MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guidecurriculum_materials.dadeschools.net/Pacing_Guides/Q… · · 2017-07-10MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide

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.




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.




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

http://www.cpalms.org/Public/PreviewCourse/Preview/13029

http://www.fsassessments.org/wp-content/uploads/2015/08/Geometry-Math-Test-Item-Specifications_Final_May-2016.pdf




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

http://www.cpalms.org/Public/PreviewStandard/Preview/5478







Module 4

4.3 MAFS.912.G-CO.3.9 MAFS.912.G-CO.4.12


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.




4.4 MAFS.912.G-CO3.9 MAFS.912.G-CO.4.12


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.




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


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


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.




INSTRUCTIONAL TOOLS














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


6, 7, 9, 11-13,

Explore 2 partially covers MAFS.912.G-CO.4.12, this standard is also covered in


MAFS.912.G-SRT.2.5 is limited to congruent triangles.




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


1-2, 4, 9, 19, 22, 23, 27

Explore partially covers MAFS.912.G-CO.4.12, this standard is also covered in





Module 6

6.1 MAFS.9.12.G-CO.4.12

MAFS.912.G-CO.4.13

MAFS.912.G-SRT.2.5


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


2, 6-10, 12, 14, 17 Problems selected may NOT require students to use the distance formula.

Module 7


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.


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


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








https://www.illustrativemathematics.org/content-standards/HSG/CO/C/10/tasks/1921




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

http://www.cpalms.org/Public/PreviewResourceAssessment/Preview/59703
























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.


INSTRUCTIONAL TOOLS













http://www.cpalms.org/Public/PreviewResourceUrl/Preview/31955













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 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.


Reason abstractly and quantitatively.


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.


Construct viable arguments and critique the reasoning of

others.


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.


Model with mathematics.


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.




MATHEMATICS FLORIDA STANDARDS

MATHEMATICAL PRACTICES

DESCRIPTION


Use appropriate tools strategically.


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.


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.


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.


Look for and express regularity in repeated

reasoning.


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.




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


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


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




Domain: GEOMETRY: CONGRUENCE

Cluster 3: Prove Geometric Theorems


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


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


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


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


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




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


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


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




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


http://www.cpalms.org/Public/PreviewResourceLesson/Preview/130852








http://www.cpalms.org/Public/PreviewResourceUpload/Preview/44482



















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










https://www.geogebra.org/m/21270

http://tube.geogebra.org/m/21277




https://www.geogebra.org/m/yTFZSVxa?doneurl=%2Fsearch%2Fperform%2Fsearch%2Fg-co.9%2Fmaterials%2F




https://www.geogebra.org/search/perform/search/g-co.13/materials/

https://www.geogebra.org/search/perform/search/g-co.10/materials/

http://www.explorelearning.com/index.cfm?method=cResource.dspDetail&ResourceID=185






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



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MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guidecurriculum_materials.dadeschools.net/Pacing_Guides/Q… · · 2017-07-10MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide