Learn how to make waves of all different shapes by adding up sines or cosines. Make waves in space and time and measure their wavelengths and periods. See how changing the amplitudes of different harmonics changes the waves. Compare different mathematical expressions for your waves.

Learn how to make waves of all different shapes by adding up sines or cosines. Make waves in space and time and measure their wavelengths and periods. See how changing the amplitudes of different harmonics changes the waves. Compare different mathematical expressions for your waves.

There is a natural (and complicated!) predator-prey relationship between the fox and rabbit populations, since foxes thrive in the presence of rabbits, and rabbits thrive in the absence of foxes. However, this relationship, as shown in the given table of values, cannot possibly be used to present either population as a function of the other. This task emphasizes the importance of the "every input has exactly one output" clause in the definition of a function, which is violated in the table of values of the two populations.

The example of rabbits and foxes was introduced in the task (8-F Foxes and Rabbits) to illustrate two functions of time given in a table. We are now in a position to actually model the data given previously with trigonometric functions and investigate the behavior of this predator-prey situation.

The example of rabbits and foxes was introduced in 8-F Foxes and Rabbits to illustrate two functions of time given in a table. The same situation was used in F-TF Foxes and Rabbits 2 to find trigonometric functions modeling the data in the table. The previous situation was somewhat unrealistic since we were able to find functions that fit the data perfectly. In this task, on the other hand, we do some legitimate modelling, in that we come up with functions that approximate the data well, but do not perfectly match, the given data.

This art history video discussion examines Fra Angelico's "The Annunciation", c. 1438-47, fresco, 230 x 321 cm (Convent of San Marco, Florence).

This art history video discussion examines Fra Angelico's "The Annunciation and Life of the Virgin" (in the predella), c. 1426, tempera on wood, 194 x 194 cm (Museo del Prado, Madrid).

This art history video discussion examines Fra Filippo Lippi's "Madonna and Child with two Angels", c. 1460-1465, tempera on panel, 95 x 63.5 cm (Galleria degli Uffizi, Florence).

This art history video discussion examines Fra Filippo Lippi's "Madonna and Child", c. 1440, tempera on panel, 79 x 51.1 cm / 31-1/8 x 20-1/8 inches (National Gallery of Art, Washington).

Match shapes and numbers to earn stars in this fractions game. Challenge yourself on any level you like. Try to collect lots of stars! The main topics of this interactive simulation include fractions, equivalent fractions, and mixed numbers.

Match shapes and numbers to earn stars in this fractions game. Challenge yourself on any level you like. Try to collect lots of stars! The main topics of this interactive simulation include fractions, equivalent fractions, and mixed numbers.

This short video and interactive assessment activity is designed to teach fifth graders about the conversion of fractions to decimals.

Explore fractions while you help yourself to 1/3 of a chocolate cake and wash it down with 1/2 a glass of orange juice! Create your own fractions using fun interactive objects. Match shapes and numbers to earn stars in the fractions games. Challenge yourself on any level you like. Try to collect lots of stars!

Explore fractions while you help yourself to 1/3 of a chocolate cake and wash it down with 1/2 a glass of orange juice! Create your own fractions using fun interactive objects. Match shapes and numbers to earn stars in the fractions games. Challenge yourself on any level you like. Try to collect lots of stars!

Explore fractions while you help yourself to 1 and 1/2 chocolate cakes and wash it down with 1/3 a glass of water! Create your own fractions using fun interactive objects. Match shapes and numbers to earn stars in the mixed number game. Challenge yourself on any level you like. Try to collect lots of stars!

This short video and interactive assessment activity is designed to teach fifth graders about fractions with common denominators - word problems.

Linear elastic and elastic-plastic fracture mechanics. Experimental methods. Microstructural effects on fracture in metals, ceramics, polymers, thin films, biological materials and composites. Toughening mechanisms. Crack growth resistance and creep fracture. Interface fracture mechanics. Fatigue damage and dislocation substructures in single crystals. Stress- and strain-life approach to fatigue. Fatigue crack growth models and mechanisms. Variable amplitude fatigue. Corrosion fatigue. Case studies of fracture and fatigue in structural, bioimplant, and microelectronic components.

This art history video discussion examines Jean-Honore Fragonard's "The Progress of Love: The Meeting", 1771-1773, oil on canvas, 317.5 x 243.8 cm (The Frick Collection, New York).

This art history video discussion examines Jean-Honore Fragonard's "The Swing", oil on canvas, 1767 (Wallace Collection, London).