Geometry

Parameters Unit 1:

Parameters of Unit 1

i. Review of Coordinate Plane

ii. Review of Formulas previously seen before

1. Distance, Midpoint, slope, Perimeter, Area, Circle

iii. Introduction of New Formula (Line Partitioning)

iv. Structure of Algebraically writing work out

1. Getting students to structure work to help them focus and not “Fear” right or wrong answers

v. Help build structure with notes, quizzes, vocabulary, assessments.

1. Notes for each lesson

2. Quiz on last day of week

3. Vocabulary Graphic organizers (Also Graphic organizers of Formulas)

4. Digital Assessments to help build towards preparation of PARCC/NJSLA

a. ixl.com

b. Deltamath

c. Schoolnet

d. All of the above assist with collection of DATA

Unit 1: Coordinate Geometry

LEVEL 1

Review of Coordinate Plane
1. X & Y axis
2. Infuse Vocabulary like Positive, Negative, units, etc
3. Quadrants
4. Plotting points
Identifying Geometric Shapes by Name
1. Naming a Point
2. Naming a Line
  1. Symbol
3. Naming a Line Segment
  1. Symbol
4. Naming an Angle
  1. Symbol
  2. Using 1 letter - Vertex
  3. Using 3 letters
5. Naming a Shape
  1. Triangle
  2. Quadrilateral
Pythagorean Theorem
1. Using Formula
  1. Identifying Parts
  2. Solving
Slope
1. Review rise over run
2. Slope Graphically
3. Slope Algebraically
4. Parallel slopes are equal
5. Perpendicular slopes are opposite reciprocal
6. Identifying & Interpreting slope
Distance
1. Identifying Vertical and Horizontal distance
  1. Pythagorean Theorem
2. Distance Formula
  1. Identifying Parts (of Points)
  2. Solving
Midpoint
1. Definition:
  1. Create 2 Equal Line Segments
  2. Half-Way point between Two Endpoints
2. Finding Midpoint Graphically
  1. Horizontal and Vertical lines
    1. Half Horizontal Distance
    2. Half Vertical Distance
  2. Diagonal lines
    1. Using Slope to Find Midpoint
Partitioning Line Segments (First “New” topic)
1. Using formula
2. Visualizing (using slope)

Level 2

Perimeter & Area
1. Definitions (Perimeter & Area)
2. Formulas (yet again)
Irregular shapes (Pentagon, trapezoid, hexagon)
Definitions of polygons
1. Triangles
  1. Scalene, Isosceles, Equilateral, Right, etc
    1. Definition of each
2. Quadrilaterals
  1. Parallelogram, Rectangle, Square, Rhombus, Kite, Trapezoid
    1. Definitions of each

Level 3

Begin to infuse Proofs (& Structure of Proofs)
1. Prove Types of triangles (Isosceles, etc)
  1. Isosceles Triangle has 2 equal sides
    1. Use Distance Formula to show distances between 2 points
      1. Show 2 sides are equal
  2. Equilateral Triangle has 3 equal sides
    1. Use Distance Formula to show distances between 2 points
      1. Show 3 sides are equal
  3. Scalene Triangle has no sides equal
    1. Use Distance Formula to show distances between 2 points
      1. Show 0 sides are equal
  4. Right Triangle has two perpendicular sides
    1. Use slope formula to show slope of two sides are opposite reciprocal
2. Prove a square is a rectangle
  1. Use distance formula to show 4 equal sides
  2. Use slope formula to show 4 opposite reciprocal slopes
3. Prove a shape is a trapezoid

Level 4 Assessment

Verbal/Written Proofs
Given a set of points on a plane, prove those sets of points is a specific classification of shape
1. Given 3 points prove those 3 points is a
  1. Scalene, Isosceles, Equilateral, Right, Right/Scalene, etc
2. Given 4 points prove those 4 points is a
  1. Parallelogram, Rectangle, Square, Rhombus, Kite, Trapezoid
Midpoint of Quadrilateral Theorem
1. Creates a parallelogram

Parameters Unit 2:

Parameters of Unit 2

i. Build Towards Creating Two-Column Proofs by Utilizing Algebraic Proofs as a basis

ii. Create a "Word Bank" of Algebraic Rules and Properties

Properties of Equality

iii. Creating Formal Proofs by Introducing Reasons in Order of Creation

Line ---> Linear Pair ---> Vertical Angles ----> Transversal ----> Corresponding Angles ----> etc

iv. Build equations using geometric properties

Builds new properties on new formal knowledge

Repetitious

v. Solve Equations using Geometric Properties

Utilize two-column proofs to show truth and relevance in practice

Unit 2: Algebraic Proofs

Level 1

Solutions
1. One solution
2. Infinite solutions
3. No solutions
Properties of Equality
1. Addition Property of Equality
  1. Adding the same amount to both sides
2. Subtraction Property of Equality
  1. Subtracting the same amount to both sides
3. Multiplication Property of Equality
  1. Multiplying same amount to both sides
4. Division Property of Equality
  1. Dividing the same amount to both sides
5. Distributive Property
  1. With Multiplication
  2. With Division
6. Square Property of Equality
  1. Squaring both sides
7. Square Root Property of Equality
  1. Square Rooting both sides
Solving Equations in One Variable with Two-Column Proofs
1. Solving One Step Equations
2. Solving Two Step Equations
3. Solving Three Step Equations
  1. Third step is Distributive Property
4. Solving Multi-Step Equations
  1. Multiple terms on one side
  2. Multiple terms on both sides

Level 2

Geometric Principles and Definitions (And Using Them in Algebraic Proofs)
1. Straight Line
  1. Definition: Straight line equals 180 degrees
    1. Construct Equations equal to 180 degrees
2. Linear Pair
  1. Supplemental Pair of Angles that Equal 180 degrees
    1. Proof:
    2. Construct Equations of Linear Pairs set equal to 180
3. Vertical Angles
  1. Definition: Opposite Equal Angles created by the Intersection of Two Lines
  2. Proof:
    1. Construct Equations of Vertical Angles set equal to one another
Transversals
1. A line that Intersects Two lines
  1. Creates 2 sets of Intersections and Specific Geometric Terms
    1. Corresponding Angles
    2. Alternate Interior Angles
    3. Alternate Exterior Angles
    4. Same Side Interior Angles
    5. Same Side Exterior Angles
If 2 Initial Lines are Parallel then Equivalent and Supplementary Angles are Created
1. Equivalent Angles
  1. Corresponding Angles
    1. Definition
  2. Alternate Interior Angles
    1. Proof:
  3. Alternate Exterior Angles
    1. Proof:
2. Supplementary Angles
  1. Same Side Interior
    1. Proof:
  2. Same Side Exterior
    1. Proof:

Level 3

Proofs Involving Transversals
1. Solve and write two-column proof for unknown angles given 1 numerical angle measure
2. Build, solve equations, and write two-column proof for unknown angles given expressions for 2 angles.

Level 4 Assessment

Building and Solving Equations Through Transversals
Building and Solving Equations Through Multiple Transversals

Parameters Unit 3:

Unit 3: Congruence

Level 1

Definitions
1. Point
  1. Name and Notation
2. Line Segment
  1. Name and Notation
3. Angle
  1. Name and Notation
4. Triangle
  1. Name and Notation
5. Rigid Motion
  1. Translation
    1. Graphically
    2. Coordinate Rules
  2. Reflection
    1. Graphically
    2. Coordinate Rules
  3. Rotation
    1. Graphically
    2. Coordinate Rules
6. Corresponding parts of Congruent shapes
  1. Naming notation
  2. Orientation
7. Definition of Congruence
  1. Congruent through Transformations
    1. Perform the Transformation. Shapes are congruent
  2. Congruent through Corresponding Parts
    1. If corresponding parts in two of the same shape are congruent then the shapes are congruent
8. Composite Transformations
  1. Only with Rigid Motions
9. Rotations when point of Rotation is not (0,0)
10. Rotations when degree of rotation is not equal to 90, 180, or 270

Level 2

Triangle Congruence Criteria

1. Side Side Side Triangle Congruence Theorem
2. Side Angle Side Triangle Congruence Theorem
3. Angle Side Angle Triangle Congruence Theorem
4. Angle Angle Side Triangle Congruence Theorem
5. Not Congruence Criteria
  1. AAA
  2. SSA

Level 3

1. Triangle Congruence Proofs
  1. 2 Separate triangles
  2. 2 Attached Triangles
    1. Key rule: Reflexive property
  3. 2 triangles with different orientation
  4. Overlapping triangles
    1. Key rule: Segment addition postulate
    2. Key rule: Angle addition postulate
2. Congruent Parts of Congruent Triangles

Level 4 Assessment

1. Perpendicular Bisector Theorem
2. Parallelogram Theorems (CPCTC)

Unit 4: Similarity

Level 1

Dilation
1. Point of Dilation
2. Scale Factor
3. Dilate a point
4. Dilate a shape
Properties of dilation
1. Proportional Side Lengths
2. Congruent Angles
3. Proportional distances from point of dilation
4. Parallel sides
Transformations
1. Rigid Motions
2. Dilation

Level 2

Similar Shapes
1. Definition
  1. Same Shape Different size
    1. Congruent Angles
    2. Proportional Sides
Corresponding Parts in Similar Triangles
1. Large Side to Large Side
2. Large Angle to Large Angle
3. Medium Side to Medium Side
4. Medium Angle to Medium Angle
5. Small Side to Small Side
6. Small Angle to Small Angle
Proportions
1. Creating proportions and solving for unknown sides
2. Building and solving equations based on similar triangles
  1. Solving for a variable in an expression that represents a missing side length
Triangle Similarity Postulates
1. Side Side Side Similarity Theorem
  1. Three proportional sides of two triangles
2. Angle Angle Similarity Theorem
  1. Two congruent angles of two triangles
    1. Third angle is assumed based on Triangle Sum Theorem
3. Side Angle Side Similarity Theorem
  1. Two proportional sides and the angle BETWEEN them of two triangles
Triangle Proportionality (Parallel Lines) Theorem
Triangle Midsegment Theorem
Angle Bisector Proportionality Theorem

Level 3

Proving Triangles are Similar
1. Formal two column proof
2. 3 Similarity Criteria and additional theorems

Level 4 Assessment

Formal 2 column proofs with triangles

Unit 5: Trigonometry

Level 1

6 Parts of a Triangle
1. 3 sides and 3 angles
2. Unit circle
  1. Radius of 1
  2. As one angle gets larger, one gets smaller
Tangent ratio
1. Leg to Leg
2. Pythagorean Theorem

Unit 6: Circles

Unit 7: 3D Shapes

Unit 8: Construction

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