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Program of Study for High School Mathematics
Students develop an understanding of the reasoning process and the concept of proof. Properties and relationships of lines, angles, and triangles are developed inductively and then verified deductively. Topics included conditional statements, syllogisms, points in space, parallel lines, transversals, planes, congruence, the Pythagorean Theorem, and similarity. Vectors, algebra, and technologies are used as tools to solve geometry problems. This course includes an integrated review of algebraic topics needed in geometry. [Standard and Advanced Studies Diploma students must pass Geometry/ Part II to received mathematics graduation credit for Geometry/ Part I.]
Some of the Geometry and some of the Algebra 1 Virginia Standards of Learning are addressed in this course. The number in parentheses following each Essential Knowledge and Skill refers to the related SOL objective. Required Program of Studies objectives that are not addressed in the Geometry Virginia SOL are in italics type.
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- Price, Cummins, et al. Glencoe Geometry: Concepts & Applications. Glencoe/McGraw-Hill. 2001.
- Burrill, et al. Glencoe Geometry: Integration, Applications, Connections. Glencoe/ McGraw-Hill. 2001.
- Larson, Boswell, & Stiff. McDougal-Littell Geometry. McDougal-Littell. 2001/2004.
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Benchmark
Students apply several alternative methods for producing deductive arguments to verify hypotheses.
Key Terms
| argument |
deductive reasoning |
properties of equality |
| biconditional |
hypothesis |
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| coordinate proof |
if-then form |
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| conclusion |
inductive reasoning |
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| conditional |
inverse |
properties of real numbers |
| conjecture |
law of detachment |
symbolic form |
| contrapositive |
law of syllogism |
two-column proof |
| converse |
paragraph proof |
valid |
| counterexample |
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Venn diagram |
Essential Knowledge and Skills/Indicators
- Evaluate algebraic expressions for a given replacement set using order of operations. (A.2)
- Justify the steps in evaluating algebraic expressions using the commutative, associative, and distributive properties. (A.3)
- Use inductive reasoning to make conjectures. (G.1)
- Write a conditional statement in if-then form. (G.1)
- Given a conditional statement, (G.1)
- identify the hypothesis and conclusion
- write the converse, inverse, and contrapositive.
- Translate short verbal arguments into symbolic form (p → q and ~p → ~q). (G.1)
- Disprove a conjecture by producing a counterexample to a proposed hypothesis.
- Use the law of syllogism and the law of detachment in deductive arguments. (G.1)
- Diagram logical arguments to represent set relationships using Venn diagrams. (G.1)
- Solve multi-step linear equations by reviewing one-step and two-step equation solving. (A.1)
- Solve multi-step equations by using: (A.1, A.3)
- Commutative properties
- Associative properties
- Distributive property
- Order of operations
- Addition and multiplication properties of equality
- Closure property
- Identity and inverse properties
- Reflexive, symmetric, and transitive properties of equality
- Substitution property of equality
- Use the properties listed above to justify steps in solving equations. (A.3)
- Solve literal equations (formulas) and implicit equations for a specified variable. (A.1)
- Solve linear inequalities in one variable. (A.1)
- Justify steps in solving linear inequalities using properties of real numbers and order. (A.3)
- Graph the solution to a linear inequality in the coordinate plane. (A.1)
- Solve linear equations and write them in if-then form ( ). (G.1)
- Justify each step in solving a linear equation with a field property of real numbers or a property of equality. (G.1)
- Present solving linear equations as a form of deductive proof. (G.1)
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II. Lines and Angles
Benchmark
Students examine concepts fundamental to the study of geometry. Through investigations they draw conclusions about relationships involving points, lines, planes, and angles.
