Geometry
Unit Plans Aligned with
The
Common Core
Content Standards
2016-2017
Compiledusing the ArkansasMathematics Standards
1
Geometry
Arkansas Mathematics Standards
Arkansas Department of Education
2016
Geometry
Unit 1: Basics Chapters 1, 2, and 3
Chapter 1 is the basic definitions and terminology used for Geometry, it is intended to get the students to correctly identify key characteristics from geometric drawings. This chapter also is an introduction to constructions.
Chapter 1: Tools of Geometry
1.1 Points, Lines, Planes, Rays, and Line Segments
1.2 Translating and Constructing Line Segments
1.3 Midpoints and Bisectors
1.4 Translating and Constructing Angles and Angle Bisectors
1.5 Parallel and Perpendicular Lines on the Coordinate Plane
1.6 Constructing Perpendicular Lines, Parallel Lines, and Polygons
1.7 Points of Concurrency
Keys terms in chapter 1: point, line, ray, plane, collinear points, noncollinear points, compass, straightedge, skew lines, endpoint of a ray, line segment, construct, sketch, draw, coplanar lines, endpoints of a line segment, congruent line segments, Distance Formula, transformation, rigid motion, pre-image, image, arc, copying a line segment, duplicating a line segment, midpoint, Midpoint Formula, segment bisector, bisecting a line segment, angle, angle bisector, copying an angle, duplicating an angle, bisecting an angle, point-slope form, perpendicular bisector, concurrent, median, centroid, altitude, point of concurrency, incenter, orthocenter
Chapter 2: Proofs
(Mainly cover the theorems and definitions out of this chapter, and use the concept of proofs in chapter 1, and throughout the entire year.)
2.1 Foundations for Proof
Key terms: induction, deduction, conditional statement, propositional form, propositional variables, hypothesis, conclusion, truth value, truth table
2.2 Special Angles and Postulates
Key terms: supplementary angles, complementary angles, adjacent angles, postulate, linear pair, vertical angles, theorem, Euclidean Geometry, Linear Pair Postulate, Angle Addition Postulate
2.3 Paragraph Proof, Two-Column Proof, Construction Proof, and Flow Chart Proof
Key Terms: Addition Property of Equality, Subtraction Property of Equality, Reflexive Property, Substitution Property, Transitive Property, flow chart proof, two-column proof, paragraph proof, construction proof, Right Angle Congruence Theorem, Congruent Compliment Theorem, Vertical Angle Theorem
2.4 Angle Postulates and Theorems
Key Terms: Corresponding Angle Postulate, conjecture, Alternate Interior Angle Theorem, Alternate Exterior Angle Theorem, Same-Side Interior Angle Theorem, Same-Side Exterior Angle Theorem
2.5 Parallel Line Converse Theorems
Key Terms: Converse, Corresponding Angle Converse Postulate, Alternate Interior Angle Converse Theorem, Alternate Exterior Angle Converse Theorem, Same-Side Interior Angle Converse Theorem, Same-Side Exterior Angle Converse Theorem
Chapter 3: Perimeter and Area of Geometric Figures on the Coordinate Plane
We will start this chapter with a review of the formulas needed to find Perimeter and Area of these figures, then transition to having the figures on the coordinate plane. Extra emphasis will be placed on section 5 of this chapter.
3.1 Using Transformations to Determine Area
3.2 Area and Perimeter of Triangles on the Coordinate Plane
3.3 Area and Perimeter of Parallelograms on the Coordinate Plane
3.4 Area and Perimeter of Trapezoids on the Coordinate Plane
Key Terms: Bases of a Trapezoid, Legs of a Trapezoid
3.5 Area and Perimeter of Composite Figures on the Coordinate Plane
Key Term: Composite Figures
***I will at this point evaluate how the pacing is going with the new standards. We should have finished all 3 of these chapters by the end of the first 9 weeks. Further modification may be necessary for the next calendar year if we find ourselves too far ahead or especially behind.***
Unit 2: Covering chapters 4, 5, 6, and 7
Chapter 4: Three Dimensional Figures
This chapter will revisit the three dimensional shapes that the students have seen before, but will now include finding the surface area and volume of them.
