Geometry, Quarter 1, Unit 1.1 Understanding Two ...cranstonmath.weebly.com/uploads/5/4/8/3/5483566/cranston-math-… · Prior Learning In grade 8, students learned that any two-dimensional

Cranston Public Schools, with process support from the Charles A. Dana Center at the University of Texas at Austin

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Geometry, Quarter 1, Unit 1.1

Understanding Two-Dimensional Congruence Through Transformations

Overview Number of instructional days: 10 (1 day = 45 minutes)

Content to be learned Mathematical practices to be integrated • Define points, lines, planes, segments, angles,

rotations, reflections, translations, parallel, perpendicular, and congruence.

• Perform transformations on the coordinate plane and determine the function rule that describes the transformation.

• Perform and describe a given geometric figure as a sequence of transformations that carry a given figure onto another.

• Determine congruence in terms of rigid motions.

• Make formal geometric constructions with a variety of tools and methods.

Model with mathematics.

• Analyze transformations to draw conclusions about congruence and parallel and perpendicular lines.

• Use constructions to understand and draw conclusions about congruence and transformations.

Use appropriate tools strategically.

• Use technological tools and available websites to deepen understanding of geometry concepts.

Attend to precision.

• Clearly communicate a transformation as a series of mappings from one set of ordered pairs to another.

• Use drawing tools to accurately construct congruent polygons.

Essential questions • What connections exist between

transformations and congruence? • How are function rules and transformations

related?

Geometry, Quarter 1, Unit 1.1 Understanding Two-Dimensional Congruence Through Transformations (10 days)


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Written Curriculum

Common Core State Standards for Mathematical Content

Congruence G-CO

Experiment with transformations in the plane

G-CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

G-CO.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

G-CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

G-CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

G-CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

Understand congruence in terms of rigid motions [Build on rigid motions as a familiar starting point for development of concept of geometric proof]

G-CO.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

Make geometric constructions

G-CO.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.

Common Core Standards for Mathematical Practice

4 Model with mathematics.

Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions



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and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their 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.

5 Use appropriate tools strategically.

Mathematically proficient students 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. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

6 Attend to precision.

Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.

Clarifying the Standards

Prior Learning

In grade 8, students learned that any two-dimensional figure can be mapped onto another figure through a series of transformations. They learned to express a transformation as a mapping from one set of coordinates to another set of coordinates. Also in grade 8, students developed an understanding that a two-dimensional figure is congruent to another if the second figure can be obtained from a sequence of rotations, reflections, and translations.

Current Learning

Students develop precise definitions of points, lines, planes, segments, and angles. They demonstrate an understanding of two-dimensional congruence through transformations. Students express a transformation mapping as a function rule and a sequence of transformations. They also develop methods for making formal geometric constructions with a variety of tools.



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Future Learning

Students will use transformations and construction throughout this course as a thought process and method for proving theorems involving congruence and similarity. They will use the correspondence between numerical coordinates and geometric points to connect geometry to algebra. Students will use definitions of geometric terms throughout this course to prove geometric theorems.

Additional Findings

Principles and Standards for School Mathematics states, “High school students should conduct increasingly independent explanations, which will allow them to develop a deeper understanding of important geometric ideas such as transformation and symmetry.” (p. 309)

The following websites provide useful information for geometry instruction:

• http://mathdl.maa.org/mathDL/47/?pa=content&sa=viewDocument&nodeId=3567 The Mathematical Sciences Digital Library Loci:Resources Geometry Playground

• www.math-drills.com/geometry.shtml (worksheets on transformations, center not always at origin)

• www.mathisfun.com (shows video of constructions)

• www.kutasoftware.com/freeige.html (worksheets on transformations)


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Proving Theorems Involving Lines and Angles


Content to be learned Mathematical practices to be integrated • Develop methods for proving parallel and

perpendicular relationships on and off the coordinate plane.

• Identify and classify angles formed by two parallel lines crossed by a transversal.

• Prove theorems involving angles formed by two parallel lines crossed by a transversal.

• Apply definitions and properties of angle relationships to solve problems. (Angle relationships include complementary, supplementary, vertical, linear pairs, adjacent angles, and angle addition postulate.)

• Apply definitions and properties of segment relationships and the segment addition postulate to solve problems.

• Derive and apply the distance formula to determine the length of a segment, and compare lengths between two or more segments.

