High School Curriculum Map Geometry - gboe.org October 2014/Geometry... · High School Curriculum Map Geometry Marking Period 1 Topic Chapters Number of Blocks Dates Tools of Geometry

Garfield High School

Aligned to the 2012 Common Core Curriculum Content Standards

ENGAGING STUDENTS • FOSTERING ACHIEVEMENT • CULTIVATING 21ST

CENTURY GLOBAL SKILLS

High School Curriculum Map

Geometry

Marking Period 1

Topic Chapters Number of Blocks Dates Tools of Geometry 1 8 9/9 – 10/2

PRE-TEST 1 9/25

Reasoning and Proof 2 7 10/4-10/22

Parallel & Perpendicular Lines

3 8 10/24 – 11/19

MP 1 ASSESSMENT 1 10/17 – 10/31

Marking Period 2

Topic Chapters Number of Blocks Dates Congruent Triangles 4 10 11/20 – 12/20

Relationships in Triangles

5 7 1/2 – 1/22

Quadrilaterals 6 7 1/23-2/12

Properties of Similarity 7 6 2/20-3/10

MP 2 ASSESSMENT 1 1/24 & 1/28

Marking Period 3

Topic Chapters Number of Blocks Dates Right Triangles and

Trigonometry 8 6 3/11 – 4/1

Transformations and Symmetry

9 5 4/2-4/15

Circles 10 6 4/23-5/6

MP 3 ASSESSMENT 1 4/2 & 4/4





Marking Period 4

Topic Chapters Number of Blocks Dates Areas of Polygons and

Circles 11 6 5/7-5/22

Extending Surface Area and Volume

12 6 5/23 – 6/5

Probability and Measurement

13 6 6/9 – 6/17

MP 4 ASSESSMENT 1 6/9 – 6/11





Unit Overview

Content Area: Math

Unit Title: Tools of Geometry

Target Course/Grade Level – Geometry

Duration: 8 Blocks

Description :

In this unit students will be introduced to several key geometry terms and concepts. Angles are classifies as

acute, right, obtuse, or straight. Vertical angles, linear pairs, complementary angles and supplementary

angles are identified. The midpoint formula is used to find the coordinates of the midpoint of a segment. The

distance formula is use to find the distance between two points on a number line as well as two points within

a plane. The area, perimeter, surface area, and volume of two and three dimensional figures will be

addressed. Students are evaluated by a unit test, quizzes, notebook, homework, and class participation

along with other alternative assessments throughout the unit.

Concepts & Understandings

Concepts

Points, lines, and planes

Linear Measure

Distance and Midpoints

Angle Measures

Angle Relationships

Two-Dimensional figures

Three-Dimensional figures

Understandings

Find distances between points and midpoints of line segments.

Identify angle relationships.

Find perimeters, area, surface area, and volumes.

Learning Targets

CPI Codes

G-CO.HS.01

G-CO.HS.12

G-GMD.HS.03

G-GPE.HS.07

G-SRT.HS.07

Math Practices

See Addendum





21st Century Themes and Skills

See Addendum

Guiding Questions

What do the terms collinear and coplanar mean?

What is the intersection of two planes?

What is a line segment?

Name the distance formulas.

How do you to find distance in coordinate geometry?

How do you find midpoint in coordinate geometry?

How do we use a protractor to measure the degree of an angle?

Describe the angle measures of an acute, obtuse, and a right angle?

What is the difference between adjacent and vertical angles?

What is a linear pair?

How can you determine if two lines are perpendicular?

What is the difference between equilateral, equiangular, and regular polygons?

What is the formula to find the area of a triangle?

What is the formula to find the surface area of a cylinder?

Unit Results

Students will ...

Identify and model points, lines, and planes.

Identify intersecting lines and planes.

Measure segments.

Calculate with measures.

Find the distance between lines and points.

Find the midpoint of a segment.

Measure and classify angles.

Identify and use congruent angles and the bisector of an angle.

Identify and use special pairs of angles.

Identify perpendicular lines.

Identify and name polygons.

Find perimeter, circumference, and area of two dimensional figures.

Identify and name three-dimensional figures.

Find surface area and volume of three-dimensional figures.





Suggested Activities

The following activities can be incorporated into the daily lessons:

Interpret geometric drawings in order to determine the different types of planes found within a figure.

Use a ruler to find the measure of a line segment in metric and customary units.

Construct a copy of a line segment.

Use the distance formula to find the distance of the hypotenuse of a triangle plotted in the coordinate plane.

Use the terms acute, right, obtuse, and straight to classify the different types of angles.

Identify adjacent, vertical, complementary and supplementary angles.

Construct perpendicular lines.

Determine whether a set of given shapes are or are not polygons.

Find the area and perimeter of a triangle.

Find the perimeter and circumference of a circle.

Find the surface area and volume of a cone.

