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Geometry and PreAP Geometry Unit 1

Geometry and PreAP Geometry Scope and Sequence with NMSI’s Laying the Foundation lessons and Teacher Led Instruction Ideas

Unit 1: Points / Lines / Planes 9 days of instruction plus assessment time – 2.5 weeks

Teacher Led Instruction (2 or 3 days) Introductory Definition & Postulates

Teacher Note: Ideas for teaching this standard include using examples/non-examples to create definitions of angle pairs and modeling point-line-plane relationships with index cards and straws. (See Demo Activity). Naming angles can be a challenge for students. (See Naming Angles Activity) ** Use Unit 1 Checkpoints: #1 – 5 and #7 following this lesson.

AL COS Common Core Standard

Common Core

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-CO1

Teacher Led Instruction (2 days) Basic Constructions Teacher Note: Ideas for teaching this lesson include using Patty Paper and compass-straight edge. Note that trisecting a line is an additional objective for Quality Core – you can find multiple methods with directions with a web search.


Common Core

12

Make formal geometric constructions with a variety of tools and methods such as compass and straightedge, string, reflective devices, paper folding, and dynamic geometric software. Constructions include 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.

G-CO12

Teacher Led Instruction (1 day) Triangle Basics Teacher Note: These Quality Core standards are not addressed within the course of study, but best fit into Unit 1. These concepts were covered in a previous grade and should be applied algebraically. The suggested checkpoints are good examples this application. **Unit 1 Checkpoints: #8 - #15 following this lesson.

Geometry and PreAP Geometry Unit 1

Teacher Led Instruction (2 days) Coordinate Geometry Teacher Note: Derive and apply formulas for finding midpoint and distance. The following resources are available in the Additional Resources folder. Midpoint Miracle: http://www.illustrativemathematics.org/illustrations/605 Unit Squares & Triangles: http://www.illustrativemathematics.org/illustrations/918 **Unit 1: Checkpoints #6 and #8 focus on coordinate applications. #8 includes two multistep Free Response questions that you might consider your students working on together within groups.


Common Core

30 Use coordinates to prove simple geometric theorems algebraically. G-GPE4

Teacher Led Instruction (1 day) Point on a Directed Line

Teacher Note: This standard is often time not included in textbooks. There are multiple resources online giving basic information about the standard. The following resources are available in the Additional Resources folder. Finding Triangle Coordinates: http://www.illustrativemathematics.org/illustrations/1685


Common Core

32 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

G-GPE6

http://www.illustrativemathematics.org/illustrations/605



Geometry and PreAP Geometry Unit 2 Page 1


Unit 2: Reasoning and Proof 10 days of instruction plus assessment time – 2 weeks

Unit Teacher Note: Check Additional Resources folder for teacher-made activities that could be helpful. These resources are available to be changed and edited as necessary to meet student needs. NMSI’s Laying the Foundation lesson: Logical Reasoning (“if-then” statements) (3 days) Teacher Note: Students will need an understanding of basic terms and parts of conditional statements for this

activity.


Common

Core

This is a foundational lesson. No specific standard listed

Teacher Led Instruction (3 days) Inductive / Deductive Reasoning Teacher Note: Your textbook should be an adequate resource. Students will need to understand inductive and deductive reasoning techniques and laws of reasoning to draw valid conclusions.

** Use Unit 2 Checkpoint: # 1 - #3 after completing this lesson.


Common

Core

This is a foundational lesson. No specific standard listed

Teacher Led Instruction (3 days) Introductory Proofs Teacher Note: Your textbook should be an adequate resource. Teachers need to focus on the quality core standards identified within this lesson prior to proving theorems about lines and angles (Vertical Angles Only). Teachers will need to review algebraic proofs with students prior to introducing geometric proofs.


Common

Core

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; and points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

G-CO9



Unit 3: Parallel and Perpendicular Lines 7 days of instruction plus assessment time – 2 weeks

NMSI’s Laying the Foundation lesson: Parallel and Perpendicular Lines—An Investigation (1.5 days) Teacher Note: Lesson has students investigate properties of parallel and perpendicular lines in regards to slope. Before the lesson, discuss the difference between parallel and perpendicular lines. Also, remind students about how to calculate slope. It is important for each group member to draw different parallel and perpendicular lines. We recommend that you only do this part of the lesson at this time. The Independent Activity will be used later in the curriculum


Common Core

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-CO1

31 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).

