1 Review the Geometry sample year-long scope and sequence associated with this unit plan.
Unit 1: Introduction to Geometric Concepts, Construction, and Proof Possible time frame: 14 days
This unit starts with familiar geometric concepts: angle, circle, perpendicular line, parallel line, and line segment. However, the students’ understanding of these topics will deepen as students move from informal definitions used in previous math courses to precise definitions of all concepts. As students reengage with geometry they will also begin modeling objects using two- and three-dimensional geometric figures to solve simple problems. Students will make formal geometric constructions (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). Students will begin to explore the idea of a formal proof as they prove theorems about lines and angles (theorems: 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). Most of these are informally developed in 7th or 8th grade and formally proven in Geometry.
Major Cluster Standards Standards Clarification
Prove geometric theorems
HSG-CO.C.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.
Apply geometric concepts in modeling situations
HSG-MG.A.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso
as a cylinder).★
★Modeling Standard
HSG-MG.A.1 Students will continue
to use geometric figures to model real-word objects and solve problems with the most emphasis in Units 7, 8, 9, and 11.
Supporting Cluster Standards Standards Clarification
Experiment with transformations in the plane
HSG-CO.A.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.
Make geometric constructions
HSG-CO.D.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.
HSG-CO.A.1 Students will continue
to use the precise definitions of geometric figures throughout the year, but most notably in Unit 10.
Applying Mathematical Practices to CCSS
MP.3 Construct viable arguments and
critique the reasoning of others. Students build proficiency with MP.3 and MP.7 as they build a mathematical system with structured statements, including postulates and proven theorems. Students should be exposed to a variety of proof styles, including flow-chart proofs, two-column proofs, and paragraph proofs, as they begin to build viable logical arguments. Again, the use of precise language, MP.6, is critical to building a logical argument.
MP.6 Attend to precision.
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What will students know and be able to do by the end of this unit?
Students will demonstrate an understanding of the unit focus and meet the expectations of the Common Core State Standards on the unit assessments.
Standards
The major cluster standards for this unit include:
HSG-CO.C.9 Prove theorems about lines and angles. HSG-MG.A.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).★
Unit Assessment
Students will demonstrate mastery of the content through assessment items and tasks requiring:
• Conceptual Understanding • Procedural Skill and
Fluency • Application • Math Practices
Objectives and
Formative Tasks
Objectives and tasks
aligned to the CCSS prepare students to meet the expectations of the unit assessments.
Concepts and Skills
Each objective is broken down into the key concepts and skills students should learn in order to master objectives.
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Sample End-of-Unit Assessment Items:
1) In the diagram below, 𝑋𝑌⃖����⃗ is a line, A is a point on the line, and B is a point not on the line. Construct a line parallel to 𝑋𝑌⃖����⃗ that passes through B. Explain how you know the line you constructed is parallel to 𝑋𝑌⃖����⃗.
2) Suppose that four lines, 𝑙1, 𝑙2, 𝑚1, and 𝑚2 lie in a given plane such that 𝑙1∥ 𝑙2, 𝑚1∥ 𝑚1, and 𝑙1is neither parallel nor perpendicular to 𝑚1.
a. Sketch (freehand) a diagram of 𝑙1, 𝑙2, 𝑚1, and 𝑚2 to illustrate the given conditions. b. In any diagram that illustrates the given conditions, how many non-overlapping angles are
formed? Count only angles that measure less than 180° and count two angles as the same only if they have the same vertex and the same sides. Among these angles, how many different angle measures are formed? Justify your answer.
Item from http://www.engageny.org/sites/default/files/resource/attachments/geometry-m1-module-overview-and-assessments.pdf
3) In the figure below, lines p and q are parallel. Lines l and m are also parallel. Prove that Angle A is congruent to Angle X. X A Y B G H E F C B D A m R S T U V W X Z
4 4) Construct 𝑃𝑄����. Then construct a perpendicular bisector of 𝑃𝑄����.
a. Explain the process that you followed to construct the segment and the perpendicular bisector. In your explanation, be sure to state which tools you used and describe how you used those tools.
b. Place a point R on the perpendicular bisector in the drawing in part a. Prove that any point R on the perpendicular bisector is equidistant from points P and Q.
5) Prove that vertical angles are congruent. Include a diagram with your proof.
6) Six objects are listed below. Circle all of the objects that could be modeled using line segments. a. A piece of paper b. A pencil c. A basketball d. A tabletop e. A piece of spaghetti f. A soda can
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Sample End of Unit Assessment Task: Constructions and Proof
1) Using the following diagram, draw a transversal. Use the transversal and a protractor to explain why these lines are parallel.
2) Copy the angle below. Then, bisect the copied angle.
3) Divide 𝐴𝐵���� into 4 segments of equal length.
Adapted from: http://www.engageny.org/resource/geometry-module-1
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End-of-unit Assessment Sample Item Responses:
Teacher Note: All drawings included in these responses may not be to scale when printed as they may change due use of different computers, printers, etc.
1) HSG-CO.A.1, HSG-CO.C.9, HSG-CO.D.12
The line constructed through point B is parallel to line XY because I used constructions to copy Angle BAY to the corresponding angle position with B as the vertex. Since the two angles are congruent, the lines are parallel.
2) HSG-CO.A.1 a.
b. There are 16 distinct angles with two different angle measures because alternate interior/exterior angles are congruent and corresponding angles are congruent.
7 3) HSG-CO.C.9
Lines p and q are transversals for lines l and m (and vice versa). Additionally, it is given that p || q and l || m. By definition of corresponding angles, Angle A and Angle E are considered corresponding angles. The Corresponding Angles Postulate says that corresponding angles are congruent, so Angle A ≅ Angle E. Another pair of corresponding angles is Angle E and Angle V. By the transitive property of congruence, if Angle A ≅ Angle E and Angle E ≅ Angle V, then Angle A ≅ Angle V. Angle V and Angle X are considered vertical angles, by definition of vertical angles. By the Vertical Angles Theorem, we know that Angle V ≅ Angle X. Again, using the transitive property of congruence, if Angle A ≅ Angle V and Angle V ≅ Angle X, then
Angle A ≅ Angle X. 4) HSG-CO.C.9, HSG-CO.D.12
a.
Using a ruler (or straightedge), draw a line segment and label the endpoints P and Q. To construct the perpendicular bisector of 𝑃𝑄����, place the point of a compass on one of the endpoints of the line segment and set the width of the compass to slightly more than half the length of the segment (the actual width does not matter). Without changing the width of the compass, draw an arc above and below the line segment. Again, without changing the width of the compass, place the point of the compass on the opposite endpoint of the line segment and draw an arc above and below the line segment so the two new arcs intersect the first two arcs. Using a straightedge (ruler), draw a line connecting the points where the arcs intersect.
Teacher Note: The description above with a diagram for each step can be viewed by
visiting
http://www.mathopenref.com/printbisectline.html.
8 b. Label the points where the set of arcs above 𝑃𝑄���� intersect as A, the point where the set of
arcs below 𝑃𝑄���� intersect as B. R can be placed anywhere on the perpendicular bisector, 𝐴𝐵����. There are two cases: 1) R is not on 𝑃𝑄����; 2) R is on 𝑃𝑄����. Prove that if R is placed anywhere on 𝐴𝐵
����, R is equidistant from P and Q.
Case 1: R is not on 𝑃𝑄����. Because 𝐴𝐵���� is a perpendicular bisector of 𝑃𝑄����, Angle PJR and Angle
QJR are both right angles which each measure 90 degrees, by definition of perpendicular. Therefore, Angle PJR is congruent to Angle QJR since congruent angles have equal measure. Also, by definition of perpendicular bisector, 𝑃𝐽��� is congruent to 𝑄𝐽���. Connect R with P with one segment and connect R with Q with one segment. This will create two triangles. Triangle PJR and Triangle QJR are both right triangles as they each include one right angle. By the Pythagorean Theorem, we know that
𝑃𝐽2+ 𝐽𝑅2= 𝑃𝑅2 and 𝑄𝐽2+ 𝐽𝑅2= 𝑄𝑅2. By definition of perpendicular bisector, we know that PJ = QJ so we can rewrite 𝑄𝐽2+ 𝐽𝑅2= 𝑄𝑅2 as 𝑃𝐽2+ 𝐽𝑅2= 𝑄𝑅2. Solving
𝑃𝐽2+ 𝐽𝑅2= 𝑃𝑅2 and 𝑃𝐽2+ 𝐽𝑅2= 𝑄𝑅2 for PR and QR respectively, we have 𝑃𝑅 = �𝑃𝐽2+ 𝐽𝑅2 and 𝑄𝑅 = �𝑃𝐽2+ 𝐽𝑅2. By the transitive property of equality,
PR = QR. If a point is equidistant from two other points, then the point is the same distance away from both points. The measure of a segment represents the distance one endpoint is from the other; therefore, if PR = QR, then R is the same distance from P and Q, and it is equidistant from P and Q.
