Download Unit 1: Points / Lines / Planes

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
1
Common Core Standard
Common
Core
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.
AL COS
12
Common Core Standard
Common
Core
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 Page 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.
AL COS
30
Common Core Standard
Use coordinates to prove simple geometric theorems algebraically.
Common
Core
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
AL COS
32
Common Core Standard
Find the point on a directed line segment between two given points that partitions
the segment in a given ratio.
Geometry and PreAP Geometry
Unit 1 Page 2
Common
Core
G-GPE6
Geometry and PreAP Geometry Scope and Sequence with NMSI’s Laying the Foundation
lessons and Teacher Led Instruction Ideas
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.
AL COS
Common Core Standard
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.
AL COS
Common Core Standard
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.
AL COS
9
Common Core Standard
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.
Geometry and PreAP Geometry
Unit 2 Page 1
Common
Core
G-CO9
Geometry and PreAP Geometry Scope and Sequence with NMSI’s Laying the Foundation
lessons and Teacher Led Instruction Ideas
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
AL COS
1
31
Common Core Standard
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 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).
Common
Core
G-CO1
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/equationsof-parallel-and-perpendicular-lines
**Use Unit 3 Checkpoint: #2, # 5 and the Free Response following this lesson.
AL COS
31
Common Core Standard
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).
Geometry and PreAP Geometry
Unit 3 Page 1
Common
Core
G-GPE5
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/anglesformed-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.
AL COS
9
Common Core Standard
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.
Common
Core
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
AL COS
12
Common Core Standard
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.
Geometry and PreAP Geometry
Unit 3 Page 2
Common
Core
G-CO12
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.
AL COS
2
4
5
Common Core Standard
Common
Core
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).
Develop definitions of rotations, reflections, and translations in terms of angles,
circles, perpendicular lines, parallel lines, and line segments.
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.2
G.CO.4
G.CO.5
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.
AL COS
2
6
Common Core Standard
Common
Core
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).
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.
Geometry and PreAP Geometry
Unit 4 Page 1
G.CO.2
G.CO.6
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.
AL COS Common Core Standard
Common
Core
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the
3
rotations and reflections that carry it onto itself.
G.CO.3
Develop definitions of rotations, reflections, and translations in terms of angles,
6
circles, perpendicular lines, parallel lines, and line segments.
G.CO.4
Verify experimentally the properties of dilations given by a center and a scale
G-SRT1
14
factor.
G-SRT1a
a. A dilation takes a line not passing through the center of the dilation to a parallel G-SRT1b
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.
Geometry and PreAP Geometry
Unit 4 Page 2
Geometry and PreAP Geometry Scope and Sequence with NMSI’s Laying the Foundation
lessons and Teacher Led Instruction Ideas
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.
AL COS
a.#14
Common Core Standard
Common
Core
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
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.
AL COS
a.#14
#15
Common Core Standard
Common
Core
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.
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-SRT1
G-SRT1a
G-SRT1b
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.
Geometry and PreAP Geometry
Unit 6 Page 1
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.
Geometry and PreAP Geometry
Unit 6 Page 2
G-SRT5
Geometry and PreAP Geometry Scope and Sequence with NMSI’s Laying the Foundation
lessons and Teacher Led Instruction Ideas
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.
AL COS Common Core Standard
Common
Core
6
Use geometric descriptions of rigid motions to transform figures and to predict
G-CO6
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.
7
Use the definition of congruence in terms of rigid motions to show that two
G-CO7
triangles are congruent if and only if corresponding pairs of sides and
corresponding pairs of angles are congruent.
8
Explain how the criteria for triangle congruence, angle-side-angle (ASA), sideG-CO8
angle-side (SAS), and side-side-side (SSS), follow from the definition of
congruence in terms of rigid motions.
10
18
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.
Use congruence and similarity criteria for triangles to solve problems and to prove
relationships in geometric figures.
Geometry and PreAP Geometry
Unit 6 Page 1
G-CO10
G-SRT5
Geometry and PreAP Geometry Scope and Sequence with NMSI’s Laying the Foundation
lessons and Teacher Led Instruction Ideas
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.
AL COS Common Core Standard
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.
Common
Core
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
AL COS
10
17
Common Core Standard
Common
Core
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
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-CO10
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.
AL COS Common Core Standard
Common
Core
18
Use congruence and similarity criteria for triangles to solve problems and to prove G-SRT5
relationships in geometric figures.
Geometry and PreAP Geometry
Unit 7 Page 1
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.
AL COS
26
Common Core Standard
Construct the inscribed and circumscribed circles of a triangle, and prove
properties of angles for a quadrilateral inscribed in a circle.
Geometry and PreAP Geometry
Unit 7 Page 2
Common
Core
G-C3
Geometry and PreAP Geometry Scope and Sequence with NMSI’s Laying the Foundation
lessons and Teacher Led Instruction Ideas
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
AL COS
21
Common Core Standard
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in
applied problems
Common
Core
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.
AL COS
12
19
21
Common Core Standard
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.
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.
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in
applied problems.
Geometry and PreAP Geometry
Unit 8 Page 1
Common
Core
G.CO.12
G.SRT.6
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.
AL COS Common Core Standard
Common
Core
21
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in
applied problems.
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.
AL COS
Common Core Standard
Common
Core
21
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in
applied problems.
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.
AL COS
Common Core Standard
Common
Core
21
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in
applied problems.
G.SRT.8
18
Use congruence and similarity criteria for triangles to solve problems and to prove
relationships in geometric figures.
G-SRT5
Geometry and PreAP Geometry
Unit 8 Page 2
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.
AL COS
Common Core Standard
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.
AL COS
1
19
21
Common Core Standard
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.
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.
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in
applied problems.
Common
Core
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.
AL COS
19
Common Core Standard
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.
Geometry and PreAP Geometry
Unit 8 Page 3
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.
AL COS Common Core Standard
7
Explain and use the relationship between the sine and cosine of complementary
angles
Common
Core
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.
AL COS
21
19
Common Core Standard
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in
applied problems
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.
Common
Core
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
23
(+) Prove the Laws of Sines and Cosines and use them to solve problems.
G-SRT.10
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.
AL COS
21
19
22
23
Common Core Standard
Common
Core
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in
applied problems.
G-SRT.8
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.
(+) 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.6
(+) Prove the Laws of Sines and Cosines and use them to solve problems.
G-SRT.10
Geometry and PreAP Geometry
Unit 8 Page 4
G-SRT.9
Geometry and PreAP Geometry Scope and Sequence with NMSI’s Laying the Foundation
lessons and Teacher Led Instruction Ideas
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)
AL COS
11
30
Common Core Standard
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
Use coordinates to prove simple geometric theorems algebraically.
Common
Core
G-CO11
G-GPE4
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.
AL COS
11
30
Common Core Standard
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
Use coordinates to prove simple geometric theorems algebraically.
Geometry and PreAP Geometry
Unit 9 Page 1
Common
Core
G-CO11
G-GPE4
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.
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Common Core Standard
Use congruence and similarity criteria for triangles to solve problems and to prove
relationships in geometric figures.
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.)
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Common Core Standard
Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
Geometry and PreAP Geometry
Unit 9 Page 2
Common
Core
G-CO13
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.
AL COS
Common Core Standard
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.
24
Prove
that all circles
are similar.
D.3.a Identify
and define
line segments associated with circles (e.g., radii, diameters,
chords, secants, tangent)
Common
Core
G-CO1
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://illuminations.nctm.org/Lesson.aspx?id=2417
http://illuminations.nctm.org/Lesson.aspx?id=3777
http://www.regentsprep.org/Regents/math/geometry/GP14/indexGP14.htm
http://www.regentsprep.org/Regents/math/geometry/GP15/indexGP15.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.
Geometry and PreAP Geometry
Unit 10 Page 1
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Common Core Standard
Common
Core
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
Identify and describe relationships among inscribed angles, radii, and chords.
