferable. Simply guess intelligently and check to get the solution: 2  3  4  5 = 120. The students should see from this demonstration that although algebra is very useful to introduce or explain some arithmetic relationships, it is not always the best method.

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x =


1 +


1 +


1 +


1 +

1 +
1 + 1 + ?1+· · ·
x2 = 1 +


1 +


1 +


1 +


1 +

1 +
1 + 1 + ?1+· · ·
x2 = 1 + x
x =  from the definition of 

It is fascinating to observe what happens when we find the powers of .
2 = 
?5 + 1
2
2
=
?5 + 3
2 =
?5 + 1
2 + 1 =  + 1
3 =   2 =  + 1 = 2 + 
=  + 1 +  = 2 + 1
4 = 2  2 =  + 1 + 1 = 2 + 2 + 1
=  + 1 + 2 + 1 = 3 + 2
5 = 3  2 = 2 + 1 + 1 = 22 + 3 + 1
= 2 + 1 + 3 + 1 = 5 + 3
6 = 3  3 = 2 + 12 + 1 = 42 + 4 + 1
= 4 + 1 + 4 + 1 = 8 + 5
7 = 4  3 = 3 + 22 + 1 = 62 + 7 + 2
= 6 + 1 + 7 + 2 = 13 + 8

A summary chart reveals a pattern among the coefficients of :
2 =  + 1
3 = 2 + 1
4 = 3 + 2
5 = 5 + 3
6 = 8 + 5
7 = 13 + 8

These are the Fibonacci numbers (see Unit 1.18).

By this time, your students are probably thinking that there is no end to the connections that one can draw to the Golden Section. Indeed, they are correct!

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=1 61803398874989484820458683436563811772030917980576

and
1  =0 61803398874989484820458683436563811772030917980576

There are lots of other interesting features of . Your students ought to be guided to develop some after you give them the proper hints. They might want to show that this infinite continued fraction has the value .
 = 1 +
1
1 +
1
1 +
1
1 +
1
1 +
1
1 +
1
1 +
1
1+· · ·

To do this, students ought to realize that nothing is lost by truncating the continued fraction at the first numerator. This will give them the following:
 = 1 +
1


which yields the Golden Ratio.

Another curious relationship is

 =


1 +


1 +


1 +

 
1
+

1 +
1 + 1 + ?1+· · ·

Taken From :Math Wonders to inspire teacher and student

Begin by having students recall the Golden Ratio:
1 ? x
x =
x
1

This gives us
x2 + x ? 1 = 0 and x = ?5 ? 1 2  for positive x

We let
?5 ? 1 2 = 1 

Not only does   1
 = 1 (obviously!), but  ? 1  = 1.

This is the only number for which this is true.

Your students may want to verify this.?

? Here is the result derived.
Since
1  = ?5 ? 1 2

then
 = 2 ?5 ? 1  ?5 + 1 ?5 + 1 = ?5 + 1 2

and
 ? 1  = ?5 + 1 2 ? ?5 ? 1 2 = 1

Taken From :Math Wonders to inspire teacher and student

Guide students to draw the appropriate rectangle with side lengths a+b and c+d. The rectangle should be partitioned into smaller rectangles, as shown in Figure 4.3. The lengths of sides and areas of regions have been

a b
c ac bc
d ad bd

Figure 4.3

labeled. As in the other cases, the area of the large rectangle equals the sum of the areas of the four quadrilaterals into which it was partitioned.

The diagram (Figure 4.3) illustrates the identity a + bc + d = ac + ad + bc + bd.

Explain to students that the method of application of areas can be used to prove most algebraic identities. The difficulty will lie in their choice of dimensions for the quadrilateral and the partitions made.

After students feel comfortable using areas to represent algebraic identities, have them consider the Pythagorean relationship, a2 + b2 = c2. Although this is not an identity, the application of areas is still appropriate. Have students draw a square of side length a + b. Show students how to partition this square into four congruent triangles and a square, as shown in Figure 4.4. The lengths of the sides have been labeled.

The diagram (Figure 4.4) illustrates:

1. Area DEFG = 4AreaGNM + AreaKLMN.
2. Therefore, a + b2 = 4 1 2ab + c2.
3. If we now substitute the identity for a+b2, which was proved above, we obtain a2 + 2ab + b2 = 2ab + c2.

