Monroe Township Schools · Monroe Township Schools Mission and Goals Mission The mission of the...

Monroe Township Schools

Curriculum Management System

Geometry Grade 10 July 2005

* For adoption by all regular education programs Board Approved: August 2005

as specified and for adoption or adaptation by all Special Education Programs in accordance with Board of Education Policy # 2220.

Table of Contents

Monroe Township Schools Administration and Board of Education Members Page 3 Acknowledgments Page 4 District Mission Statement and Goals Page 5 Introduction/Philosophy/Educational Goals Pages 6 National and State Standards Page 7 Scope and Sequence Page 8-9 Goals/Objectives/Instructional Tools/Activities Pages 10-50 Benchmarks Page 51

2

MONROE TOWNSHIP SCHOOL DISTRICT

ADMINISTRATION

Dr. Ralph P. Ferrie, Superintendent

Dr. Christopher H. Tienken, Assistant Superintendent

BOARD OF EDUCATION

Ms. Kathy Kolupanowich, President Mr. Lew Kaufman, Vice President

Mr. Marvin Braverman Ms. Carol Haring

Mr. Joseph Homoki Mr. John Leary

Ms. Kathy Leonard Mr. Harold Pollack Ms. Amy Speizer

JAMESBURG REPRESENTATIVE

Ms. Patrice Faraone

Student Board Members

Mr. Darren Goldberg

Mr. Brian Hackett Mr. George Leonard

3

Acknowledgments

The following individuals are acknowledged for their assistance in the preparation of this Curriculum Management System: Writers Names: Susan Patikowski Supervisor Name: Robert O’Donnell, Supervisor of Mathematics and Educational Technology Technology Staff: Al Pulsinelli Reggie Washington Bill Wetherill Secretarial Staff: Debby Gialanella Geri Manfre Gail Nemeth

4

MMoonnrrooee TToowwnnsshhiipp SScchhoooollss Mission and Goals

Mission

The mission of the Monroe Township School District, a unique multi-generational community, is to collaboratively develop and facilitate programs that pursue educational excellence and foster character, responsibility, and life-long learning in a safe, stimulating, and challenging environment to empower all individuals to become productive citizens of a dynamic, global society.

Goals

To have an environment that is conducive to learning for all individuals. To have learning opportunities that are challenging and comprehensive in order to stimulate the intellectual, physical, social and emotional development of the learner. To procure and manage a variety of resources to meet the needs of all learners. To have inviting up-to-date, multifunctional facilities that both accommodate the community and are utilized to maximum potential. To have a system of communication that will effectively connect all facets of the community with the Monroe Township School District. To have a staff that is highly qualified, motivated, and stable and that is held accountable to deliver a safe, outstanding, and superior education to all individuals. 5

INTRODUCTION, PHILOSOPHY OF EDUCATION, AND EDUCATIONAL GOALS

Philosophy

Monroe Township Schools are committed to providing all students with a quality education resulting in life-long learners who can

succeed in a global society. The mathematics program, grades K-12, is predicted on that belief and is guided by the following six principals as stated by the National Council of Teachers of Mathematics (NCTM) in the Principles and Standards for School Mathematics, 2000. First, a mathematics education requires equity. All students will be given worthwhile opportunities and strong support to meet high mathematical expectations. Second, a coherent mathematics curriculum will effectively organize, integrate, and articulate important mathematical ideas across the grades. Third, effective mathematics teaching requires the following: a) knowing and understanding mathematics, students as learners, and pedagogical strategies, b) having a challenging and supportive classroom environment and c) continually reflecting on and refining instructional practice. Fourth, students must learn mathematics with understanding. A student’s prior experiences and knowledge will actively build new knowledge. Fifth, assessment should support the learning of important mathematics and provide useful information to both teachers and students. Lastly, technology enhances mathematics learning, supports effective mathematics teaching, and influences what mathematics is taught.

As students begin their mathematics education in Monroe Township, classroom instruction will reflect the best thinking of the day. Children will engage in a wide variety of learning activities designed to develop their ability to reason and solve complex problems. Calculators, computers, manipulatives, technology, and the Internet will be used as tools to enhance learning and assist in problem solving. Group work, projects, literature, and interdisciplinary activities will make mathematics more meaningful and aid understanding. Classroom instruction will be designed to meet the learning needs of all children and will reflect a variety of learning styles.

In this changing world those who have a good understanding of mathematics will have many opportunities and doors open to them throughout their lives. Mathematics is not for the select few but rather is for everyone. Monroe township Schools are committed to providing all students with the opportunity and the support necessary to learn significant mathematics with depth and understanding. This curriculum guide is designed to be a resource for staff members and to provide guidance in the planning, delivery, and assessment of mathematics instruction.

Educational Goals Geometry is the second course of the college preparatory sequence. It is designed to provide an in-depth analysis of the real world

system and expand upon the geometrical knowledge attained in the middle schools as well as expand the use of algebra topics learned in Algebra I. Topics included are: lines, angles, planes, logical reasoning, conditional statements, parallel and perpendicular lines, triangle classification, congruent triangles, triangle relationships, similar figures, right triangle trigonometry, quadrilaterals, transformations, circles, polygons, three dimensional figures, and geometric probability.

6

New Jersey State Department of Education Core Curriculum Content Standards

A note about Mathematics Standards And Cumulative Progress Indicators. The New Jersey Core Curriculum Content Standards for Mathematics were revised in 2002. The Cumulative Progress Indicators (CPI's) referenced in this curriculum guide refer to these new standards and may be found in the Curriculum folder on the district servers. A complete copy of the new Core Curriculum Content Standards for Mathematics may also be found at: http://www.nj.gov/njded/cccs/s4_math.htm

7

http://www.nj.gov/njded/cccs/s4_math.htm

Geometry

Scope and Sequence

Quarter I I. Lines, Angles, and Planes a. Identify points, lines, and planes b. Distance and midpoint – on a line graph and a coordinate plane. c. Segment bisectors d. Classify angles e. Angle bisectors f. Special Angle Pairs – adjacent, vertical, linear pair, complementary,

supplementary, perpendicular lines.

II. Logical Reasoning and Conditional Statements a. Patterns – next three terms and finding the nth term. b. Conditional Statements – conjectures, if-then statements, truth

values, and counterexamples. c. Postulates about points, lines, and segments. d. Algebraic proofs. e. Segment and angle proofs.

III. Parallel and Perpendicular Lines a. Angles formed by a transversal b. Angle relationships formed by parallel lines and a transversal c. Use slope to determine parallel or perpendicular d. Slope-Intercept form e. Identify angle relationships that produce parallel lines and proofs.

Quarter II IV. Congruent Triangles a. Classify triangles – angles and sides b. Properties of isosceles triangles c. Exterior angles and interior angles – angle sum theorem and exterior

angle theorem d. Congruent triangles and corresponding parts - CPCTC e. Congruent triangles – SSS, SAS, AAS, ASA theorems f. Congruent right triangles – LL, HA, LA, HL – emphasize HL g. Proofs using congruent triangles.

V. Triangle Relationships a. Lines in a triangle – median, altitude, angle bisector, perpendicular

bisector b. Triangle Inequalities – triangle inequality theorem, SAS Inequality,

SSS Inequality c. Determine if triangle is acute, right, or obtuse using the Pythagorean

Theorem

VI. Proportions and Similarity a. Use proportions to solve problems b. Scale factor and similarity postulates – AA, SSS, SAS c. Determining similar polygons d. Proportional parts of triangles e. Fractals (OPTIONAL)

8

Quarter III

VII. Right Triangles and Trigonometry a. Simplify Radicals b. Geometric Mean c. Pythagorean Theorem d. Special right triangles - 45-45-90 and 30-60-90 e. Trig Ratios – SOHCAHTOA f. Angles of Elevation and Depression g. Law of Sines and Law of Cosines

VIII. Quadrilaterals a. Properties of parallelograms and ensuring parallelograms b. Special Parallelograms – Rectangle, Rhombus, Square c. Trapezoids and medians (midsegments) d. Other quadrilaterals – kites, properties of quadrilaterals

IX. Transformations a. Translations and vector addition b. Reflections c. Rotations d. Tessellations (OPTIONAL) e. Dilations f. Vectors – vector sum, magnitude, direction

Quarter IV X. Circles a. Parts of a circle b. Circumference and Area c. Arcs, angles, and sectors d. Properties of chords – inscribed angles, inscribed quadrilaterals e. Properties of tangent lines and secant lines f. Equation of a circle (OPTIONAL)

XI. Polygons a. Identify and name b. Interior and exterior angles c. Area d. Geometric probability e. Networks (vertex-edge graphs)

XII. Three Dimensional Figures a. Nets b. Prisms and Cylinders – lateral area and surface area c. Prisms and Cylinders – volume d. Pyramids and Cones – lateral area and surface area e. Pyramids and Cones – volume

9

10

Topic: Lines, Angles, and Planes Curriculum Management System Grade Level/Subject: 10 / Geometry

Goal 1: The student will be able to identify and use parts and types of lines, angles, and planes in problems solving.

