MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing...
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MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide
Division of Academics - Department of Mathematics Page 1 of 17 Topic II First Nine Weeks
GEOMETRY Course Code: 120631001
Topic II: Lines, Angles, and Triangles
Pacing Date(s) Traditional 28 09/15/17 – 10/26/17
Block 14 09/15/17 – 10/26/17
Topic II Assessment Window Administered after Topic III
MATHEMATICS FLORIDA STATE STANDARDS (MAFS) & MATHEMATICAL PRACTICES (MP)
ESSENTIAL CONTENT OBJECTIVES (from Item Specifications)
MAFS.912.G-CO.3.9: Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: 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. (MP.2, MP.3, and MP.5) Achievement Level Description
MAFS.912.G-CO.4.12: Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). 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. (MP.5, and MP.6) Achievement Level Description Also assesses:
MAFS.912.G-CO.4.13: Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. (MP.5, and MP.6) Achievement Level Description
MAFS.912.G-GPE.2.5: Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems. (MP.2, MP.3, MP.6, MP.7) Achievement Level Description
A. Lines and Angles 1. Angles Formed by Intersecting
Lines 2. Transversals and Parallel Lines 3. Proving Lines Are Parallel
a) Constructing Parallel Lines
4. Perpendicular Lines a) Constructing Perpendicular
Lines b) Constructing Perpendicular
Bisectors
5. Equations of Parallel and Perpendicular Lines
I can:
Prove theorems about lines*
Prove theorems about angles*
Use theorems about lines to solve problems
Use theorems about angles to solve problems
Identify the result of a formal geometric construction
Determine the steps of a formal geometric construction
Prove the slope criteria for parallel lines*
Prove the slope criteria for perpendicular lines*
Find equations of lines using the slope criteria for parallel and perpendicular lines
*Proofs include: narrative proofs, flow-chart
proofs, two-column proofs, or informal proofs.
MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide
Division of Academics - Department of Mathematics Page 2 of 17 Topic II First Nine Weeks
GEOMETRY Course Code: 120631001
MATHEMATICS FLORIDA STATE STANDARDS (MAFS) & MATHEMATICAL PRACTICES (MP)
ESSENTIAL CONTENT OBJECTIVES (from Item Specifications)
MAFS.912.G-CO.2.7: Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. (Assessed with G-CO.2.6) (MP.3) Achievement Level Description
MAFS.912.G-CO.2.8: Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. (Assessed with G-CO.2.6) (MP.2, and MP.3) Achievement Level Description
MAFS.912.G-CO.3.10: Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: 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. (MP.2, MP.3, and MP.5) Achievement Level Description
MAFS.912.G-SRT.2.5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. (MP.3, MP.4, MP.6, MP.7)
Achievement Level Description
B. Triangle Congruence Criteria 1. Exploring What Makes Triangles
Congruent 2. ASA Triangle Congruence 3. SAS Triangle Congruence 4. SSS Triangle Congruence
C. Applications of Triangle Congruence
1. Justifying Constructions 2. AAS Triangle Congruence 3. HL Triangle Congruence
D. Properties of Triangles 1. Interior and Exterior Angles 2. Isosceles and Equilateral
Triangles 3. Triangle Inequalities
I can:
Explain triangle congruence using the definition of congruence in terms of rigid motions
Apply congruence to solve problems
Use congruence to justify steps within the context of a proof
Prove theorems about triangles*
Use theorems about triangles to solve problems
Use congruence criteria for triangles to solve problems
Use congruence criteria for triangles to prove relationships in geometric figures
Use similarity criteria for triangles to solve problems
Use similarity criteria for triangles to prove relationships in geometric figures
*Proofs include: narrative proofs, flow-chart
proofs, two-column proofs, or informal proofs.
MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide
Division of Academics - Department of Mathematics Page 3 of 17 Topic II First Nine Weeks
GEOMETRY Course Code: 120631001
INSTRUCTIONAL TOOLS
RECOMMENDED INSTRUCTIONAL DESIGN AND PLANNING CONTINUUM
Before During After
Prior to the lesson:
Outline content standard(s).
Determine learning targets.