Then clearly a2 + b2 = c2, which is, of course, the Pythagorean theorem.

b
b
b
b
c
c
c
c
a
a
a
a
M
N
F
D E
G
K
L
Figure 4.4

Taken From :Math Wonders to inspire teacher and student

The concept that an algebraic identity can be demonstrated geometrically by using areas to show, for example, a + b2 = a2 + 2ab + b2.

To begin, have students draw a square of side length a + b. The square should then be partitioned into various squares and rectangles, as shown in Figure 4.1. The lengths of the various sides are appropriately labeled.

a b a a2 ab b ab b2

Figure 4.1

Students can easily determine the area of each region. Since the area of the large square equals the sum of the areas of the four quadrilaterals into which it was partitioned, students should get

a + b2 = a2 + ab + ab + b2 = a2 + 2ab + b2

A more rigorous proof can be found in Euclid’s Elements, Proposition 4, Book 11.

Next, illustrate geometrically the identity ab + c = ab + ac. To begin, have students draw a rectangle whose adjacent sides are of lengths a and b + c. The rectangle should then be partitioned into smaller rectangles, as shown in Figure 4.2. The lengths of these sides are also labeled.

b c
a ab ac

Figure 4.2

Students can easily determine the area of each region. Elicit from students that since the area of the large rectangle equals the sum of the areas of the two quadrilaterals into which it was partitioned, the diagram illustrates

ab + c = ab + ac.

Taken From :Math Wonders to inspire teacher and student

Let’s look at a situation that lends itself to a few nifty alternatives. Consider the problem of finding the sum of the following series:
1
1  2 +
1
2  3 +
1
3  4 +· · ·+
1
49  50

One way to begin is to see if there is any visible pattern.
1
1  2 =
1
2
1
1  2 +
1
2  3 =
2
3
1
1  2 +
1
2  3 +
1
3  4 =
3
4
1
1  2 +
1
2  3 +
1
3  4 +
1
4  5 =
4
5

From this pattern, we guess the following pattern:
1
1  2 +
1
2  3 +
1
3  4 +
1
4  5 +· · ·+
1
49  50 =
49
50

Another pattern for this series can be obtained by representing each fraction in the series as a difference in the following way:
1
1  2 =
1
1 ?
1
2
1
2  3 =
1
2 ?
1
3
1
3  4 =
1
3 ?
1
4
1
49  50 =
1
49 ?
1
50

Adding these equations, the left-hand side gives us our sought-after sum,
and on the right-hand side almost all the fractions drop out, leaving
1
1 ? 1
50 = 49
50 .

These very surprising illustrations of useful patterns will get a reaction
from your students, “Oh, I would never be able to do this on my own,”
but this should be an unacceptable response, for “practice makes perfect”!

Taken From :Math Wonders to inspire teacher and student

1  2 = 2
2  3 = 6
3  4 = 12
4  5 = 20
5  6 = 30

you now have the option of having your students try to justify the following relationships by using algebra or by convincing themselves through examples that they are, in fact, true. Remember, only a general proof will show they hold for all cases.

An oblong number can be expressed as the sum of consecutive even integers, beginning with 2:

2 + 4 + 6 + 8 = 20

An oblong number is twice a triangular number:

15  2 = 30

The sum of two consecutive squares and the square of the oblong between them is a square?:

9 + 16 + 122 = 169 = 132

? This is a tricky one so it is provided for you here. Represent the statement algebraically as n2 + n + 12 + nn + 12. Expanding and collecting like terms gives us
n2 + n2 + 2n + 1 + n2 + n2 = 2n2 + 2n + 1 + n4 + 2n3 + n2
= n4 + 2n3 + 3n2 + 2n + 1 = n2 + n + 12

Taken From :Math Wonders to inspire teacher and student

Consider the following table:
Triangular Square Pentagonal Hexagonal
numbers numbers numbers numbers
1 1 1 1 1
2 3 4 5 6
3 6 9 12 15
4 10 16 22 28
5 15 25 35 45
6 21 36 51 66
7 28 49 70 91
8 36 64 92 120
9 45 81 117 153
10 55 100 145 190
11 66 121 176 231
12 78 144 210 276
13 91 169 247 325
14 105 196 287 378
15 120 225 330 435
16 136 256 376 496
17 153 289 425 561
18 171 324 477 630
19 190 361 532 703
20 210 400 590 780
n nn+1?
2
n2n?0
2 = n2 n3n?1
2
n4n?2
2

Based on the pattern developing across the nth row, have your students determine what the general forms of the heptagonal, octagonal, decagonal, etc., numbers are.

