MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide

GEOMETRY HONORS Course Code: 1206320

Office of Academics and Transformation Page 1 of 10 Topic V_Second Nine Weeks

Topic V: Quadrilaterals Properties

COMMON CORE STATE STANDARD(S) &

MATHEMATICAL PRACTICE (MP)

NEXT GENERATION SUNSHINE STATE STANDARD(S)

ESSENTIAL CONTENT OBJECTIVES

MACC.912.G-CO.3.11: Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. (MP.2, MP.3, MP.5) MACC.912.G-GPE.2.4: Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). (MP.2, MP.3, MP.7) MACC.912.G-GPE.2.5: Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). (MP.3, MP.8)

Standard 3: Quadrilaterals MA.912.G.3.1: Describe, classify, and compare relationships among quadrilaterals including the square, rectangle, rhombus, parallelogram, trapezoid, and kite.(WI) MA.912.G.3.2: Compare and contrast special quadrilaterals on the basis of their properties. (WI)

MA.912.G.3.3: Coordinate geometry is used while students prove quadrilaterals to be congruent, similar, or regular. Coordinate geometry is used to prove properties of quadrilaterals. (EOC).

MA.912.G.3.4: Prove theorems involving quadrilaterals. (WI, EOC)

Standard 8: Mathematical Reasoning and Problem Solving MA.912.G.8.2: Use a variety of problem‐solving strategies, such as drawing a diagram, guess‐and‐check, solving a simpler problem, writing an equation, and working backwards.

MA.912.G.8.4: Make conjectures with justifications about geometric ideas. Distinguish between information that supports a conjecture and the proof of a conjecture. (FI, WI, EOC)

MA.912.G.8.5: Write geometric proofs, including proofs by contradiction and proofs involving coordinate geometry. Use and compare a variety of ways to present deductive proofs, such as flow charts, paragraphs, two‐column, and indirect proofs. (WI)

MA.912.G.8.6: Perform basic constructions using straightedge and compass, and/or drawing programs describing and justifying the procedures used. Distinguish between sketching, constructing, and drawing geometric figures.

A. Special Quadrilaterals Definitions 1. Trapezoid and Kite 2. Parallelogram, Rhombus,

Rectangle, and Square. B. Kites Properties

1. Angles 2. Diagonals 3. Diagonal Bisector 4. Angle Bisector

C. Trapezoids Properties 1. Consecutive Angles 2. Isosceles Trapezoid

a. Base Angles b. Diagonals

D. Mid-segment Properties 1. Triangle Mid-segments 2. Trapezoid Mid-segment

E. Properties of Parallelograms 1. Angles, Sides, and

Diagonals F. Special Parallelograms

Properties 1. Rectangle 2. Rhombus 3. Square

G. Proving Quadrilaterals Properties

H. Applications in the Real-World

NGSSS State the properties of kites, trapezoids,

parallelograms, rectangles, and rhombi

Calculate the length of a mid-segments in triangles and trapezoids

Write flowchart and paragraph proofs explaining the relationship between the sides and angles in a polygon

Describe and compare the relationship among parallelograms, rectangles, rhombi, squares, kites, and trapezoids

Use coordinate Geometry to prove properties of quadrilaterals.

Practice construction skills

Develop reasoning, problem-solving skills, and cooperative behavior tools for construction

Key: FI - Fall Interim WI - Winter Interim EOC – Geometry EOC

Pacing Date(s) Traditional 22 days 10/24/13 - 11/27/13

Block 11 days 10/24/13 - 11/27/13

MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide

GEOMETRY HONORS Course Code: 1206320

Office of Academics and Transformation Page 2 of 10 Topic V_Second Nine Weeks

INSTRUCTIONAL TOOLS

Core Text Book: Discovering Geometry (4th

Edition)

Benchmark Suggested Lessons Teacher Notes

MA.912.G.3.1

MA.912.G.3.2

MA.912.G.3.3

MA.912.G.3.4

MA.912.G.8.2

MA.912.G.8.4

MA.912.G.8.5

MA.912.G.8.6

1.6

5.3 – 5.7

UYAS page 135

UYAS page 167

Notes: Use Geometry EOC Item Specifications as a resource document to help define grade-level content.