G-GPE5

Teacher Led Instruction (1.5 days) Proving Lines Parallel / Perpendicular

Teacher Note: The following links may help you: https://www.khanacademy.org/math/trigonometry/graphs/parallel_perpendicular/v/parallel-lines-3 https://www.khanacademy.org/math/algebra/linear-equations-and-inequalitie/more-analytic-geometry/v/equations-of-parallel-and-perpendicular-lines **Use Unit 3 Checkpoint: #2, # 5 and the Free Response following this lesson.


Common Core

31 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).

G-GPE5

https://www.khanacademy.org/math/trigonometry/graphs/parallel_perpendicular/v/parallel-lines-3

https://www.khanacademy.org/math/algebra/linear-equations-and-inequalitie/more-analytic-geometry/v/equations-of-parallel-and-perpendicular-lines

https://www.khanacademy.org/math/algebra/linear-equations-and-inequalitie/more-analytic-geometry/v/equations-of-parallel-and-perpendicular-lines


Teacher Led Instruction(2 days) Parallel Lines Cut by a Transversal Teacher Note: Your textbook should be an adequate resource. See the graphic organizer in Additional Resource file. Also, Patty Paper Geometry by Michael Serra (See transformational proof of angle relationships).

The following links may also be helpful: https://www.khanacademy.org/math/geometry/parallel-and-perpendicular-lines/old_angles/v/angles-formed-between-transversals-and-parallel-lines https://www.khanacademy.org/math/geometry/parallel-and-perpendicular-lines/old_angles/v/angles-of-parallel-lines-2 https://www.khanacademy.org/math/geometry/parallel-and-perpendicular-lines/old_angles/v/the-angle-game ** Use Unit 3 Checkpoint: #1, #3, #4, and #6 following this lesson.


Common Core

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; and points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

G-CO9

Teacher Led Instruction (2 days) Constructing Parallel / Perpendicular Teacher Note: The following links may help you: http://www.mathsisfun.com/geometry/construct-perponline.html http://www.mathsisfun.com/geometry/construct-perpnotline.html http://www.mathsisfun.com/geometry/construct-paranotline.html


Common Core

12 Make formal geometric constructions with a variety of tools and methods such as compass and straightedge, string, reflective devices, paper folding, and dynamic geometric software. Constructions include 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.

G-CO12

https://www.khanacademy.org/math/geometry/parallel-and-perpendicular-lines/old_angles/v/angles-formed-between-transversals-and-parallel-lines

https://www.khanacademy.org/math/geometry/parallel-and-perpendicular-lines/old_angles/v/angles-formed-between-transversals-and-parallel-lines

https://www.khanacademy.org/math/geometry/parallel-and-perpendicular-lines/old_angles/v/angles-of-parallel-lines-2

https://www.khanacademy.org/math/geometry/parallel-and-perpendicular-lines/old_angles/v/the-angle-game

http://www.mathsisfun.com/geometry/construct-perponline.html

http://www.mathsisfun.com/geometry/construct-perpnotline.html

http://www.mathsisfun.com/geometry/construct-paranotline.html


Geometry and PreAP Geometry Scope and Sequence with NMSI’s Laying the Foundation

lessons and Teacher Led Instruction Ideas

Unit 4: Transformations

5 days of instruction plus assessment time – 1.5 weeks

Unit Teacher Note: Students should have been exposed to translation and reflection of linear, quadratic, and piecewise functions in the coordinate plane.

NMSI’s Laying the Foundation lesson- Linear Reflections (1 day)

Teacher Note: Patty paper may be helpful. Check Additional Resources folder for teacher-made activities that could be helpful. These resources are available to be changed and edited as necessary to meet student needs. Additional websites are also cited. ** Use Unit 4 Checkpoints: #1 – # 4 following this lesson.

NMSI’s Laying the Foundation: Transformation of Piecewise Functions (2days)

Teacher Note: Calculate slopes and lengths of line segments, graph translations and reflections in function notation, write transformational functions from verbal descriptions. ** Use Unit 4 Checkpoint: #5 following this lesson.


Common Core

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.2

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.

G.CO.6


Common Core

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.2

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

G.CO.4

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.

G.CO.5


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.