Sugg

este

d da

ys o

f In

stru

ctio

n

Objectives / Cluster Concepts / Cumulative Progress Indicators (CPI's) The student will be able to:

Essential Questions Sample Conceptual Understandings

Instructional Tools / Materials / Technology / Resources / Learning Activities / Interdisciplinary Activities / Assessment Model

11 1.1. Identify and use points, lines, and planes in space. (CPI 4.2.12 A1)

1.2. Find the length and midpoint of a

segment. (CPI 4.2.12 C1) 1.3. Identify and use segments, midpoints, and

segment bisectors. (CPI 4.2.12 C1) Topic 1: Continued Next Page

• Why do chairs sometimes wobble? • Identify collinear and coplanar points. • Name points, lines, line segments, and

planes both in words and by symbols • Coordinate Geometry: identify the x-axis, y-

axis, origin, quadrants, and plot points. • Introduce z-axis (mention) • Two lines intersect to form a point • Two planes intersect to form a line. • Skew lines are in two different planes, never

intersect, but are not parallel. • I want to make a garden that is 12 feet by 4

feet. If the border pieces are 4 feet long, how many pieces will I need?

• Name distance in words and by symbols. • Find the distance on a number line (whether

horizontal or vertical)

PROJECT IDEA: Plotting a picture. Students draw a picture on a coordinate plane, marking points on the grid. Students make a list of 50-100 points that outline their drawing. The list should be in order, so that they could give it to a friend in class and the friend could plot the points and draw the original picture. RESOURCES: • Geometry, Glencoe, © 2005 -Review of plotting points pg. 728 -1.1 pg. 6-12 HSPA: • Amsco: MATHEMATICS: Preparing for the

New Jersey HSPA, Grade 11 (navy blue binding – paperback)

- Cluster 2.A.1Geometric Terms – pg. 23-26 • Prentice Hall: New Jersey HSPA

Mathematics Comprehensive Review (textbook - purple binding – paperback)

- 10.1 Points, Lines, Planes, and Segments pg. 206-210 RESOURCES: • Geometry, Glencoe, © 2005 -1.3 pg. 21-28 -Activity – The Pythagorean Theorem pg. 28



Sugg

este

d da

ys o

f In

stru

ctio

n




Objectives 1.2 and 1.3 continued. Topic 1: Continued Next Page

• Distance formula –

( ) ( )2 22 1 2 1D x x y y= − + −

• Pythagorean theorem: 2 2 2

2 2 2

a b cleg leg hyp

+ =

+ =• Midpoint formulas:

Number Line: 1 2

2x x+

Coordinate Plane: 1 2 1 2,2 2

x x y y+ +⎛ ⎞⎜ ⎟⎝ ⎠

Emphasize - coordinates. EX 1: Find AB if A(3, 2) and B(3, -5) EX 2: Find AB if A(-5, 3) and B(7, -2)

If M is the Midpoint of AB . EX 1: AM = 4x – 5, MB = 11 + 2x, find x, AB. EX 2: AB = 14x + 2, AM = 6-3x, find x, MB. EX 3: A(2, -2), B(2,8), find the coordinates of M. EX 4: A(-1,2), M(6,1), find the coordinates of B. • Segment addition postulate – see geometer

sketchpad activity pg. 101 • Mark segments congruent with slash marks. • Properties of the Midpoint: Midpoint is exactly in middle. Midpoint cuts the whole segment in half. The distance from the midpoint to either

endpoint is equal.

TECHNOLOGY: • Prentice Hall Mathematics: - TI – 83/84

PLUS Activities for Algebra, Geometry, and Algebra II (resource workbook) -Perpendicular Bisectors – pg. 53

11

12



Sugg

este

d da

ys o

f In

stru

ctio

n




1.4. Measure and Classify Angles. (CPI 4.2.12 C1) 1.5. Identify and use congruent angles in

problem solving. (CPI 4.2.12 C1) Topic 1: Continued Next Page

• Why when it is the brightest outside does no one use their visors in their cars, yet in the early morning or evening they do?

• Identify the sides, vertex, interior, and exterior of an angle.

• Name a ray in words and by symbols. • Name an angle by symbols, using one letter

(the vertex), three letters, and a number. • Know the difference between the symbols:

ABC and m ABC • Measure an angle using a protractor

(optional) • Angles are measured in degrees:

Emphasize - students must have degree signs next to angle measures.

• Classify Angles as acute, right, obtuse, or straight.

• Identify and label congruent angles • Use Algebra to find angle measurements. • Identify and use properties of angle

bisectors: Bisector is exactly in middle. Bisector cuts the whole angle in half. The measure of each angle on either

side of the bisector is equal.

EX. BDuuur

is the angle bisector of , If (8 10)OABD X= − and

, find x,

ABC

(10 20)oDBC x= − m ABC• Angle addition postulate

RESOURCES: • Geometry, Glencoe, © 2005 - 1.4 pg. 29-36 HSPA: • Prentice Hall: New Jersey HSPA


- 10.2 Rays and Angles pg. 211-214 TECHNOLOGY: • Prentice Hall Mathematics: - TI – 83/84

PLUS Activities for Algebra, Geometry, and Algebra II (resource workbook) - Angle Bisectors – pg. 55



Sugg

este

d da

ys o

f In

stru

ctio

n




1.6. Identify and use special types of angles and pairs of angles. (CPI 4.2.12 C1)

End, Topic 1

• Streets intersect to form many different types of angles, discuss the angels of roads around Monroe and Jamesburg.

• Identify and name adjacent angles • Identify vertical angles - congruent • Complementary angles - two angles whose

sum is 90o • Supplementary angles – two angles whose

sum is 180o • Linear Pair of angles form a line; they are

supplementary and their sum is 180o • Perpendicular lines form 4 right angles. • Use Algebra to find angle measurements

using each angle type listed above.

RESOURCES: • Geometry, Glencoe, © 2005 - 1.5 pg. 37-43 HSPA: • Amsco: MATHEMATICS: Preparing for the

New Jersey HSPA, Grade 11 (navy blue binding – paperback) - Cluster 2.A.3 – pg. 33-36


PLUS Activities for Algebra, Geometry, and Algebra II (resource workbook) - Linear Pairs – pg. 57 - Vertical Angles – pg. 59

13

14

Topic: Logical Reasoning and Conditional Statements Curriculum Management System Grade Level/Subject: 10 / Geometry

Goal 2: The student will be able to use logical reasoning and conditional statements to solve problems.

Sugg

este

d da

ys o

f In

stru

ctio

n




18 2.1. Identify the next three terms in a pattern. (CPI 4.3.12 A1)

2.2. Find the nth term of an arithmetic pattern. (CPI 4.2.13 A1)

2.3. Use and analyze conditional statements.

(CPI 4.2.12 A2, 4.3.12 A2, 4.5.12 D5) Topic 2: Continued Next Page

• If you start a savings account with $500, and add $20 each week, how much money will you have after 1 year?

• Visual patterns – repeating shapes. • Number patterns, arithmetic, geometric,

Fibonacci • Repeating patterns: what is the 38th term in

GEOMETRYGEO…? What is the 47th digit

in 17

? Or what is the units digit in ? 253

• In the pattern 4, 6, 8, 10, … what are the next three terms? What is the 40th term? What is the nth term?

• Think about some recent advertising

campaigns, what is the purpose of a marketing campaign? Why is it important to determine if a statement is true or false?

• Make a conjecture with given information. • State a counterexample for a false

conditional statement. EX. If you live in New Jersey, then you live in Monroe. Counterexample - Jamesburg or Old Bridge.

• Write conditional statements in if-then form. • Identify the hypothesis and the conclusion in

an if-then statement. • Identify a conditional statement, the inverse

and the converse of a conditional statement and determine the truth values of each.

