K. Geometry

Because major areas of study at postsecondary institutions have different prerequisites, certain mathematics benchmarks are marked with an asterisk (*). These asterisked benchmarks represent content that is recommended for all students, but is required for those students who plan to take calculus in college, a requisite for mathematics and many mathematics-intensive majors. The high school graduate can:

K1. Understand the different roles played by axioms, definitions and theorems in the logical structure of mathematics, especially in geometry:

K1.1. Identify, explain the necessity of and give examples of definitions, axioms and theorems.

K1.2. State and prove key basic theorems in geometry such as the Pythagorean theorem, the sum of the angles of a triangle is 180 degrees, and the line joining the midpoints of two sides of a triangle is parallel to the third side and half its length.

K1.3. Recognize that there are geometries, other than Euclidean geometry, in which the parallel postulate is not true.

Example:

On a globe the lines of longitude intersect at both the North and South Poles creating a closed figure with only two sides; this is an example of a situation that cannot occur in Euclidean geometry but does occur in spherical geometry.

K2. Identify and apply the definitions related to lines and angles and use them to prove theorems in (Euclidean) geometry, solve problems, and perform basic geometric constructions using a straight edge and compass:

K2.1. Identify and apply properties of and theorems about parallel lines and use them to prove theorems such as two lines parallel to a third are parallel to each other and to perform constructions such as a line parallel to a given line through a point not on the line.

K2.2. Identify and apply properties of and theorems about perpendicular lines and use them to prove theorems such as the perpendicular bisectors of line segments are the set of all points equidistant from the two end points and to perform constructions such as the perpendicular bisector of a line segment.

K2.3. Identify and apply properties of and theorems about angles and use them to prove theorems such as two lines are parallel exactly when the alternate interior angles they make with a transversal are equal and to perform constructions such as the bisector of an angle.

K3. Know the basic theorems about congruent and similar triangles and use them to prove additional theorems and solve problems.

Example:

When you set a projector 12 feet from the screen, the image on the screen measures 8 feet across. What will the width of the image be if you move the projector 3 feet further from the screen?

K4. Know the definitions and basic properties of a circle and use them to prove basic theorems and solve problems.

(Associated Postsecondary Assignment: #1)

Example:

A line tangent line to a circle is perpendicular to the line segment from the center of the circle to the point of tangency.

K5. Apply the Pythagorean theorem, its converse and properties of special right triangles to solve problems.

(Associated Postsecondary Assignment: #1)

Example:

Given the lengths of two sides of a right triangle, find the length of the third side.

Example:

Given a triangle with side lengths of 12 and 13 inches, identify the triangle as acute, right, obtuse or not a triangle at all for various lengths of the third side such as 4, 5, 6, 18 or 26 inches. Justify your answers.

Example:

Determine the lengths of the sides of the special right triangle with angles 30, 60 and 90 degrees and the special right triangle with angles 45, 45 and 90 degrees if the length of the smallest side in each case is 1 meter.

K6. Use rigid motions (compositions of reflections, translations and rotations) to determine whether two geometric figures are congruent and to create and analyze geometric designs.

Example:

Prove the side-angle-side criterion for showing that two triangles are congruent.

Example:

Analyze tessellations of the plane.

K7. Know about the similarity of figures and use the scale factor to solve problems.

Example:

Read and extract information from scale drawings; compute lengths and areas from scale drawings.

K8. Know that geometric measurements (length, area, perimeter, volume) depend on the choice of a unit and that measurements made on physical objects are approximations; calculate the measurements of common plane and solid geometric figures:

K8.1. Understand that numerical values associated with measurements of physical quantities must be assigned units of measurement or dimensions; apply such units correctly in expressions, equations and problem solutions that involve measurements; and convert a measurement using one unit of measurement to another unit of measurement.

(Associated Workplace Tasks: #1 and 2)

(Associated Postsecondary Assignment: #2)

Example:

Convert feet per second to miles per hour, and use dimensional analysis to verify that the calculation yields the appropriate measurement unit .

Example:

Confirm that the distance traveled in 45 minutes at the rate of 2.4 meters per second is 6.48 kilometers.

Example:

Convert speed of 150 meters per second to miles per hour.