G.CO.5

Teacher Led Instruction (2 days) Transformation of Shapes Teacher Note: There are NCTM Illuminations activities in the Additional Resources folder that will address these standards.


Common Core

3

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

G.CO.3

6

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

G.CO.4

14

Verify experimentally the properties of dilations given by a center and a scale factor. a. A dilation takes a line not passing through the center of the dilation to a parallel line and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

G-SRT1 G-SRT1a G-SRT1b



Unit 5: Similarity 5 days of instruction plus assessment time – 1 week

Teacher Led Instruction (½ day)

Dilating a Line

Teacher Note: Students should explore the task Dilating a Line from the Illustrative Mathematics website. http://www.illustrativemathematics.org/illustrations/602 This task will help the teacher assess the students’ prior knowledge and address the vocabulary necessary for this unit.


Common Core

#14

Verify experimentally the properties of dilations given by a center and a scale factor. a. a. A dilation takes a line not passing through the center of the dilation to a

parallel line and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

G-SRT1

G-SRT1a G-SRT1b

NMSI’s Laying the Foundation lesson: Similar Triangles (1 day)

Teacher Note: In this lesson, students discover in similar triangles the lengths of the sides are proportional and the corresponding angles are congruent. The activity builds so that when students complete number 5, they will have addressed the AL COS # 14 requirement. A discussion of center and scale factor should be incorporated into this lesson. Also to address AL COS #15, the teacher will need to include the concept of transformations as they relate to similarity.


Common

Core

#14

Verify experimentally the properties of dilations given by a center and a scale factor. a. a. A dilation takes a line not passing through the center of the dilation to a

parallel line and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

G-SRT1

G-SRT1a G-SRT1b

#15 Given two figures, use the definition of similarity in terms of similarity

transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

G-SRT2

**Unit 5 Checkpoint Problems: #6 - #9 address transformations using the coordinate plane. These problems could be used to begin the discussion on similarity or combined with other problems to access students’ understanding of transformations.



Teacher Led Instruction (½ day) Similar Triangles Teacher Note: This task specifically addresses AL COS # 16. If the teacher is having trouble incorporating transformations into his/her lessons, then this task will help. http://www.illustrativemathematics.org/illustrations/1422

#16 Use the properties of similarity transformations to establish the angle-angle (AA) criterion for two triangles to be similar.

G-SRT3

Teacher Led Instruction (2 days) Solving Problems using Congruence & Similarity for Triangles Teacher Note: A variety of tasks that address this standard are on the Illustrative Mathematics website. http://www.illustrativemathematics.org/HSG-SRT.B

** Unit 5 Checkpoint: #1 – #5 require students to apply similarity criteria to solve the problems. These problems can be included in the 2 days set aside for solving problems using similarity for triangles.

#18 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

G-SRT5


http://www.illustrativemathematics.org/HSG-SRT.B



Unit 6: Triangle Congruence 8 days of instruction plus assessment time – 2 weeks

Teacher Led Instruction (8 days) Proving Triangles Congruent

Teacher Note: Use models of congruent/non-congruent triangles with angles/sides labeled to demonstrate the theorems that work and the combinations that do not prove triangles congruent. (See Congruent Triangles Demo in Additional Resources). Give each student a triangle and have them find their partner with the same three attributes measured. Pairs determine if triangles are congruent or not. Debrief with Intro to Congruent Triangles (See Additional Resources). Practice naming congruent triangles and the theorem used to prove congruence before progressing to 2-column proofs. Most textbooks have excellent sets of practice for the proving of congruent triangles.


Common Core

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.

G-CO6

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-CO7

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

G-CO8

10 Prove theorems about triangles. Theorems include measures of interior angles of a triangle sum to 180º, 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, and the medians of a triangle meet at a point.

G-CO10

18 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

G-SRT5



Unit 7: Relationships in Triangles 12 days of instruction plus assessment time - 2.5 weeks

NMSI’s Laying the Foundation lesson: Minimum Cable Length (3 days)

Teacher Note: Real world application of standard.


Common Core

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; and points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

G-CO9

Teacher Led Instruction (2 days) Points of Concurrency Teacher Note: Standard can be easily demonstrated with Patty Paper and verified using dynamic geometry software and/or graphing calculator. Include centroid, orthocenter, incenter and circumcenter. See graphic organizer and Illustrative Math lesson in Additional Resources folder. Joining Midpoints of Sides of a Triangle: http://www.illustrativemathematics.org/illustrations/1095


Common Core

10

Prove theorems about triangles. Theorems include measures of interior angles of a triangle sum to 180º, 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, and the medians of a triangle meet at a point

G-CO10

17

Prove theorems about triangles. Theorems include a line parallel to one side of a triangle divides the other two proportionally, and conversely; and the Pythagorean Theorem proved using triangle similarity.

G-SRT4

Teacher Led Instruction (1 day) Parallel Lines Cut by a Transversal Teacher Note: Specifically focus on the proportionality theorem based on parallel lines cut by two transversals.


Common Core


G-SRT5



Teacher Led Instruction (3 days) Inscribed/Circumscribed Triangle Constructions Teacher Note: Build on the points for concurrency definitions to construct using incenter and circumcenter. Meeting for Lunch lesson from NCTM is an excellent real world application for this problem. Dividing a Town into Pizza Delivery Regions http://illuminations.nctm.org/Lesson.aspx?id=2688 Another practical application of circumcenter . Both lessons can be found in the Additional Resources folder.


Common Core

26 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

G-C3

http://illuminations.nctm.org/Lesson.aspx?id=2688



Unit 8: Right Triangles

12 days of instruction plus assessment time – 3 weeks

Unit Teacher Note: Check Additional Resources folder for teacher-made activities that could be helpful. These resources are available to be changed and edited as necessary to meet student needs. NMSI’s Laying the Foundation lessons: Pythagorean Theorem Application (1 day)

Teacher Note: Complete #1-6 and #11-12 Students should know Pythagorean Triples. This lesson can be used for whole group or small group instruction. ** Use Unit 8 Checkpoints: #1- 3 and #7 following this lesson


Common Core

21 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems

G-SRT.8

NMSI’s Laying the Foundation lessons: Hold that Fold to a Minimum (1 day)

Teacher Note: Introduction to right triangles. Teachers may need to lead students through the discovery of similar triangles. Students should be familiar with the use of rulers. **Use Unit 8 Checkpoints: #5 – 6, #14, Free Response # 1 after this lesson.


Common Core

12

Make formal geometric constructions with a variety of tools and methods such as compass and straightedge, string, reflective devices, paper folding, and dynamic geometric software. Constructions include 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.

G.CO.12

19 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

G.SRT.6

21 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

G.SRT.8


NMSI’s Laying the Foundation lessons: Radical Probability - Simplifying Radical Expressions (1 day)

Teacher Note: Use as an introduction to the Pythagorean Theorem. Students must have experience with simple and compound probability and simplifying radicals. This lesson extends from Algebra I and introduces Algebra II standards.

NMSI’s Laying the Foundation lessons: Introduction to Related Rates: Pythagorean Theorem (1 day)

Teacher Note: Students should be able to apply the Pythagorean Theorem. This lesson is an excellent introduction to angles of elevation and angles of depression.


Common Core


G.SRT.8

Fishermen, Balloons, Lasers, and Bicycles: Related Rates- New Lesson Teacher Note: #1 is a fisherman on a cliff and introduces the dr/dt and reinforces Pythagorean theorem. #2 is about a hot air balloon and discussed dh/dt and the Pythagorean theorem. #3 is a patient that is having a laser procedure with same correlation. #4 is about two cyclists traveling apart and how their distance is changing. All of these situations are found in related rate problems in AP Calculus.


Common Core


G.SRT.8

23 (+) Prove the Laws of Sines and Cosines and use them to solve problems. G-SRT.10

Shadows and Ladders- Related Rates Teacher Note: #1 is about a shadow with graphs and connections to dh/dt. The idea of similar triangles of triangle formed by light and total shadow and person and shadow is introduced.#2 is the ladder sliding down the house problem with reference to the rate of change of the distance the ladder is from the ground. These are two famous problems in AP Calculus in Related Rates.


Common Core


G.SRT.8


G-SRT5


NMSI’s Laying the Foundation lessons: The Pythagorean Theorem in Motion (1 day)

Teacher Note: This lesson reinforces material taught in the 8th grade. It provides a discovery lesson for the Pythagorean Theorem. ** Use Unit 8 Checkpoints: Free Response 2 AND Free Response 3 following this lesson.


Common Core

NMSI’s Laying the Foundation lessons: A Piece of Pythagorean Pi (1 day)

Teacher Note: Complete #1-12b. Discover the type of triangle based on angles. Make sure to allow time to find the patterns. ** Use Unit 8 Checkpoints: #8 - 9 following this lesson.


Common Core

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.1


G.SRT.6


G.SRT.8

NMSI’s Laying the Foundation lessons: Introduction to Trigonometric Ratios with Special

Right Triangles (1 day)

Teacher Note: Notes: Students will develop the unit circle. Students should be familiar with special right triangles and six trig ratios. Arc length formula can be used to introduce radians in this lesson if desired. Good for developing the unit circle. ** Use Unit 8 Checkpoints: #10, #12, #15 following this lesson.


Common Core


G.SRT.6


NMSI’s Laying the Foundation lessons: Introduction to Related Rates: Trigonometric Ratios (1.5 day)

Teacher Note: Students should be familiar with trig ratios.


Common Core

7 Explain and use the relationship between the sine and cosine of complementary angles

G.SRT.7

NMSI’s Laying the Foundation lessons: Trigonometric Applications (1.5 day)

Teacher Note: Omit # 11 ** Use Unit 8 Checkpoints: #11, #13 following this lesson.


Common Core

21 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems

G-SRT.8


G-SRT.6

22 (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

G-SRT.9


NMSI’s Laying the Foundation lessons: Law of Sines (1 day)

Teacher Note: Omit # 11, Derivation of the Law of Sines. Students should be familiar with inscribed angles.


Common Core

21

Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

G-SRT.8

19 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle leading to definitions of trigonometric ratios for acute angles.

G-SRT.6

22 (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

G-SRT.9




Unit 9: Polygons 8 days of instruction plus assessment time – 2.5 weeks

NMSI’s Laying the Foundation lessons: Coordinate Geometry and Proof (3-4 days) Teacher Note: This activity uses slope, distance and midpoint to prove properties of quadrilaterals. Use graphic organizer and chart to debrief characteristics. **Unit 9 Checkpoint: #1 (would make a good entrance or exit ticket on parallelograms) **Unit 9 Checkpoint: #2, #3, Free Response 1, and Free Response 2 (coordinate proof problems—may be beneficial to complete in groups) **Unit 9 Checkpoint: #4 (would make a good group problem for rhombi)


Common Core

11 Prove theorems about parallelograms. Theorems include opposite sides are congruent, opposite angles are congruent; the diagonals of a parallelogram bisect each other; and conversely, rectangles are parallelograms with congruent diagonals. D.2.g. Identify and classify quadrilaterals, including parallelograms, rectangles, rhombi, squares, kites, trapezoids, and isosceles trapezoids, using their properties

G-CO11


NMSI’s Laying the Foundation lessons: Parallel and Perpendicular Lines (Independent Assignment) (1 Day- Optional) Teacher Note: This activity has students verify a shape is a rectangle. Students are then challenged to create a parallelogram that is not a rectangle or square. This activity does not investigate diagonals. Teachers may want to develop an additional investigation/activity to look at properties of diagonals.


Common Core

11 Prove theorems about parallelograms. Theorems include opposite sides are congruent, opposite angles are congruent; the diagonals of a parallelogram bisect each other; and conversely, rectangles are parallelograms with congruent diagonals

G-CO11



Teacher Led Instruction (2 days) Interior / Exterior Angle Sums in Polygons Teacher Note: The textbook should do a nice job with this. See Angles of a Polygon activity in Additional Resources folder.


Common Core


G-SRT5

NMSI’s Laying the Foundation lessons: Polygons I (2 days)

Teacher Note: Pre-requisite: calculate area and perimeter of general polygons, calculate circumference and area of circles and have experience with special right triangles. The following links may also be helpful for compass/straightedge constructions http://www.mathopenref.com/constequilateral.html http://www.mathopenref.com/printsquare.html http://www.mathopenref.com/constinhexagon.html Polygons I is an excellent reinforcement of trigonometry as well as an introduction to circles. Graphing calculator required. (Omit column for area which will be addressed later.)