HSPA: • Amsco: MATHEMATICS: Preparing for the

New Jersey HSPA, Grade 11 (navy blue binding – paperback) - Cluster 2.A.4 Inductive and Deductive Reasoning – pg. 36-38 - Cluster 4.A.1 Patterns – pg. 177-181 - Cluster 4.A.2 Sequences and Series – pg. 181-187 - Cluster 4.A.3 Representation of Relationships and Patterns – pg. 188-189

• Prentice Hall: New Jersey HSPA Mathematics Comprehensive Review (textbook - purple binding – paperback)

- 13.1 Patterns and Sequences pg. 298-301

RESOURCES: • Geometry, Glencoe, © 2005 -2.1 pg. 62-66 -2.3 pg. 75-81



Sugg

este

d da

ys o

f In

stru

ctio

n




2.4. Identify and use basic postulates about points, lines, and planes. (CPI 4.2.12 A4, 4.5.12 D3, 4.5.12 D5)

Topic 2: Continued Next Page

• Identify and use postulates about points, lines and planes:

- Through any two points there is exactly one line. - Through any three points there is exactly one plane. - A line contains at least two points. - A plane contains at least three points not on the same line. - If two points lie in a plane, then the entire line containing those points lies in that plane. - If two lines intersect then their intersection is exactly one point. - If two planes intersect then their intersection is a line. • Discuss parallel lines and skew lines use

two pens or a ruler and a pointer to show skew lines. Illustrate with the edges of a box.

• Discuss how to prove something is true, have students give paragraph proofs, verbally.

RESOURCES: • Geometry, Glencoe, © 2005 - 2.5 pg. 89-93

15

16



Sugg

este

d da

ys o

f In

stru

ctio

n




2.5. Complete algebraic proofs (CPI 4.2.12 A4, 4.5.12 D3, 4.5.12 D5)

2.6. Use two-column proofs to prove segment

relationships. (CPI 4.2.12 A4) Topic 2: Continued Next Page

• Review properties of Algebra: - Associative - Commutative - Reflexive - Symmetric - Transitive - Addition and Subtraction - Multiplication and Division - Substitution - Distributive • Solve an algebra problem such as 3(x-2)=42

and state which property is used each step. • Introduce a two-column proof with

statements and reasons. • Complete two-column algebraic proofs. • Discuss which of above properties hold true

for segments and angles (reflexive, symmetric, and transitive)

• Lawyers must prove each piece of evidence

for or against their clients. Can things in math be proven in the same way?

• Review segment addition postulate, midpoint theorem, reflexive, symmetric, and transitive properties with relation to segment congruence.

• Complete two column proofs with segment congruence.

RESOURCES: • Geometry, Glencoe, © 2005 - 2.6 pg. 94-100 RESOURCES: • Geometry, Glencoe, © 2005 - 2.7 pg. 101-106



Sugg

este

d da

ys o

f In

stru

ctio

n




2.7. Use two-column proofs to prove angle relationships. (CPI 4.2.12 A4, 4.5.12 D3, 4.5.12 D5)

End, Topic 2.

• Review angle addition postulate, angle bisectors, linear pairs, vertical angles, supplementary and complementary angles, perpendicular angles, reflexive, symmetric, and transitive properties with relation to angle congruence.

• Complete two column proofs with angle congruence.

RESOURCES: • Geometry, Glencoe, © 2005 - 2.7 pg. 107-114 ASSESSMENT MODEL: Assessment questions should be open ended and should follow the general format illustrated in the Essential Questions/Sample Conceptual Understandings section.

17

Topic: Parallel and Perpendicular Lines Curriculum Management System

Grade Level/Subject: 10 / Geometry

Goal 3: The student will be able to use angle relationships with parallel and perpendicular lines to solve problems.

Sugg

este

d da

ys o

f In

stru

ctio

n




13 3.1. Identify angles formed by two lines and a transversal. (CPI 4.2.12 A3, 4.2.12 A4, 4.2.12 C1)

3.2. Identify and use angle relationships

formed by two parallel lines and a transversal. (CPI 4.2.12 A3, 4.2.12 C1, 4.5.12 C1, 4.5.12 D3)


• When a house is built, Construction workers must up walls straight. They use beams in the walls called studs to stabilize the wall. How can the workers determine if the studs are parallel?

• Discuss parallel lines and parallel planes. • Introduce types of angles: Alternate

exterior, Alternate interior, Corresponding, Consecutive (Same-Side Interior)

• Identify angles when three lines cross (not parallel), and when four lines cross (two sets of parallel lines).

• A railroad train travels along two rails. The

two rails must be the same distance apart along the entire track. How can we make sure the train will not derail?

• Identify the alternate interior, alternate exterior, and corresponding angles, and identify the congruent angles

• Identify the consecutive angles and the supplementary angles, show relationship is supplementary.

• Congruent angles can be identified by making a “Z”, zigzagging across the transversal.

• Perpendicular transversal theorem. • Use algebra to solve problems involving

parallel lines and a transversal.

RESOURCES: • Geometry, Glencoe, © 2005 - 3.1 pg. 126-131 - Geometer sketchpad activity p. 132 HSPA: • Prentice Hall: New Jersey HSPA


- Intersecting, Perpendicular, and Parallel Lines pg. 215-219 RESOURCES: • Geometry, Glencoe, © 2005 - Geometer sketchpad activity pg. 132 - 3.2 pg. 133-138 TECHNOLOGY: • Prentice Hall Mathematics: - TI – 83/84

PLUS Activities for Algebra, Geometry, and Algebra II (resource workbook) - Parallel Lines, Related Angles – pg. 61

• Exploring Geometry with The Geometer’s Sketchpad (resource workbook)

- Angles formed by Intersecting Lines– pg.15 - Properties of Parallel Lines 17

18

Topic: Parallel and Perpendicular Lines Curriculum Management System Grade Level/Subject: 10 / Geometry

Goal 3: The student will be able to use angle relationships with parallel and perpendicular lines to solve problems.

Sugg

este

d da

ys o

f In

stru

ctio

n




3.3. Use the slope of a line to determine whether two lines are intersecting, parallel, or perpendicular. (CPI 4.2.12 C1, 4.3.12 B2)

3.4. Write the equation of a line in slope-

intercept form. (CPI 4.2.12 C1, 4.3.12 B2, 4.3.12 C1)

3.5. Identify angle relationships that produce

parallel lines. (CPI 4.2.12 A3, 4.2.12 A4, 4.2.12 C1, 4.3.15 B2, 4.5.12 C1)

End, Topic 3.

• When driving in the mountains, or a very hilly area, there are signs posted with pictures of a truck and percentages on them. What are these used for, what does the percentage mean?

• Review slope formula: 2 1

2 1

x xy y−−

• Review relationships of slopes of parallel and perpendicular lines.

• Review slope-intercept form of an equation. • Use slope to graph a line: given the slope

and a point, given slope-intercept form. • Review slope-intercept form. • Review point-slope form. • Write the equation of a line, given one point

and the slope, or given two points. • This is a review lesson from algebra, not an

introductory lesson. • Use two-column proofs and paragraph

proofs to prove angle relationships between parallel lines.

RESOURCES: • Geometry, Glencoe, © 2005 - 3.3 pg. 139-144 - pg. 741 – Review graphing using intercepts and slopes. HSPA: • Amsco: MATHEMATICS: Preparing for the

New Jersey HSPA, Grade 11 (navy blue binding – paperback) - Cluster 2.B.5 Coordinate Geometry – pg. 58-61

RESOURCES: • Geometry, Glencoe, © 2005 - 3.4 pg. 145-150 RESOURCES: • Geometry, Glencoe, © 2005 - 3.5 pg. 151-158 ASSESSMENT MODEL: Assessment questions should be open ended and should follow the general format illustrated in the Essential Questions/Sample Conceptual Understandings section.

19

Topic: Congruent Triangles Curriculum Management System


Goal 4: The student will be able to use triangle classifications and congruent triangles to solve problems.

Sugg

este

d da

ys o

f In

stru

ctio

n




18 4.1. Classify triangles by sides and angles (CPI 4.2.12 A3, 4.2.12 A4)

4.2. Use the exterior and interior angles of a

triangle to solve problems. (CPI 4.2.12 A3, 4.2.12 A4) Topic 4: Continued Next Page

• Construction workers use triangles when building houses, office buildings, and bridges, name some locations where you’ve seen triangles used in construction. Why do you think this is?

• Identify the vertices, angles, and sides of a triangle.

• Side Classifications – scalene, isosceles, and equilateral

• Angle Classifications – acute, obtuse, right, and equiangular

• Emphasize: In a right triangle, there is at most 1 right angle. And, in an obtuse triangle, there is at most 1 obtuse angle.

• In a right triangle, identify the right angle, the legs and the hypotenuse.

• In an isosceles triangle, identify the base, the vertex, and the base angles.

• Use the distance formula to prove triangles scalene, isosceles, and equilateral (optional)

• Use algebra to prove triangles isosceles. • Angle Sum Theorem, the sum of the angles

inside an triangle is 180o • Exterior Angles Theorem, the exterior angle

is equal to the sum of the two remote interior angles.

• Solve problems involving interior and exterior angles.