Key Terms
| acute angle |
intersecting lines |
plane |
| adjacent angles |
line |
point |
| alternate exterior angles |
linear pair |
ray |
| alternate interior angles |
median |
right angle |
| altitude |
midpoint |
segment |
| angle |
midsegment |
skew lines |
| angle bisector |
oblique lines |
slope |
| complementary angles |
obtuse angle |
straight angle |
| congruent angles |
parallel lines |
supplementary angles |
| consecutive interior angles |
perpendicular bisector |
transversal |
| corresponding angles |
perpendicular lines |
vertical angles |
| equidistant |
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Vectors
| column matrix |
matrix |
tail |
| equivalent vectors |
matrix addition |
terminal point |
| head |
resultant vector |
vector |
| initial point |
scalar multiplication |
zero vector |
| magnitude (norm) |
standard position |
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Essential Knowledge and Skills/Indicators
- Solve a system of linear equations algebraically to find the ordered pair that satisfies both equations simultaneously and solves a system. Use the substitution or elimination techniques to solve the system. (A.9)
- Solve a system of linear equations graphically by graphing the equations on the same set of axes and finding the coordinates of the point of intersection. (A.9)
- Identify types of angle pairs: (G.3)
- complementary angles
- supplementary angles
- vertical angles
- linear pairs of angles
- alternate interior angles
- consecutive interior angles
- corresponding angles
- Use inductive reasoning to determine the relationship between complementary angles, supplementary angles, vertical angles, and linear pairs of angles. (G.3)
- Define and identify parallel lines. (G.3)
- Determine if a system of linear equations has one solution, no solution, or infinitely many solutions. (A.9)
- Describe slope as a constant rate of change. (A.7)
- Recognize and describe a line with a slope that is positive, negative, or zero. (A.7)
- Characterize the changes in the graph of the line as translations, reflections, and/or dilations. (A.6)
- Calculate the slope of a line given: (A.7)
- the coordinates of two points on the line
- the equation of the line
- the graph of the line
- Recognize and describe a line with a slope that is undefined. (A.7)
- Given the equation of a line, solve for the dependent variable and graph: (A.6)
- using transformations
- using the slope and y-intercept
- using the x-intercept and y-intercept
- Graph an inequality in two variables. (A.6)
- Write the equation of a line given: (A.8)
- the coordinates of two points on the line
- the slope and the y-intercept of the line
- one point on a vertical line
- one point on a horizontal line
- the graph of a line
- Compare the slopes of lines with their relative positions in a coordinate plane, including parallel, perpendicular, horizontal, and vertical.
- Find the slope of a line given the graph of the line, the equation of the line, or the coordinates of two points on the line. Investigate the relationship between the slopes of parallel lines. (G.2)
- Explore the relationship between alternate interior angles, consecutive interior angles, and corresponding angles when they occur as a result of parallel lines being cut by a transversal. (G.3)
- State these angle relationships as conditional statements. (G.3)
- Solve practical problems involving these angle relationships. (G.3)
- Use the converses of the conditional statements about the angles associated with two parallel lines cut by a transversal to show necessary and sufficient conditions for parallel lines. (G.4)
- Verify the converses using deductive arguments, coordinate, and algebraic methods. (G.4)
- Construct the following (G.11):
- a line segment congruent to a given segment
- the bisector of a line segment
- an angle congruent to a given angle
- the bisector of a given angle
- a perpendicular to a given line from a point not on the given line
- a perpendicular to a given line at a point on the given line.
- Identify, draw, and label vectors using appropriate notation in two (and three) dimensional space representations.
- Express the addition and scalar multiplication of vectors both graphically and algebraically.
- Represent vectors as matrices and determine resultants by scalar addition.
- Apply vectors to practical situations.
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III. Triangles
Benchmark
Students identify properties and investigate relationships of triangles using symmetry, congruence, and transformation.