Vocabulary: Kite, Vertex Angles, Non-vertex Angles, Trapezoid, Bases, Base Angles, Isosceles Trapezoid, Rhombus, Rectangle, Square, Midsegment.

Instructional Strategies: Example/Non-Example, Definition Section of Notebook, Similarities/Differences, and Graphic Organizer

Some students may believe that a construction is the same as a sketch or drawing. Emphasize the need for precision and accuracy when doing constructions. Stress the idea that a compass and straightedge are identical to a protractor and ruler. Explain the difference between measurement and construction.

Discuss the role of algebra in providing a precise means of representing a visual image.

Use slopes and the Euclidean distance formula to solve problems about figures in the coordinate plane such as:

• Given three points, are they vertices of an isosceles, equilateral, or right triangle?

• Given four points, are they vertices of a parallelogram, a rectangle, a rhombus, or a square?

MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide

GEOMETRY HONORS Course Code: 1206320

Office of Academics and Transformation Page 3 of 10 Topic V_Second Nine Weeks

COMMON CORE STATE STANDARDS

MATHEMATICAL PRACTICES

DESCRIPTION

MACC.K12.MP.1 (back to top)

Make sense of problems and persevere in solving them.

Mathematically proficient students will be able to:

Explain the meaning of a problem and looking for entry points to its solution.

Analyze givens, constraints, relationships, and goals.

Make conjectures about the form and meaning of the solution and plan a solution pathway.

Consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution.

Monitor and evaluate their progress and change course if necessary.

Explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends.

Check answers to problems using a different method, and continually ask, “Does this make sense?”

Identify correspondences between different approaches.

MACC.K12.MP.2 (back to top)

Reason abstractly and quantitatively.

Mathematically proficient students will be able to:

Make sense of quantities and their relationships in problem situations.

Decontextualize—to abstract a given situation and represent it symbolically.

Contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols

Create a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them.

Know and be flexible using different properties of operations and objects.

MACC.K12.MP.3 (back to top)

Construct viable arguments and critique the reasoning of

others.

Mathematically proficient students will be able to:

Understand and use stated assumptions, definitions, and previously established results in constructing arguments.

Make conjectures and build a logical progression of statements to explore the truth of their conjectures.

Analyze situations by breaking them into cases, and can recognize and use counterexamples.

Justify their conclusions, communicate them to others, and respond to the arguments of others.

Reason inductively about data, making plausible arguments that take into account the context from which the data arose.

Compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is.

Determine domains to which an argument applies.

MACC.K12.MP.4 (back to top)

Model with mathematics.

Mathematically proficient students will be able to:

Apply the mathematics they know to solve problems arising in everyday life, society, and the workplace.

Use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another.

Apply what they know and feel comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later.

Identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas.

Analyze relationships mathematically to draw conclusions.

Interpret mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide

GEOMETRY HONORS Course Code: 1206320

Office of Academics and Transformation Page 4 of 10 Topic V_Second Nine Weeks

COMMON CORE STATE STANDARDS

MATHEMATICAL PRACTICES

DESCRIPTION

MACC.K12.MP.5 (back to top)

Use appropriate tools strategically.

Mathematically proficient students will be able to:

Consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software.

Make sound decisions about when each of the tools appropriate for their grade or course might be helpful, recognizing both the insight to be gained and their limitations. Example: High school students analyze graphs of functions and solutions using a graphing calculator.

Detect possible errors by strategically using estimation and other mathematical knowledge.

Know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data.

Identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems.