RESOURCES: • Geometry, Glencoe, © 2005 - 4.1 pg. 178-183 HSPA: • Prentice Hall: New Jersey HSPA


-10.4 Triangles RESOURCES: • Geometry, Glencoe, © 2005 - 4.2 pg. 216-221

20

21

Topic: Congruent Triangles Curriculum Management System Grade Level/Subject: 10 / Geometry


Sugg

este

d da

ys o

f In

stru

ctio

n




4.3. Use properties of isosceles triangles to solve problems. (CPI 4.2.12 A3,

4.2.12 A4) 4.4. Name congruent triangles and identify

corresponding parts. (CPI 4.2.12 A3, 4.2.12 A4)


• Review the vertex, the base, and the base angels.

• If the triangle is isosceles, then the base angles are congruent, (If sides then angles).

• If the base angles are congruent then the triangle is isosceles. (If angles then sides).

• Given one angle in an isosceles triangle, find the other two angels.

• Complete two-column proofs regarding isosceles triangles.

• Use algebra to solve isosceles triangle problems.

EX. In isosceles triangle ABC, where A is the vertex, AB = 4x – 5, AC = 11 + 2x, BC = 3X. Find x, BC. • I need to get to the other side of the river,

but I need to know how long of a bridge to make, how can I use markings on this side of the river to see if I have enough?

• Review reflexive, symmetric, and transitive properties.

• Identify all congruent angles and all congruent sides of congruent triangles.

• Make congruence statements • Emphasize: order of letters matters. • CPCTC

RESOURCES: • Geometry, Glencoe, © 2005 - 4.6 pg. 216-221 TECHNOLOGY: • Exploring Geometry with The Geometer’s

Sketchpad (resource workbook) - Properties of Isosceles Triangles – pg. 68 - Constructing Isosceles Triangles – pg. 70 RESOURCES: • Geometry, Glencoe, © 2005 - 4.3 pg. 192-198 TECHNOLOGY: • Prentice Hall Mathematics: - TI – 83/84

PLUS Activities for Algebra, Geometry, and Algebra II (resource workbook) - Exterior Angle of a Triangle – pg. 63


- Triangle Sum – pg. 65

22



Sugg

este

d da

ys o

f In

stru

ctio

n




4.5. Prove triangles congruent by using SSS, SAS, AAS, and ASA theorems and postulates. (CPI 4.2.12 A3, 4.2.12 A4, 4.2.12 C1)

4.6. Prove right triangles congruent using LL,

HA, LA, and HL theorems and postulates. (CPI 4.2.12 A3, 4.2.12 A4)


• Complete constructions using a ruler and protractor. Have students compare their triangles. Or teacher complete construction on overhead and pass overhead around classroom, everyone’s triangle should match exactly.

• Side-Side-Side and Side-Angle-Side • Angle-Side-Angle and Angle-Angle-Side • Show students why SSA does not work –

can be acute or obtuse triangle. • Third Angle Theorem – if two angles in one

triangle are congruent to two angles in another triangle, then the third angles in each triangle are congruent.

• Identify the postulate to be used given drawings.

• Utilize vertical angles and shared sides, students must identify vertical angles and shared sides as congruent parts.

• Make congruence statements: emphasize order of letters matters.

• Discuss: if a triangle is a right triangle, how

many other pieces of information are needed?

• Introduce LL, HA, LA • Emphasize HL, (and why SSA works for

right triangles only)

RESOURCES: • Geometry, Glencoe, © 2005 - 4.4 pg. 200-206 (SSS, SAS) - 4.5 pg. 207-213 (ASA, AAS) HSPA: • Amsco: MATHEMATICS: Preparing for the

New Jersey HSPA, Grade 11 (navy blue binding – paperback)

- Cluster 2.B.1 Congruence – 44-46 • Prentice Hall: New Jersey HSPA


- 10.7 Congruence and Similarity pg. 236-241

RESOURCES: • Geometry, Glencoe, © 2005 - Geometry Activity pg. 215



Sugg

este

d da

ys o

f In

stru

ctio

n




4.7. Complete two column proofs using congruent triangles. (CPI 4.2.12 A3, 4.2.12 A4)

End, Topic 4

• Complete various two column proofs using SSS, SAS, ASA, AAS, and HL.

• Include proofs using vertical angles, shared sides, and parallel lines.

• Include proofs using CPCTC.

RESOURCES: • Geometry, Glencoe, © 2005 - 4.4 pg. 200-206 (SSS, SAS) - 4.5 pg. 207-213 (ASA, AAS) - Geometry Activity pg. 215 ASSESSMENT MODEL: Assessment questions should be open ended and should follow the general format illustrated in the Essential Questions/Sample Conceptual Understandings section.

23

24

Topic: Triangle Relationships Curriculum Management System Grade Level/Subject: 10 / Geometry

Goal 5: The student will be able to use the relationships of sides and angles in triangles to solve problems.

Sugg

este

d da

ys o

f In

stru

ctio

n




14 5.1. Use lines in triangles to solve problems. (CPI 4.2.12 A4, 4.2.12 C1)


• Triangles come in many shapes and sizes (different combinations of side lengths). If I wanted to draw a circle inside of a triangle; If I wanted to draw a circle that contained all the vertices of a triangle; If I wanted to balance a triangle on the tip of my pencil; where would the center of the circle need to be?

• Draw, Identify and Use lines in triangles: altitude, median, angle bisector, and perpendicular bisector

• Draw and label lines in triangles (see reading mathematics on pg. 246)

• Use algebra to solve problems involving lines in triangles.

EX: In ∆ABC, AM is a median. If BM = 3x – 12, MC = x, and AC = 2x – 3, find x, BC, AC. • Theorems: • Any point on the angle bisector is equidistant

from the sides of the angle. (optional) • Any point on the perpendicular bisector is

equidistant from the endpoints. (optional) • The centroid is the intersection of the

medians, and is the balancing point of a triangle. (optional)

RESOURCES: • Geometry, Glencoe, © 2005 - Geometry Activity pg. 236 - 5.1 pg. 238-246 - Reading Mathematics pg. 246 TECHNOLOGY: • Prentice Hall Mathematics: - TI – 83/84

PLUS Activities for Algebra, Geometry, and Algebra II (resource workbook) - Angle Bisectors in Triangles I – pg. 67 - Segment Bisectors in Triangles – pg. 69 - Perpendicular Bisectors in Triangles – pg. 71


- Medians in a Triangle - 71 - Perpendicular Bisectors in a Triangle – 73 - Altitudes in a Triangle – 75 - Angle Bisectors in a Triangle – 77



Sugg

este

d da

ys o

f In

stru

ctio

n




5.2. Recognize and Apply triangle inequalities. (CPI 4.2.12 A3, 4.3.12 C1, 4.5.12 A3)


• Without using a protractor, How can you tell which angle is the biggest angle?

• Angle-Side Relationships: The angle opposite the longest side is the largest angle. The angle opposite the shortest side is the shortest angle and vice versa.

• Determine side-angle relationships when there are two adjacent triangles.

• Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

EX. Is it possible for a triangle to have side lengths of 3, 5, and 8? 2, 4, and 5? 3, 3, 10? • Given three side lengths, determine whether

the sides will form a triangle. • Given two side lengths, find the range of

lengths for the 3rd side. EX. If two sides of a triangle are 10 and 13, what is the range of sizes for the third side? • Understand that the shortest distance from a

point to a line is on a perpendicular segment • Inequalities involving two triangles. • SAS Inequality (hinge theorem) • SSS Inequality • Use inequality theorems to determine which

angle or which side is larger. Make inequality statements comparing two angles or two segments.

• Some using algebra.

RESOURCES: • Geometry, Glencoe, © 2005 - 5.2 pg. 247-254 - 5.4 pg. 261-266 - 5.5 pg. 267-273 TECHNOLOGY: • Prentice Hall Mathematics: - TI – 83/84

PLUS Activities for Algebra, Geometry, and Algebra II (resource workbook)

- Inequalities in Triangles – pg. 75

25



Sugg

este

d da

ys o

f In

stru

ctio

n




5.3. Use side lengths to determine if a triangle is acute, right, or obtuse (CPI 4.2.12 A1, 4.2.12 E1)

End, Topic 5.

• If I am not given any angle measurements, how can I tell if a triangle is acute, right, or obtuse?

• Use Pythagorean Theorem. If hypotenuse is smaller than what c should be according to the theorem, then the triangles is acute, if hypotenuse is larger than what c should be, then the triangle is obtuse.

EX. Is a triangle with side lengths 3, 4, 6 is acute, right, or obtuse?