Unit Overview

Content Area: Math

Unit Title: Reasoning and Proof

Target Course/Grade Level: Geometry

Duration: 7 Blocks

Description:

The goals of this chapter include recognizing, analyzing, and writing conditional statements as well as writing

postulates using conditional statements. Students recognize and use definitions and biconditional

statements. They use symbolic notation to represent logical statements and use laws of logic to draw

conclusions from arguments. Students use properties from algebra and geometry to measure and justify

segment and angle relationships and congruence. Students also prove statements about segments and

angles using congruence. Students are evaluated by a unit test, quizzes, notebook, homework, and class

participation along with other alternative assessments throughout the unit.


Concepts

Inductive reasoning and conjecture

Logic

Understandings

Make conjectures and find counterexamples for statements.





Conditional statements

Deductive reasoning

Postulates and paragraph proofs

Algebraic proof

Proving segment relationships

Proving angle relationships

Use deductive reasoning to reach valid conclusions.

Write proofs involving segment and angle theorems.

Learning Targets

CPI Codes

G-CO.HS.09

G-CO.HS.10

G-CO.HS.11

G-CO.HS.12

G-MG.HS.03

Math Practices

See Addendum


See Addendum

Guiding Questions

What is a conjecture?

What is the purpose of creating a counterexample?

How can we tell the difference between of conjunction and a disjunction?

Is the negation of a statement always false?

What is the Hypothesis and conclusion?

What does it mean for two statements to be logically equivalent?

What is the key word for biconditional statement?

Given certain statements tell whether there is enough information to evaluate deductive reasoning?

Given various congruent line segments, will you be able to tell all the valid statements about the given information?

What is the difference between a postulate and an axiom?

Given various congruent lines, will you be able to solve for the variable to find the missing length of the line segment?

Given two or more angles will you be able to prove properties of angle congruence?





Unit Results

Students will ...

Make conjectures based on inductive reasoning.

Find counterexamples.

Determine truth values of negations, conjunctions, and disjunctions, and represent them using Venn diagrams.


Analyze statements in if-then form.

Write the converse, inverse, and contra-positive of if-them statements.

Use the Law of Detachment.

Use the Law of Syllogism.

Identify and use basic postulates about points, lines, and planes.

Write paragraph proofs.

Use algebra to write two-column proofs.

Use properties of equality to write geometric proofs.

Write proofs involving segment addition.

Write proofs involving segment congruence.

Write proofs involving supplementary and complementary angles.

Write proofs involving congruent and right angles.



Make conjectures from a set of data.


Determine truth values of conjunctions and disjunctions, negations, and true statements.

Construct truth tables.

Create a Venn diagram.

Write conditional statements in if-then form.

Construct a truth table for conditional statements.

Analyze statements using postulates.

Justify each step when solving equations using the properties of real numbers.

Use the segment addition postulate.

Prove complementary and supplementary angles.





Unit Overview

Content Area: Math

Unit Title: Perpendicular and Parallel Lines


Duration: 8 Blocks

Description:

In this chapter, Students investigate the relationships between lines and angles on a plane and in space. They will

study the angles formed when two lines are cut by a transversal. They will learn how to write flow proofs and use these

proofs along with two-column and paragraph proofs and use these proofs along with two-column and paragraph proofs

to prove theorems about perpendicular and parallel lines. Students will apply properties of parallel lines to solve real-life

problems. They will also use a straightedge and a compass to construct parallel lines. Students will find the slopes of

lines and use slope to identify parallel and perpendicular lines in a coordinate plane. They will write equations of parallel

and perpendicular lines in a coordinate plane. The student will get evaluation by quizzes and by tests after completion

of sections and the chapter.


Concepts

Parallel lines

Transversals

Angles

Parallel lines

Slopes of lines

Equations of lines

Proving lines parallel

Perpendiculars and distance

Understandings

Investigate the relationships between lines and angles

on a plane and in space. Study the angles formed when two lines are cut by a

transversal.

learn how to write flow proofs and use these proofs

along with two-column and paragraph proofs and use these proofs along with two-column and paragraph

proofs to prove theorems about perpendicular and

parallel lines. Apply properties of parallel lines to solve real-life

problems. –

Find the slopes of lines and use slope to identify parallel

and perpendicular lines in a coordinate plane. Write equations of parallel and perpendicular lines in a

coordinate plane.

Learning Targets

CPI Codes

G-CO.HS.01

G-CO.HS.09

G-CO.HS.12

G-GPE.HS.05

G-MG.HS.03





Math Practices

See Addendum


See Addendum

Guiding Questions

How can you identify the relationships between lines and angles?

How can you write and prove results about perpendicular lines?

What is a transversal?

How can you use properties of parallel lines to solve real-life problems?

How can you prove lines are parallel?

How can you use Properties of Parallel Lines?

How to Finding slopes of lines and use slope to identify parallel lines in a coordinate plane?

How to Use slope to identify perpendicular lines in a coordinate plane?

Unit Results

Students will ...

Identify the relationship between two lines or two planes.

Name angle pairs formed by parallel lines and transversals.

Use theorems to determine the relationships between specific pairs of angles.

Use algebra to find angle measurements.

Find slopes of lines.

Use slope to identify parallel and perpendicular lines.

Write an equation of a line given information about the graph.

Solve problems by writing equations.

Recognize angle pairs that occur with parallel lines.

Prove that two lines are parallel.

Find the distance between a point and a line.

