# Project Mathematics! (2 Parts)

Synopsis
A series of seven videos for teaching mathematics using computer animation. Further information on the series is given online at: http://www.projectmathematics.com/index.htm
Language
English
Country
United States
Year of production
1993
Documentation
Workbooks included with each programme
Uses
A-level maths students
Subjects
Mathematics
Keywords
pi; polynomials; Pythagoras’ theorem; similarity; sines; cosines; trigonometry

### Distribution Formats

Type
VHS
Format
PAL
Price
£35.00 each
Availability
Sale
Duration/Size
22-29 minutes each
Year
2004

### Sections

Title
Story of Pi, The
Synopsis
The programe defines pi as the ratio of circumference to diameter of a circle, and shows how pi appears in a variety of formulae, many of which have nothing to do with circles. After discussing the early history of pi, the program invokes similarity to explain why the ratio of circumference to diameter is the same for all circles, regardless of size. This ratio, a fundamental constant of nature, is denoted by the Greek letter pi, so that 2 pi r represents the circumference of a circle of radius r.

Two animated sequences show that a circular disk of radius r can be dissected to form a rectangle of base pi r and altitude r, so the area of the disk is pi r squared, a result known to Archimedes. Animation shows the method used by Archimedes to estimate pi by comparing the circumference of a circle with the perimeters of inscribed and circumscribed polygons.

The next segment describes different rational estimates for pi obtained by various cultures, points out that pi is irrational and demonstrates the appearance of pi in probability problems. The concluding segment explains that major achievements in estimating pi represent landmarks of important advances in the history of mathematics.

Video extracts and supporting information available on line at http://www.projectmathematics.com/storypi.htm
Duration
24 mins

Title
Theorem of Pythagoras
Synopsis
The video examines Pythagoras’ theorem and emphasises its use in everyday practical problems.It presents three real-life versions of the same mathematical problem - how do you find the length of one side of a right-angled triangle if the lengths of the other two sides are known. The problem is solved by a computer-animated derivation of the Pythagorean theorem (based on similar triangles).

The algebraic formula a2 + b2 = c2 is interpreted geometrically in terms of areas of squares, and is then used to solve the three real-life problems posed earlier. Historical context is provided through stills showing Babylonian clay tablets and various editions of Euclid’s Elements. Several different computer-animated proofs of the Pythagorean theorem are presented, and the theorem is extended to 3-space.

Video extracts and supporting information available on line at www.projectmathematics.com/pythag.htm
Duration
22 mins

Title
Polynomials
Synopsis
The programme opens with examples of polynomial curves that appear in real life, including parabolic trajectories and the use of cubic splines in designing sailing boats.

Polynomials are systematically classified by degree. Linear polynomials are discussed - their graphs are straight lines of various slope, then quadratic polynomials - their graphs are parabolas, the prototype being the graph of y = x2. Animation shows how the Cartesian equation changes when the curve is translated vertically or horizontally or subjected to a vertical change of scale. Cubic polynomials are treated next, with discussion of zeros, local maxima and minima, and points of inflection. There are three prototypes y = x3, y = x3 + x, and y = x3 - x. All cubics can be obtained by horizontal or vertical translation or by horizontal or vertical change of scale, or by taking mirror images. A similar discussion is given for quartics and higher degree polynomials, all of which have infinitely many prototypes.

Video extracts and supporting information available online at http://www.projectmathematics.com/polynom.htm
Duration
26 mins

Title
Similarity
Synopsis
Examines similarity and looks at its implications. To change size without changing shape, scaling is introduced. Scaling multiplies lengths of all line segments by the same number and produces a similar figure. Similarity preserves angles and ratios of lengths of corresponding line segments. Applications show how Thales might have used similarity to find the height of a column and of a pyramid by comparing lengths of shadows. Another application of similarity explains why the sum of the angles in any triangle is a straight angle. Similarity is discussed for more general polygons and for three-dimensional objects. Animation shows what happens to perimeters, areas, and volumes under scaling, with illustrations from real life.
Similarity is the basis of all measurement. It reveals the secret of map making and scale drawings, and also explains some aspects of photographic images. Similarity helps explain why a hummingbird’s heart beats so much faster than a human heart, and why it is impossible for a small creature such as a praying mantis to become as large as a horse.

Video extracts and supporting information available online at http://www.projectmathematics.com/similar.htm
Duration
26 mins

Title
Sines and Cosines, Part 1: Periodic Functions
Synopsis
Shows how sines and cosines arise in different contexts: As the rectangular coordinates of a point moving on a unit circle, as graphs related to vibrating motion (illustrated by musical instruments), and as ratios of sides of right triangles. Reflecting the sine curve about various lines reveals simple properties of the sine function, for example, sin(-t) = - sin t, sin(pi - t) = sin t, sin(pi + t) = - sin t. Reflection of the sine curve about the line t = pi/4 generates a new curve, called a cosine curve, given by cos t = sin(pi/2 - t). Periodic waves are discussed, and the tape illustrates Fourier’s remarkable discovery that all periodic functions are linear combinations of sines and cosines. Historical background of trigonometry is included.

Video extracts and supporting information available online at http://www.projectmathematics.com/sincos1.htm
Duration
28 mins

Title
Sines and Cosines, Part 2: Trigonometry
Synopsis
Focuses on the use of sines and cosines in trigonometry, with special emphasis on the law of cosines and the law of sines. They enable us to find all parts of a triangle if three parts are known, and at least one of them is a side. Applications are described in astronomy, navigation, and surveying by triangulation. One of the major triumphs of surveying by triangulation is the Survey of India, which took more than a century to complete. The programme describes how the survey was done and how it determined the height of Mt. Everest. The program also outlines a brief history of surveying instruments, from the dioptra of ancient times to orbiting satellites of modern times.

Video extracts and supporting information available online at http://www.projectmathematics.com/sincos2.htm
Duration
29 mins

Title
Sines and Cosines, Part 3: Addition formulae
Synopsis
Relates the sine and cosine of an angle with lengths of chords of a circle, as expounded in Claudius Ptolemy’s Almagest. This leads to simple derivations of the addition formulas for determining the sine and cosine of a sum of two angles. One application shows that a combination of a sine wave with a cosine wave of the same frequency is another sine wave, possibly shifted. This property plays an important role in the study of simple harmonic motion. The programme also outlines a brief history of the city of Alexandria, founded by Alexander the Great in 331 B.C. His successors created a centre of Hellenistic culture in Alexandria that attracted many of the greatest mathematical scholars of antiquity, including Euclid, Apollonius, Archimedes, Eratosthenes, Hero, Pappus, and Claudius Ptolemy.

Video extracts and supporting information available online at http://www.projectmathematics.com/sincos3.htm
Duration
29 mins

Name

### Distributor

Name

#### Boulton-Hawker Films Ltd

Email
sales@boultonhawker.co.uk
Web
http://www.boultonhawker.co.uk/ External site opens in new window
Phone
01449 616 200
Fax
01449 677 600