26

Topic: Proportions and Similarity Curriculum Management System


Goal 6: The student will be able to use proportions to determine similarity of triangles.

Sugg

este

d da

ys o

f In

stru

ctio

n




14 6.1. Use properties of proportions to solve problems. (4.1.12 B1, 4.3.12 B4)


• I am thinking about rearranging my bedroom, but I am not sure everything will fit and still give me room to walk around, how can I find out without moving all the furniture?

• Simplify ratios

• Solve proportions: 2 243 x=

• Algebraic proportions: 3 5 13

4 2x − −

=

• Word problems: making a ratio, using ratios: • The ratio of the angles in a triangle is 2:3:4,

find the measure of each angle. • A softball player scored 9 runs in 12 games,

what is the ratio of runs to games

RESOURCES: • Geometry, Glencoe, © 2005 - 6.1 pg. 282-288

27

28

Topic: Proportions and Similarity Curriculum Management System Grade Level/Subject: 10 / Geometry


Sugg

este

d da

ys o

f In

stru

ctio

n




6.2. Use proportions to determine whether polygons are similar. (CPI 4.2.12 A3, 4.2.12 E1)

6.3. Use proportions and similarity postulates to prove triangles similar. (CPI 4.2.12 A3, 4.2.12 E1)


• If I wanted to know the height of the Great Pyramid in Egypt but don’t have anything that can measure that big, is there another way?

• Similar Figures – same shape, different size • Find the scale factor of similar figures • Write similarity statements for similar figures • Decide whether figures are similar by finding

corresponding sides proportional. • Angle measurements in similar figures are

equal (common mistake: many students want to make angles proportional as well)

• Similar Triangles: • Theorems: AA Similarity (review third angle

theorem), SSS Similarity, SAS Similarity • Identify which theorem to use, include

triangles with vertical angles and parallel lines, students must identify these congruent parts.

RESOURCES: • Geometry, Glencoe, © 2005 - 6.2 pg. 289-297 - 6.3 pg. 298-306 HSPA: • Prentice Hall: New Jersey HSPA


- 10.7 Congruence and Similarity pg. 236-241 - 11.3 Ratio and Proportion in Geometry pg. 256-260 TECHNOLOGY: • Exploring Geometry with The Geometer’s

Sketchpad (resource workbook) - Similar Polygons - 169 - Similar Triangles – AA Similarity - 170 - Similar Triangles – SSS, SAS, SSA - 171

29



Sugg

este

d da

ys o

f In

stru

ctio

n




6.4. Use proportional parts of triangles to solve problems. (CPI 4.2.12 A3)


• Triangle Proportionality Theorem pg. 307 • Midsegment of a triangle and Midsegment

theorem. • If a line is parallel to one side of a triangle

and intersects the other two sides, then it separates these sides into segments of proportional lengths.

EX. In ∆ABC, E lies on AC and F lines on AB EF BC , if AE=5, EC = 8, AF = 12, find FB.

• If three or more parallel lines intersect two transversals, then they cut off the transversals proportionally.

• Determine that the perimeter of similar figures have the same proportionality as the sides.

• Solve problems with similar triangles.

RESOURCES: • Geometry, Glencoe, © 2005 - 6.4 pg. 307-315 - 6.5 pg. 316-323 HSPA: • Amsco: MATHEMATICS: Preparing for the

New Jersey HSPA, Grade 11 (navy blue binding – paperback) - Cluster 2.B.2 Similarity – pg. 47-48


PLUS Activities for Algebra, Geometry, and Algebra II (resource workbook) - Parallel Segments in Triangles – pg. 83



Sugg

este

d da

ys o

f In

stru

ctio

n




6.5. Recognize and describe characteristics of fractals. (OPTIONAL) (CPI 4.2.12 A3, 4.2.12 B4)

End, Topic 6 MIDTERM EXAM

• In nature, many objects are self-similar, what are some self-similar objects? (snowflakes, leaves)

• Sierpinski Triangle Activity pg. 324 • Define and identify characteristics: iteration,

fractal, and self-similar.

RESOURCES: • Geometry, Glencoe, © 2005 - 6.6 pg. 325-331 PROJECT IDEA: Fractal Project Students identify objects that are self similar. Students must find 5 objects that have characteristics of fractals – at least 1 in nature and 1 artificial. Use the resources such as the internet and the media center. Students must present their results by making a poster, writing a report, or give a verbal presentation of the results. HSPA: • Amsco: MATHEMATICS: Preparing for the

New Jersey HSPA, Grade 11 (navy blue binding – paperback) - Cluster 3.E.1 Recursion, Iteration, and Fractals – pg. 154-157

TECHNOLOGY: • Exploring Geometry with The Geometer’s

Sketchpad (resource workbook) - Creating a Hat Curve Fractal - 202 - Creating a Sierpinski Gasket Fractal - 204 ASSESSMENT MODEL: Assessment questions should be open ended and should follow the general format illustrated in the Essential Questions/Sample Conceptual Understandings section.

30

31

Topic: Right Triangles and Trigonometry Curriculum Management System Grade Level/Subject: 10 / Geometry

Goal 7: The student will be able to use right triangle trigonometry to solve problems.

Sugg

este

d da

ys o

f In

stru

ctio

n




18 7.1. Simplify Radicals (CPI 4.1.12 B1, 4.1.12 B2) 7.2. Find the geometric mean between two

numbers. (CPI 4.2.12 A1, 4.2.12 C1) Topic 7: Continued Next Page

• Sometimes we need measurements, so we must round, what are some instances where we need exact answers? (Rounding error)

• See Skills Review pg. 744 • This is a review lesson from Algebra I • Emphasize the need for exact answers, the

need for simplifying radicals.

• Simplify these:

64

18

25163012

• Given two numbers, how can I find one

number that is proportional to both of the given numbers?

• Find the geometric mean between 2 and 32. 2

32x

x= , x2 = 64, x = 8.

RESOURCES: • Geometry, Glencoe, © 2005 - pg. 744 Skills Review • Algebra 1, McDougal Littel, © 2004 -9.1 Solving Quadratic Equations by Finding-Square Roots (pg. 503 – 510) -9.2 Simplifying Radicals (pg. 511-516) -12.2 Operations with Radical Expressions (pg. 716-721) – only multiplying and dividing as in Example 2 and 3, or Problem #’s 7, 10, 12, 31, 32, 40, 41, 43 • Prentice Hall: Algebra - Tools for a Changing

World (textbook - purple binding) -9.4 Simplifying Radicals (pg. 430-434) RESOURCES: • Geometry, Glencoe, © 2005 - 7.1 pg. 342-348



Sugg

este

d da

ys o

f In

stru

ctio

n




7.3. Use the Pythagorean Theorem to solve problems. (CPI 4.2.12 A1, 4.2.12 E1)

7.4. Use properties of 45-45-90 and 30-60-90

right triangles to solve problems. (CPI 4.2.12 A1)


• What whole number side lengths always form right triangles, Pythagorean Triples? How can I use these to solve problems?

• Leg2 + Leg2 = Hypotenuse2 • Find leg • Find hypotenuse • If two sides of a right triangle are 3 and 4,

what are the possible side lengths for the third side?

• Isosceles and equilateral triangles have

special properties, how about isosceles right triangles, how is the hypotenuse related to the legs, it can’t be double?

• Use the diagonal of a square to prove 45-45-90, the sides are: a, a, a 3

• Altitude of Equilateral triangle to prove 30-60-90, the sides are: a, a 3, 2a

• Given one side of a special right triangle, find the other two sides.

RESOURCES: • Geometry, Glencoe, © 2005 - Geometry Activity pg. 349 - 8.2 pg. 350-356 HSPA: • Amsco: MATHEMATICS: Preparing for the

New Jersey HSPA, Grade 11 (navy blue binding – paperback) - Cluster 2.C.6 Pythagorean Theorem – pg. 87-88


Sketchpad (resource workbook) - The Pythagorean Theorem - 153 - Visual Demonstration of the Pythagorean Theorem – 156 - Pythagorean Triples – 157 RESOURCES: • Geometry, Glencoe, © 2005 - 7.3 pg. 357-363 TECHNOLOGY: • Exploring Geometry with The Geometer’s

Sketchpad (resource workbook) - The Isosceles Right Triangle - 159 - The 30 – 60 Right Triangle – 161

32

33



Sugg

este

d da

ys o

f In

stru

ctio

n




7.5. Use trigonometric ratios to solve problems. (CPI 4.2.12 E1)

7.6. Solve problems involving angles of

elevation and angles of depression. (4.2.12 E1)


If a handicap ramp can have a maximum 4

grade of inclination, and the door is 3 feet above ground level, how far away from the

p start?