They are
n5n ? 3
2

n6n ? 4
2

n7n ? 5
2

? This comes from n1n??1
2 = nn+1
2 .

Taken From :Math Wonders to inspire teacher and student

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Have students experiment with a calculator and discover the following conjecture on their own:

One plus the sum of the squares of any three consecutive odd numbers is always divisible by 12.

The beauty and the instructional benefit of this are manifested in the
simplicity of the procedure used to prove this statement. First, establish a way to represent an odd and an even number. For any integer n, 2n will always be even and 2n + 1 must then be odd.

We begin by letting 2n+1 be the middle number of the three consecutive odd numbers under consideration. Then 2n + 1 ? 2 = 2n ? 1 is the next smaller odd number and 2n + 1 + 2 = 2n + 3 is the next larger odd number. We are now ready to represent the relationship we are seeking to prove.

2n ? 12 + 2n + 12 + 2n + 32 + 1 = 12n2 + 12n + 12
= 12n2 + n + 1 = 12M

where M represents some integer.??

We can then conclude that this sum of squares plus 1 is always divisible by 12. This should be merely a springboard to other similar algebraic investigations into number theory.

? Named for Christian Goldbach (1690–1764) and transmitted in a letter to the famous mathematician Leonhard Euler in 1742.
?? Since n is an integer, n2 is also an integer, so the sum n2 + n + 1 must also be an integer. Represent that integer as M.

Taken From :Math Wonders to inspire teacher and student

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Let’s use algebra to clear up this very strange result, established above by example. Point out to students that we will be using algebra to help us understand this mathematical quirk.

We typically represent a two-digit number as 10t + u, where t represents the tens digit and u represents the units digit. Then the sum of the digits is t + u and the product of the digits is tu.

The number meeting the above conditions = 10t + u
= tu + t + u
10t = tu + t
9t = tu
u = 9 for t = 0?

This discussion should evoke a curiosity among students about numbers with more than two digits. For example:

109 = 10  9 + 10 + 9
119 = 11  9 + 11 + 9
129 = 12  9 + 12 + 9

Here the digits to the left of the 9 are considered as a number and treated just as we treated the tens digit above. The results are the same.

This can be extended to any number of digits as long as the units digit is a 9. ? In this case, the rule also holds if t = 0.

Taken From :Math Wonders to inspire teacher and student

Any two-digit number ending in 9 can be expressed as the sum of
the product of the digits and the sum of the digits.

More simply stated:

Any two-digit number ending in 9
= product of digits + sum of digits

One of the real advantages of algebra is the facility with which, through its use, we can justify many mathematical applications. Why is it possible to represent a number ending in a 9 in the following way?

9 = 0  9 + 0 + 9
19 = 1  9 + 1 + 9
29 = 2  9 + 2 + 9
39 = 3  9 + 3 + 9
49 = 4  9 + 4 + 9
59 = 5  9 + 5 + 9
69 = 6  9 + 6 + 9
79 = 7  9 + 7 + 9
89 = 8  9 + 8 + 9
99 = 9  9 + 9 + 9

Taken From :Math Wonders to inspire teacher and student

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Have the students work individually with the following oral instructions:

Select any three-digit number with all digits different from one
another. Write all possible two-digit numbers that can be formed
from the three-digits selected earlier. Then divide their sum by the
sum of the digits in the original three-digit number.

Students should all get the same answer, 22. There ought to be a big
resulting “Wow!”

For example, consider the three-digit number 365. Take the sum of all
the possible two-digit numbers that can be formed from these three digits: 36+35+63+53+65+56 = 308. The sum of the digits of the original number is 3 + 6 + 5 = 14. Then 308 14 = 22.

To analyze this unusual result, we will begin with a general representation of the number: 100x + 10y + z. Algebraic intertainments 101 We now take the sum of all the two-digit numbers taken from the three digits:

10x+y+10y+x+10x+z+10z+x+10y+z+10z+y
=102x+2y+2z+2x+2y+2z
=112x+2y+2z
=22x+y+z

which, when divided by the sum of the digits, x + y + z, is 22.
These illustrations show the value of algebra in explaining simple arithmetic phenomena.

Taken From :Math Wonders to inspire teacher and student

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We could employ factoring the difference of two squares: x2 ? y2 = x ? yx + y. That would give us

362 ? 352 = 36 ? 3536 + 35 = 171 = 71

Your students will be quite amazed at how this common form of factoring has reduced this multiplication problem to a trivial situation.