Make sense of problems and persevere in solving them.

• Plan a solution pathway by analyzing angle relationships (i.e., determine if angles are complementary, supplementary, vertical angles, etc., to solve problems).

• Make/test conjectures involving special angle pairs to determine if two lines are parallel.

• Check for the sensibility of the solution involving angle measurements and lengths of segments.

Construct viable arguments and critique the reasoning of others.

• Use stated assumptions, definitions, and previously established results to prove theorems.

• Distinguish correct logic or reasoning from flawed logic or reasoning.


• Calculate distance and slope accurately and efficiently.

• Use clear and concise language to construct a logical argument.

Essential questions • How do you use logic and reasoning to build

an understanding of angle and segment relationships, parallel lines, and perpendicular lines?

• How do you use coordinate geometry to prove that segments are congruent and lines are parallel and perpendicular?

Geometry, Quarter 1, Unit 1.2 Proving Theorems Involving Lines and Angles (10 days)


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Written Curriculum


Congruence G-CO

Experiment with transformations in the plane

G-CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

Prove geometric theorems [Focus on validity of underlying reasoning while using variety of ways of writing proofs]

G-CO.9 Prove theorems about lines and angles. 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.

Expressing Geometric Properties with Equations G-GPE

Use coordinates to prove simple geometric theorems algebraically [Include distance formula; relate to Pythagorean theorem]

G-GPE.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).


1 Make sense of problems and persevere in solving them.

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can 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. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.



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3 Construct viable arguments and critique the reasoning of others.

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to 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. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.




Prior Learning

In grade 4, parallel and perpendicular lines were introduced to students (4.G.1). In grade 7, students used facts about supplementary, complementary, vertical, and adjacent angles in a multistep problem to write and solve simple equations for an unknown angle in a figure (7.G.5).

In grade 8, students applied/derived the slope between two points (8.EE.6), and they discovered angle relationships when parallel lines are crossed by a transversal (8.G.5). Students also applied the Pythagorean theorem to find the distance between two points in a coordinate system (8.G.8).

Current Learning

Students reinforce definitions of parallel and perpendicular lines, and they define and verify angle relationships (complementary, supplementary, vertical angles, etc.) Students apply segment and angle addition postulates to solve problems. They develop formal and informal proofs of angle relationships formed by two parallel lines crossed by a transversal. Students derive the distance formula using their prior knowledge of the Pythagorean theorem.



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Future Learning

Students will use logical arguments developed in this course as a core skill to prove theorems throughout the year. They will represent angles and lengths of sides with algebraic expressions and use those expressions to write an equation to solve for an unknown.

Additional Findings

None at this time.

Writing Team Notes

Students often mistake lengths of sides with angle measure (i.e., for ΔABC with AB = BC, if AB = 2x, BC = 4x – 3, then students often say 2x = 180°).

When applying the distance formula, students often square a negative number incorrectly, and their result stays negative. Remind students that there are no negative lengths.


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Proving Triangle Congruence with Rigid Motions


Content to be learned Mathematical practices to be integrated • Understand the definition of congruent

triangles.

• Reinforce the meaning of corresponding sides and angles of triangles.

• Determine if two triangles are congruent by definition.

• Discover and apply postulates (ASA, SAS, SSS) to prove triangles congruent.

• Determine that all corresponding parts of congruent triangles are congruent (CPCTC).

• Use organizational formats for proving theorems (i.e., two-column proofs, paragraph proofs, flowchart proofs, etc.)


• Understand and use prior learning of transformations in constructing arguments to prove triangles congruent.

• Justify conclusions, communicate them to others, and respond to the arguments of others regarding triangle congruence.

Model with mathematics.

• Verify triangle congruence postulates through transformations.

• Analyze congruent triangles to draw conclusions on corresponding parts.


• Use clear and concise language to construct a logical argument.

Essential questions • How do you use logic and reasoning to build an

understanding of triangle congruence? • How are logical arguments developed and

organized to prove triangle congruence?

Geometry, Quarter 1, Unit 1.3 Proving Triangle Congruence with Rigid Motions (5 days)


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Written Curriculum


Congruence G-CO

Understand congruence in terms of rigid motions [Build on rigid motions as a familiar starting point for development of concept of geometric proof]

G-CO.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.

G-CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.



Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to 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. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

4 Model with mathematics.

Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their 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.