Anticipate student understanding and misconceptions.
Determine prerequisite skills.
Plan for learning experiences that target Rigor o Conceptual Understanding o Procedural Fluency o Application
Determine the task students will demonstrate to reach the desired learning targets.
Plan instructional delivery methods that will maximize initial engagement and sustain it throughout the lesson.
Decide how students will reflect upon, self-assess, and set goals for their future learning.
During the lesson:
Activate (or supply) prior knowledge and/or spiral back o Warm ups, Bell Ringers, Openers, etc.
Tailor lesson experiences to the different needs and ability of the learners.
Clarify vocabulary and mathematical notation.
Incorporate a variety of higher order questions to encourage and increase critical thinking skills.
Continuously check for student understanding and provide feedback.
Provide opportunities for students to develop self-assessment and to reflect about their understanding and work.
Bring closure to the lesson so that the students can articulate what they have learned.
After the lesson:
Analyze evidence of student learning to develop intervention, enrichment, and future instruction.
Discuss results of assessments with students.
Engage students in reflective processes and goal setting.
Engage in self-reflection to adapt/modify teaching strategies to improve instruction.
Geometry Core – H.M.H. Resources Geometry Intensive Math – H.M.H. Resources
Unit Resources Unit Resources
Unit Tests – A, B, and C Unit Test Modified
Performance Assessment
Module Resources Module Resources
Module Test B Module Test Modified
Common Core Assessment Readiness
RTI T2 – Strategic Intervention
Advanced Learners – Challenge Worksheets Skills Pre-Test, Skills Post Test, Skills Worksheets
RTI T3 – Intensive Intervention Worksheets
Lesson Resources Lesson Resources
Lessons – Work text/Interactive Student Edition Practice and Problem Solving: D (modified)
Practice and Problem Solving: A/B
RTI T1 – Lesson Intervention Worksheets
Advanced Learners - Practice and Problem Solving: C Reteaching
Reading Strategies
Success for English Learners
PMT Preferences: Auto-assign for intervention and enrichment: NO Auto-assign for intervention and enrichment: NO
PMT Preferences: Auto-assign for intervention and enrichment: YES Auto-assign for intervention and enrichment: NO
Auto-assign for intervention and enrichment: YES Test and Quizzes Daily Intervention
Homework Standard-Based Intervention
Course Intervention
Pacing Date(s) Traditional 28 09/15/17 – 10/26/17
Block 14 09/15/17 – 10/26/17
Topic II Assessment Window Administered after Topic III
Core Text Book: Houghton Mifflin Harcourt - Geometry
Geometry Course Description
Geometry EOC Item Specifications
INSTRUCTIONAL TOOLS
MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide
Division of Academics - Department of Mathematics Page 4 of 17 Topic II First Nine Weeks
GEOMETRY Course Code: 120631001
STANDARDS MODULES TEACHER NOTES
MAFS.912.G-CO.2.7
MAFS.912.G-CO.2.8
MAFS.912.G-CO.3.9
MAFS.912.G-CO.3.10
MAFS.912.G-CO.4.12
MAFS.912.G-CO.4.13
MAFS.912.G-GPE.2.5
MAFS.912.G-SRT.2.5
Module 4
Module 5
Module 6
Module 7
Geometry Core Block Schedule – Suggested Pace
Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7 Day 8 Day 9 Day 10 Day 11 Day 12 4.1- 4.2 4.3 4.4 4.5 5.1 5.2 5.3 5.4 6.1 6.2 6.3 7.1
Day 13 Day 14 7.2 7.3
Geometry Intensive Math - RTI T3 – Intensive Intervention Worksheets
Module 4 - Building Block Skills: 7, 15, 16, 22, 23, 42, 46, 53, 56, 66, 71, 87, 95, 98, 102, 103 Module 5 - Building Block Skills: 8, 16, 46, 48, 53, 56, 71, 74, 98, 102, 103 Module 6 - Building Block Skills: 8, 16, 48, 71, 74, 98, 103, 104 Module 7 - Building Block Skills: 7, 8, 10, 11, 15, 16, 27, 38, 45, 48, 53, 56, 66, 69, 70
74, 95, 98, 100, 102, 104 Topic II Assessment: Administered after Topic III
Reporting Category: Congruency, Similarity, Right Triangles, and Trigonometry
% of Test Average % Correct 2015
2016
Average % Correct 2016 Average % Correct 2017
46% 46% 36% 33%
Reporting Category: Circles, Geometric Measurement, and Geometric Properties with Equations
% of Test Average % Correct 2015 Average % Correct 2016 Average % Correct 2017
38% 39% 28% 27%
MODULE LESSON STANDARDS SUGGESTED PROBLEMS
BY TEACHERS FOR TEACHERS* NOTES / RESOURCES
Module 4
4.1 MAFS.912.G-CO.3.9 Homework and Practice
1, 5, 7, 13 -15,17,19
The standard may also assess relationships between congruent supplements and congruent complements.
