(16.) Alternative Mathematics.

(a.) General idea of number as a quantity having unlimited positive
and negative range; rational and irrational numbers.

Quantities that can be wholly represented by numbers—e.g., length,
cost, area, angle, time. Relationships between such quantities treated
graphically. Ideas of variation, variable, and function. Graphical work
with two variables to give practice in curve tracing only. Ideas of continuity
and limit. Linear relation between two variables—i.e., the function
$Y = aX + b$, treated symbolically and graphically with oblique as well as
rectangular axes.

Dimensions of length, area, and volume functions. Simple functions
stated in algebraic symbols, and commonly known as formulæ. Practice
in evaluation, including cases where it is necessary to change the independent
variable in formulæ such as $a = \frac{v - u}{t}$, $T = 2\pi \sqrt{\frac{l}{g}}$.

Simple manipulative exercises in the addition, subtraction, multiplication,
division, and factorization of algebraic expressions, limited, as far
as possible, to trinomials of no higher degree than the second. The
methods used must invariably show work written in lines, with a full
understanding of the meaning and use of the sign of equality. Simple
equations involving one or two unknown quantities, and problems thereon,
provided that the examples set must bear the stamp of reality, whether
they be expressed in symbols or stated in words.

(b.) Ideas of direction and of similarity of space. The straight line
defined as one of uniform direction, and its minimum treated as a consequence
of its “directness.” All straight lines similar to one another: application
to making of straight edges. Plane surfaces: application to surface plates.
The angle treated as a quantitative change in direction. Rotational
proofs of the simple angular properties of figures, especially figures of three
and four sides.

Parallels treated as straight lines in the same direction. Angular
properties of figures with parallel sides.

(c.) Ideas of symmetry and superposition. Uniqueness of figure formed
by three given straight lines in the same plane. Congruence of triangles
deduced from this result, and consequently such geometrical results in
reference to triangles and parallelograms as are usually proved in an
elementary course from congruent triangles.

Use of scales: scale ratio. Simple cases of proportionality, including
similarity of triangles and other plane figures, treated with reference to
scale ratio. The recognition and use of the sine, cosine, and tangent of
an angle, regarded both as a function of the angle and as a ratio between
the sides. Graphs and functional scales of sine, cosine, tangent.

Areas of rectangles, triangles, and parallelograms. Proofs by dissection,
combined with symbolic expression, as in the geometrical truths
illustrated by the expressions—

Area of triangle = $\frac{1}{2}bh = \frac{1}{2}bc \sin A = \frac{1}{2}ca \sin B$
$(a + b + c)x = ax + bx + cx$
$(a + x)x = ax + x^2$
$(a \pm b)^2 = a^2 \pm 2ab + b^2$
$(a + b)(a - b) = a^2 - b^2$

Areas of similar figures proportional to the squares of the ratios of
corresponding sides. Pythagoras’ theorem deduced from this result together
with the fact that the perpendicular from the right angle to the hypotenuse
divides the triangle into two parts both similar to it.

(d.) The simplest treatment of directed quantities, such as displacements,
to show that certain quantities cannot be fully represented by numbers
alone. Ideas of addition and subtraction for such quantities applied in
simple diagrams drawn to scale, such as are usually met with in elementary
practical questions on heights and distances.

(17.) Hygiene.

(a.) The Home.—General considerations of site, aspect, construction,
&c., in order to obtain hygienic conditions. Value of a garden, fresh air,
and sunlight. Heating and lighting. Number and arrangement of rooms.
Storage of food. Water-supply (quantity of water required for each person,
methods of collection and distribution; effects of insufficient or impure
water-supply; methods of purification). Sanitary arrangements (good
systems of domestic drainage; water and earth). Dust and dirt. Disposal
of refuse. Necessity for cleanliness of home and surroundings.

(b.) Personal Cleanliness.—Cleanliness, washing and bathing; structure
of skin; care of the mouth and teeth; care of the hair; care of the feet;