Subtest II: Mathematics Study Guide for the CSET Multiple Subjects Test

Measurement and Geometry

Geometric Objects

Geometric objects include 2-dimensional (flat) figures known as plane figures, and 3-dimensional solid figures.

Characteristics

Know the characteristics of common plane and solid geometric figures. Some, but not all, are listed below. Also, see the term congruence in the next section.

Isosceles triangle— a triangle with two congruent sides (also two congruent angles)

Right triangle— a triangle with a right angle

Sphere— informally, a ball-shaped object; formally, the set of all points a given distance in space from a fixed point (the center)

Quadrilateral— a plane figure with four sides

Square—a quadrilateral with four congruent sides (also four congruent angles)

Parallelogram— a quadrilateral with both pairs of opposite sides parallel

Rectangular prism— a solid figure with six rectangles for faces

Cube— a rectangular prism with all square faces

Object Comparisons

The list below shows different ways geometric figures can be compared to each other. Congruence is by far the most used of these concepts in geometry.

Congruence—Congruent means identical in every way. Congruent segments or angles have the same measure. Congruent figures fit exactly on top of each other.

Congruent Figures:

Similarity—Two figures are similar if they have the same shape, though one may be bigger. For example, all squares are similar.

Similar Figures:

Symmetry—The three types of symmetry are line symmetry, point symmetry, and rotational symmetry. Line symmetry is very common. Your face is a pretty good example. If you draw a line down the center of your face, the left side and right side match up, point for point. Of course it won’t be perfect on a real face, but picture it done to a capital A. An A has line symmetry, and the line you draw is called the line of symmetry..

Translation—Picture a paper triangle, or other figure, sliding across your desk to a new location without any rotation. That movement is called a translation.

Rotation—Picture a paper triangle, or other figure, on your desk. Stick a pin through it into your desk. Spin it. The triangle can’t go anywhere but it can rotate. That’s rotation. Easy. The point where you stuck the pin is the center of rotation. In the diagram below, D is the center of rotation.

Reflection— Picture a trapeze performer doing a handstand on a trapeze bar.and then swinging down to hang below the bar. That kind of a flip is called a reflection. If you can picture a triangle doing the same kind of flip over a line, you have the right idea.

The Pythagorean Theorem

In a right triangle with legs a and b and hypotenuse c , it is true that:

\[a^{2}+b^{2}=c^{2}\]

Normally, you would be given two of the sides and asked to find the third one.

The converse is also true. If you have some triangle with sides a,b, and c, and you can show that \(a^{2}+b^{2}=c^{2}\) is true, then you know the triangle is a right triangle.

Parallel Lines

Parallel lines will never intersect.

If a third line (a transversal) intersects them, eight angles will be formed. In the figure below, some pairs of angles are defined like this and are congruent

In the figure above, these things are true:

A and C (as well as B/D, E/G, and F/H) are vertical angles
A and E (as well as D/H, B/F, and C/G) are corresponding angles
D and F, as well as C and E are alternate interior angles
A and G, as well as B and H are alternate exterior angles

Other pairs of angles are not congruent, but have a relationship that helps you determine their measure. The measures of these pairs of angles add to 180 degrees.

A and B (as well as C/D, E/F, G/H, A/D, E/H, B/C, and F/G): adjacent angles
A and H (as well as B/G): exterior angles on the same side of a transversal
D and E (as well as C/F): interior angles on the same side of a transversal

Representing Geometric Objects

Strictly speaking, geometric objects have a perfection that can exist only in the mind. However, our minds need a lot of help visualizing them, so we resort to drawing lines on paper to approximate the figures or using an \(x-y\) coordinate system to define them. Sometimes, physical models can help, especially with solid figures.

Using Tools

Common tools of geometry include a compass and a straightedge. With these, many geometric figures can be exactly copied using construction techniques, and certain operations can be performed, such as bisecting angles or line segments to name a couple. For approximately measuring angles and segments, a protractor and ruler are used.