• o

•

• ngle

•

•

st foot,

•

•

door must the ram• SOH-CAH-TOA Find sine, cosine, and tangent of given

angles. Find missing sides given an ameasurement and one side.

Use inverse trig functions to find angle measurements given two sides.

As a plane begins its final approach to an airport, it is 8.8 miles from the airport. It will descent at a 15o angle. To the nearewhat is the plane’s current altitude?

ession. • Identify angles of elevation and depr Use trig functions to solve problems

pression involving angles of elevation and de Focus on real life problems such as

essential question.


New Jersey HSPA, Grade 11 (navy blue binding – paperback) - Cluster 2.C.7 Trigonometric Ratios – pg. 89-90


- 11.5 Right triangle trigonometry – pg. 266-269 TECHNOLOGY: • Exploring Geometry with The Geometer’s

Sketchpad (resource workbook) - Trigonometric Ratios - 195 - Modeling a Ladder Problem – 195 RESOURCES: • Geometry, Glencoe, © 2005 - 7.5 pg. 371-376



Sugg

este

d da

ys o

f In

stru

ctio

n




7.7. Use the Law of Sines and Law of Cosines to solve problems. (CPI 4.2.12 E1)

End, Topic 7

• Find angle measurements, find side lengthWe already proved that any two triangles that have the same three side lengths must be congruent (SSS). This means that they must have the same angle measurements as well. Can we find the angle measurements for any triangle, not just right

s. •

•

or unior year.

of Sines.

Solve problems using the Law of Cosines

triangles? This is an introductory lesson. It will be covered more in depth in Trig/Math Analysis, but students need this introduction fphysics classes in their j

• Introduce Law of Sines. • Solve problems using the Law• Introduce the Law of Cosines •

RESOURCES: • Geometry, Glencoe, © 2005 - 7.6 pg. 377-384 - 7.7 pg. 385-390 ASSESSMENT MODEL: Assessment questions should be open ended and should follow the general format illustrated in the Essential Questions/Sample Conceptual Understandings section.

34

35

Topic: Quadrilaterals Curriculum Management System Grade Level/Subject: 10 / Geometry

Goal 8: The student will be able to use properties of quadrilaterals to solve problems.

Sugg

este

d da

ys o

f In

stru

ctio

n




13 8.1. Recognize and apply the properties of parallelograms. (4.2.12 A3)

8.2. Recognize and apply the properties that ensure a quadrilateral is a parallelogram. (CPI 4.2.12 A3)


• There are many quadrilaterals in nature, where do we see quadrilaterals, and where to we see parallel lines? Why are these so special in architecture?

• Define the five properties of a parallelogram: two pairs of parallel sides, opposite sides are congruent, opposite angles are congruent, consecutive angles are supplementary, and diagonals bisect each other.

• Name the parallelogram in words and by symbols.

• Label parallelogram with arrows to indicate parallel lines and with slashes to indicate congruence.

• Use algebra to solve parallelogram problems • Apply properties that ensure a quadrilateral

is a parallelogram – the five properties of a parallelogram, and if one side of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram.

• Use algebra to create conditions that ensure a quadrilateral is a parallelogram.

EX. In ABCD , AB = 15, CD = 2x + 3 Find x. • Do many board examples giving 3 angles in

a parallelogram, students must find the rest of the angles.

RESOURCES: • Geometry, Glencoe, © 2005 - 8.2 pg. 411-416 - 8.3 pg. 417-419 HSPA: • Amsco: MATHEMATICS: Preparing for the New Jersey HSPA, Grade 11 (navy blue binding – paperback) - Cluster 2.A.2 Properties of Geometric Figures pg. 27-33 • Prentice Hall: New Jersey HSPA


-10.5 Quadrilaterals pg. 226-231 TECHNOLOGY: • Exploring Geometry with The Geometer’s Sketchpad (resource workbook) - Properties of Parallelogram – 91



Sugg

este

d da

ys o

f In

stru

ctio

n




8.3. Recognize and apply the properties of special parallelograms – rectangles, rhombi, and squares. (CPI 4.2.12 A3)


• The floor and the walls of a house must be rectangular in order for everything else in the house to be plum. How do construction workers ensure that the floor and the walls are rectangular?

• Understand that Rectangles, Rhombi, and Squares are still parallelograms, therefore all the properties of parallelograms still hold for these shapes.

• Special Properties: • Rectangles: All angles are equal (90o) Diagonals are equal • Rhombi All sides are equal Diagonals are perpendicular Diagonals bisect opposite angles. • Square: Is a parallelogram, a rhombus, and a rectangle, and has all the properties of each. • Solve problems involving special

parallelograms, using algebra. EX. In rhombus ABCD, AB=15, BC=3x -6, find x. • Use the Pythagorean Theorem when solving

a Rhombus. EX. In rhombus ABCD, BD=15, AC=20, find AB. • Do many board examples given 1 angle in a

special parallelogram, and students must find other angles.

RESOURCES: • Geometry, Glencoe, © 2005 - 8.4 pg. 424-430 - 8.5 pg. 431-437 TECHNOLOGY: • Exploring Geometry with The Geometer’s Sketchpad (resource workbook) - Properties of Rectangles - 93 - Properties of Rhombuses – 95

36



Sugg

este

d da

ys o

f In

stru

ctio

n




8.4. Recognize and apply the properties of trapezoids. (CPI 4.2.12 A3)

8.5. Recognize and apply the properties of

other quadrilaterals. (CPI 4.2.12 A3) End, Topic 8

• Define properties of trapezoids: one pair of parallel sides, define sides, bases and legs, angles connecting the bases are consecutive, therefore are supplementary.

• Define isosceles trapezoids: congruent legs, base angles are congruent and diagonals are congruent.

• Median of a trapezoid, use the formula:

22 • median = base + base as students relate to this better than using a fraction.

1

• Solve various problems using algebra with trapezoids – angles, sides in isosceles triangles and medians.

• Introduce other types of quadrilaterals such

as kites. • Review the sum of the interior angles must

be 360o.

RESOURCES: • Geometry, Glencoe, © 2005 - 8.6 pg. 439-446 - Hierarchy of Polygons (Quadrilaterals) TECHNOLOGY: • Exploring Geometry with The Geometer’s Sketchpad (resource workbook) - Properties of Isosceles Trapezoids - 97 - Midsegments of a Trapezoid and a Triangle pg. 100 - Summarizing Properties of Quadrilaterals. RESOURCES: • Geometry, Glencoe, © 2005 - pg. 438 Geometry Activity - Kites ASSESSMENT MODEL: Assessment questions should be open ended and should follow the general format illustrated in the Essential Questions/Sample Conceptual Understandings section.

37

Topic: Transformations Curriculum Management System


Goal 9: The student will be able to recognize and apply properties of transformations.

Sugg

este

d da

ys o

f In

stru

ctio

n




17 9.1. Recognize and apply properties of Translations. (CPI 4.2.12 B1)

9.2. Use vector addition to represent transformations. ( CPI 4.2.12 B1,

4.3.12 B3) Topic 9: Continued Next Page

• Objects in life move and change constantly, how can one tell if the objects have really changed, or only changed position or orientation?

• Discuss types of transformations: students have done transformations in middle school, review names as translations (slides), reflections (flips), and rotations.

• Introduce the image and the pre-image. • Discuss orientation, and which

transformations preserve orientation. • Define isometry (preserve size) • Plot points on a coordinate plane and

perform translations to relocate the figure. • Use vector notation to indicate how the

translation is taking place. EX. A(3,-5) is being translated by: ( ) ( ), 2 describe the transformation in words, and find the image of A.

, 3x y x y→ + −

• Given vector notation, and a point of the pre-image, find the point of the image.

• Given a point on the pre-image and the corresponding point on the image, determine the vector notation of the translation.

PROJECT IDEA: Investigating Symmetry Students identify objects that have symmetry and analyze transformations that preserve shape and size. Students find 5 objects with symmetry – at least 1 in nature and 1 artificial. Next, identify a way to transform the object to display the symmetry, labeling all angles used. Students present their results with a poster, a report, or a verbal presentation of the results. See Explorations and Projects workbook by McDougal Littell pg. 80-81 RESOURCES: • Geometry, Glencoe, © 2005 - Geometry Activity pg. 462 - 9.2 pg. 470-475 - 9.6 pg. 498-505 HSPA: • Amsco: MATHEMATICS: Preparing for the

New Jersey HSPA, Grade 11 (navy blue binding – paperback) - Cluster 2.B.3 – Transformations – pg. 49-34 - Cluster 2.B.4 – Tessellation – pg. 56-57


Sketchpad (resource workbook) - Introducing Transformations – pg. 31 - Translations in the Coordinate Plane–pg.39 - Symmetry in Regular Polygons – 50

38

Topic: Transformations Curriculum Management System Grade Level/Subject: 10 / Geometry


Sugg

este

d da

ys o

f In

stru

ctio

n




9.3. Recognize apply properties of reflected images. (CPI 4.2.12 B1, 4.3.12 B3)

9.4. Recognize and apply properties of

rotations. (CPI 4.2.12 B1, 4.3.12 B3) Topic 9: Continued Next Page

• Introduce reflection and line of reflection. • Discuss how a reflection differs from a

translation. (orientation) • Reflect figures from side to side, and those

that overlap the line of reflection. • Plot figures on coordinate planes and reflect

them over the x-axis and y-axis. • Draw lines of symmetry of a figure. • Introduce rotations, angle of rotation, and

center of rotation. • Discuss how it is different from the

translations and reflections. • Find the rotational symmetry of a figure by

the degrees of rotation.