Common Core

13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. G-CO13

http://www.mathopenref.com/constequilateral.html

http://www.mathopenref.com/printsquare.html

http://www.mathopenref.com/constinhexagon.html


Geometry and Pre-AP Geometry Scope and Sequence with NMSI’s Laying the Foundation

lessons and Teacher Led Instruction

Unit 10: Circles

15 days of instruction plus assessment time-3 weeks

Unit Teacher Note: Check Additional Resources folder for teacher-made activities that could be helpful. These resources are available to be changed and edited as necessary to meet student needs.

NMSI’s Laying the Foundation lesson: Approximating Pi with Measurements (2 days)

Teacher Note: Circumference, histograms, and scatterplots are used in this lesson. The students should have worked with these concepts in 6, 7, 8th grades and Algebra I. This lesson will serve as a great review of these concepts. This lesson will lend easily to proving that all circles are similar.


Common Core

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.

D.3.a Identify and define line segments associated with circles (e.g., radii, diameters,

chords, secants, tangent)

G-CO1

24 Prove that all circles are similar. G-C1

NMSI’s Laying the Foundation lesson: Limiting Cases in Geometry ( 8 days)

Teacher Note: This lesson involves understanding measures of angles and segment lengths created by tangent lines and/or secant lines in circles. Students will need an understanding of the theorems related to the measures of arcs, angles and length of the segments related to circles prior to beginning this lesson. Students are asked to find perimeter and area of rectangles and triangles in this lesson. This will be a review of concepts taught in 7th and 8th grade math. You will only use exercises 1-4 and 7-8 in this lesson.

Additional Resources for this lesson and be found:



http://www.regentsprep.org/Regents/math/geometry/GP14/indexGP14.htm


http://alex.state.al.us/lesson_view.php?id=6690

**Unit 10 Checkpoint: #2-5 following this lesson.

**Unit 10 Free Response question could be used as a small group activity.





http://alex.state.al.us/lesson_view.php?id=6690



Common Core

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-CO1

25

Identify and describe relationships among inscribed angles, radii, and chords.

Include the relationship between central, inscribed, and circumscribed angles;

inscribed angles on a diameter are right angles; the radius of a circle is perpendicular

to the tangent where the radius intersects the circle.

D.3.b Determine the measure of central and inscribed angles and their intercepted

arcs.

D.3.c Find segment lengths, angles measures, and intercepted are measures formed

by chords, secants, and tangents intersecting inside and outside circles.

G-C2

26 Construct the inscribed and circumscribed circles of a triangle, and prove properties

of angles for a quadrilateral inscribed in a circle.

D.3.d Solve problems using inscribed and circumscribed polygons.

G-C3

NMSI’s Laying the Foundation lesson: Equations of Tangent Lines (5 days)

Teacher Note: This lesson combines concepts of parallel and perpendicular lines with intersection of equations, writing equations of lines and coordinate geometry. Students should have an understanding of theorems related to lines tangent to a circle. Students should be introduced to the equation of a circle prior to beginning this lesson.

Additional Resources for this lesson can be found:

http://www.regentsprep.org/Regents/math/geometry/GCG6/indexGCG6.htm

** Checkpoint Unit 10: #1 use during this lesson.


Common Core

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-CO1

27 (+) Construct a tangent line from a point outside a given circle to the circle.

G-C4

29

Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

G-GPE1

30

Use coordinates to prove simple geometric theorems algebraically.

G-GPE4

http://www.regentsprep.org/Regents/math/geometry/GCG6/indexGCG6.htm



Unit 11: Geometric Measurement in Two Dimensions 12 days of instruction plus assessment time - 3 weeks

NMSI’s Laying the Foundation lesson: Accumulating Area (1 day)

Teacher Notes: This lesson can be used as a discovery lesson to briefly introduce the concept of area under the graph of a function. Although the concept of area is explored, accumulation of area is the focus of this lesson. Since the equation for lines are given as the boundary of the figure, students will get to review how to graph a line. ** Use Unit 11 Checkpoints: #1,#6,#8, and #9 following this lesson

#33

Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.*

G-GPE7

#41

Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost, working with typographic grid systems based on ratios).*

c.

G-MG3

NMSI’s Laying the Foundation lesson: Using Coordinate Geometry to Determine the Area of a Triangle (1 day) Teacher Notes: This activity uses multiple methods to find the area of a triangle in the coordinate plane.