Use technological tools to explore and deepen their understanding of concepts

MACC.K12.MP.6

(back to top)

Attend to precision.

Mathematically proficient students will be able to:

Communicate precisely to others.

Use clear definitions in discussion with others and in their own reasoning.

State the meaning of the symbols they choose, including using the equal sign consistently and appropriately.

Be careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem.

Calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context.

MACC.K12.MP.7

(back to top)

Look for and make use of structure.

Mathematically proficient students will be able to:

Discern a pattern or structure. Example: In the expression x2 + 9x + 14, students can see the 14 as 2 × 7 and the 9 as 2 + 7.

Recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. Step back for an overview and shift perspective.

See complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. Example: They can see 5 – 3(x – y)2 as

5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

MACC.K12.MP.8 (back to top)

Look for and express regularity in repeated

reasoning.

Mathematically proficient students will be able to:

Notice if calculations are repeated, and look both for general methods and for shortcuts. Example: Noticing the regularity in the way terms cancel when expanding (x-1)(x+1),(x-1)(x2+x+1),and(x-1)(x3 +x2+x+1)might lead them to the general formula for the sum of a geometric series.

Maintain oversight of the process, while attending to the details as they work to solve a problem.

Continually evaluate the reasonableness of their intermediate results.

MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide

GEOMETRY HONORS Course Code: 1206320

Office of Academics and Transformation Page 5 of 10 Topic V_Second Nine Weeks

NEXT GENERATION SUNSHINE STATE STANDARDS GEOMETRY BODY OF KNOWLEDGE

Standard 3: Quadrilaterals

Classify and understand relationships among quadrilaterals (rectangle, parallelogram, kite, etc.). Relate geometry to algebra by using coordinate geometry to determine regularity, congruence, and similarity. Use properties of congruent and similar quadrilaterals to solve problems involving lengths and areas, and prove theorems involving quadrilaterals..

BENCHMARK CODE BENCHMARK

MA.912.G.3.1 (back to top)

Describe, classify, and compare relationships among quadrilaterals including the square, rectangle, rhombus, parallelogram, trapezoid, and kite.

Remarks/Examples: This benchmark examines properties of quadrilaterals one at a time.

Example: Explore a trapezoid through manipulatives, drawings, and/or technology. Draw the diagonals and determine whether they are perpendicular. Give a convincing argument that your judgment is correct.

Cognitive Complexity/Depth of Knowledge Rating:: Moderate

Clarification (EOC): Assessed with MA.912.G.3.4

Content Limits (EOC): Assessed with MA.912.G.3.4

MA.912.G.3.2 (back to top)

Compare and contrast special quadrilaterals on the basis of their properties.

Remarks/Examples: This benchmark examines similarities and differences between different types of quadrilaterals. Example: Explain the similarities and differences between a rectangle, rhombus, and kite. Create a Venn diagram to match your explanation.

Cognitive Complexity/Depth of Knowledge Rating: Moderate

Clarification (EOC): Assessed with MA.912.G.3.4

Content Limits (EOC): Assessed with MA.912.G.3.4

MA.912.G.3.3 (back to top)

MC

Use coordinate geometry to prove properties of congruent, regular, and similar quadrilaterals. Remarks/Examples: Coordinate geometry is used while students prove quadrilaterals to be congruent, similar, or regular. Coordinate geometry is used to prove properties of quadrilaterals. Example 1: Is rectangle ABCD with vertices at A(0, 0), B(4, 0), C(4, 2), D(0, 2) congruent to rectangle PQRS with vertices at P(‐2, ‐1), Q(2, ‐1), R(2,

1), S(‐2, 1)? Justify your answer.

Cognitive Complexity/Depth of Knowledge Rating: High

Clarification (EOC): Students will use coordinate geometry and geometric properties to justify measures and characteristics of congruent, regular, and similar quadrilaterals. Content Limits (EOC)

Items may include statements and/or justifications to complete formal and informal proofs.