Include the relationship between central, inscribed, and circumscribed angles;
25
G-C2
inscribed angles on a diameter are right angles; the radius of a circle is perpendicular
to the tangent where the radius intersects the circle.
26
Construct the inscribed and circumscribed circles of a triangle, and prove properties
G-C3
D.3.b Determine the measure of central and inscribed angles and their intercepted
of angles for a quadrilateral inscribed in a circle.
arcs.
D.3.d Solve problems using inscribed and circumscribed polygons.
D.3.c Find
lengths,
anglesEquations
measures, and
intercepted
are measures
NMSI’s Laying
the segment
Foundation
lesson:
of Tangent
Lines
(5 days) formed
by chords, secants, and tangents intersecting inside and outside circles.
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.
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29
Common Core Standard
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.
(+) Construct a tangent line from a point outside a given circle to the circle.
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.
Use coordinates to prove simple geometric theorems algebraically.
30
Geometry and PreAP Geometry
Unit 10 Page 2
Commo
n Core
G-CO1
G-C4
G-GPE1
G-GPE4
Geometry and PreAP Geometry Scope and Sequence with NMSI’s Laying the Foundation
lessons and Teacher Led Instruction Ideas
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
#41
Use coordinates to compute perimeters of polygons and areas of triangles and
rectangles, e.g., using the distance formula.*
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-GPE7
G-MG3
c.
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
Use coordinates to compute perimeters of polygons and areas of triangles and
rectangles, e.g., using the distance formula.*
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
#41
Use coordinates to compute perimeters of polygons and areas of triangles and
rectangles, e.g., using the distance formula.*
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).*
Geometry and PreAP Geometry
Unit 11 Page 1
G-GPE7
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 Use coordinates to compute perimeters of polygons and areas of triangles and
G-GPE7
#33
rectangles, e.g., using the distance formula.*
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
#28
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.
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-GMD1
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.
Geometry and PreAP Geometry
Unit 11 Page 2
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
#28
Use congruence and similarity criteria for triangles to solve problems and to prove
relationships in geometric figures.
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-SRT5
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
S-MD6
number generator).
#43
(+) Analyze decisions and strategies using probability concepts (e.g., product testing,
S-MD7
medical testing, pulling a hockey goalie at the end of a game).
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
#43
(+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random
number generator).
(+) Analyze decisions and strategies using probability concepts (e.g., product testing,
medical testing, pulling a hockey goalie at the end of a game).
S-MD6
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).*
Geometry and PreAP Geometry
Unit 11 Page 3
G-MG2
Geometry and PreAP Geometry Scope and Sequence with NMSI’s Laying the Foundation
lessons and Teacher Led Instruction Ideas
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
AL COS Common Core Standard
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
G-GMD1
arguments, Cavalieri’s principle, and informal limit arguments.
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.
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Common Core Standard
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.*
Identify the shapes of two-dimensional cross-sections of three-dimensional objects,
and identify three-dimensional objects generated by rotations of two-dimensional
objects.
Common
Core
G-GMD3
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.
AL COS
Common Core Standard
Geometry and PreAP Geometry
Common
Core
Unit 12 Page 1
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.
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Common Core Standard
36
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.*
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.
Common
Core
G-GMD3
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
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38
Common Core Standard
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.*
Identify the shapes of two-dimensional cross-sections of three-dimensional objects,
and identify three-dimensional objects generated by rotations of two-dimensional
objects.
Common
Core
G-GMD3
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.
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Common Core Standard
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.*
Common
Core
G-GMD3
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.
AL COS
36
Common Core Standard
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.*
Geometry and PreAP Geometry
Unit 12 Page 2
Common
Core
G-GMD3
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
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37
Common Core Standard
Determine the relationship between surface areas of similar figures and volumes of
similar figures.
Common
Core
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.
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40
41
Common Core Standard
Use geometric shapes, their measures, and their properties to describe objects (e.g.,
modeling a tree trunk or a human torso as a cylinder).*
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).*
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).*
Geometry and PreAP Geometry
Unit 12 Page 3
Common
Core
G-MG1
G-MG2
G-MG3