Geometry #33


G-GPE7

NMSI’s Laying the Foundation lesson: Graphing and Finding Areas of Bounded Regions

(1 day) Teacher Notes: Students will need to know how to graph a line in order to complete this activity. It is a good activity to follow Accumulating Area because it requires students to find the area of triangles, rectangles, and trapezoids. ** Use Unit 11 Checkpoints: #4 , FR 2 and FR 3 following this lesson

#33


G-GPE7

#41

Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost, working with typographic grid systems based on ratios).*

G-MG3


NMSI’s Laying the Foundation lesson: Exploration of Rectangles with Set Perimeter (1 day) Teacher Notes: This is an open-ended activity that is designed to encourage students to conjecture questions that they might be able to answer about areas of rectangles with a perimeter of 32 cm.

Geometry #33


G-GPE7

NMSI’s Laying the Foundation lesson: Infinity in Geometry (problems 1 and 2) (1 day) Teacher Notes: Students should know the basic area formulas. These problems illustrate ways to use concepts of infinity to intuitively justify area formulas in geometry. ** Use Unit 11 Checkpoints: #2 following this lesson

#35

Give an informal argument for the formulas for the circumference of a circle; area of a circle; and volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.

G-GMD1

Teacher Led Instruction (1 day) Sectors of Circles http://map.mathshell.org/materials/lessons.php?taskid=441&subpage=concept

Teacher Notes: This lesson is intended to help teachers assess how well students are able to solve problems involving area and arc length of a sector of a circle using radians. It assumes familiarity with radians and should not be treated as an introduction to the topic. This lesson is intended to help teachers identify and assist students who have difficulties in computing perimeters, areas, and arc lengths of sectors using formulas and in finding the relationships between arc lengths, and areas of sectors after scaling. ** Use Unit 11 Checkpoints: #3, #10-15 and FR 1 following this lesson

#35

Give an informal argument for the formulas for the circumference of a circle; area of a circle; and volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.

G-GMD1

#28

Derive, using similarity, the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

G-C5

Teacher Led Instruction (1 day) Inscribing and Circumscribing Right Triangles http://map.mathshell.org/materials/lessons.php?taskid=403&subpage=problem Teacher Notes: This lesson is intended to help teachers assess how well students are able to use geometric properties to solve problems. Students will decompose complex shapes into simpler ones in order to solve a problem, bring together several geometric concepts to solve a problem, and find the relationship between radii of inscribed and circumscribed circles of right triangles.

#34 Determine areas and perimeters of regular polygons, including inscribed or circumscribed polygons, given the coordinates of vertices or other characteristics.

http://map.mathshell.org/materials/lessons.php?taskid=441&subpage=concept

http://map.mathshell.org/materials/lessons.php?taskid=403&subpage=problem


NMSI’s Laying the Foundation lesson: Area as a Function of an Angle (1 day) Teacher Notes: Students use trigonometric ratios and special right triangle relationships to calculate characteristics of triangles.

#18

Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

G-SRT5

NMSI’s Laying the Foundation lesson: Special Right Triangle Applications (Problems 1 – 8) (1 day) Teacher Notes: In this lesson, students use the properties of special right triangles to find perimeters, areas, surface areas, and volumes. Problems 1 – 8 are for two dimensions. ** Use Unit 11 Checkpoints: #5 and #7 following this lesson

#18

Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

G-SRT5

#28

Derive, using similarity, the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

G-C5

NMSI’s Laying the Foundation lesson: Carnival Probability (1 ½ days) Teacher Notes: Students will use simulations to experiment with probability and then apply the areas of geometric figures to compute theoretical probability.

#42 (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).

S-MD6

#43 (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

S-MD7

NMSI’s Laying the Foundation lesson: Geometric Probability (1 ½ days) Teacher Notes: Students will explore probability problems. ** Use Unit 11 Checkpoints: #16,#17,#18 following this lesson

#42 (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).

S-MD6

#43 (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

S-MD7

Teacher Led Instruction (2 days) Real World Application of Area Teacher Notes: This website has several options to address this standard.

http://ccssmath.org/

#40 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, British Thermal Units (BTUs) per cubic foot).*

G-MG2

http://ccssmath.org/



Unit 12: Geometric Measurements in Three Dimensions 18 days of instruction plus assessment time – 4 weeks

Teacher Led Instruction (2 days) Deriving Volume Formulas

Teacher Note: Use 3-D models of cylinders, cones, pyramids and prisms with same height and base to allow students to discover the relationships between the formulas. Use Cavalieri’s Principle to Compare Aquarium Volumes: https://www.illustrativemathematics.org/illustrations/530


Common

Core

35 Give an informal argument for the formulas for the circumference of a circle; area

of a circle; and volumes of a cylinder, pyramid, and cone. Use dissection

arguments, Cavalieri’s principle, and informal limit arguments.