Using the facts about supplementary, complementary, vertical, and adjacent angles is a pre-requisite skill, MAFS.7.G.2.5.
Using informal arguments to establish facts about angles created when parallel lines are cut by a transversal is a pre-requisite skill, MAFS.8.G.1.5.
When completing proofs refer to Achievement Level Descriptions to scaffold the instruction.
students may struggle with finding the most efficient process
students may have gaps in their reasoning and miss key steps
4.2 MAFS.912.G-CO.3.9
Homework and Practice
1, 9-14, 16, 18, 19, 22
Lesson Performance Task
(Part a) only
INSTRUCTIONAL TOOLS
Topic Resources PowerPoint Available in Learning Village
MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide
Division of Academics - Department of Mathematics Page 5 of 17 Topic II First Nine Weeks
GEOMETRY Course Code: 120631001
MODULE LESSON STANDARDS SUGGESTED PROBLEMS
BY TEACHERS FOR TEACHERS* NOTES / RESOURCES
Module 4
4.3 MAFS.912.G-CO.3.9 MAFS.912.G-CO.4.12
Homework and Practice
7-11, 13, 15
Explain 2 partially covers MAFS.912.G-CO.4.12, this standard is also covered in
Topic III, IV, V, and VII.
Using informal arguments to establish facts about angles created when parallel lines are cut by a transversal is a pre-requisite skill, MAFS.8.G.1.5.
When completing proofs refer to Achievement Level Descriptions to scaffold the instruction.
students may struggle with finding the most efficient process
students may have gaps in their reasoning and miss key steps
4.4 MAFS.912.G-CO3.9 MAFS.912.G-CO.4.12
Homework and Practice
1, 4, 12, 14, 15, 18
Explore and Elaborate partially covers MAFS.912.G-CO.4.12, this standard is
also covered in Topic III, IV, V, and VII.
Understating and applying Pythagorean Theorem is pre-requisite skill MAFS.8.G.2.
When completing proofs refer to Achievement Level Descriptions to scaffold the instruction.
students may struggle with finding the most efficient process
students may have gaps in their reasoning and miss key steps
4.5 MAFS.912.G-GPE.2.5 Homework and Practice
6,10,11,16,19
Problems may NOT ask the students to provide only the slope of a parallel or perpendicular line.
Being familiar with slope-intercept form, standard form, and point-slope from of a line is a pre-requisite skill MAFS.912.A-CED.1.2.
Module 5
5.1 MAFS.912.G-CO.2.7 Homework and Practice
1-10, 15
Problems selected may NOT require students to use the distance formula.
Throughout the lesson ask students to identify the rigid motion that is applicable to determine if two figures are congruent.
students may confuse congruence with equality, stress that congruence refers to figures, while equality refers to values and distances
5.2
MAFS.912.G-CO.2.8
MAFS.912.G-CO.2.7
MAFS.912.G-CO.3.10
MAFS.912.G-SRT.2.5
Homework and Practice
7-11,12-15, 20
Facts about angle sum of a triangle is a pre-requisite skill MAFS.8.G.1.5.
Problem 12-13 partially covers MAFS.912.G-CO.4.12, this standard is also
covered in Topic III, IV, V, and VII.
MAFS.912.G-SRT.2.5 is limited to congruent triangles.