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Case 2: R is on 𝑃𝑄����. If R is on the perpendicular bisector and on 𝑃𝑄����, then R must be located at
the midpoint of 𝑃𝑄���� because the perpendicular bisector of a segment intersects the segment at its midpoint. By definition, the midpoint of a line segment divides the segment into two equal parts. Therefore, R is equidistant from P and Q.
Teacher Note: Students will not have learned the postulates for triangle congruence; therefore, they cannot use that information in a proof. This proof draws upon their knowledge of the Pythagorean Theorem from Grade 8 and their ability to manipulate equations. This is a formal proof. Teacher discretion may allow for a less formal proof with solid logical reasoning. In a less formal proof, students may not account for both cases and there may be missing steps from the proof above. This would provide an opportunity for rich discussion following the assessment to understand better the reasoning needed to write a thorough proof.
10 5) HSG-CO.C.9
Given: Angle 1 and angle 2 are vertical angles. Prove: Angle 1 ≅ Angle 2
Statements Reasons
1. Angle 1 and angle 2 are vertical angles. 1. Given 2.
2. Angle Addition Postulate
3. 3. Substitution property of equality
4. 4. Reflexive property of equality
5. 5. Subtraction property of equality
6. 6. Definition of congruence
Teacher Note: This is a sample proof. Some students will use different reasons so all should be checked for accuracy and reasoning.
6) HSG-MG.A.1
Solution: Answer choices: b. A pencil
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Sample End of Unit Assessment Task Responses: Constructions and Proof
1) HSG-CO.C.9
After drawing in the transversal, I measured the angles and found the measures of the angles shown. Since the alternate interior angles have the same measure, they are congruent. If two lines are cut by a transversal such that the alternate interior angles are congruent hen the lines are parallel. Thus, the lines above are parallel.
2) HSG-CO.D.12
Teacher Note: For the steps to copy an angle visit: http://www.mathopenref.com/printcopyangle.html For the steps to bisect an angle visit:
http://www.mathopenref.com/printbisectangle.html
12 3) HSG-CO.D.12
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Possible Pacing and Sequence of Standards
Content and Practice Standards Possible Pacing and Sequence
Prove geometric theorems
HSG-CO.C.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. Apply geometric concepts in modeling situations HSG-MG.A.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).★
Experiment with transformations in the plane HSG-CO.A.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.
Make geometric constructions
HSG-CO.D.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;
Days 1-4 Objectives:
Students will be able to provide precise definitions for angles, circles, perpendicular lines, parallel lines, and line segments.
Students will use geometric shapes, their measures, and their properties to solve a real world problem. Concepts and Skills:
• Define angles, circles, perpendicular lines, parallel lines, and line segments.
• Locate real-world objects to represent angles, circles, perpendicular lines, parallel lines, and line segments.
• Identify various shapes as they appear in real-world objects and explain how the items they selected meet the outlined criteria.
• Use geometric shapes to describe a real world object. Sample Tasks:
1. Students will complete a scavenger hunt for geometric shapes. Provide students with a rubric to show them what type of objects/shapes they will need to find. If possible, provide groups of students with a digital camera. Allow students to explore the campus and take pictures of the real-world objects that they want to use. Students will need one day to find the shapes and prepare a presentation of the shapes, and one day to share their presentations of the shapes to the class. Students may need to work outside of class to prepare their presentations.
2. See Illustrative Mathematics for a sample task which could be used during instruction. Illustrative Math Paper Clip Task
14 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.
Possible Connections to Standards for Mathematical Practices
MP.3 Construct viable arguments and critique the reasoning of others.
Students will construct viable arguments as they create proofs of the theorems in this unit. The proofs students will create will be based on their reasoning about the figures and the properties of those figures. Students should be given opportunities to discuss the proofs they create with other students and crirtique others’ reasoning.
MP.4 Model with mathematics.
Students are modeling with mathematics as they identify objects which model geometric shapes, their properties, and measures.
MP.5 Use appropriate tools strategically. Students will learn to use the tools available to create formal constructions. Then when solving real-world problems, students will be able to make strategic choices about those tools.
MP.6 Attend to precision.
Students will have to use precise vocabulary and definitions when constructing valid proofs.
Days 5-7 Objectives:
Students will make formal geometric constructions with a variety of tools and methods. Concepts and Skills:
• Use a compass and a straightedge to copy a segment, copy and angle, bisect a segment, bisect an angle, construct perpendicular lines, construct a perpendicular bisector of a line segment, construct a line parallel to a given line through a point not on the line, and construct an equilateral triangle. • Use patty paper and paper folding to find the midpoint of a segment, construct parallel lines,
construct perpendicular lines, construct a perpendicular bisector, and bisect an angle. Sample Tasks:
Critter Constructions (pages 1-32): This activity requires students to use patty paper for geometric constructions.
Days 8-10 Objectives:
Students will prove theorems about lines and angles. Concepts and Skills:
• Prove that vertical angles are congruent.
• Prove that when a transversal crosses parallel lines, alternate interior angles and corresponding angles are congruent and the converse.
• Show that points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints and the converse.
Sample Task:
Parking Lot Mathematics: This real-world activity allows students to explore the relationships between parallel lines, alternate interior angles, and corresponding angles.
15 MP.7 Look for and make use of structure.
Students will look for structure in various figures to determine how to proceed with proofs of the theorems.
Days 11-12 Objective:
Students will apply the understandings of constructions and proof to solve a real-world problem. Application Task Map Construction
The student will use the relationships between angles formed by two lines cut by a transversal to determine whether two lines are parallel; verify the parallelism, using algebraic and coordinate methods as well as deductive proofs.
16 Application Task:
Teacher Note: The directions below indicate students should use a poster board; however, this task can be completed on a piece of paper as well. The size of the triangle may need to be adjusted if the task is changed to be submitted on a sheet of paper.
Directions for Map Construction:
Note: All streets (lines) constructed should be extended to “run off” the poster board.
1. Begin by sketching a compass to indicate the directions north, south, east, and west. Draw this in the upper left hand corner of the poster board.
2. Construct an equilateral triangle in the center of your poster board. The sides of the triangle should each measure 4 inches. Use a straightedge to extend the lines including the sides of each triangle so the lines “run off” the poster board. Label the vertices A, B, and C. This will give you three streets: 𝐴𝐵����, 𝐵𝐶����, and 𝐶𝐴����.
3. Construct a street parallel to street 𝐵𝐶����. Name this street 𝐴𝐷����.
4. Construct a street perpendicular to street 𝐴𝐷���� so that it lies to the east of the triangle but does not pass through ∆ 𝐴𝐵𝐶. Label the intersection of the perpendicular street and street 𝐴𝐷���� as point E. Label the intersection of the perpendicular street and street 𝐵𝐶���� as point F.
5. Label the intersection of streets 𝐴𝐵���� and 𝐸𝐹���� as point G.
6. Construct a street perpendicular to street 𝐵𝐶���� so that it lies to the east of street 𝐸𝐹����. Label the intersection of this perpendicular street and street 𝐴𝐷���� as point H. Label the intersection of the perpendicular street and street 𝐵𝐶���� as point J.
7. Label the intersection of streets 𝐴𝐵���� and 𝐻𝐽���� as point K.
17 Answer the following questions based on the map you created. Use complete sentences in your
explanations and justifications. For any proofs, you may choose the type of proof you create. Be sure to include logical reasoning to avoid leaving relevant information out of the proof.
1. State the measures of the following angles. Explain how you know the measures of the angles without using a protractor. Verify the angle measures.
a. measure of Angle BAC b. measure of Angle DAB c. measure of Angle ABF d. measure of Angle BGF
2. What does it mean for two lines to be parallel? Prove that streets 𝐴𝐷���� and 𝐵𝐶���� are parallel. 3. Are there any other pairs of streets that are parallel? Explain your reasoning using the
properties of angles and parallel lines.
4. Imagine the section of the city depicted by your map has a subway that runs entirely underground and directly beneath street 𝐸𝐹����.
a. Is the subway parallel to 𝐸𝐹����? Explain.
b. Is the subway parallel to another street? Explain.
c. Since the subway runs underneath the streets on this map, it will never intersect with any of the streets shown on the map. Does this mean the subway is parallel to some or all of these streets? Explain.
18 Application Task Exemplar Response:
Teacher Note: This picture is missing the intersection labeled K. Also, this is not drawn to the scale identified in the task. The construction marks were also removed. Point N was placed to aid in explanations in part two of the task.
19 Answer the following questions based on the map you created. Use complete sentences in your
explanations and justifications. For any proofs, you may choose the type of proof you create. Be sure to include logical reasoning to avoid leaving relevant information out of the proof.