4.1 Rotating Two-Dimensional Figures through Space
Key Term: Disc
This section will help the students grasp the idea if three dimensional figures by showing them how to create them from two-dimensional figures.
4.2 Translating and Stacking Two-Dimensional Figures
Key Terms: Isometric Paper, Right Triangular Prism, Oblique Triangular Prism, Right Rectangular Prism, Oblique Rectangular Prism, Right Cylinder, Oblique Cylinder
4.3 Cavalieri’s Principal
Key Term: Cavalieri’s Principal
Cavalieri’s Principal is mentioned NUMEROUS times in the Common Core Standards, therefore, I feel it must be explained until all the students have a grasp if what is saying.
4.4 Volume of Cones and Pyramids
Key Formulas: Volume of a cone, volume of a pyramid
4.5 Volume of Sphere
Key Terms: Sphere, radius of a sphere, diameter of a sphere, great circle of a sphere, hemisphere, annulus
Key Formula: Volume of a sphere
4.6 Using Volume Formulas
This section will have been covered by the previous work done in this chapter, we just have to include real world examples in the assignments
4.7 Cross Sections
This section can also be covered by using examples during the previous sections, and does not need to be a section all to itself
4.8 Diagonals in Three Dimensions
Using the Pythagorean Theorem
Chapter 5: Properties of Triangles
5.1 NO STANDARD FOR THIS SECTION, OMIT?
5.2 Triangle Sum, Exterior Angle, and Exterior Angle Inequality Theorems
Key Terms/ Theorems: Triangle Sum Theorem, remote interior angles of a triangle, Exterior Angle Theorem, Exterior Angle Inequality Theorem
5.3 The Triangle Inequality Theorem
Key Theorem: Triangle Inequality Theorem
5.4 and 5.5 45°-45°-90° and 30°-60°-90° Triangles
Key Terms: 45-45-90 Triangle, 30-60-90 Triangle
Chapter 6: Similarity through Transformations
6.1 Dilating Triangles to Create Similar Triangles
Key Term: Similar Triangles
6.2 Similar Triangle Theorems
Key Terms/ Theorems: Angle-Angle Similarity Theorem, Side-Side-Side Similarity Theorem, Included Angle, Included Side, Side-Angle-Side Similarity Theorem
6.3 Theorems about Proportionality
Key Terms/ Theorems: Angle Bisector/Proportional Side Theorem, Triangle Proportionality Theorem, Converse of the Triangle Proportionality Theorem, Proportional Segments Theorem, Triangle Midsegment Theorem
I have found that this section is particularly tough to get the students to understand and be able to apply. It may take more than a day or two to teach this, and at some point I will have to move on just because we have bogged down at this point.
6.4 More Similar Triangles
Key Theorems: Right Triangle/Altitude Similarity Theorem, Geometric Mean, Right Triangle Altitude/Hypotenuse Theorem, Right Triangle Altitude/Leg Theorem
This section is about the same as 6.3 in terms of difficulty to get the students to understand.
6.5 Proving the Pythagorean Theorem and the Converse of the Pythagorean Theorem
6.6 Applications of Similar Triangles
Key Term: Indirect Measurement
Chapter 7: Congruence through Transformations
7.1 Translating, Rotating, and Reflecting Geometric Figures
7.2 Congruent Triangles
7.3 Side-Side-Side Congruence Theorem
7.4 Side-Angle-Side Congruence Theorem
7.5 Angle-Side-Angle Congruence Theorem
7.6 Angle-Angle-Side Congruence Theorem
7.7 Using Congruent Triangles
This chapter is more about proving that these theorems are true, and then doing two column proofs that use one of the new theorems. The difficult part will be getting the students to use the two column proof method.
At this point, we will again re-evaluate the unit plans. If the units go according to the planned pace, we will finish chapter 7 just in time to review and have semester testing.