Find the distance between parallel lines.



Identify parallel and skew lines. Explore angles which are formed by parallel lines and transversals.





Use the corresponding angles postulate. Prove the alternate interior angle theorem. Investigate slope and its relationship between real-world quantities. Classify the different types of slope. Find the equations of lines using the slope-intercept and point-slope form. Prove two lines are parallel. Use angle relationships like the alternate exterior angle and consecutive interior angle theorem to find the value

of x. Construct a perpendicular line. Find the distance from a point on a line to a point in the plane.

Unit Overview

Content Area: Math

Unit Title: Congruent Triangles


Duration: 10 Blocks

Description :

This chapter introduces students to proving triangles congruent and using congruent triangles in real-life problems.

Students first classify triangles and find angle measures. They identify congruent figures and corresponding parts of

figures, and learn to correctly name angles and triangles. Students prove triangles are congruent using SSS, SAS, ASA,

and AAS. They use congruence postulates to solve real-life problems. They use congruent triangles to plan and write

proofs and to prove the validity of constructions. They use properties of isosceles, equilateral, and right triangles.

Finally they learn how to place geometric figures in a coordinate plane to prove statements about the figures. The

student will get evaluation by quizzes and by tests after completion of sections and the chapter.


Concepts

Classifying triangles

Angles of triangles

Congruent triangles

SSS and SAS congruence

ASA and AAS congruence

Isosceles and Equilateral Triangles

Congruence transformations

Triangles and Coordinate Proofs

Understandings

Apply special relationships about the interior and

exterior angles of a triangle. Identify congruent figures and corresponding parts.

Prove that two triangles are congruent are congruent.

Use congruence postulates in real-life problems

Use congruence postulates and theorem to prove

triangles congruent.

Use properties of right triangles.

Special properties of isosceles and equilateral triangles.

Learning Targets





CPI Codes

G-CO.HS.06

G-CO.HS.07

G-CO.HS.10

G-CO.HS.12

G-GPE.HS.04

G-SRT.HS.05

Math Practices

See Addendum


See Addendum

Guiding Questions

How can we go about classifying different types of triangles?

What are the different classifications for a triangle based on their angle and side length measures?

What role do auxiliary lines play in proving the Triangle Angle-Sum Theorem?

What is a congruent polygon?

What is the angle addition postulate?

How do the theorems and postulates of triangle congruence apply to right triangles?

How can we prove that two triangles are congruent?

Can you use the lengths of the hypotenuse and leg to show that two right triangles are congruent?

Which two angles in an isosceles triangle are equal?

Name three words that can be used to describe the why a figure it moved.

What is a coordinate proof?

How can we use a coordinate proof to find the missing coordinates of a triangle found in the coordinate plane?

Unit Results

Students will ...

Identify and classify triangles by angle measures.

Identify and classify triangles by side measures.

Apply the triangle angle – sum theorem.

Supply the exterior angle theorem.

Name and use corresponding parts of congruent polygons.

Prove triangles congruent using the definition of congruence.

Use the SSS postulate to test for triangle congruence.





Use the SAS postulate to test for triangle congruence.

Use the ASA postulate to test for triangle congruence.

Use the AAS theorem to test for congruence.

Use properties of isosceles triangles.

Use properties of equilateral triangles.

Identify reflections, translations, and rotations.

Verify congruence after a congruence transformation.

Position and label triangles for use in coordinate proofs.

Write coordinate proofs.



Make formal geometric constructions with a variety of tools and methods. Classify triangles based on their angles measure. Classify triangles based on the measure of their sides. Complete geometry lab to find special relationships among the angles of triangles. Prove triangle angle-sum theorem. Define and determine the meaning of congruent polygons. Identify corresponding congruent parts. Prove that two triangles are congruent. Use SSS to prove triangle congruence. Use SAS to prove triangle congruence. Construct congruent triangles. Use ASA to prove triangle congruence. Use AAS to prove triangle congruence. Apply triangle congruence theorems to right triangles. Identify congruent segments and angles. Find the missing values in two congruent triangles. Apply triangle congruence to real-world situations.

Unit Overview

Content Area: Math

Unit Title: Relationships in Triangles


Duration: 7 Blocks

Description :

This chapter covers various properties of triangles. Students begin by studying perpendicular bisectors and angle

bisectors in general and then they relate these to specific triangles. Students learn that the perpendicular bisectors of a





triangle are concurrent. Students then study medians and altitudes and learn that these three segments associated

with a triangle are also concurrent. Students study the Midsegment Theorem and learn about various triangle

inequalities. The chapter concludes with a lesson indirect proof. The student will get evaluation by quizzes and by tests

after completion of sections and the chapter.


Concepts

Bisectors of Triangles

Medians and Altitudes of Triangles

Inequalities in one triangle

Indirect proof

Triangle inequality

Inequalities in two triangles

Understandings

Studying perpendicular bisectors and angle bisectors in

general and then they relate these to specific triangles.

Learn that the perpendicular bisectors of a triangle are

concurrent. – Study medians and altitudes and learn that these three

segments associated with a triangle are also concurrent.