1. State the measures of the following angles. Explain how you know the measures of the angles without using a protractor. Verify the angle measures.
a. measure of Angle BAC
The measure of Angle BAC is 60. Angle BAC is an angle in the equilateral triangle. All angles in an equilateral triangle are 60 degrees.
b. measure of Angle DAB
The measure of Angle DAB is also 60. Angle DAN is a copy of Angle ACB so it has the same measure of Angle ACB which is 60. Angle DAN, Angle CAB, and Angle DAB are adjacent and their non-common rays form a line which means all the sum of the measures of the angles is 180. Since the sum of Angle CAB and Angle DAN is 120, the measure of Angle DAB is also 60.
c. measure of Angle ABF
The measure of Angle ABF is 120. Angle ABF and Angle ABC are adjacent and their non-common rays form a line, so the sum of the measures of the two angles is180. Angle ABC is an angle in the equilateral triangle therefor its measure is 60. 180 – 60 = 120.
d. measure of Angle BGF
The measure of Angle BGF is 30. Angle FBG is 60 degrees because Angle ABC and Angle FBG are vertical angles and vertical angles are congruent which means they have the same measure. Angle BFG is 90 degrees because street 𝐸𝐹���� is perpendicular to street 𝐵𝐶���� at point F which means the angles formed at the intersection (labeled point F) are right angles. Right angles are 90 degree angles. Together, Angle FBG, Angle BFG, and Angle BGF are three angles in right triangle BFG. The sum of the measures of the three angles of a triangle is 180. 180- (60 + 90) = 30.
2. What does it mean for two lines to be parallel? Prove that streets 𝐴𝐷���� and 𝐵𝐶���� are parallel. If two lines are parallel, they lie in the same plane and they do not intersect. I constructed 𝐴𝐷���� by copying Angle ACB to construct Angle NAD. Copying an angle creates two congruent angles so Angle ACB is congruent to Angle NAD. Street 𝐴𝐶���� is a transversal of streets 𝐴𝐷���� and 𝐵𝐶���� which makes Angle ACB and Angle NAD corresponding angles by the definition of corresponding angles. If two lines are cut by a transversal so that two corresponding angles are congruent, then the lines are parallel. Therefore streets 𝐴𝐷���� and 𝐵𝐶���� are parallel.
3. Are there any other pairs of streets that are parallel? Explain your reasoning using the properties of angles and parallel lines.
Yes, streets 𝐸𝐹���� and 𝐻𝐽���� are parallel. Street 𝐸𝐹���� was constructed to be perpendicular to street 𝐴𝐷����. All four angles formed at the intersection labeled E are right angles, and all right angles are congruent. Because 𝐸𝐹���� is a transversal intersecting streets 𝐴𝐷���� and 𝐵𝐶����,.and they are parallel, street 𝐸𝐹���� must be perpendicular to street 𝐵𝐶����. Street 𝐻𝐽���� was constructed to be perpendicular to street 𝐵𝐶����. By the same reasoning used earlier, street 𝐻𝐽���� must be perpendicular to street 𝐴𝐷����. If two lines are perpendicular to the same line then the lines are parallel. Therefore streets 𝐸𝐹���� and 𝐻𝐽
20 4. Imagine the section of the city depicted by your map has a subway that runs entirely
underground and directly beneath street 𝐸𝐹����. a. Is the subway parallel to 𝐸𝐹����? Explain.
Yes the subway is parallel to 𝐸𝐹����. Even though the subway is not on the street level, a plane can be formed between the street and the subway. According to the map, both street 𝐸𝐹���� and the subway would be running north and south. Since they will never intersect, the street and the subway are parallel.
b. Is the subway parallel to another street? Explain.
Yes the subway is also parallel to 𝐻𝐽����. Even though the subway is not on the street level, a plane can be formed between the street and the subway. According to the map, both street 𝐻𝐽���� and the subway would be running north and south. Since they will never intersect, the street and the subway are parallel.
c. Since the subway runs underneath the streets on this map, it will never intersect with any of the streets shown on the map. Does this mean the subway is parallel to some or all of these streets? Explain.
The subway is only parallel to those streets which run north and south. Streets that run any other direction (on the map they would intersect streets 𝐸𝐹���� and 𝐻𝐽����) would not be in any same plane as the subway. Two lines that do not intersect and are not in the same plane are considered skew lines, not parallel. Therefore, the subway would be considered skew to all streets that do not run north and south.