Unit 3: Covering chapters 8, 9, and 10
Chapter 8: Using Congruence Theorems
8.1 Right Triangle Congruence Theorems
Key Theorems: Hypotenuse-Leg (HL) Theorem, Leg-Leg (LL) Theorem, Hypotenuse-Angle (HA) Theorem, Leg-Angle (LA) Congruence Theorem
8.2 Corresponding Parts of Congruent Triangles are Congruent
Key Terms/Theorems: Corresponding Parts of Congruent Triangles are Congruent (CPCTC), Isosceles Triangle Base Angle Theorem, and Isosceles Triangle Base Angle Converse Theorem
8.3 Isosceles Triangle Theorems
Key Terms/Theorems: Vertex Angle of an Isosceles Triangle, Isosceles Triangle Base Theorem, Isosceles Triangle Vertex Angle Theorem, Isosceles Triangle Perpendicular Bisector Theorem, Isosceles Triangle Altitude to Congruent Sides Theorem, Isosceles Triangle Angle Bisector to Congruent Sides Theorem
8.4 Inverse, Contrapositive, Direct Proof, and Indirect Proof
Key Terms/Theorems: Inverse, Contrapositive, Direct Proof, Indirect Proof (or proof by contradiction), Hinge Theorem, Hinge Converse Theorem
Chapter 9: Trigonometry
9.1 Introduction to Trigonometry
Key Terms: Reference Angle, Opposite Side, Adjacent Side
9.2 Tangent Ratio, Cotangent Ratio, and Inverse Tangent
Key Terms: Rationalize the denominator, Tangent (tan), Cotangent (cot), Inverse Tangent
9.3 Sine Ratio, Cosecant Ratio, and Inverse Sine
Key Terms: Sine (sin), Cosecant (csc), Inverse Sine
9.4 Cosine Ratio (cos), Secant Ratio (sec), and Inverse Cosine
Key Terms: Cosine Ratio, Secant Ratio, Inverse Cosine
9.5 Complement Angle Relationships
9.6 Deriving the Triangle Area Formula, the Law of Sines, and the Law of Cosines
Key Terms: Law of Sines, Law of Cosines, the formula for area of a triangle {A=1/2 ab(sin C)}
We can lump together sections 3, 4, and 5. Also, it will take a few days for the students to fully grasp the Laws of Sine and Cosine, so I will plan more days for section 9.6
Chapter 10: Properties of Quadrilaterals
10.1 Squares and Rectangles
Key Theorem: Perpendicular/Parallel Line Theorem
10.2 Properties of Parallelograms and Rhombi
Key Theorem: Parallelogram/Congruent-Parallel Side Theorem
10.3 Properties of Kites and Trapezoids
Key Terms/Theorems: Base Angles of a Trapezoid, Isosceles Trapezoid, Biconditional Statement, Midsegment, Trapezoid Midsegment Theorem
10.4 Sum of the Interior Angle Measures of a Polygon
Key Term: Interior Angle of a Polygon
10.5 Sum of the Exterior Angle Measures of a Polygon
Key Term: Exterior Angle of a Polygon
10.6 Classifying Quadrilaterals Based on Their Properties
10.7 Classifying Quadrilaterals on the Coordinate Plane
This chapter should end right at the end of the 3rd nine weeks if the pacing is correct.