When completing proofs refer to Achievement Level Descriptions to scaffold the instruction.
students may struggle with finding the most efficient process
students may have gaps in their reasoning and miss key steps
INSTRUCTIONAL TOOLS
MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide
Division of Academics - Department of Mathematics Page 6 of 17 Topic II First Nine Weeks
GEOMETRY Course Code: 120631001
MODULE LESSON STANDARDS SUGGESTED PROBLEMS
BY TEACHERS FOR TEACHERS* NOTES / RESOURCES
Module 5
5.3
MAFS.912.G-CO.2.8
MAFS.912.G-CO.2.7
MAFS.912.G-CO.3.10
MAFS.912.G-SRT.2.5
Homework and Practice
6, 7, 9, 11-13,
Explore 2 partially covers MAFS.912.G-CO.4.12, this standard is also covered in
Topic III, IV, V, and VII.
MAFS.912.G-SRT.2.5 is limited to congruent triangles.
When completing proofs refer to Achievement Level Descriptions to scaffold the instruction.
students may struggle with finding the most efficient process
students may have gaps in their reasoning and miss key steps
5.4
MAFS.912.G-CO.2.8
MAFS.912.G-CO.2.7
MAFS.912.G-CO.3.10
MAFS.912.G-SRT.2.5
Homework and Practice
1-2, 4, 9, 19, 22, 23, 27
Explore partially covers MAFS.912.G-CO.4.12, this standard is also covered in
Topic III, IV, V, and VII.
When completing proofs refer to Achievement Level Descriptions to scaffold the instruction.
students may struggle with finding the most efficient process
students may have gaps in their reasoning and miss key steps
Module 6
6.1 MAFS.9.12.G-CO.4.12
MAFS.912.G-CO.4.13
MAFS.912.G-SRT.2.5
Homework and Practice
2, 4-10, 12, 17, 18 MAFS.912.G-SRT.2.5 is limited to congruent triangles.
6.2 MAFS.912.G-SRT.2.5 Homework and Practice
2-4, 7, 12, 13, 16, 17, 21
Explain 3 reviews MAFS.912.G-GPE.2.7, this standard is also covered in Topic
IV.
6.3 MAFS.912.G-SRT.2.5
MAFS.912.G-CO.2.8
Homework and Practice
2, 6-10, 12, 14, 17 Problems selected may NOT require students to use the distance formula.
Module 7
7.1 MAFS.912.G-CO.3.10 Homework and Practice
2, 10, 13-16, 27
Explore 2 and Explain 1 is outside the scope of the standard.
Using informal arguments to establish about facts about the angle sum and exterior angle of triangles is a pre-requisite skill, MAFS.8.G.1.5.
7.2 MAFS.912.G-CO.3.10 Homework and Practice
1, 3-7, 9, 13-15, 18, 19
Focusing on constructing triangles from three measures of angles or sides,
noticing when the conditions determine a unique triangle is pre-requisite skill, MAFS.7.G.1.2.
Illustrative Mathematics Task(s): Congruent angles in isosceles triangles
7.3 MAFS.912.G-SRT.2.5
Your Turn
13-14
Homework and Practice
5, 6, 9, 11, 14, 18, 20
Explain 4 is the Hinge Theorem.
*Problems were suggested by M-DCPS teachers during May and June 2017 Geometry PDs.
INSTRUCTIONAL TOOLS
MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide
Division of Academics - Department of Mathematics Page 7 of 17 Topic II First Nine Weeks
GEOMETRY Course Code: 120631001
Vocabulary: vertical angles, transversal, parallel lines, corresponding angles, alternative interior angles, alternative exterior angles, indirect proof,
hypotenuse, interior angle, auxiliary line, exterior angle, isosceles triangle, equilateral triangle, equiangular triangle, congruent triangles, SSS triangle
congruence postulate, SAS triangle congruence postulate, ASA triangle congruence postulate, AAS triangle congruence postulate, HL right triangle
congruence
Standard – Name MAFS.912.G-CO.2.7 Congruence Implies Congruent Corresponding Parts
Students are asked to prove two triangles congruent given that all pairs of corresponding sides and angles are congruent
Proving Triangles Using Corresponding Parts Students are asked to prove two triangles congruent given that all pairs of corresponding sides and angles are congruent.