The distributive property is always useful, as for multiplying 8  67.
Replacing 67 with 70 ? 3 allows us to rewrite the multiplication as
870 ? 3 = 870 ? 83 = 560 ? 24 = 536.

Or to multiply 36  14, we can rewrite this as 3610 + 4 = 3610 + 364 = 360 + 144 = 504. In the absence of a calculator, this is a much more efficient way to look at multiplication.

Multiplying two numbers with a difference of 4 can also be simply
done by first inspecting the situation in general terms (i.e., algebraically):

The two numbers can be (x + 2) and (x ? 2). These have a difference of 4. Their product is x + 2x ? 2 = x2 ? 4. Thus, we must find the average of the two numbers, x, and square it and subtract 4.

For example, to use this notion to multiply 67 by 71, we find the average, 69. Then square 69 to get 4,761 and subtract 4 to get 4,757. It may not always be easier to do the multiplication, but it will give the students a sense of “usefulness” of some of the algebra they know.

Multiplying two consecutive numbers uses the property xx + 1 = x2 + x. This applied to 23  24 = 232 + 23 = 529 + 23 = 552 provides a

refreshing alternative to the usual multiplication algorithm. Again, it must be stressed that this is not a replacement for the calculator—something students would clearly not accept.

By this time, students might be motivated to discover or establish their own shortcut algorithms. It ought to be done for fun—though not a calculator replacement!

Taken From :Math Wonders to inspire teacher and student

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Show your students how algebra can be used to shed new light on and a deeper appreciation for mathematical relationships. It will entertain them while demonstrating the beauty of algebraic processes.

Taken From :Math Wonders to inspire teacher and student

In a drawer, there are 8 blue socks, 6 green socks, and 12 black
socks. What is the minimum number of socks Henry must take
from the drawer, without looking, to be certain that he has two
socks of the same color?

The phrase “


certain


two socks of the same color” is the key to the problem. The problem does not specify which color, so any of the three would be correct. To solve this problem, have your students reason from a “worst case scenario.” Henry picks one blue sock, one green sock, and then one black sock. He now has one of each color, but no matching pair. (True, he might have picked a pair on his first two selections, but the problem calls for “certain.”) Notice that as soon as he picks the fourth sock, he must have a pair of the same color.

Consider a second problem:

In a drawer, there are 8 blue socks, 6 green socks, and 12 black
socks. What is the minimum number of socks that Evelyn must
take from the drawer, without looking, to be certain that she has
two black socks?

Although this problem appears to be similar to the previous one, there
is one important difference. In this problem, a specific color has been
specified. Thus, it is a pair of black socks that we must guarantee being selected. Again, let’s use deductive reasoning and construct the “worst case scenario.” Suppose Evelyn first picks all of the blue socks (8). Next she picks all of the green socks (6). Still not one black sock has been chosen. She now has 14 socks in all, but none of them is black. However, the next two socks she picks must be black, since that is the only color remaining. To be certain of picking two black socks, Evelyn must select 8 + 6 + 2 = 16 socks in all. Have students create similar problems and present solutions.

Taken From :Math Wonders to inspire teacher and student

Now, since the area of parallelogram ABCD and the area of parallelogram EFGD are both equal to twice the area of triangle EDC, the areas of the two parallelograms must be equal. Thus, the area of parallelogram EFGD equals 20 square units.

Although the solution method that we have just shown is not often used, it is effective and efficient.

Nevertheless, this problem can be solved quite elegantly by solving a
simpler analogous problem (without loss of generality). Recall that the
original given conditions were that the two parallelograms had to have a common vertex D, and a vertex of one had to be on the side of the other as shown with points E and C. Now, let us suppose that C coincided with G, and E coincided with A. This satisfies the given condition of the original problem and makes the two parallelograms coincide. Thus, the area of parallelogram EFGD = 20 square units.

We could also look at this last solution as one of using extremes. That is, we might consider point E on AB, yet placed at an extreme, such as on point A. Similarly, we could place C on G and satisfy all the conditions of the original problem. Thus, the problem is trivial, in that the two parallelograms coincide. This point is one of the more neglected techniques for solving problems. It ought to be emphasized now.

Remember how difficult you perceived the problem at the start?

Taken From :Math Wonders to inspire teacher and student

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