Unit 4: Covering chapters 11, 12, and 13
Chapter 11: Circles
11.1 Introduction to Circles
Key Terms: radius, diameter, center of a circle, chord, secant of a circle, tangent of a circle, point of tangency, central angle, inscribed angle, are, major arc, minor arc, semicircle
11.2 Central Angles, Inscribed Angles, and Intercepted Arcs
Key Terms/Theorems: degree measure of an arc, adjacent arcs, Arc Addition Postulate, intercepted arc, Inscribed Angle Theorem, Parallel Lines-Congruent Arcs Theorem
11.3 Measuring Angles Inside and Outside of Circles
Key Theorems: Interior Angles of a Circle Theorem, Exterior Angles of a Circle Theorem, Tangent to a Circle Theorem
11.4 Chords
Key terms/Theorems: Diameter-Chord Theorem, Equidistant Chord Theorem, Equidistant Chord Converse Theorem, Congruent Chord-Congruent Arc Theorem, Congruent Chord-Congruent Arc Converse Theorem, segments of a chord, Segment-Chord Theorem
11.5 Tangent and Secants
Key Terms/Theorems: tangent segment, Tangent Segment Theorem, secant segment, external secant segment, Secant Segment Theorem, Secant Tangent Theorem
Chapter 12: Arcs and Sectors of Circles
12.1 Inscribed and Circumscribed Triangles and Quadrilaterals
Key Terms/Theorems: inscribed polygon, Inscribed Right-Triangle-Diameter Theorem, Inscribed Right Triangle-Diameter Converse Theorem, circumscribed polygon, Inscribed Quadrilateral-Opposite Angles Theorem
12.2 Arc Length
Key Terms: arc length, radian
12.3 Sectors and Segments of a Circle
Key Terms: concentric circles, sector of a circle, segment of a circle
12.4 Circle Problems
Key Terms: Linear Velocity, Angular Velocity
Chapter 13: Circles and Parabolas
13.1 Circles and Polygons on the Coordinate Plane
13.2 Deriving the Equation for a Circle\
13.3 Determining Points on a Circle
13.4 Equation of a Parabola
Key Terms: locus of points, parabola, focus of a parabola, directrix of a parabola, general form of a parabola, standard form of a parabola, axis of symmetry, vertex of a parabola, concavity
13.5 More with Parabolas
Course Title: Geometry
Course/Unit Credit: 1
Course Number: 431000
Teacher Licensure: Please refer to the Course Code Management System ( the most current licensure codes.
Grades: 9-12
Prerequisite: Algebra Ior Algebra A/B
Course Description: “The fundamental purpose of the course in Geometry is to formalize and extend students’ geometric experiences from the middle grades. Students explore more complex geometric situations and deepen their explanations of geometric relationships, moving towards formal mathematical arguments. Important differences exist between this Geometry course and the historical approach taken in Geometry classes. For example, transformations are emphasized early in this course. Close attention should be paid to the introductory content for the Geometry conceptual category found in the high school AMS.
This document was created to delineate the standards for this course in a format familiar to the educators of Arkansas. For the state-provided Algebra A/B, Algebra I, Geometry A/B, Geometry, and Algebra II documents, the language and structure of the ArkansasMathematics Standards(ASM) have been maintained. The following information is helpful to correctly read and understand this document.
“Standards define what students should understand and be able to do.
Clusters are groups of related standards. Note that standards from different clusters may sometimes be closely related, because mathematics is a connected subject.
Domains are larger groups of related standards. Standards from different domains may sometimes be closely related.”-
Standards do not dictate curriculum or teaching methods. For example, just because topic A appears before topic B in the standards for a given grade, it does not necessarily mean that topic A must be taught before topic B. A teacher might prefer to teach topic B before topic A, or might choose to highlight connections by teaching topic A and topic B at the same time. Or, a teacher might prefer to teach a topic of his or her own choosing that leads, as a byproduct, to students reaching the standards for topics A and B.
The standards in this document appear exactly as written in the ASM. Italicized portions of the standards offer clarification.The Plus Standards (+) from the ArkansasMathematics Standards may be incorporated into the curriculum to adequately prepare students for more rigorous courses (e.g., Advanced Placement, International Baccalaureate, or concurrent credit courses).