RESOURCES: • Geometry, Glencoe, © 2005 - 9.1 pg. 463-469 TECHNOLOGY: • Prentice Hall Mathematics: - TI – 83/84

PLUS Activities for Algebra, Geometry, and Algebra II (resource workbook) - Reflections – pg. 91


- Properties of Reflection – pg. 36 - Reflections in the Coordinate Plane – pg.38 - Reflections over Two Parallel Lines – pg.44 - Symmetry in Regular Polygons – 50 RESOURCES: • Geometry, Glencoe, © 2005 - 9.3 pg. 476-482

39



Sugg

este

d da

ys o

f In

stru

ctio

n




9.5. Identify characteristics of tessellations (OPTIONAL) (CPI 4.2.12 B1, 4.3.12 B3)

9.6. Recognize and apply the properties of

dilations. (CPI 4.2.12 B1, 4.3.12 B3) Topic 9: Continued Next Page

• Define a tessellation, discuss characteristics. • Identify tessellations • Create own tessellation. • Identify the differences between dilations

and other transformations. (size) • Relate to similar figures. • Relate to art – perspective drawings.

PROJECT IDEA: Making a tessellation. Students must design and color their own tessellation made up of 1, 2, or 3 repeating shapes. Students may need to research the internet, or textbooks to find ideas for patterns. All tessellations must be original and hand drawn, not computer generated. Creativity is the focus of the grade. PROJECT IDEA: M.C. Escher Project Students work in pairs or on their own to study the works of M.C. Escher and find out how he created some of his drawings or patterns. Students should use the media center or the internet to research. Each student picks one work to analyze – finding geometric shapes, dividing the work into a diagram, and interesting features. Students present analysis and information about M.C. Escher in the form of a report, a poster, or a presentation. RESOURCES: • Geometry, Glencoe, © 2005 - 9.4 pg. 483-489 RESOURCES: • Geometry, Glencoe, © 2005 - 9.5 pg. 490-497

40



Sugg

este

d da

ys o

f In

stru

ctio

n




9.7. Find the magnitude and direction of a vector. (CPI 4.2.12 B1, 4.3.12 B3)

End, Topic 9

• When a plane flies across the country, it hits wind resistance that pushes the plane in different directions, how do pilots know they are going to end up at the right destination?

• Find the angle sum, review vector addition. (Find the diagonals of the parallelogram and add the components.)

• Find the vector magnitude (distance) • Find the direction of the vector (angle) using

trig functions. EX 1: A boat is going upstream at a still water speed of 45 knots. The current downstream is 3 knots. At what speed is the boat actually traveling? EX 2: A plane flies north at a speed of 240 miles per hour, and a wind is blowing from the east at 70 miles per hour, at what speed and in what direction is the plane actually flying?


New Jersey HSPA, Grade 11 (navy blue binding – paperback) - Cluster 2.B.6 –Vectors – pg. 63-65


- 12.1 Reflections and Symmetry pg. 282-285 - 12.2 Translations and Rotations pg. 296-290 - 12.3 Dilations pg. 290-292 - 12.4 Vectors pg. 293-295 ASSESSMENT MODEL: Assessment questions should be open ended and should follow the general format illustrated in the Essential Questions/Sample Conceptual Understandings section.

41

Topic: Circles Curriculum Management System


Goal 10: The student will be able to use and apply properties of lines and angles in circles.

Sugg

este

d da

ys o

f In

stru

ctio

n




17 10.1. Identify parts of a circle (CPI 4.2.12 A3) 10.2. Find the circumference and area of a

circle. (CPI 4.2.12 A3) 10.3. Identify and use arcs, angles, and sectors

to solve problems. (CPI 4.2.12 A1, 4.2.12 A3) Topic 10: Continued Next Page

• Define center, radius, diameter, chord • Use radius, diameter, circumference, and

area to find each other. • Use inscribed triangles and rectangles to

find parts of circles. • C = πd = 2πr • A = πr2 • If r = 10, find d, C, A, leave in terms of pi. • If C = 15π, find r, d, A, leave in terms of pi. • Find the measure of central angles and arcs. • Define and name minor arcs, major arcs,

and semicircles. • Use arc measure to find arc length. • Use central angles to find the area of a

sector.

RESOURCES: • Geometry, Glencoe, © 2005 - 10.1 pg. 522- 528 - 11.3 pg. 610-616 TECHNOLOGY: • Exploring Geometry with The Geometer’s

Sketchpad (resource workbook) - Introducing Circles – 119 - The Circumference / Diameter Ratio - 127 PROJECT IDEA: Circle Graph – Friend Survey Students must ask 50 people a question. (favorite color, type of food, etc.) After recording the answers, students must hand draw a circle graph after finding the appropriate central angle for each answer category. Students must display their project in a poster and have a creative display. RESOURCES: • Geometry, Glencoe, © 2005 - 10.2 pg. 529-535 - 11.5 pg. 622-627 TECHNOLOGY: • Exploring Geometry with The Geometer’s

Sketchpad (resource workbook) - Arcs and Angles - 125

42

Topic: Circles Curriculum Management System Grade Level/Subject: 10 / Geometry


Sugg

este

d da

ys o

f In

stru

ctio

n




10.4. Recognize and apply properties of chords. (CPI 4.2.12 A1, 4.2.12 A3)


• Theorems about chords: • Two minor arcs are congruent IFF their

corresponding chords are congruent. • If a diameter or a radius is perpendicular to a

chord, then it bisects the chord. • If two chords are parallel, then the arcs

connecting the chords are congruent. • Solve problems involving chords. • Inscribed angles, triangles, and

quadrilaterals. • Find the measure of an inscribed angle, and

compare the measure to the arc and the central angle.

• Theorems: • If two inscribed angles intercept the same

arc, then the angles are congruent • If an inscribed angle intercepts a semicircle,

the angle is a right angle (makes a right triangle with hypotenuse as the diameter.)

• If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.

RESOURCES: • Geometry, Glencoe, © 2005 - 10.3 pg. 536-543 - 10.4 pg. 544-551 TECHNOLOGY: • Exploring Geometry with The Geometer’s

Sketchpad (resource workbook) - Chords in a Circle – 121 • Prentice Hall Mathematics: - TI – 83/84

PLUS Activities for Algebra, Geometry, and Algebra II (resource workbook) - Triangles and Circles – pg. 79 - Inscribed Angles – pg. 89

43



Sugg

este

d da

ys o

f In

stru

ctio

n




10.5. Recognize and apply properties of tangent lines and secant lines.

(CPI 4.2.12 A1, 4.2.12 A3) Topic 10: Continued Next Page

• Introduce tangent lines and points of tangency.

• Theorems: • Tangent line is perpendicular to radius at

point of tangency. (vice versa) • Intersecting tangent lines – segments from

point of tangency to intersection – are congruent.

• Solve problems involving right triangles and tangent lines and circumscribed polygons.

• Investigate and find the measures of these angle types:

• Secant lines intersecting inside a circle. (or chords)

• Secant lines intersecting outside a circle. • A secant line and a tangent line intersecting

on the circumference of the circle • A secant line and a tangent line intersecting

outside the circle. • Tangent lines intersecting outside the circle. • Solve problems involving angle types above,

use numbers and use algebra. • Also find segment lengths for intersecting

segments: (introductory – not in depth) • Chord-Chord • Secant-Tangent • Tangent-Tangent

RESOURCES: • Geometry, Glencoe, © 2005 - 10.5 pg. 552-558 - 10.6 pg. 561-568 - 10.7 pg. 569-574 TECHNOLOGY: • Exploring Geometry with The Geometer’s

Sketchpad (resource workbook) - Tangents to a Circle - 123 - Tangent Segments – 124

44



Sugg

este

d da

ys o

f In

stru

ctio

n




10.6. Sketch the graph of a circle given the equation and write the equation of a circle given the graph. (OPTIONAL)

(CPI 4.2.12 A3, 4.5.12 E1) End, Topic 10

• Equation of a circle:

( ) ( )2 2 2x h y k r− + − =

• Center at (h, k), notice pos / neg switch. • Radius = r • Sketch the graph of a circle given the

equation • Write the equation of a circle given the

graph.

