Unit Plans Aligned With

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

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

• Angle
• Line segment
• Circle
• Perpendicular lines
• Parallel lines

• 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).

• 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.

• 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.

• Conditional Statements (inverse, converse, and contrapositive)

• Venn Diagrams

Domain: Similarity, Right Triangles, and Trigonometry