Download 2013–2014 Geometry Scope and Sequence ()

Geometry
This represents a general outline of the course. It details which standards will be covered in what
order, and provides limited explanatory material about the purpose and content of each
section. The theorems and constructions listed are intended to be a minimum expectation for
students. Other theorems and constructions may be appropriate. Implicit within each section is
the expectation that students refer back to previous material. For example, in the section on
polygons, students should also be asked to verify properties using the distance formula. Also
implicit is the goal that students solve real-world problems and problems that require application
of multiple standards.
Standards listed in purple are my own standards. Standards listed in green refer to both the
Common Core State Standards and the Washington State 2008 standards.
This course aims to follow the Van Hiele model of learning geometry. The course as a whole aims
to move from Level 0–1 to Level 3–4. Additionally, each section should also be taught in a way
that moves from Level 0–1 to Level 3–4.
Whole Course Concepts
The following points are intended to be developed and practiced throughout the entire course.
1. Determine the coordinates of a point that is described geometrically. Standard 25 —
G.4.B
2. Use inductive and deductive reasoning to make conjecture and write proofs. Standards
51–52 — G.1.B–C
3. Identify errors or gaps in a mathematical argument and develop counterexamples to
refute invalid statements. Standard 54 — G.1.E
4. Use coordinates to prove geometric theorems algebraically. Standard 55 — G.GPE.4
5. Identify and use appropriate degrees of precision in measurement. Standard 56 — G.6.E
6. Apply estimation strategies. Standard 57 — G.6.E
7. Use geometric methods, shapes, their measures, and their properties to describe objects
and solve problems. Use concepts of density based on area and volume in modeling
situations. Standards 59–60 — G.MG.1–3; G.6.C
8. The Common Core Standards for Mathematical Practice:

Make sense of problems and persevere in solving them.

Reason abstractly and quantitatively.

Construct viable arguments and critique the reasoning of others.

Model with mathematics.

Use appropriate tools strategically.
Geometry Course Outline based on WA 2008 and CCSS Standards
Last updated 8/15/2013

Attend to precision.

Look for and make use of structure.

Look for and express regularity in repeated reasoning.
Course Outline
1. Early definitions and undefined terms. Making a case for the importance of precision in
language and proof. Standard 1 — G.CO.1

List of undefined terms: point, line, plane.

List of definitions: collinear, coplanar, ray, line segment, parallel, perpendicular,
skew, triangle, circle.

Describe the intersections of lines in the plane and in space, of lines and planes,
and of planes in space. Standard 3 — G.2.D
2. Introduction to constructions. Beginning to write informal mathematical proofs, and
practicing simple constructions. Generally, we are operating at Van Hiele Level 1 and
moving toward Level 2. Standard 36 — G.CO.12; G.2.C

List of constructions: copy a line segment (non-collapsible compass); an
equilateral triangle; the perpendicular bisector of a line segment.
3. Algebra review and proof. Continuing to develop mathematical proofs, now in the
context of algebra proofs.

Angle and segment addition postulates. Standards 26–27

Solve problems with complementary and supplementary angles. Standard 28

Review slope, writing equations of lines (including parallel and perpendicular),
and solving equations and systems. Standard 23 — G.GPE.5; G.2.A; G.4.A
4. Transformations. Developing transformations. Defining and using congruence, similarity,
and symmetry. Standards 2, 4–6 — G.CO.2–6; G.SRT.1–2; G.5.A–D
5. Introduction to analytic geometry. Developing the midpoint formula and reviewing the
Pythagorean theorem for use in the distance formula.

Solve problems using the midpoint formula. Standard 29 — G.GPE.6; G.4.B

Prove the Pythagorean theorem using transformations and use it to solve
problems. Standard 16 — G.SRT.4, 8; G.3.D

Derive, and solve problems using, the distance formula. Standard 30 — G.GPE.7;
G.3.D

Locate the point on a line segment that partitions the segment into a given ratio.
Standard 24 — G.GPE.6
6. Introduction to logic. Developing the basics of logic as it will be used in this class.
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
Distinguish between, and use, inductive and deductive logic. Standards 50–52 —
G.1.A–C

Write the converse, inverse, and contrapositive of a statement and determine
their validity. Standard 53 — G.1.D
7. Lines and angles. Establishing results related to lines and angles, writing more formal
proofs of the theorems, and applying those theorems to solve problems. Exploring more
constructions.