Study the Midsegment Theorem and learn about various

triangle inequalities. Relationships between the sides and angles of triangles.

Writing indirect proofs.

Learning Targets

CPI Codes

G-CO.HS.07

G-CO.HS.10

G-MG.HS.03

G-SRT.HS.04

Math Practices

See Addendum


See Addendum

Guiding Questions

What is true about any point on the perpendicular bisector of a segment?

What is true about any point on the bisector of the angle?

What happens when perpendicular bisectors of a triangle intersect, where do they intersect?

What is a circumcenter of the triangle?

What is the in center of the triangle?





If you want to balance a triangular model on the tip of a pencil, where would you place the pencil?

What is a centroid?

What is the orthocenter of the triangle?

What is a midsegment of a triangle?

What is special about the midsegment of a triangle?

What angles in a triangle are used in the exterior angle inequality?

If given the sides and the measures of the interior angles of a triangle, can you state what you know using the

triangle inequality theorem? What is the first step of writing an indirect proof?

Unit Results

Students will ...

Identify and use perpendicular bisectors in triangles.

Identify and use angles bisectors in triangles.

Identify and use the medians in triangles.

Identify and use altitudes in triangles.

Recognize and apply properties of inequalities to the measures of the angles of a triangle.

Recognize and apply properties of inequalities to the relationships between the angles and sides of a triangle.

Write indirect algebraic proofs.

Write indirect geometric proofs.

Use the triangle inequality theorem to identify possible triangles.

Prove triangle relationships using the triangle inequality theorem.

Apply the hinge theorem or its converse to make comparisons in two triangles.

Prove triangle relationships using the Hinge theorem or its converse.



Construct a perpendicular bisector of the side of a triangle. Use the perpendicular bisector theorem. Finding the circumcenter of a triangle. Using the incenter of a triangle. Constructing medians and altitudes. Using the centriod of the triangle to find the missing lengths. Finding the othrocenter. Proving the exterior angle inequality theorem. Ordering the sides of a triangle from smallest to largest or largest to smallest. Ordering the measure of a triangle from smallest to largest or largest to smallest. Writing indirect algebraic and geometric proofs. Proving triangle equalities in one triangle and two triangles.





Unit Overview

Content Area: Math

Unit Title: Quadrilaterals


Duration: 7 Blocks

Description:

The students will learn how to classify special quadrilaterals and how to use their properties. They will write proofs

about special quadrilaterals. They will learn how to find areas of triangles and quadrilaterals. Students are evaluated by

a unit test, quizzes, notebook, homework, and class participation along with other alternative assessments throughout

the unit.


Concepts

Angles of polygons

Parallelograms

Tests for parallelograms

Rectangles

Rhombi and squares

Trapezoids and kites

Understandings

Find and use the sum of the measures of the interior

and exterior angles of a polygon.

Use properties of parallelograms.

Recognize and apply properties of quadrilaterals.

Compare quadrilaterals.

Find areas of triangles and quadrilaterals.

Write proofs about special quadrilaterals.

Learning Targets

CPI Codes

G-CO.HS.11

G-GPE.HS.04

G-MG.HS.01

G-MG.HS.03

G-GPE.HS.07

Math Practices

See Addendum


See Addendum

Guiding Questions





How many sides are there in the following figure?

What's the name of the polygon if the figure has 5 angles?

Which two angles are the same in a parallelogram?

What did you learn about consecutive angles in a parallelogram?

Describe the properties of parallelogram.

What is the sum of the interior angles of a parallelogram?

Which sides are congruent in a kite?

Describe the properties of a trapezoid.

Describe the properties of a kite.

Describe the properties of a rhombus.

What classifies a trapezoid as isosceles?

What’s the area of a square?

What's the area of a quadrilateral?

Which angles are congruent in a kite?

Unit Results

Students will ...

Find and use the sum of the measures of the interior angles of a polygon.

Find and use the sum of the measures of the exterior angles of a polygon.

Recognize and apply properties of the sides and angles of parallelograms.

Recognize and apply properties of the diagonals of parallelograms.

Recognize the conditions that ensure quadrilateral is a parallelogram.

Prove that a set of points forms a parallelogram in the coordinate plane.

Recognize and apply properties of rectangles.

Determine whether parallelograms are rectangles.

Recognize and apply the properties of rhombi and squares.

Determine whether quadrilaterals are rectangles, rhombi, or squares.

Apply properties of trapezoids.

Apply properties of kites.



Find the measure of the interior angles of a polygon. Recognize the given polygon and apply its properties. Practice measuring sides and angles in order to classify the different types of polygons.





Unit Overview

Content Area: Math

Unit Title: Properties of Similarity


Duration: 6 blocks

Description:

In this chapter, students explore the concept of similarity. Students will first simplify ratios; solve proportions using the

cross product property, and use properties of proportions to solve real-life problems. Similar polygons are then

introduced and their properties are used to solve real-life problems. Next, similar triangles are explored in greater

depth. Students will learn to prove that two triangles are similar using the AA Similarity Postulate, the SSS Similarity

Theorem, and the SAS Similarity Theorem. Similar triangles are used to solve indirect measurement problems.