RESOURCES: • Geometry, Glencoe, © 2005

- 10.8 pg. 575-580 ASSESSMENT MODEL: Assessment questions should be open ended and should follow the general format illustrated in the Essential Questions/Sample Conceptual Understandings section.

45

46

Topic: Polygons Curriculum Management System Grade Level/Subject: 10 / Geometry

Goal 11: The student will be able to use properties of polygons to solve problems.

Sugg

este

d da

ys o

f In

stru

ctio

n




15 11.1. Identify and name polygons (CPI 4.2.12 E2, 4.3.12 B3) 11.2. Find the sum of the measures of interior

and exterior angles of a polygon. (CPI 4.2.12 A3)


• Relate to driver’s ed. Discuss different signs drivers see on the road, what types of polygons are formed by these different signs?

• Identify if a figure is a polygon based on its shape.

• Determine if the figure is concave or convex. • Name polygons based on number of sides: 3

sides up to 10 sides and 12 sides, otherwise n-gon. (triangle, quadrilateral, decagon, dodecagon, 13-gon, etc.)

• find perimeter • Regular polygon – all sides and all angles

equal. • Use algebra to solve word problems about

perimeter. • Divide into triangles using diagonals from 1

vertex for many different polygons, students discover 180(n – 2).

• Review properties of exterior angles in triangles; discuss what it means to be an exterior angle. Sum = 360o

• Solve problems finding angle measures inside and outside of a polygon. Use numbers and use algebra.


New Jersey HSPA, Grade 11 (navy blue binding – paperback) - Cluster 2.A.2 Properties of Geometric Figures – pg. 27-33


- 10.6 Polygons pg. 232-235. - 11.1 Perimeter and Circumference pg. 244-249 RESOURCES: • Geometry, Glencoe, © 2005 - 8.1 pg. 404-410 TECHNOLOGY: • Prentice Hall Mathematics: - TI – 83/84

PLUS Activities for Algebra, Geometry, and Algebra II (resource workbook) - Exterior Angle Sums – pg. 65


- Exterior Angles in a Polygon - 109 - Polygon Angle Measure Sums - 112



Sugg

este

d da

ys o

f In

stru

ctio

n




11.3. Find the area of polygons. (CPI 4.2.12 A3, 4.2.12 A4, 4.3.12 C1)

11.4. Find the geometric probability of an event.

(CPI 4.4.12 B2) Topic 11: Continued Next Page

• Find the area of polygons: • Parallelograms • Triangles, Trapezoids, Rhombi • Regular Polygons and Circles • Irregular Figures • Shaded Areas – wasted area of a pizza box • Review area of shaded regions and sectors. • Find geometric probability using real life

situations. • Use circle graphs / spinners with congruent

sectors, and with non-congruent sectors. • shaded areas • circle graphs / spinners

RESOURCES: • Geometry, Glencoe, © 2005 - 11.2 pg. 595-601 - 11.3 pg. 601-610 - 11.4 pg. 611-616 - 11.5 pg. 617-621 HSPA: • Amsco: MATHEMATICS: Preparing for the

New Jersey HSPA, Grade 11 (navy blue binding – paperback) - Cluster 2.C.1 – Perimeter and Circumference – pg. 69-71 -Cluster 2.C.2 – Area – pg. 72-75


- 11.2 Area pg. 250-255 RESOURCES: • Geometry, Glencoe, © 2005 - 11.5 pg. 622-627 • Look in probability section of any HSPA

book for extra problems, they are mixed in with all probability problems.

47



Sugg

este

d da

ys o

f In

stru

ctio

n




11.5. Use vertex-edge graphs or networks to solve problems

End, Topic 11.

• Use polygons as outlines for networks / vertex edge graphs.

• Determine if nodes are odd or even (the number of edges coming out of a node)

• Determine if the network is traversable or traceable (same definition – can go over every edge once and only once.) – at most 2 odd nodes.

• Use networks to solve problems such as the handshake problem.

• Make matrices to represent the network paths.

HSPA: • Amsco: MATHEMATICS: Preparing for the

New Jersey HSPA, Grade 11 (navy blue binding – paperback) - Cluster 3.D.2 – Networks – pg. 151-154


-13.3 Vertex Edge Graphs pg. 304-307 ASSESSMENT MODEL: Assessment questions should be open ended and should follow the general format illustrated in the Essential Questions/Sample Conceptual Understandings section.

48

49

Topic: Three Dimensional Figures. Curriculum Management System Grade Level/Subject: 10 / Geometry

Goal 12: The student will be able to find the lateral area, surface area, and volume of three dimensional figures.

Sugg

este

d da

ys o

f In

stru

ctio

n




12 12.1. Identify the three-dimensional shape that a net will form. (CPI 4.2.12 A2, 4.2.12 E2)

12.2. Find the Lateral Area and Surface Area of Right Prisms and Cylinders. (CPI 4.2.12 A2)

12.3. Find the Volume of Prisms and Cylinders.

(CPI 4.2.12 E2) Topic 12: Continued Next Page

• What shape of box will take the least amount of material to make while maximizing the volume?

• Determine whether a net will form an open or a closed figure, and the figure it will form: prism, pyramid, cylinder, or cone.

• Identify and label the bases, lateral faces, lateral edges, all edges, and vertices of prisms and cylinders.

• LA = PerimeterBase * height • SA = LA + 2 AreaBase • Solve problems involving right prisms with

regular polygons as bases, and cylinders. • Find the volume of a right prism or cylinder • V = AreaB * height

RESOURCES: • Geometry, Glencoe, © 2005 - 12.1 pg. 636-642 - 12.2 pg. 643-648 - 12.3 pg. 649-654 - 12.4 pg. 655-659 HSPA: • Amsco: MATHEMATICS: Preparing for the

New Jersey HSPA, Grade 11 (navy blue binding – paperback) - Cluster 2.A.5 Spatial Relationships – pg. 38-41 - Cluster 2.C.4 Surface Area – pg. 79-81


- 11.6 Prisms and Cylinders pg. 270-275


New Jersey HSPA, Grade 11 (navy blue binding – paperback) - Cluster 2.C.3 – Volume – pg. 76-78

Topic: Three Dimensional Figures. Curriculum Management System Grade Level/Subject: 10 / Geometry

Goal 12: The student will be able to find the lateral area, surface area, and volume of three dimensional figures.

Sugg

este

d da

ys o

f In

stru

ctio

n




12.4. Find the Lateral Area and Surface Area of Right Pyramids and Cones. (CPI 4.2.12 E2)

12.5. Find the Volume of Pyramids and Cones.

(CPI 4.2.12 E2) End, Topic 12 FINAL EXAM

• Identify the base, edges, lateral faces, lateral edges, height, and slant height.

• B1LA= Perimeter • slant height2

• SA = LA + AreaB • Find the LA and SA of a right pyramid with

bases shaped as regular polygons, and cones.

• How much ice can fit in a snow cone? • Find the volume of any right pyramid or cone

• B1V= Area • height3

RESOURCES: • Geometry, Glencoe, © 2005 - 12.5 pg. 660-665 - 12.6 pg. 666-670 HSPA: • Prentice Hall: New Jersey HSPA


- 11.7 Pyramids, Cones, and Spheres pg. 276-280


50

Geometry

COURSE BENCHMARKS

1. The student will be able to identify and use parts and types of lines, angles, and planes in problems solving. 2. The student will be able to use logical reasoning and conditional statements to solve problems. 3. The student will be able to use angle relationships with parallel and perpendicular lines to solve problems. 4. The student will be able to use triangle classifications and congruent triangles to solve problems. 5. The student will be able to use the relationships of sides and angles in triangles to solve problems. 6. The student will be able to use proportions to determine similar figures. 7. The student will be able to use right triangle trigonometry to solve problems. 8. The student will be able to use properties of quadrilaterals to solve problems. 9. The student will be able to recognize and apply properties of transformations. 10. The student will be able to use and apply properties of lines and angles in circles. 11. The student will be able to use properties of polygons to solve problems. 12. The student will be able to find the lateral area, surface area, and volume of three dimensional figures.

51

Monroe Township Schools · Monroe Township Schools Mission and Goals Mission The mission of the...

Documents

Transcript of Monroe Township Schools · Monroe Township Schools Mission and Goals Mission The mission of the...