C&I

Curriculum and Instruction

Mathematics

Crosswalk: Comparison of the 1999 Core Curriculum and 2005 Core Curriculum for High School Mathematics
September 2005


Geometry is intended to be the second course in mathematics for high school students. There is no other school mathematics course that offers students the opportunity to act as mathematicians.  Within this course, students will have the opportunity to make conjectures about geometric situations and prove in a variety of ways, both formal and informal, that their conclusion follows logically from their hypothesis.  This course is meant to employ an integrated approach to the study of geometric relationships. Integrating synthetic, transformational, and coordinate approaches to geometry, students will justify geometric relationships and properties of geometric figures. Congruence and similarity of triangles will be established using appropriate theorems. Transformations including rotations, reflections, translations, and glide reflections and coordinate geometry will be used to establish and verify geometric relationships.  A major emphasis of this course is to allow students to investigate geometric situations. Properties of triangles, quadrilaterals, and circles should receive particular attention.  It is intended that students will use the traditional tools of compass and straightedge as well as dynamic geometry software that models these tools more efficiently and accurately, to assist in these investigations.  Geometry is meant to lead students to an understanding that reasoning and proof are fundamental aspects of mathematics and something that sets it apart from the other sciences.  

The following chart lists the concepts and skills in Geometry (2005 Core) and indicates where it was included in the 1999 Core.

Geometric Relationships / Constructions / Locus / Informal and Formal Proofs /
Transformational Geometry / Coordinate Geometry

Geometric Relationships

2005 Core Curriculum

1999 Core Curriculum

Performance
Indicator

Concept/Skill

Key Idea

Concept/Skill

G.G.1

A line perpendicular to each of two intersecting lines at their point of intersection, is perpendicular to the plane determined by them

 

Not addressed

G.G.2

Through a given point there passes one and only one plane perpendicular to a given line

 

Not addressed

GG.3

Through a given point there passes one and only one plane perpendicular to a given line

 

Not addressed

G.G.4

Two lines perpendicular to the same plane are coplanar

 

Not addressed

G.G.5

Two planes are perpendicular to each other if and only if one plane contains a line perpendicular to the second plane

 

Not addressed

 

2005 Core Curriculum

1999 Core Curriculum

Performance
Indicator

Concept/Skill

Key Idea

Concept/Skill

G.G.6

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

 

Not addressed

G.G.7

If a line is perpendicular to a plane then every plane containing the line is perpendicular to the given plane

 

Not addressed

G.G.8

If a plane intersects two parallel planes, then the intersection is two parallel lines

 

Not addressed

G.G.9

Two planes perpendicular to the same line are parallel.

 

Not addressed

GG.10

The lateral edges of a prism are congruent and parallel

 

Not addressed

G.G.11

Two prisms have equal volumes if their bases have equal areas and their altitudes are equal

 

Not addressed

G.G.12

The volume of a prism is the product of the area of the base and the altitude

Math B – 5H

Derive formulas to find measures such as length, area, and volume in real-world context

G.G.13

Apply the properties of a regular pyramid, including:

  • 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

Math B – 5H

 

 

 

 

Derive formulas to find measures such as length, area, and volume in real-world context

 

2005 Core Curriculum

1999 Core Curriculum

Performance
Indicator

Concept/Skill

Key Idea

Concept/Skill

G.G.14

Apply the properties of a cylinder, including:

  • 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

Math B 5H

 

 

Derive formulas to find measures such as length, area, and volume in real-world context

G.G.15

Apply the properties of a right circular cone, including:

  • 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 it base and its altitude

 

Math B – 5H

 

 

 

 

 

Derive formulas to find measures such as length, area, and volume in real-world context

G.G.16

Apply the properties of a sphere, including:

  • 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 4πr2
  • Volume is 4/3 πr3

 

 

 

 

 

 

 

 

 

 

Derive formulas to find measures such as length, area, and volume in real-world context

 