Showing Congruence Using Corresponding Parts – Part 1
Students are given two triangles in which all pairs of corresponding parts are congruent and are asked to use the definition of congruence in terms of rigid motion to show the triangles are congruent.
Showing Congruence Using Corresponding Parts – Part 2
Students are given two triangles in which all pairs of corresponding parts are congruent and are asked to use the definition of congruence in terms of rigid motion to show the triangles are congruent.
Showing Triangles Congruent Using Rigid Motion
Students are asked to use the definition of congruence in terms of rigid motion to show that two triangles are congruent in the coordinate plane.
MAFS.912.G-CO.2.8
Justifying ASA Congruence Students are asked to use rigid motion to explain why the ASA pattern of congruence ensures triangle congruence.
Justifying HL Congruence Students are asked to use rigid motion to explain why the HL pattern of congruence ensures right triangle congruence.
Justifying SAS Congruence Students are asked to use rigid motion to explain why the SAS pattern of congruence ensures triangle congruence.
Justifying SSS Congruence Students are asked to use rigid motion to explain why the SSS pattern of congruence ensures triangle congruence.
MAFS.912.G-CO.3.10
An Isosceles Trapezoid Problem Students are asked to explain why the sum of the lengths of the diagonals of an isosceles trapezoid is less than its perimeter.
Interior Angles of a Polygon Students are asked to explain why the sum of the measures of the interior angles of a convex n-gon is given by the formula (n – 2)180°.
Isosceles Triangle Proof Students are asked to prove that the base angles of an isosceles triangle are congruent.
Locating the Missing Midpoints Students are given a triangle in which the midpoints of two sides are shown and are asked to describe a method for locating the midpoint of the remaining side using only a straight edge and pencil.
Median Concurrence Proof Students are asked to prove that the medians of a triangle are concurrent.
Proving the Triangle Inequality Theorem Students are asked to prove the Triangle Inequality Theorem.
The Measure of an Angle of a Triangle Students are given the measure of one interior angle of an isosceles triangle and are asked to find the measure of another interior angle.
The Third Side of a Triangle Students are given the lengths of two sides of a triangle and asked to describe all possible lengths of the remaining side.
Mathematics Formative Assessment System (MFAS)
INSTRUCTIONAL TOOLS
MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide
Division of Academics - Department of Mathematics Page 8 of 17 Topic II First Nine Weeks
GEOMETRY Course Code: 120631001
INSTRUCTIONAL TOOLS
Standard – Name
MAFS.912.G-CO.3.10 (Cont.)
Triangle Midsegment Proof Students are asked to prove that the segment joining the midpoints of two sides of a triangle is parallel to the third side of the triangle and half of its length
Triangle Sum Proof Students are asked prove that the measures of the interior angles of a triangle sum to 180°.
Triangles and Midpoints Students are asked to explain why a quadrilateral formed by drawing the midsegments of a triangle is a parallelogram and to find the perimeter of the triangle formed by the midsegments.
MAFS.912.G-CO.4.12
Bisecting a Segment and Angle Worksheet The students are asked to construct the bisectors of a given segment and a given angle and to justify one of the steps in each construction.
Constructing a Congruent Angle Students are asked to construct an angle congruent to a given angle.
Constructing a Congruent Segment Students are asked to construct a line segment congruent to a given line segment.
Constructions for Parallel Lines Students are asked to construct a line parallel to a given line through a given point.
Constructions for Perpendicular Lines Students are asked to construct a line perpendicular to given line (1) through a point not on the line and (2) through a point on the line.
MAFS.912.G-CO.4.13
Construct the Center of a Circle Students are asked to construct the center of a circle.
Equilateral Triangle in a Circle Students are asked to construct an equilateral triangle inscribed in a circle.
Regular Hexagon in a Circle Students are asked to construct a regular hexagon inscribed in a circle.
Square in a Circle Students are asked to construct a square inscribed in a circle.
MAFS.912.G-GPE.2.5 Finding Equations of Parallel and Perpendicular Lines
Students are asked to assess the relationship between the slopes of parallel and perpendicular lines.