Prove and apply theorems about angles: angles created by two parallel lines cut
by a transversal; vertical angles; congruent complements/supplements. Standard
19 — G.CO.9; G.2.B

Prove and apply theorems about lines: if two intersecting lines form a linear pair of
congruent angles, the lines are perpendicular; if two lines are
parallel/perpendicular to a third line, they are parallel/perpendicular to each
other; if a line is perpendicular to one of two parallel lines, then it is perpendicular
to the other line. Standard 20 — G.CO.9; G.2.A

List of constructions: a line parallel/perpendicular to a given line through a point
not on the given line; a line perpendicular to a given line through a point on the
given line; copy a segment (collapsible compass); copy an angle; bisect an
angle; a square. Standard 36 — G.CO.12–13; G.2.C
8. Triangles. Establishing results related to triangles, writing more formal proofs of the
theorems, and applying those theorems to solve problems. Exploring more constructions.

Develop triangle congruence and similarity theorems using transformations. Prove
and apply these theorems. Standard 8 — G.CO.7–8; G.SRT.3, 5; G.3.B

Prove and apply theorems about triangles: the measure of an exterior angle of a
triangle is equal to the sum of the measures of the two remote interior angles; the
triangle inequality theorem; the hinge theorem; the largest angle of a triangle is
opposite the longest side, and the smallest angle is opposite the shortest side; the
interior angle measures of a triangle sum to 180°; the base angles theorem; the
triangle midsegment theorem; a line parallel to one side of a triangle divides the
other two sides proportionally; a triangle is equilateral iff it is equiangular.
Standard 9 — G.CO.10; G.SRT.4; G.3.A; G.4.C

Prove and apply theorems about the centroid of a triangle: the medians of a
triangle are concurrent at the centroid; the centroid divides the medians in the
ratio 2: 1. Standard 10 — G.CO.10; G.SRT.4; G.3.A

Prove and apply theorems about the circumcenter of a triangle: the
perpendicular bisectors of a triangle are concurrent at the circumcenter; the
circumcenter is equidistant from the vertices of the triangle; the position of the
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circumcenter can be used to classify a triangle as acute, right, or obtuse.
Standard 11 — G.CO.10; G.SRT.4; G.3.A

Prove and apply theorems about the incenter of a triangle: the angle bisectors of
a triangle are concurrent at the incenter; the incenter is equidistant from the sides
of the triangle. Standard 13 — G.CO.10; G.SRT.4; G.3.A

Prove and apply theorems about the orthocenter of a triangle: the altitudes of a
triangle are concurrent at the orthocenter; the position of the orthocenter can be
used to classify a triangle as acute, right, or obtuse. Standard 12 — G.CO.10;
G.SRT.4; G.3.A

List of constructions: the circumscribed circle for a given triangle; the inscribed
circle for a given triangle. Standard 36 — G.C.3; G.CO.12; G.3.I

List of theorems: a point in the interior of an angle lies on the angle bisector iff it is
equidistant from the sides of the angle; a point lies on the perpendicular bisector
of a line segment iff it is equidistant from the endpoints of the segment.
9. Polygons. Establishing results related to polygons, writing more formal proofs of the
theorems, and applying those theorems to solve problems. Exploring more constructions.

Prove and apply theorems about parallelograms: a quadrilateral is a
parallelogram iff its opposite sides (angles) are congruent; a quadrilateral is a
parallelogram iff its consecutive interior angles are supplementary; a quadrilateral
is a parallelogram iff its diagonals bisect each other; a parallelogram is a
rectangle iff its diagonals are congruent; a parallelogram is a rectangle iff at least
one angle is a right angle; a parallelogram is a rhombus iff its diagonals are
perpendicular; a parallelogram is a rhombus iff its diagonals bisect pairs of
opposite angles; a parallelogram is a rhombus iff one pair of consecutive sides is
congruent. Standard 21 — G.CO.11; G.3.F; G.4.C

Prove and apply theorems about polygons: if a figure is a trapezoid then
consecutive angles between a pair of parallel lines are supplementary; if the legs
of a trapezoid are congruent then it is isosceles; a trapezoid is isosceles iff its
diagonals are congruent; the midsegment of a trapezoid is parallel to the bases
and half the sum of their lengths; if a figure is a kite then the diagonals are
perpendicular; if a figure is a kite then exactly one pair of opposite angles are
congruent; the sum of one set of exterior angles of a polygon is 360°; the sum of
the interior angles of a polygon is 180(𝑛 − 2)°. Standard 22 — G.3.G; G.4.C