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Prior Learning

In grade 8, students learned that congruence of triangles is preserved through transformations (8.G.2).

Earlier in Unit 1.1, students reinforced their knowledge of congruence and basic geometric terms.

Current Learning

Students define congruent triangles and prove triangle congruence through postulates. They develop logical reasoning and organizing skills to write a proof.

Future Learning

Logical reasoning and proof writing will be used throughout this course to prove theorems and advance student knowledge. Applications of congruent triangles will be used to solve real-world problems.

Additional Findings

None at this time.

Writing Team Notes

Postulates ASA, SAS, and SSS are in the standard, but you may want to include the AAS and HL theorems.

The following website provides useful information for geometry instruction: www.kutasoftware.com/freeige.html (worksheets on applying postulates)



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Proving Theorems Involving Triangles


Content to be learned Mathematical practices to be integrated • Review classification of triangles and all

polygons. (Note: Not included in the CCSS.)

• Construct and define the median and centroid of a triangle.

• Identify how the centroid partitions the median of a triangle into a specific ratio.

• Prove and apply theorems on

1. the sum of interior angles of a triangle.

2. the sum of interior and exterior angles in any polygon. (Note: Not included in the CCSS.)

3. inequalities in one triangle and two triangles (triangle inequality theorem, SSS and SAS inequality theorems, etc.) (Note: Not included in the CCSS.)

4. base angles of an isosceles triangle.

5. a segment that joins the midpoints of two sides of a triangle, is parallel to the third side, and is half the length of the third side.

Make sense of problems and persevere in solving them.

• Plan a solution pathway by analyzing theorems involving triangles.

• Make/test conjectures involving interior and exterior angles to derive theorems.

• Check for the sensibility of the solution involving angle measurements and lengths of segments.


• Justify conclusions, communicate them to others, and respond to the arguments of others regarding theorems involving triangles.

Essential questions • How is the sum of interior angles of a triangle

proved?

• How is the sum of interior/exterior angles related to the number of sides of a polygon?

(Note: Not included in the CCSS.)

• What is the relationship between the distance of the centroid to the vertices of a triangle?

• What is the relationship between the angles and sides of one triangle? (Note: Not included in the CCSS.)

• What is the relationship between the angles and sides of two triangles? (Note: Not included in the CCSS.)

Geometry, Quarter 1, Unit 1.4 Proving Theorems Involving Triangles (10 days)


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Written Curriculum


Congruence G-CO

Prove geometric theorems [Focus on validity of underlying reasoning while using variety of ways of writing proofs]

G-CO.10 Prove theorems about triangles. 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.

Expressing Geometric Properties with Equations G-GPE

Use coordinates to prove simple geometric theorems algebraically [Include distance formula; relate to Pythagorean theorem]

G-GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.


1 Make sense of problems and persevere in solving them.

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can 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. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.


Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient



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students are also able to 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. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.


Prior Learning

In grade 4, students classified two-dimensional figures based on the presence or absence of parallel or perpendicular lines or the presence or absence of angles of a specified size. They recognized right triangles as a category and identified right triangles. (4.G.2)

In grade 5, students classified two-dimensional figures into categories based on their properties. (5.G.3)

In grade 8, students used informal arguments to establish facts about the angle sum and exterior angle of triangles (8.G.5).

Current Learning

Students create formal proofs involving the angle sum of any triangle, base angles of isosceles triangles, inequalities for one and two triangles, and the midsegment of a triangle. Students locate the centroid of a triangle and prove the associated theorem to solve problems. They also compare sides and angles of triangles using inequalities.

Future Learning

Students will use logical reasoning and proof writing throughout the course to prove theorems and advance student knowledge. They will use applications of theorems involving triangles to solve real-world problems (e.g., Unit 2.3, solving problems using trigonometry).

Additional Findings

None at this time.

Writing Team Notes

Teachers will need to define a median and altitude of a triangle and review perpendicular bisectors. Common Core State Standards do not clearly specify which theorems involving triangles are included in this unit. The number of days devoted to this unit may need to be changed to allow enough time to cover all of the material.

Teachers may find it helpful to use coordinate geometry to determine the location of the centroid of a triangle.



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Documents

Geometry, Quarter 1, Unit 1.1 Understanding Two ...cranstonmath.weebly.com/uploads/5/4/8/3/5483566/cranston-math-… · Prior Learning In grade 8, students learned that any two-dimensional