Items may include the use of coordinate planes.

MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide

GEOMETRY HONORS Course Code: 1206320

Office of Academics and Transformation Page 6 of 10 Topic V_Second Nine Weeks

Standard 3: Quadrilaterals

Classify and understand relationships among quadrilaterals (rectangle, parallelogram, kite, etc.). Relate geometry to algebra by using coordinate geometry to determine regularity, congruence, and similarity. Use properties of congruent and similar quadrilaterals to solve problems involving lengths and areas, and prove theorems involving quadrilaterals..

BENCHMARK CODE BENCHMARK

MA.912.G.3.4 (back to top)

MC/ FR

Prove theorems involving quadrilaterals Remarks/Examples: . Example: Prove that the diagonals of a rectangle are congruent. Cognitive Complexity/Depth of Knowledge Rating: High

Clarification (EOC): Students will use geometric properties to justify measures and characteristics of quadrilaterals. (Also assesses: MA.912.D.6.4, MA.912.G.3.1, MA.912.G.3.2, and MA.912.G.8.5.) Content Limits (EOC)

Items may require statements and/or justifications to complete formal and informal proofs.

Standard 8: Mathematical Reasoning and Problem Solving

In a general sense, mathematics is problem solving. In all mathematics, students use problem-solving skills: they choose how to approach a problem, they explain their reasoning, and they check their results. At this level, students apply these skills to making conjectures, using axioms and theorems, constructing logical arguments, and writing geometric proofs. They also learn about inductive and deductive reasoning and how to use counterexamples to show that a general statement is false.

BENCHMARK CODE BENCHMARK

MA.912.G.8.2 (back to top)

Use a variety of problem‐solving strategies, such as drawing a diagram, making a chart, guess‐and‐check, solving a simpler problem, writing an

equation, and working backwards. Remarks/Examples:

Example: How far does the tip of the minute hand of a clock move in 20 minutes if the tip is 4 inches from the center of the clock? Cognitive Complexity/Depth of Knowledge Rating: Moderate Clarification (EOC): Embedded throughout. Content Limits (EOC): Embedded throughout

MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide

GEOMETRY HONORS Course Code: 1206320

Office of Academics and Transformation Page 7 of 10 Topic V_Second Nine Weeks

Standard 8: Mathematical Reasoning and Problem Solving

In a general sense, mathematics is problem solving. In all mathematics, students use problem-solving skills: they choose how to approach a problem, they explain their reasoning, and they check their results. At this level, students apply these skills to making conjectures, using axioms and theorems, constructing logical arguments, and writing geometric proofs. They also learn about inductive and deductive reasoning and how to use counterexamples to show that a general statement is false.

MA.912.G.8.4 (back to top)

MC

Make conjectures with justifications about geometric ideas. Distinguish between information that supports a conjecture and the proof of a conjecture. Remarks/Examples:

Example: Calculate the ratios of side lengths in several different-sized triangles with angles of 90°, 50°, and 40°. What do you notice about the ratios? How might you prove that your observation is true (or show that it is false)? Cognitive Complexity/Depth of Knowledge Rating: High Clarification (EOC): Students will provide statements and/or reasons in a formal or informal proof or distinguish between mere examples of a

geometric idea and proof of that idea. Content Limits (EOC)

Items must adhere to the content limits stated in other benchmarks.

Items may include proofs about congruent/similar triangles and parallel lines.

MA.912.G.8.5 (back to top)

Write geometric proofs, including proofs by contradiction and proofs involving coordinate geometry. Use and compare a variety of ways to present deductive proofs, such as flow charts, paragraphs, two‐column, and indirect proofs. Remarks/Examples: Example 1: Prove that the sum of the measures of the interior angles of a triangle is 180°.

Example 2: Prove that the perpendicular bisector of line segment AB is the set of all points equidistant from the endpoints A and B.