List of constructions: the regular hexagon inscribed in a given circle; the square
inscribed in a given circle. Standard 36 — G.CO.12–13
10. Trigonometry. Developing the trigonometric ratios and using them to solve problems.
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
Explain and use sine, cosine, and tangent to solve problems. Explain and use the
relationship between the sine and cosine of complementary angles. Standard 14
— G.SRT.6–8; G.3.E

Derive and apply the properties of special right triangles to solve problems.
Standard 15 — G.3.C
+
1
Derive and find the area of a triangle using the formula 𝐴 = 𝑎𝑏 sin 𝑐. Standard 17
2
— G.SRT.9
+
Prove and use the Laws of Sines and Cosines to find unknown measures in right
and non-right triangles. Standard 18 — G.SRT.10–11
11. Circles. Establishing results related to circles, writing more formal proofs of the theorems,
and applying those theorems to solve problems. Exploring more constructions.

Prove and apply theorems about circles: two chords are congruent iff they are
equidistant from the center of the circle; two minor arcs are congruent iff their
corresponding chords are congruent; if the diameter of a circle is perpendicular
to a chord, then the diameter bisects the chord and the arc intercepted by the
chord; if two secants intersect in the interior of a circle, then the sum of the
measures of the two vertical angles formed is equal to the sum of the measures of
the corresponding intercepted arcs; a line is tangent to a circle iff it is
perpendicular to the radius drawn to the point of tangency; all circles are similar;
if a quadrilateral is inscribed in a circle, then opposite angles are supplementary.
Standard 31 — G.C.1–2; G.3.H

Prove and apply theorems about central, inscribed, and circumscribed angles:
two arcs are congruent iff their central angles are congruent; if two inscribed
angles in a circle intercept the same arc, then they have the same measure; an
angle inscribed in a circle is a right angle iff its corresponding arc is a semicircle,
and the longest side of the resulting triangle is a diameter of the circle; the
measure of an inscribed angle in a circle is half the measure of the intercepted
arc; the measure of a central angle is equal to the measure of the intercepted
arc; the measure of a circumscribed angle is 180° minus the measure of the
central angle that forms the same arc. Standard 35 — G.C.2; G.3.H

Derive the standard form for the equation of a circle. Write the equation of a
circle described analytically. Describe the center and radius of a circle given as
an equation. Standard 32 — G.GPE.1; G.4.D

Solve systems of equations to find the point(s) of intersection of a circle and a line.
Standard 33 — G.4.D

Derive and apply formulas for arc length and area of a sector of a circle. Define
the radian and convert between radians and degrees. Standard 34 — G.C.5;
G.6.A
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
List of constructions: the diameter of a given circle; the center of a given circle
given two chords; a line tangent to a given circle though a given point
on/outside the circle. Standard 36 — G.C.4; G.CO.12; G.3.I
12. The third dimension. Developing formulas for volume and surface area. Describing and
reasoning about three-dimensional objects.

Derive and apply formulas for: the circumference of a circle; area of a circle;
volume of a cylinder; volume of a cone; volume of a pyramid; volume of a
sphere. Standard 37 — G.GMD.1–3; G.6.C

Identify two-dimensional cross-sections of three-dimensional objects. Identify
three-dimensional objects formed by rotating two-dimensional figures. Solve
problems involving these relationships. Standard 38 — G.GMD.4; G.3.K

Describe prisms, pyramids, parallelepipeds, tetrahedra, and regular polyhedra in
terms of their faces, edges, and properties. Know and apply Euler’s formula for
convex polyhedra. Standard 39 — G.3.J

Analyze and apply distance and angle measures on a sphere. Standard 40 —
G.6.B
13. Measurement. Things that don’t fit in anywhere else…

Determine and apply the effect that changing one, two, or three linear
dimensions has on perimeter, area, volume, or surface area on two- and threedimensional figures. Standard 7 — G.6.D

Use proportional relationships (dimensional analysis) to solve problems involving
conversions within and between measurement systems. Standard 58 — G.6.F
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