Proving Slope Criterion for Parallel Lines – 1 Students are asked to prove that two parallel lines have equal slopes
Proving Slope Criterion for Parallel Lines – 2 Students are asked to prove that two lines with equal slopes are parallel.
Proving Slopes Criterion for Perpendicular Lines – 1 Students are asked to prove that the slopes of two perpendicular lines are both opposite and reciprocal.
Proving Slopes Criterion for Perpendicular Lines – 2 Students are asked to prove that if the slopes of two lines are both opposite and reciprocal, then the lines are perpendicular.
Writing Equations for Parallel Lines Students are asked to identify the slope of a line parallel to a given line and write an equation for the line given a point.
Writing Equations for Perpendicular Lines Students are asked to identify the slope of a line perpendicular to a given line and write an equation for the line given a point.
Mathematics Formative Assessment System (MFAS)
INSTRUCTIONAL TOOLS
MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide
Division of Academics - Department of Mathematics Page 9 of 17 Topic II First Nine Weeks
GEOMETRY Course Code: 120631001
Standard – Name MAFS.912.G-SRT.2.5
Basketball Goal Students are asked to decide if a basketball goal is regulation height and are given enough information to determine this using similar triangles.
County Fair Students are given a diagram of a county fair and are asked to use similar triangles to determine distances from one location of the fair to another.
Prove Rhombus Diagonals Bisect Angles Students are asked to prove a specific diagonal of a rhombus bisects a pair of angles.
Similar Triangles – 1 Students are asked locate a pair of similar triangles in a diagram, explain why they are similar, and use the similarity to find two unknown lengths in the diagram.
Similar Triangles – 2 Students are asked locate a pair of similar triangles in a diagram, explain why they are similar, and use the similarity to find two unknown lengths in the diagram.
STEM Lessons - Model Eliciting Activity
STEM Lessons
N/A
CPALMS Perspectives Videos
Professional/Enthusiasts
N/A
Expert
N/A
INSTRUCTIONAL TOOLS
Mathematics Formative Assessment System (MFAS)
MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide
Division of Academics - Department of Mathematics Page 10 of 17 Topic II First Nine Weeks
GEOMETRY Course Code: 120631001
MATHEMATICS FLORIDA STANDARDS
MATHEMATICAL PRACTICES
DESCRIPTION
MAFS.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.
MAFS.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.
MAFS.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.
MAFS.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
Division of Academics - Department of Mathematics Page 11 of 17 Topic II First Nine Weeks
GEOMETRY Course Code: 120631001
MATHEMATICS FLORIDA STANDARDS
MATHEMATICAL PRACTICES
DESCRIPTION
MAFS.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
MAFS.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.
MAFS.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.
MAFS.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
Division of Academics - Department of Mathematics Page 12 of 17 Topic II First Nine Weeks
GEOMETRY Course Code: 120631001
Domain: GEOMETRY: CONGRUENCE
Cluster 2: Understanding Congruence in Terms of Rigid Motion
MAFS.912.G-CO.2.7 (Assessed with MAFS.912.G-CO.2.6)
Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. Context Complexity: Level 1: Recall
Level 2 Level 3 Level 4 Level 5
identifies corresponding parts of two congruent triangles
shows that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent using the definition of congruence to solve problems; applies congruence to solve problems; uses rigid motions to show ASA, SAS, SSS, or HL is true for two triangles
shows and explains, using the definition of congruence in terms of rigid motions, the congruence of two triangles; uses algebraic descriptions to describe rigid motion that will show ASA, SAS, SSS, or HL is true for two triangles
justifies steps of a proof given algebraic descriptions of triangles, using the definition of congruence in terms of rigid motions that the triangles are congruent using ASA, SAS, SSS, or HL
MAFS.912.G-CO.2.8 (Assessed with MAFS.912.G-CO.2.6)
Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse-Leg) follow from the definition of congruence in terms of rigid motion. Context Complexity: Level 2: Basic Application of Skills & Concepts
Level 2 Level 3 Level 4 Level 5
Same as MAFS.912.G-CO.2.7 Same as MAFS.912.G-CO.2.7 Same as MAFS.912.G-CO.2.7 Same as MAFS.912.G-CO.2.7
Cluster 2: Prove Geometric Theorems
MAFS.912.G-CO.3.9 Assessed
Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: 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. Context Complexity: Level 3: Strategic Thinking and Complex Reasoning
Level 2 Level 3 Level 4 Level 5
uses theorems about parallel lines with one transversal to solve problems; uses the vertical angles theorem to solve problems
completes no more than two steps of a proof using theorems about lines and angles; solves problems using parallel lines with two to three transversals; solves problems about angles using algebra
completes a proof for vertical angles are congruent, alternate interior angles are congruent, and corresponding angles are congruent
creates a proof, given statements and reasons, for points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints
MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide
Division of Academics - Department of Mathematics Page 13 of 17 Topic II First Nine Weeks
GEOMETRY Course Code: 120631001
Domain: GEOMETRY: CONGRUENCE
Cluster 3: Prove Geometric Theorems
MAFS.912.G-CO.3.10 Assessed
Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: 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.