Example 3: Prove that two lines are parallel if and only if the alternate interior angles the lines make with a transversal are equal. Cognitive Complexity/Depth of Knowledge Rating: High Clarification (EOC): Assessed with MA.912.G.3.4 and MA.912.G.4.6.

Content Limits (EOC): Assessed with MA.912.G.3.4 and MA.912.G.4.6

MA.912.G.8.6 (back to top)

Not Assessed

Perform basic constructions using straightedge and compass, and/or drawing programs describing and justifying the procedures used. Distinguish between sketching, constructing, and drawing geometric figures. Remarks/Examples: Example: Construct a line parallel to a given line through a given point not on the line, explaining and justifying each step. Cognitive Complexity/Depth of Knowledge Rating: High

MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide

GEOMETRY HONORS Course Code: 1206320

Office of Academics and Transformation Page 8 of 10 Topic V_Second Nine Weeks

TECHNOLOGY TOOLS

FLORIDA FOCUS http://focus.florida-achieves.com

Sign-in and Password: First 4 letters of Last Name and Last 4 Digits Employee Number. Example: abcd1234

MA.912.G.3.3

MA.912.G.3.4

GIZMO CORRELATION

GIZMO TITLE

Classifying Quadrilaterals - Activity B

Parallelogram Conditions

Special Parallelograms

GEOMETER’S SKETCHPAD ACTIVITIES

SKETCHPAD TITLE TITLE

Properties of Kites

Properties of the Midsegment of a Trapezoid

Properties of Parallelograms

Properties of Special Parallelograms

VIDEO TITLE

Section A: General Definitions

Quadrilaterals

Coordinate Geometry Problems

Coordinate Geometry and Quantitative Comparisons Review

TOPIC V

VIDEO TITLE

IMAGE

TOPIC V

DISCOVERY EDUCATION CORRELATION

VIDEO TITLE

Quadrilaterals

MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide

GEOMETRY HONORS Course Code: 1206320

Office of Academics and Transformation Page 9 of 10 Topic V_Second Nine Weeks

TOPIC V

DISCOVERY EDUCATION CORRELATION

MODEL LESSON

Congruence and Proof: Session 4: Prove Theorems about Parallelograms Coordinate Geometry and How It's Used: Session 2: Parallel and Perpendicular Lines Coordinate Geometry and How It's Used: Session 3: Classifying Polygons Coordinate Geometry and How It's Used: Session 4: Equidistant to a Point MATH OVERVIEW

Geometry: Special Parallelograms

Geometry: Properties of Parallelograms

Geometry: Quadrilaterals

Geometry: Trapezoids and Kites

MATH EXPLANATION TITLE

Geometry: Properties of Parallelograms: Opposite Angles in Parallelograms

Geometry: Properties of Parallelograms: Parallelogram Proofs

Geometry: Properties of Parallelograms: Making Conclusions About Parallelograms

Geometry: Properties of Parallelograms: Variables in Parallelograms

Geometry: Properties of Parallelograms: Determining a Parallelogram

Geometry: Proving that a Quadrilateral is a Parallelogram: Proofs for Parallelograms

Geometry: Proving that a Quadrilateral is a Parallelogram: Parallelograms and Flow Proofs

Geometry: Special Parallelograms: Finding Rhombus Angles

MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide

GEOMETRY HONORS Course Code: 1206320

Office of Academics and Transformation Page 10 of 10 Topic V_Second Nine Weeks

Date Pacing Guide

Standards Data Driven Standard(s)

Activities Assessment(s) Strategies

Traditional 22 days

Block

11 days

10/24/13 - 11/27/13

(back to top)

Standard 3: Quadrilaterals

MA.912.G.3.1

MA.912.G.3.2

MA.912.G.3.3

MA.912.G.3.4

Standard 8: Mathematical Reasoning

and Problem Solving

MA.912.G.8.2

MA.912.G.8.4

MA.912.G.8.5

MA.912.G.8.6