Proportionality theorems involving parallel lines, angle bisectors, and transversals are examined and used to calculate

segment lengths. Finally students identify dilations and use properties of dilations in real-life applications. The student

will get evaluation by quizzes and by tests after completion of sections and the chapter.


Concepts

Ratios and proportions

Similar polygons

Similar triangles

Parallel lines and perpendicular parts

Parts of similar triangles

Similarity Transformations

Scale drawings and models

Understandings

Identify similar polygons and use ratios and

proportions to solve problems.

Identify and apply similarity transformations.

Use scale models and drawings to solve problems.

Identify types of polygons

Identify similar polygons

Use corresponding parts of similar figures.

Learning Targets

CPI Codes

G-GPE.HS.05

G-MG.HS.03

G-SRT.HS.01

G-SRT.HS.02

G-SRT.HS.03

G-SRT.HS.04

G-SRT.HS.05

Math Practices

See Addendum






See Addendum

Guiding Questions

What's the second way to write the Ratio? For any Ratio can denominator be zero? What is the cross product property?

Is it possible to switch the denominator on both sides?

When can you say two triangles are similar?

How many conditions do you need to show to prove two triangles similar?

How do you show the triangles are similar, when you are given compound figures?

When can you say that two triangles are similar?

What is the SSS similarity theorem?

What is dilation?

How can we determine the dilation of a given figure?

Unit Results

Students will ...

Write ratios.

Write and solve proportions.

Use proportions to identify similar polygons.

Solve problems using the properties of similar polygons.

Identify similar triangles using the AA Similarity postulate and the SSS and SAS Similarity Theorems.

Use the similar triangles to solve problems.

Use proportional parts within triangles.

Use proportional parts with parallel lines.

Recognize and use proportional relationships of corresponding angle bisectors, altitudes, and medians of similar

triangles.

Use the triangle bisector theorem.

Identify similarity transformations.

Verify similarity transformations.

Interpret scale models.

Use scale factors to solve problems.







Writing and simplifying ratios. Use extended ratios. Using cross products to solve proportions. Create similarity statements. Identifying similar polygons. Using similar figures to find missing measurements. Proving the AA similarity postulate. Identifying parts of similar triangles. Finding the midsegment of a triangle. Using special segments to prove similar triangles. Identifying dilations. Using a scale drawing.

Unit Overview

Content Area: Math

Unit Title: Right Triangles and Trigonometry


Duration: 6 blocks

Description:

The goals of this chapter include solving problems involving similar right triangles using the geometric mean

and indirect measurement Students prove the Pythagorean Theorem and use it and its converse to solve

problems. Students find the lengths of sides of special right triangles and use them to solve real-life

problems. They find the sine, cosine, and tangent ratios and use them to solve real-life problems. Finally,

students find the magnitude and direction of vectors and add vectors. The student will get evaluation by

quizzes and by tests after completion of sections and the chapter.


Concepts

Geometric mean The Pythagorean Theorem and its

converse Special right triangles

Trigonometry Angles of elevations and depression Laws of sines Laws of cosines Vectors

Understandings

Use the Pythagorean Theorem. Use properties of special right triangles. Use trigonometry to find missing of triangles. Identify four ways to prove triangles are similar. Give information about their sides and angles. Identify how to use similar polygons to solve real

life problems.





Learning Targets

CPI Codes

G-GPE.HS.06

G-MG.HS.03

G-SRT.HS.04

G-SRT.HS.05

G-SRT.HS.06

G-SRT.HS.07

G-SRT.HS.08

G-SRT.HS.09

G-SRT.HS.10

Math Practices

See Addendum


See Addendum

Guiding Questions

What is the geometric mean of the numbers 14 and 86? Given the altitude of a right triangle can you find the similar triangles? Define the Pythagorean triple and give an example. Find the length of the second leg of a right triangle with a given the values of a leg and the

hypotenuse. Describe how to classify a triangle with side lengths 6,9,10. Describe how to use the converse of the Pythagorean Theorem. How can we use the Pythagorean Theorem to classify different kinds of triangles? Find the ratio of the lengths of the sides of a 30, 60, 90 triangle and a 45, 45, 90 triangle.

Can the side lengths of a 45, 45, 90 triangle form a Pythagorean triple? What does the term SOHCAHTOA mean? What is the component form of a vector? What kind of operations can we perform using vectors? How do we add vectors? How do we find the magnitude and direction of a given vector?





Unit Results

Students will ...

Find the geometric mean between two numbers. Solve problems involving relationships between parts of a right triangle and the altitude to its

hypotenuse.

Use the Pythagorean Theorem. Use the converse of the Pythagorean Theorem. Use the properties of 45-45-90 triangles. Use the properties of 30-60-90 triangles. Find trigonometric ratios using right triangles.

Use trigonometric ratios to find angle measures in right triangles. Solve problems involving angles of elevation and depression. Use angles of elevation and depression to find the distance between two objects. Use the Laws of Sines to solve triangles. Use the Law of Cosines to solve triangles. Perform vector operations geometrically. Perform vector operations on the coordinate plane.



