A rough Unit Plan for the topic Area.

- Mathematics > General
- Mathematics > Geometry

- Grade 9
- Grade 10
- Grade 11
- Grade 12

Know and apply that if a line is perpendicular to each of two intersecting lines at their point of intersection, then the line is perpendicular to the plane determined by them

Know and apply that through a given point there passes one and only one plane perpendicular to a given line

Know and apply that through a given point there passes one and only one line perpendicular to a given plane

Know and apply that two lines perpendicular to the same plane are coplanar

Know and apply that two planes are perpendicular to each other if and only if one plane contains a line perpendicular to the second plane

Know and apply that if a line is perpendicular to a plane, then any line perpendicular to the given line at its point of intersection with the given plane is in the given plane

Know and apply that if a line is perpendicular to a plane, then every plane containing the line is perpendicular to the given plane

Know and apply that if a plane intersects two parallel planes, then the intersection is two parallel lines

Know and apply that if two planes are perpendicular to the same line, they are parallel

Know and apply that the lateral edges of a prism are congruent and parallel

Know and apply that two prisms have equal volumes if their bases have equal areas and their altitudes are equal

Know and apply that the volume of a prism is the product of the area of the base and the altitude

lateral edges are congruent

lateral faces are congruent isosceles triangles

volume of a pyramid equals one-third the product of the area of the base and the altitude

bases are congruent

volume equals the product of the area of the base and the altitude

lateral area of a right circular cylinder equals the product of an altitude and the circumference of the base

lateral area equals one-half the product of the slant height and the circumference of its base

volume is one-third the product of the area of its base and its altitude

the intersection of a plane and a sphere is a circle

a great circle is the largest circle that can be drawn on a sphere

two planes equidistant from the center of the sphere and intersecting the sphere do so in congruent circles

surface area is 4 pi r²

volume is 4/3 pi r³

Construct a bisector of a given angle, using a straightedge and compass, and justify the construction

Construct the perpendicular bisector of a given segment, using a straightedge and compass, and justify the construction

Construct lines parallel (or perpendicular) to a given line through a given point, using a straightedge and compass, and justify the construction

Construct an equilateral triangle, using a straightedge and compass, and justify the construction

Investigate and apply the concurrence of medians, altitudes, angle bisectors, and perpendicular bisectors of triangles

Solve problems using compound loci

Graph and solve compound loci in the coordinate plane

Determine the negation of a statement and establish its truth value

Know and apply the conditions under which a compound statement (conjunction, disjunction, conditional, biconditional) is true

Identify and write the inverse, converse, and contrapositive of a given conditional statement and note the logical equivalences

Write a proof arguing from a given hypothesis to a given conclusion

Determine the congruence of two triangles by using one of the five congruence techniques (SSS, SAS, ASA, AAS, HL), given sufficient information about the sides and/or angles of two congruent triangles

Identify corresponding parts of congruent triangles

Investigate, justify, and apply theorems about the sum of the measures of the angles of a triangle

Investigate, justify, and apply the isosceles triangle theorem and its converse

Investigate, justify, and apply theorems about geometric inequalities, using the exterior angle theorem

Investigate, justify, and apply the triangle inequality theorem

Determine either the longest side of a triangle given the three angle measures or the largest angle given the lengths of three sides of a triangle

Determine if two lines cut by a transversal are parallel, based on the measure of given pairs of angles formed by the transversal and the lines

Investigate, justify, and apply theorems about the sum of the measures of the interior and exterior angles of polygons

Investigate, justify, and apply theorems about each interior and exterior angle measure of regular polygons

Investigate, justify, and apply theorems about parallelograms involving their angles, sides, and diagonals

Investigate, justify, and apply theorems about special parallelograms (rectangles, rhombuses, squares) involving their angles, sides, and diagonals

Investigate, justify, and apply theorems about trapezoids (including isosceles trapezoids) involving their angles, sides, medians, and diagonals

Justify that some quadrilaterals are parallelograms, rhombuses, rectangles, squares, or trapezoids

Investigate, justify, and apply theorems about geometric relationships, based on the properties of the line segment joining the midpoints of two sides of the triangle

Investigate, justify, and apply theorems about the centroid of a triangle, dividing each median into segments whose lengths are in the ratio 2:1

Establish similarity of triangles, using the following theorems: AA, SAS, and SSS

Investigate, justify, and apply theorems about similar triangles

Investigate, justify, and apply theorems about proportional relationships among the segments of the sides of the triangle, given one or more lines parallel to one side of a triangle and intersecting the other two sides of the triangle

the altitude to the hypotenuse of a right triangle is the mean proportional between the two segments along the hypotenuse

the altitude to the hypotenuse of a right triangle divides the hypotenuse so that either leg of the right triangle is the mean proportional between the hypotenuse and segment of the hypotenuse adjacent to that leg

Investigate, justify, and apply the Pythagorean theorem and its converse

perpendicular bisectors of chords

the relative lengths of chords as compared to their distance from the center of the circle

a perpendicular to the tangent at the point of tangency

two tangents to a circle from the same external point

common tangents of two non-intersecting or tangent circles

inside the circle (two chords)

on the circle (tangent and chord)

outside the circle (two tangents, two secants, or tangent and secant)

Investigate, justify, and apply theorems about arcs of a circle cut by two parallel lines

along two tangents from the same external point

along two secants from the same external point

along a tangent and a secant from the same external point

along two intersecting chords of a given circle

Define, investigate, justify, and apply isometries in the plane (rotations, reflections, translations, glide reflections) Note: Use proper function notation.

Investigate, justify, and apply the properties that remain invariant under translations, rotations, reflections, and glide reflections

Identify specific isometries by observing orientation, numbers of invariant points, and/or parallelism

Justify geometric relationships (perpendicularity, parallelism, congruence) using transformational techniques (translations, rotations, reflections)

Define, investigate, justify, and apply similarities (dilations and the composition of dilations and isometries)

Investigate, justify, and apply the properties that remain invariant under similarities

Identify specific similarities by observing orientation, numbers of invariant points, and/or parallelism

Investigate, justify, and apply the analytical representations for translations, rotations about the origin of 90º and 180º, reflections over the lines x = 0, y = 0, and y = x, and dilations centered at the origin

Find the slope of a perpendicular line, given the equation of a line

Determine whether two lines are parallel, perpendicular, or neither, given their equations

Find the equation of a line, given a point on the line and the equation of a line perpendicular to the given line

Find the equation of a line, given a point on the line and the equation of a line parallel to the desired line

Find the midpoint of a line segment, given its endpoints

Find the length of a line segment, given its endpoints

Find the equation of a line that is the perpendicular bisector of a line segment, given the endpoints of the line segment

Investigate, justify, and apply the properties of triangles and quadrilaterals in the coordinate plane, using the distance, midpoint, and slope formulas

Solve systems of equations involving one linear equation and one quadratic equation graphically

Write the equation of a circle, given its center and radius or given the endpoints of a diameter

Write the equation of a circle, given its graph Note: The center is an ordered pair of integers and the radius is an integer.

Find the center and radius of a circle, given the equation of the circle in center-radius form

Graph circles of the form (x - h)² + (j - k)² = r²