Constructions

2005 Core Curriculum

1999 Core Curriculum

Performance
Indicator

Concept/Skill

Key Idea

Concept/Skill

 G.G.17

Bisect a given angle using a straightedge and compass, and justify the construction

Math A – 4B

 

Math 7/8- 4J

Justify the procedures for basic geometric constructions

Bisect an angle, using a compass and a straightedge

G.G.18

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

Math A – 4B

 

Math 7/8– 4J

Justify the procedures for basic geometric constructions

 

 

Construct the perpendicular bisector of a line segment

G.G.19

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

Math A – 4B

Justify the procedures for basic geometric constructions

G.G.20

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

Math A – 4B

Justify the procedures for basic geometric constructions

 

Locus

2005 Core Curriculum

1999 Core Curriculum

Performance
Indicator

Concept/Skill

Key Idea

Concept/Skill

G.G.21

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

 

Not specifically addressed

G.G.22

Compound loci

Math A – 4D

Develop and apply the concept of basic loci to compound loci

G.G.23

Graph and solve compound loci in the coordinate plane

Math A – 4D

Not specifically related to the coordinate plane


 

Informal and Formal Proofs

2005 Core Curriculum

1999 Core Curriculum

Performance
Indicator

Concept/Skill

Key Idea

Concept/Skill

G.G.24

Determine the negation of a statement and establish its truth value

Math A –1A

                1B

Construct valid arguments

Follow and judge the validity of arguments

G.G.25

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

Math A –1A

                1B

Construct valid arguments

Follow and judge the validity of arguments

G.G.26

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

Math A –1A

                1B

Construct valid arguments

Follow and judge the validity of arguments

G.G.27

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

Math B – 1A

Math B – 1B

Math B – 7H

Construct proofs based on deductive reasoning

Construct indirect proofs

Apply axiomatic structure to geometry

  • Geometric proof

G.G.28

Determine the congruence of two triangles using SSS, SAS, ASA, AAS, HL

Math A – 4B

Justify the procedures for basic geometric constructions

G.G.29

Identify corresponding parts of congruent triangles

Math A – 4B

Justify the procedures for basic geometric constructions

G.G.30

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

Math A – 4A

Sum of the measures of angles of a triangle

G.G.31

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

Math A – 4A

Base angles of an isosceles triangle

G.G.32

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

Math A – 4A

Exterior angle of a triangle

2005 Core Curriculum

1999 Core Curriculum

Performance
Indicator

Concept/Skill

Key Idea

Concept/Skill

G.G.33

Investigate, justify, and apply the triangle inequality theorem

Math A – 4A

Triangular inequality

G.G.34

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

Math A – 4A

Triangular inequality

G.G.35

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

Math A – 4A

Alternate interior and exterior angles and corresponding angles

G.G.36

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

Math A – 4A

Sum of interior and exterior angles of a triangle

G.G.37

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

 

Not directly addressed

G.G.38

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

Math A – 4A

Study of quadrilaterals:  properties of parallelograms

G.G.39

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

Math A – 4A

Study of quadrilaterals:  properties of rectangles, rhombi, squares, and trapezoids

G.G.40

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

Math A – 4A

Study of quadrilaterals:  properties of trapezoids

2005 Core Curriculum

1999 Core Curriculum

Performance
Indicator

Concept/Skill

Key Idea

Concept/Skill

G.G.41

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

Math A – 4A

Study of quadrilaterals

G.G.42

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

 

Not addressed

G.G.43

Investigate, justify, and apply theorems about the centroid of a triangle diving each median into segment whose lengths are in the ratio2:1

 

Not addressed

G.G.44

Similarity of triangles (AA, SAS, and SSS)

Math A – 4B

Comparison of triangles :  congruence

 

G.G.45

Investigate, justify, and apply theorems about similar triangles

 

Not specifically addressed

G.G.46

Investigate, justify, and apply theorems about proportional relationships among the segments of the sides of the triangle, given one or more lines 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

 