Finding geometric mean. Students will be able to solve problems involving similar right triangles formed by altitude drawn to

the hypotenuse of a right triangle. Identifying similar right triangles. Determine whether or not a triangle is a right triangle using the Pythagorean Theorem. Use the converse of the Pythagorean Theorem to classify the different kinds of triangles. Finding the length of the hypotenuse in a 45-45-90 triangle. Finding the length of the hypotenuse in a 30-60-90 triangle. Finding sine, cosine, and tangent ratios. Use special right triangles to find trigonometric ratios. Finding angle measures using inverse Trigonometric Ratios. Solving a right triangle. Finding the angle of elevation and the angle of depression. Using the laws of sines and cosines. Represent vectors geometrically. Find the resultant vector of two vectors. Writing vectors in component form. Finding the magnitude and direction of a vector. Completing operations with vectors.





Unit Overview

Content Area: Math

Unit Title: Transformations and Symmetry


Duration: 5 blocks

Description:

In this chapter, student will identify reflections, rotations, translations, and the characteristics of an isometric.

They will solve problems involving rigid transformations in the coordinate plane and will prove theorems about

reflections, rotations, and translations. They will also use software to investigate double reflections in the

coordinate plane Students will describe translations using vectors and will identify vector components.

Students will also learn how two or more translations produce a composition and they will identify glide

reflections in a plane. In the final lesson, they will use transformations to classify, identify, and draw frieze

patterns. Then they will examine how to use frieze patterns to create decorative borders for real-life objects.

Throughout the chapter, students will apply what they learn to real-life applications such as stenciling

designs, carpentry, surveying, molecular chemistry, log design, navigation, and land architecture. The student

will get evaluation by quizzes and by tests after completion of sections and the chapter.


Concepts

Reflections

Translations

Rotations

Compositions of Transformations

Symmetry

Dilations

Understandings

Identifying types of rigid transformations.

Using properties of reflections.

Relating reflections and line symmetry.

Relating rotations and rational systems.

Name and draw figures that have been reflected, translated, rotated, or dilated.

Recognize and draw compositions of transformations.

Identify symmetry in two- and three- dimensional figures.

Using properties of translations.

Using the properties of glide reflections.

Classifying frieze patterns.

Learning Targets

CPI Codes





G-CO.HS.01

G-CO.HS.02

G-CO.HS.03

G-CO.HS.04

G-CO.HS.05

G-CO.HS.06

G-CO.HS.07

G-SRT.HS.01

Math Practices

See Addendum


See Addendum

Guiding Questions

What is a transformation?

What is the image?

What is the preimage?

What is a reflection?

How can we graph a reflection in a coordinate plane?

What is a line of reflection?

What is a rotation?

Where is the fixed point of rotation?

What is the angle of rotation?

How to identify and use translations in the plane?

How to use vectors in real-life situations?

How to identify and use reflections, translations, rotations, and freeze patterns?

How to use the transformations in real-life?

What is symmetry?

What is rotational symmetry?

What is a dilation?

Describe how to create a dilation of a given figure with a scale factor of ½.

Unit Results

Students will ...





Draw reflections.

Draw reflections in the coordinate plane.

Draw translations.

Draw translations in the coordinate plane.

Draw rotations.

Draw rotations in the coordinate plane.

Draw glide reflections and other compositions of isometries in the coordinate plane.

Draw compositions of reflections in parallel and intersecting lines.

Identify line and rotational symmetries in two-dimensional figures.

Identify plane and axis symmetries in three-dimensional figures.

Draw dilations.

Draw dilations in the coordinate plane.



Naming transformations

Identifying Isometrics.

Reflecting a figure in a line.

Drawing translations in the coordinate plane.

Drawing rotations.

Graphing a glide reflection.

Reflecting a given figure in two lines.

Identifying lines of symmetry.

Identifying rotational symmetry.

Determining symmetry in a three-dimensional figure.

Drawing a dilation.

Finding the scale factor of a dilation.

Creating dilations in the coordinate plane.

Preserving length and angle measures.

Identify and use reflection in a plane

Identify relationships between

Identify rotations in a plane

Rotations in a coordinate plane.

Use properties of Translations

Finding the image of a composition

Describe a composition





Unit Overview

Content Area: Math

Unit Title: Circles


Duration: 6 blocks

Description

The goals of this chapter include identifying segments and lines related to circles, using properties of a

tangent to a circle, using properties of arcs and chords of circles, using inscribed angles and

properties of inscribed polygons to solve problems related to circles, and finding angles and arc

measures related to circles. Students find the lengths of the segments of chords, tangents, and

secants They find the equation of a circle and use it to graph and solve problems. Finally, students

draw loci that satisfy given conditions. The student will get evaluation by quizzes and by tests after

completion of sections and the chapter.


Concepts

Circles and Circumference Measuring angles and arcs

Arcs and chords

Inscribed angles

Tangents

Secants, Tangents, and Angle Measures

Special Segments in a Circle

Equations of Circles.

Understandings

Use properties of circle.

Write the equation of a circle.

Use properties of inscribed polygons.

Use the relationships between central angles, arcs, and inscribed angles in a circle.