Context Complexity: Level 3: Strategic Thinking and Complex Reasoning
Level 2 Level 3 Level 4 Level 5
uses theorems about interior angles of a triangle, exterior angle of a triangle
completes no more than two steps in a proof using theorems (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) about triangles; solves problems about triangles using algebra; solves problems using the triangle inequality and the Hinge theorem
completes a proof for theorems about triangles; solves problems by applying algebra using the triangle inequality and the Hinge theorem; solves problems for the midsegment of a triangle, concurrency of angle bisectors, and concurrency of perpendicular bisectors
completes proofs using the medians of a triangle meet at a point; solves problems by applying algebra for the midsegment of a triangle, concurrency of angle bisectors, and concurrency of perpendicular bisectors
Cluster 4: Make Geometric Constructions
MAFS.912.G-CO.4.12 Assessed
Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). 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.
Context Complexity: Level 2: Basic Application of Skills & Concepts
Level 2 Level 3 Level 4 Level 5
chooses a visual or written step in a construction
identifies, sequences, or reorders steps in a construction: 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
identifies sequences or reorders steps in a construction of an equilateral triangle, a square, and a regular hexagon inscribed in a circle
explains steps in a construction
MAFS.912.G-CO.4.13 (Assessed with MAFS.912.G-CO.4.12
Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Context Complexity: Level 2: Basic Application of Skills & Concepts
Level 2 Level 3 Level 4 Level 5
Same as MAFS.912.G-CO.4.12 Same as MAFS.912.G-CO.4.12 Same as MAFS.912.G-CO.4.12 Same as MAFS.912.G-CO.4.12
MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide
Division of Academics - Department of Mathematics Page 14 of 17 Topic II First Nine Weeks
GEOMETRY Course Code: 120631001
Domain: GEOMETRY: EXPRESSING GEOMETRIC PROPERTIES WITH EQUATIONS
Cluster 2: Use Coordinates to Prove Simple Geometric Theorems Algebraically
MAFS.912.G-GPE.2.5 Assessed
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). Context Complexity: Level 2: Basic Application of Skills and Concepts
Level 2 Level 3 Level 4 Level 5
identifies that the slopes of parallel lines are equal slope and y-intercept for the given line contains a variable
creates the equation of a line that is parallel given a point on the line and an equation, in slope-intercept form, of the
parallel line or given two points (coordinates are integral) on the line that is parallel; creates the equation of a line that is perpendicular given a point on the line and an equation of a line, in slope-intercept form
creates the equation of a line that is parallel given a point on the line and an equation, in a form other than slope-intercept; creates the equation of a line that is perpendicular that passes through a specific point when given two points
or an equation in a form other than slope-intercept
proves the slope criteria for parallel and perpendicular lines; writes equations of parallel or perpendicular lines when the coordinates are written using variables or the slope and y-intercept for the given line contains a variable
Domain: GEOMETRY: SIMILARITY, RIGHT TRIANGLES, & TRIGONOMETRY
Cluster 2: Prove Theorems Involving Similarity
MAFS.912.G-SRT.2.5 Assessed
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Context Complexity: Level 3: Strategic Thinking & Complex Reasoning
Level 2 Level 3 Level 4 Level 5
finds measures of sides and angles of congruent and similar triangles when given a diagram
criteria
solves problems involving triangles, using congruence and similarity criteria; provides justifications about relationships using congruence and similarity criteria
completes proofs about relationships in geometric figures by using congruence and similarity criteria for triangles
proves conjectures about congruence or similarity in geometric figures, using congruence and similarity criteria
MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide
Division of Academics - Department of Mathematics Page 15 of 17 Topic II First Nine Weeks
GEOMETRY Course Code: 120631001
CPALM RESOURCES
LESSON PLANS
Analyzing Congruence Proofs
Exploring Congruence Using Transformations
Match That!
Analyzing Congruence Proofs
Back to the Basics: Constructions
Congruent Trapezoids
Parallel Lines
WORKSHEET
VIRTUAL MANIPULATIVE
Congruent Triangles
Circumscribe a Circle About a Triangle
PROBLEM-SOLVING TASK
Reflections and Equilateral Triangles II
Are the Triangles Congruent?
When Does SSA Work to Determine Triangle Congruence?
Why Does ASA Work?
Why Does SAS Work?
Angle bisection and midpoints of line segments
TUTORIALS
MAFS.912.G-CO.3.9
Angles Formed by Parallel Lines and Transversals
Figuring Out Angles Between Transversal and Parallel Lines
Finding the measure of vertical angles
Introduction to vertical angles
Parallel lines and transversal lines
Parallel lines and transversals
Parallel lines, transversals and triangles
Proof: Vertical Angles Are Equal
Proving vertical angles are equal
Using Algebra to Find Measures of Angles Formed from Transversal
TECHNOLOGY TOOLS
MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide
Division of Academics - Department of Mathematics Page 16 of 17 Topic II First Nine Weeks
GEOMETRY Course Code: 120631001
CPALM RESOURCES
MAFS.912.G-GPE.2.5
Parallel Lines
Perpendicular Lines
MAFS.912.G-CO.2.8
Congruent Triangles and SSS
Finding congruent triangles
Triangle congruence postulates
Using SSS in a proof
MAFS.912.G-CO.4.13
Constructing Polygons Inscribed in Circles
MAFS.912.G-CO.3.10
Proof: Sum of Measures of Angles in a Triangle Are 180
Triangle Angle Example 1
GEOGEBRA
APPLET TITLE
Copying a Segment
Copying an Angle
Vertical Angles
Corresponding Angles Congruent
Angle Relationships Parallel Lines Cut by a Transversal
Prove Parallel Lines
Exploring Parallel and Perpendicular Lines
Parallel and Perpendicular Lines
Parallel Lines Exploration
Equilateral Triangles
Mid-Segments
GIZMOS CORRELATION
APPLET TITLE
Investigating Angle Theorems
Triangle Angle Sum
Triangle Inequalities
MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide
Division of Academics - Department of Mathematics Page 17 of 17 Topic II First Nine Weeks
GEOMETRY Course Code: 120631001
GIZMOS CORRELATION
APPLET TITLE
Constructing Congruent Segments and Angles
Constructing Parallel and Perpendicular Lines
TOPIC II DISCOVERY EDUCATION CORRELATION
VIDEO TITLE
Basic Geometric Constructions
MATH OVERVIEW
Geometry: Parallel Lines Cut by a Transversal
Geometry: Proving Triangles Are Congruent: SSS and SAS Postulates
Geometry: Proving Triangles Are Congruent: ASA Postulate and the AAS Theorem
MATH EXPLANATION TITLE
Geometry: Congruent Triangles and Congruence Transformations: Identifying Congruent Triangles
Geometry: Congruent Triangles and Congruence Transformations: Naming Congruent Angles and Sides
Geometry: Angles of Triangles: Using Properties of Congruent Triangles to Find Missing Parts
Geometry: Introduction to Proofs: Developing Proofs
Geometry: Triangle Inequality Theorems: Write an Inequality
Geometry: Angles of Triangles: Using the Angle Sum Theorem