Key Terms
| acute triangle |
converse of Pythagorean theorem |
perimeter |
| angle of depression |
cosine |
Pythagorean Theorem |
| angle of elevation |
distance formula |
radical |
| area |
equiangular triangle |
ratio |
| base |
equilateral triangle |
right triangle |
| base angles |
exterior angle |
scale factor |
| centroid |
geometric mean |
scalene triangle |
| circumcenter |
hypotenuse |
similarity |
| circumscribe |
incenter |
sine |
| concurrent |
inscribe |
tangent |
| congruence |
isosceles triangle |
triangle |
| congruence postulates |
legs |
triangle inequality theorem |
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midpoint formula |
trigonometry |
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obtuse triangle |
vertex angle |
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orthocenter |
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Essential Knowledge and Skills/Indicators
- Investigate and identify congruent figures. (G.5)
- Define congruent figures. (G.5)
- Map corresponding parts (angles and sides) of congruent figures onto each other. (G.5)
- Discuss applications of congruence such as rubber stamps, manufacturing, and patterns. (G.5)
- Understand the structure of Euclidean geometry: (G.5)
- undefined terms
- defined terms
- postulates
- theorems
- Verify that triangles are congruent using the following postulates: (G.5)
- angle-angle-side (AAS)
- angle-side-angle (ASA)
- hypotenuse-leg (HL)
- side-angle-side (SAS)
- side-side-side (SSS)
- Write a plan for a geometric proof. (G.5)
- Write deductive arguments as well as coordinate and algebraic demonstrations that triangles are congruent. (G.5)
- Use the definition of congruent triangles (corresponding parts of congruent triangles are congruent) to plan and write proofs. (G.5)
- Investigate and identify the constraints on the lengths of the sides of a triangle to develop the triangle inequality. (G.5)
- Use the triangle inequality to determine if three given segment lengths will form a triangle. (G.5)
- Apply the triangle inequality to relationship between the angle measures in triangles and the lengths of the sides opposite those angles. (G.6)
- Given side lengths in a triangle, identify the angles in order from largest to smallest or vice versa. (G.6)
- Given angle measures in a triangle, identify the sides in order from largest to smallest or vice versa. (G.6)
- Use indirect proof (proof by contradiction) to argue that all but one possible case in a given situation is impossible. (G.6)
- Use properties of proportions to solve practical problems. (G.6)
- Investigate and identify similar polygons. (G.6)
- Define similar polygons. (G.5)
- Use the following postulates to verify that triangles are similar. Deductive arguments as well as algebraic and coordinate methods may be used. (G.5)
- angle-angle (AA)
- side-angle-side (SAS)
- side-side-side (SSS)
- Find the coordinates of the midpoint of a line segment. (G.2)
- Verify the Pythagorean Theorem and its converse using deductive arguments as well as algebraic and coordinate methods. (G.7)
- Solve practical problems involving the Pythagorean Theorem and its converse. Use a calculator to find decimal approximations of solutions. (G.7)
- Use the Pythagorean Theorem to derive the distance formula. (G.2)
- Use the distance formula to find the length of line segments when given the coordinates of the endpoints. (G.2)
- Investigate the side lengths of isosceles right triangles and 30-60-90 triangles. Use inductive reasoning to conjecture about the relationships among the side lengths. (G.7)
- Use the properties of special right triangles to solve practical problems. Use a calculator to find decimal approximations of solutions. (G.7)
- Define sine, cosine, and tangent as trigonometric ratios in a right triangle. (G.7)
- Discuss exact values for trigonometric ratios and decimal approximations. (G.7)
- Use right triangle trigonometry to solve right triangles. (G.7)
- Use right triangle trigonometry to solve practical problems. Use a calculator to find decimal approximations of solutions. (G.7)
- Determine perimeter and area of triangles.
- Identify and investigate medians, altitudes, perpendicular bisectors of sides, angle bisectors, and transversals parallel to one side of a triangle.
- Construct the medians, altitudes, perpendicular bisectors, and angle bisectors and their respective points of concurrency in a triangle.
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David Van Vleet
Mathematics Specialist
DVanVleet@fcps.edu
703-846-8650
Donald Lacey Instructional Center
3705 Crest Drive
Annandale, VA 22003
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