Define and use secants and tangents. Use an equation to identify or describe a

circle.

Learning Targets

CPI Codes

G-CO.HS.01

G-CO.HS.12

G-GPE.HS.01

G-GPE.HS.06

G-MG.HS.03

GC.HS.01

GC.HS.02

GC.HS.03

GC.HS.04





GC.HS.05

Math Practices

See addendum


See Addendum

Guiding Questions

Name the different parts of a circle (i.e. radius, diameter, and chords).

What is chord?

How do we find the circumference of a circle?

How can we go about finding the central angle of a circle?

Which is the longest chord in a circle?

What is an arc?

Can you name any two properties of a circle?

What is an inscribed angle?

How can we find the length of a segment tangent to a circle?

What is the point of tangency?

What is a secant line?

How can we find the measure of the angle created by two secant line segments?

Unit Results

Students will ...

Identify and use parts of circles.

Solve problems involving the circumference of a circle.

Identify the central angles, major arcs, minor arcs, and semicircles, and find their measures.

Find arc lengths.

Recognize and use relationships between arcs and chords.

Recognize and use relationships between arcs, chords, and diameters.

Find measure of inscribed angles.

Find measures of angles of inscribed polygons.

Use properties of tangents.

Solve problems involving circumscribed polygons.





Find measures of angles formed by lines intersecting on or inside a circle.

Find measures of angles formed by lines intersecting outside the circle.

Find measures of segments that intersect in the interior of a circle.

Find the measure of segments that intersect in the exterior of a circle.

Write the equation of a circle.

Graph a circle on the coordinate plane.



Identify segments in a circle.

Find ratios and diameter.

Find measures in intersecting circles.

Finding measures of central angles.

Classifying arcs and find arc measures.

Using arc addition to find measures of arcs.

Use congruent chords to find arc measure.

Use congruent arcs to find chord lengths.

Using inscribed angles to find measures.

Using inscribed angles in proofs.

Identifying common tangents.

Using tangents to find missing values.

Using congruent tangents to find missing measurements.

Using intersecting chords or secants.

Using tangent sand secants that intersect outside a circle.

Using the intersection of two chords.

Identify segments and lines, & tell whether the line or segment is best described as a chord, secant, a tangent, a diameter, or a radius of a given circle.

Recognize relationships between arcs and central angles of circles, and chords and diameters of circles

Solve problems or find variable values using the measures of inscribed angles and intercepted arcs.

Create simple proofs involving the measures of inscribed angles.





Unit Overview

Content Area: Math

Unit Title: Area of Polygons and Circles


Duration: 6 blocks

Description :

Explores many topics which relate to the area of polygons and circles. The chapter begins by finding the measures of the

interior and exterior angles of polygons. These angle measures are then used to find the area of an equilateral triangle

and other polygons. Perimeters and areas of similar figures are investigated. Next, students are introduced to the

circumference of a circle and the length of a circular arc. This leads to finding the area of a circle and the area of a sector

of a circle. Finally geometric probability is introduced. Throughout the chapter, real-life problems involving area, perimeter,

circumference, arc length, and geometric probability are solved. The student will get evaluation by quizzes and by tests

after completion of sections and the chapter.


Concepts

Areas of Parallelograms and Triangles

Areas of Trapezoids, Rhombi, and Kites

Areas of Circles and Sectors

Areas of Regular Polygon and Composite Figures

Areas of Similar Figures

Understandings

Finding areas of polygons

Solve problems involving areas and sectors of circles

Find scale factors using similar figures

To understand the properties of special

parallelograms and find the area of polygons and

circles.

Learning Targets

CPI Codes

G-GMD.HS.01

G-GPE.HS.07

G-MG.HS.01

G-MG.HS.03

GC.HS.05b

Math Practices

See Addendum


See Addendum





Guiding Questions

State the types of polygons.

Apply the theorem 11.2 to the given question.

What is a regular polygon?

What is an equilateral and equiangular polygon?

When can you say the figures are similar?

What is scale factor?

Can a circle be a polygon?

What is circumference?

Describe the difference between circumference and perimeter.

How do you find the area of two concentric circles?

How do you find the area and perimeter of a triangle?

Describe the steps that need to be taken in order to find the area of a composite figure.

How do you find the area of a sector of a circle?

What is a regular polygon?

Unit Results

Students will ...

Find perimeters and areas of parallelograms.

Find perimeters and areas of triangles.

Find areas of trapezoids.

Find areas of rhombi and kites.

Find areas of circles.

Find areas of sectors of a circle.

Find areas of regular polygons.

Find areas of composite figures.

Find areas of similar figures by using scale factors.

Find scale factors or missing measures given the areas of similar figures.



Finding the perimeter and area of a parallelogram. Prove the area congruence postulate. Finding the perimeter and area of a triangle. Using area to find missing measures. Finding the area of a trapezoid. Finding the area of a Rhombus and Kite. Finding the area of a circle. Using the area of a circle to find the missing measurement.





Finding the area of a sector. Identify segments and angles in regular polygons. Finding the area of a regular polygon. Using the formulas for the area of regular polygons. Find the area of a composite figure by adding and subtracting. Finding areas of similar figures. Using areas of similar figures.