G-GMD1

NMSI’s Laying the Foundation lesson: Using Linear Equations to Define Geometric Solids (2 days)

Teacher Note: Defines geometric solids by revolving regions about horizontal and vertical lines. Take note of “Teacher Notes” which encourage modeling of revolutions. It is also helpful to practice procedure of drawing solids on the coordinate plane before beginning activity. Activity includes cylinders and cones as well as compositions. **Unit 13 Checkpoints #1 - #3, #5, #8 could be used BEFORE the activity to check for basic application of volume formulas.


Common

Core

36 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.* G-GMD3

38 Identify the shapes of two-dimensional cross-sections of three-dimensional objects,

and identify three-dimensional objects generated by rotations of two-dimensional

objects.

G-GMD4

Teacher Led Instruction (2 days) Calculating Surface Area and Volume Quality Core Objectives to be covered with this unit. Most textbooks include excellent practice problems. Graphic Organizer in Additional Resources can be helpful with formulas.


Common Core

https://www.illustrativemathematics.org/illustrations/530


NMSI’s Laying the Foundation lesson: Special Right Triangle Application (1 day) Teacher Note: (#1 – #8 were completed during 2-D Unit.) #9 of this activity will use a cross-section of a cube to form a pyramid. #10 – 13 define 3-D figures based on rotations and find the surface area and volume of resulting figure. Students will apply the Special Right Triangle formulas.


Common

Core

36 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.*

G-GMD3



objects.

G-GMD4

NMSI’s Laying the Foundation lesson: Volumes of Revolution (3 days) Teacher Note: Use as an extension after Using Linear Equations to Define Geometric Solids. Figures include cylinder with cylinder removed, cone, sphere, and rotation of regions bounded by parabolas. ** Unit 12 Checkpoints: #4, #6, #7, #9, #10 can be used after this lesson


Common

Core




objects.

G-GMD4

NMSI’s Laying the Foundation lesson: Infinity in Geometry ( ½ day) Teacher Note: (#1 – #2 were completed during 2-D Unit.) #3 of this activity will apply the volume of a sphere to derive the surface area of a sphere.


Common

Core


NMSI’s Laying the Foundation lesson: Working with Formulas and Function Notation ( ½ day) Teacher Note: This activity can be used as a review of 2-D and 3-D formulas during which students will manipulate formulas using function notation.


Common

Core



NMSI’s Laying the Foundation lesson: Introduction to Related Rates: Areas, Volumes and Similar Figures (3 days) Teacher Note: Focus on areas and volumes (surface area not included). But, during debriefing teacher could discuss relationship between similar areas / similar surface areas. This activity would be an excellent opportunity for students to work together and then present a specifically assigned problem to the class. ** Unit 12 Checkpoint: Free Response 1 – 5 could be used after this lesson


Common

Core 37 Determine the relationship between surface areas of similar figures and volumes of

similar figures. AL

Teacher Led Instruction (4 days) Modeling Problems that simulate 3-D Objects Teacher Note: http://www.illustrativemathematics.org/HSG-MG.A 20 different modeling/design lessons to choose from at this Illustrative site. Assign student groups different problems as a project to solve and present to the class.


Common

Core

39 Use geometric shapes, their measures, and their properties to describe objects (e.g.,

modeling a tree trunk or a human torso as a cylinder).* G-MG1

40 Apply concepts of density based on area and volume in modeling situations (e.g.,

persons per square mile, British Thermal Units (BTUs) per cubic foot).* G-MG2

41

Apply geometric methods to solve design problems (e.g., designing an object or

structure to satisfy physical constraints or minimize cost, working with typographic

grid systems based on ratios).*

G-MG3

http://www.illustrativemathematics.org/HSG-MG.A

9 days of instruction plus assessment time 2.5 weeks€¦ · · 2015-08-21**unit 1: checkpoints...

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