Not specifically addressed

G.G.47

Investigate, justify, and apply theorems about mean proportionality:

  • attitude to the hypotenuse of a right triangle

 

Not specifically addressed

 

2005 Core Curriculum

1999 Core Curriculum

Performance
Indicator

Concept/Skill

Key Idea

Concept/Skill

G.G. 48

Pythagorean theorem and its converse

Math B – 5H

Math 7/8 -7I

Pythagorean Theorem

Develop and apply the Pythagorean principle in the solution of problems

G.G.49

Investigate, justify and apply theorems regarding chords of a circle

Math B 5D

Prove and apply theorems related to lengths of segments in a circle

G.G.50

Investigate, justify, and apply theorems about tangent lines to a circle

Math B – 5D

Prove theorems related to lengths of line segments in a circle

 

G.G.51

Investigate, justify, and apply theorems about the arcs determined by the rays of angles formed by two lines intersecting a circle

Math B – 5D

Prove theorems related to lengths of line segments in a circle

 

G.G.52

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

Math B – 5D

Prove theorems related to lengths of line segments in a circle

 

G.G.53

Investigate, justify, and apply theorems regarding segments intersected by a circle

Math B – 5D

Prove theorems related to lengths of line segments in a circle

 

Transformational Geometry

2005 Core Curriculum

1999 Core Curriculum

Performance
Indicator

Concept/Skill

Key Idea

Concept/Skill

G.G.54

Define, investigate, justify, and apply isometries in the plane

Math A – 4C Math B – 7L

 

 

 

 

Math B –7M

Use transformations in the coordinate plane

Use basic transformations to demonstrate similarity and congruence of figures

  • Transformations that provide similarity
  • Direct isometries
  • Opposite isometries

Identify and differentiate between direct and indirect isometries

G.G.55

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

Math A – 4C

Use transformations in the coordinate plane

G.G.56

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

 

Not specifically addressed

G.G.57

Justify geometric relationships using transformational techniques

Math B – 3C

Use transformations on figures in the coordinate plan

G.G.58

Define, investigate, justify and apply similarities

Math B – 7L

Use transformations to demonstrate similarity of figures

G.G.59

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

Math B – 7L

Use basic transformations to demonstrate similarity of figures

G.G.60

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

 

Not specifically addressed

 

2005 Core Curriculum

1999 Core Curriculum

Performance
Indicator

Concept/Skill

Key Idea

Concept/Skill

G.G.61

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

 

Not addressed

 

Coordinate Geometry

2005 Core Curriculum

1999 Core Curriculum

Performance
Indicator

Concept/Skill

Key Idea

Concept/Skill

G.G.62

Slope of a perpendicular line, given the equation the a line

 

Not addressed

G.G.63

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

 

Not addressed

G.G.64

Equation of a line given a point on the line and the equation of a line perpendicular to the given line

 

Not addressed

G.G.65

Find the length of a line segment, given its endpoints

 

Not addressed

G.G.66

Midpoint of a line segment

 

Not addressed

G.G.67

Length of a line segment

 

Not addressed

G.G.68

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

 

Not addressed

G.G.69

Properties of triangles and quadrilaterals in the coordinate plane, using the distance, midpoint, and slope formulas

 

Not addressed

G.G.70

Graphic solutions of systems of equations involving one linear equation and one quadratic equation

Math A –7A

Graphic solution of systems of linear equations, inequalities, and quadratic-linear pair

 

2005 Core Curriculum

1999 Core Curriculum

Performance

Indicator

Concept/Skill

Key Idea

Concept/Skill

G.G.71

Equation of a circle, given its center ad radius or the endpoints of a diameter

 

Not addressed

G.G.72

Equation of a circle given its graph (center is an ordered pair of integers and the radius is an integer)

 

Not addressed

G.G.73

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

 

Not addressed

G.G.74

Graph circles of the form (x – h)2 + (y – k)2 = r2

 

Not addressed

 


 

Last Updated: March 28, 2011