Combining and Dissecting Objects

Sometimes it’s helpful to imagine dissecting figures into smaller simpler figures. Imagine splitting a rectangle into two triangles, for example, or dissecting a hexagon into six triangles to find its area. Or, you could go the other way and imagine combining simpler figures into larger ones. A rectangle with identical right triangles added to each end could be a parallelogram or a trapezoid, depending on how you do it.

Measuring Geometric Objects

An amazing number of things in this world can be, and are, measured. Research scientists, for example, go to extraordinary lengths to accurately measure things like time, distance, force, temperature, mass, and a lot more. In geometry, we stick to length and angle measures, and use those to calculate these:

perimeter
area
surface areas of solids
volumes of solids

Especially when sketching your own geometric figures, you should be able to estimate angle measures and comparable lengths.

Formulas for Measuring

Perimeter—There are formulas for some figures, but basically you just add up all the side lengths.

Area— Some common area formulas:

Rectangles and parallelograms: \(A=hw\)
Triangles: \(A=\frac{1}{2}bh\)
Circles: \(A=\pi r^{2}\)

Surface Area of Solids—Some common surface area formulas:

Rectangular prism: S.A. = \(2lw+2hw+2hl\)
Cube: S.A. = \(6s^{2}\)
Cylinder: S.A. = \(2 \pi r^{2} +2 \pi rh\)
Sphere: S.A. = \(4\pi r^{2}\)

Volume— Some common volume formulas:

rectangular prism— \(V=lwh\)
cone—\(V=\frac {1}{3} \pi r^{2}h\)
cylinder—\(V= \pi r^{2}h\)
sphere— \(\frac{4}{3} \pi r^{3}\)

Comparing Measurement Systems

The metric system relies on base units, prefixes and powers of ten to identify its various units of length, mass, and volume. Complete listings showing all prefixes can easily be found, and here we will just show a few basic ones being used with the gram unit.

\[\begin{array}{|c|c|c|c|c|c|} \hline \text{milligram} & \text{centigram} & \text{ decigram} & \text{ } & \text{kilogram} & \text{megagram} \\ \hline \text{mg} & \text{cg} & \text{dg} & \text{g} & \text{kg} & \text{Mg} \\ \hline 10^{-3} g & 10^{-2} g & 10^{-1} g & 10^{0} g & 10^{3} g & {10^{6} g} \\ \hline \end{array}\]

Converting between units in the metric system is just a matter of moving the decimal. If you are going to a bigger unit, the decimal point moves to the left. If you are going to a smaller unit, the decimal point moves to the right. Subtract the powers of ten to get the number of places to move (bigger unit minus smaller unit).

For example: change 5 kg to cg.
Centigrams are smaller than kilograms so the decimal goes right.
Subtracting exponents: \(3-(-2)=5\) places to move the decimal.
5kg = 500,000 cg

U.S. customary units have no overall system of converting between, say, pounds and ounces or miles and feet. See the next section for using proportions to do those conversions

To estimate rough conversions between the two systems these may help.

1 liter is close to a quart.
1 meter is close to a yard.
1 kg is a little over 2 pounds.
\(\frac{5}{8}\) mile is close to a kilometer.

Using Proportions

Remember that a proportion is a statement of equality between two ratios.

Such as: \(\frac{x}{4} = \frac{7}{12}\)

A common use of proportions is to convert from one measurement unit to another. Say you need to convert 30 inches to feet. You would use this proportion:

\(\frac {1}{12}=\frac {30}{x}\) Solving this would give you \(x=2.5\) inches.

In the case of making scale drawings, like blueprints, or models, you would do the same sort of thing.

For example, using a scale of \(\frac{1}{48}\) how long would you draw a 80 foot wall?

\(\frac{x}{80} = \frac{1}{48}\) would give you a length of \(1\frac{2}{3}\) ft, or \(20\) inches.

Using Other Measurement Forms

Basic measurement units can be combined to give useful new units.

For example, take a length unit divided by a time unit and get speed:

\[\frac{miles}{hour}\]

\[\frac{meters}{second}\]

Weight (force) divided by area gives a pressure unit:

\[\frac {pounds}{square \hspace{0.1 cm}inch}\]

(psi for those who check their tire pressures)