******Proofs will be infused throughout the entire school year and notjust taught as a stand-alone chapter.******
Geometry
DomainCluster Course Emphases
Congruence1. Investigate transformations in the plane / Supporting
2. Understand congruence in terms of rigid motions / Major
3. Apply and prove geometric theorems / Major
4. Make geometric constructions / Supporting
5. Logic and Reasoning
Similarity, Right Triangles, and Trigonometry
6. Understand similarity in terms of similarity transformations / Major
7. Apply and prove theorems involving similarity / Major
8. Define trigonometric ratios and solve problems involving right triangles / Major
9. Apply trigonometric to general triangles
Circles
10. Understand and apply theorems about circles / Additional
11. Find arc lengths and areas of sectors of circles / Additional
Expressing Geometric Properties with Equations
12. Translate between the geometric description and the equation of a conic section / Additional
13. Use coordinates to prove simple geometric theorems algebraically / Major
Geometric measurement and dimension
14. Explain volume formulas and use them to solve problems / Additional
15. Visualize relationships between two-dimensional and three-dimensional objects / Additional
Modeling with Geometry
16. Apply geometric concepts in modeling situations / Major
Asterisksidentify potential opportunities to integrate content with the modeling practice
Domain: Congruence
Cluster(s): 1. Investigate transformations in the plane
2. Understand congruence in terms of rigid motions
3. Apply and prove geometric theorems
4. Make geometric constructions
5. Logic and Reasoning
HSG.CO.A.1Chapter 1 / 1 / Based on the undefined notions of point, line, plane, distance along a line, and distance around a circular arc, define:
- Angle
- Line segment
- Circle
- Perpendicular lines
- Parallel lines
HSG.CO.A.2
Chapter 1 / 1 /
- Represent transformations in the plane (e.g. using transparencies, tracing paper, geometry software, etc.).
- Describe transformations as functions that take points in the plane as inputs and give other points as outputs.
- Compare transformations that preserve distance and angle to those that do not. (e.g., translation versus dilation).
HSG.CO.A.3
Chapter 7 / 1 / Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and/or reflections that carry it onto itself. / Supporting
HSG.CO.A.4
Chapter 7 / 1 / Develop definitions of rotations, reflections, and translations in termsof angles, circles, perpendicular lines, parallel lines, and line segments. / Supporting
HSG.CO.A.5
Chapter 7 / 1 /
- Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure, (e.g., using graph paper, tracing paper, miras, geometry software, etc.).
- Specify a sequence of transformations that will carry a given figure onto another.
HSG.CO.B.6
Chapter 7 / 2 /
- Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure
- Given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
HSG.CO.B.7
Chapter 7 / 2 / Use the definition of congruence in terms of rigid motions to showthat two triangles are congruent if and only if corresponding pairs ofsides and corresponding pairs of angles are congruent. / Major
HSG.CO.B.8
Chapters 6 and 7 / 2 / Explain how the criteria for triangle congruence (ASA, SAS, and SSS)follow from the definition of congruence in terms of rigid motions.Investigate congruence in terms of rigid motion to develop the criteria for triangle congruence (ASA, SAS, AAS, SSS, and HL)
Note: The emphasis in this standard should be placed on investigation / Major
HSG.CO.C.9
Chapter 2 / 3 / Apply and prove theorems about lines and angles.
Theorems include but are not limited to: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.
Note: Proofs are not an isolated topic and therefore should be integrated throughout the course. / Major
HSG.CO.C.10
Chapters 6,7 and 8 / 3 / Apply and prove theorems about triangles.
Theorems include but are not limited to: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
Note: Proofs are not an isolated topic and therefore should be integrated throughout the course. / Major
HSG.CO.C.11
Chapter 10 / 3 / Apply and prove theorems about quadrilaterals.
Theorems include but are not limited to relationships among the sides, angles, and diagonals of quadrilaterals and the following theorems concerning parallelograms: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
Note: Proofs are not an isolated topic and therefore should be integrated throughout the course. / Major
HSG.CO.D.12
Chapter 1, continues / 4 / Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).
Constructions may include but are not limited to: copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
Note: Constructions are not an isolated topic and therefore should be integrated throughout the course. / Supporting
HSG.CO.D.13
Chapter 1 / 4 / Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
Note: Constructions are not an isolated topic and therefore should be integrated throughout the course. / Supporting
HSG.CO.E.14
Chapter 2 / 5 / Apply inductive reasoning and deductive reasoning for making predictions based on real world situations using:
- Conditional Statements (inverse, converse, and contrapositive)
- Venn Diagrams
Domain: Similarity, Right Triangles, and Trigonometry