The homology is, potentially, our greatest form for discovering and teaching. Although ancient -- going back to Euclid, and earlier -- yet it is foolishly neglected. And it is trivialized in the multiple choice puerilities of standardized tests.

Here's a humdrum homology, used in tests -- wheel: vehicle:: foot: animal. "Wheel plays the relation to vehicle that foot plays to animal".

Yes, it's sometimes called an "analogy" -- by confusion. Here, we have both RELATA and RELATIONS. Above, the RELATA are "wheel", "vehicle", "foot", "animal". The RELATIONS are implied. For, wheel has the RELATION of PROPELLOR to vehicle; and foot also has the RELATION of PROPELLOR to an animal. The RELATIONS are the SAME, while the RELATA are DIFFERENT.

Hence, this form is an ANALOGY when you emphasize the RELATA; a HOMOLOGY, when you emphasize the RELATION(S).

If you rewrite the previous homological example as, "_____ is to vehicle as foot is to animal", the reader realizes (or discovers) that the implied propelling process of foot to animal is fulfilled by wheel to vehicle.

The homology gave rise to the CONCEPT of RATIOS and RATIONAL NUMBERS in MATHEMATICS. Consider the test form, "___ is to 6 and 5 is to 10". We realize that "5 is half of 10", so the form asks, "What is half of 6?" -- answer: 3. Writing it another way, "3:6::5:10" gives rise to the RATIONAL form, "3/6 = 5/10", whereby the solidus, "/", replaces the colon, and equality, "=", replaces double colons.

My point is that you can rehearse or exercise teens in verbal and word or rhetorical forms of the homology to induce their ingenuity.

Then you can guide them to parse out a homology -- beginning with oneself and one's experience -- and complete it by a seemingly comparable role in situation with a schoolmate or neighbor. Thus, Betty realizes that her new bangs preen her the way that Purlie's cornrows preen her. (Betty: bangs::Purlie:cornrows)

Or start with the "side" of the OTHER, and try to find something comparable with SELF. Thus, when Jimmy tries to understand how Juanito felt when schoolkids called him a "Spic", Tim remembers how he he felt when schoolkids called him "a bastard" (Jimmy:bastard::Juanito:spic).

As a sidetrip, teens can be shown how homologies and ratios explain different regimes of the physical world. The homology, human:gravity::fly:watery_surface_tension, means that a fly trying to get a drink of water from a pool is in the same danger as a human leaning over a cliff to pick a flower. For, while you leave a bath or pool with a minute fraction of water on you, a mouse leaves with water equal to its own weight & a fly leaves with several times its weight -- due to the surace tension of water.

Such ideas can prepare teens to understand homology as a powerful tool of EDUCATION, alternatively of DISCOVERY, as shown in a TABLE.