K8.2. Determine the perimeter of a polygon and the circumference of a circle; the area of a rectangle, a circle, a triangle and a polygon with more than four sides by decomposing it into triangles; the surface area of a prism, a pyramid, a cone and a sphere; and the volume of a rectangular box, a prism, a pyramid, a cone and a sphere.

(Associated Workplace Task: #1)

(Associated Postsecondary Assignment: #1)

Example:

How much material is removed when you drill a hole with a diameter of 2 cm through a block of metal that is 3 cm thick?

K8.3. Know that the effect of a scale factor k on length, area and volume is to multiply each by k, k² and k³, respectively.

Example:

Know that a 16" (diameter) pizza has four times as much pizza as an 8" (diameter) pizza.

K9. Visualize solids and surfaces in three-dimensional space when given two-dimensional representations (e.g., nets, multiple views) and create two-dimensional representations for the surfaces of three-dimensional objects.

K10. Represent geometric objects and figures algebraically using coordinates; use algebra to solve geometric problems:

(Associated Postsecondary Assignment: #1)

K10.1. Express the intuitive concept of the "slant" of a line in terms of the precise concept of slope, use the coordinates of two points on a line to define its slope, and use slope to express the parallelism and perpendicularity of lines.

K10.2. Describe a line by a linear equation.

Example:

Find an equation for the line containing the points (32, 0) and (212, 100). If the first coordinate of a point on this line is 98.6, what is the second coordinate? Identify the point on this line where the two coordinates are the same.

K10.3. Find the distance between two points using their coordinates and the Pythagorean theorem.

K10.4. * Find an equation of a circle given its center and radius and, given an equation of a circle, find its center and radius.

(Associated Postsecondary Assignment: #1)

Example:

The circle with radius 5 and center at (1, 0) has equation (x – 1)² + y² = 25.

Example:

Transform the quadratic equation x² + 2x + y² – 4y = 4 into the form (x + 1)² + (y – 2)² = 9 by completing the square; realize that the graph of the equation is a circle with center at (–1, 2) and with radius 3.

K11. Understand basic right-triangle trigonometry and apply it to solve problems:

K11.1. Understand how similarity of right triangles allows the trigonometric functions sine, cosine and tangent to be defined as ratios of sides and be able to use these functions to solve problems.

(Associated Postsecondary Assignment: #1)

K11.2. Apply the trigonometric functions sine, cosine and tangent to solve for an unknown length of a side of a right triangle, given one of the acute angles and the length of another side.

Example:

Safety regulations require that the angle between a ladder and the wall should be between 25 and 30 degrees. What is the range of safe placements (distance from the wall) for the bottom of a 12-foot ladder? Where should the base of a 20-foot ladder be placed to satisfy the same safety regulation?

K11.3. Use the standard formula for the area of a triangle, , to explain the area formula, where a and b are the lengths of two sides of a triangle and C is the measure of the included angle formed by these two sides, and use it to find the area of a triangle when given the lengths of two of its sides and the included angle.

K12. * Know how the trigonometric functions can be extended to periodic functions on the real line, derive basic formulas involving these functions, and use these functions and formulas to solve problems:

K12.1. * Know that the trigonometric functions sine and cosine, and thus all trigonometric functions, can be extended to periodic functions on the real line by defining them as functions on the unit circle, that radian measure of an angle between 0 and 360 degrees is the arc length of the unit circle subtended by that central angle, and that by similarity, the arc length s of a circle of radius r subtended by a central angle of measure t radians is s = rt.

K12.2. * Know and use the basic identities, such as sin² (x) + cos² (x) = 1 and and formulas for sine and cosine, such as addition and double angle formulas.

Example:

Use the identity sin² (x) + cos² (x) = 1 to determine the sine of an angle when its cosine is known. Example: Use the addition formula to find the amplitude, period and phase shift of a cos(wt) + b sin(wt) by expressing it as c sin (wt + d) for some constants c and d.

K12.3. * Graph sine, cosine and tangent as well as their reciprocals, secant, cosecant and cotangent; identify key characteristics.

K12.4. * Know and use the law of cosines and the law of sines to find missing sides and angles of a triangle.