Unit Overview

Content Area: Math

Unit Title: Extending Surface Area and Volume


Duration: 6 blocks

Description:

In this chapter students will investigate the surface area and volume of solids. First, they will learn to distinguish

polyhedra from other solids and to classify polyhedra. They will identify the Platonic solids and use Euler's Theorem.

Students will use nets to help them explore the surface area of prisms and cylinders. They will use the Pythagorean

theorem to identify the surface area of pyramids, and use proportions involving circles to identify the surface area of

cones. They will then develop methods for finding the volume of prisms and cylinders, including using Cavalieri's Principle.

Students will find the volume of pyramids and cones. After learning to find the surface area and volume of spheres,

students will investigate similar solids, including scale factors and how surface area and volume relate to the dimensions of

similar solids. The student will get evaluation by quizzes and by tests after completion of sections and the chapter.


Concepts

Representations of Three-Dimensional Figures

Surface Areas of Prisms and Cylinders

Surface Areas of Pyramids and Cones

Volumes of Prisms and Cylinders

Volumes of Pyramids and Cones

Surface Area and Volume of Spheres

Spherical Geometry

Congruent and Similar Solids

Understandings

Find lateral areas, surface areas, and volumes of

various solid figures

Investigate Euclidean and spherical geometries

Use properties of similar solid figures.

Learning Targets

CPI Codes





G-CO.HS.13

G-GMD.HS.01

G-GMD.HS.02

G-GMD.HS.03

G-GMD.HS.04

G-GPE.HS.01

Math Practices

See Addendum


See Addendum

Guiding Questions

What is a three-dimensional figure?

What is lateral area?

What three things do you need to know in order to find the surface area of a pyramid?

What formula is needed to find the lateral area of a cylinder?

What is the volume of a sphere whose radius is 22 ft?

Describe Euclidean geometry.

Describe spherical geometry.

Compare and contrast Euclidean and spherical geometries.

What information do you need to know in order to determine whether or not two solids are congruent?

What information do you need to know in order to determine whether or not two solids are similar?

Unit Results

Students will ...

Draw isometric views of three-dimensional figures.

Investigate cross sections of three-dimensional figures.

Find lateral areas and surface areas of prisms.

Find lateral areas and surface areas of cylinders.

Find lateral areas and surface areas of pyramids.

Find lateral areas and surface areas of cones.

Find volumes of prisms.

Find volumes of cylinders.

Find volumes of pyramids.

Find volumes of cones.

Find surface areas of spheres.

Find volumes of spheres.

Describe sets of points on a sphere.





Compare and contrast Euclidean and spherical geometries.

Identify congruent of similar solids.

Use properties of similar solids.



Use dimensions of a solid to sketch a solid. Finding the lateral area of a prism. Finding the surface area of a prism. Finding the lateral area and surface area of a cylinder. Find the lateral area of a regular prism. Find the surface area of a square pyramid. Finding the surface area of a cone. Find the volume of a prism and a cylinder. Comparing volumes of solids. Finding real world volumes. Use great circles to find surface area. Find surface area of a sphere. Describe sets of points on a sphere. Compare plane Euclidean and spherical geometries. Identify lines in spherical geometry. Identify similar and congruent solids.





Unit Overview

Content Area: Math

Unit Title: Probability and Measurement


Duration: 6 blocks

Description:

In this chapter, students will investigate probability and measurement. They will first learn to represent sample space by

creating a tree diagram, an organized list, and a table. They will identify the differences between permutations and

combinations as well as be able to use them to solve probability. Students will find probabilities using length and area.

They will identify the difference between single and compound events. Students will find the probabilities of compound

events. The student will get evaluation by quizzes and by tests after completion of sections and the chapter.


Concepts

Sample Space

Permutations and Combinations

Geometric Probabilities

Simulations

Independent and Dependent events

Mutually Exclusive Events

Understandings

Use the Fundamental Counting Principle

Use permutations and combinations to solve

probability Design simulations to estimate probabilities

Summarize data from simulations

Apply physical models, graphs, and networks to

develop solutions in applied contents.

Learning Targets

CPI Codes

S.CP.9

S.MD.7

G.MG.3

S.MD.6

S.CP.2

S.CP.3

S.CP.4

S.CP.6

S.CP.1

S.CP.7

Math Practices

See Addendum






See Addendum

Guiding Questions

What is probability?

What does it mean for a coin to be fair?

How can the Fundamental Counting Principle be used to represent a sample space?

Explain the difference between permutations and combinations.

What makes a circular permutation different?

How can probability be used in geometry?

What is a simulation?

How can we design a simulation using a geometric model?

What is an independent event? A dependent event?

Explain the difference between independent and dependent events?

How can we define conditional probability?

Explain what it means for an event to be mutually exclusive.

Unit Results

Students will ...

Represent sample spaces.

Use the Fundamental Counting Theorem to count the number of possible outcomes.

Use permutations to solve probability.

Use combinations to solve probability.

Find probabilities by using lengths and areas.

Design simulations to estimate probabilities.

Summarize data from simulations.

Find probabilities of independent and dependent events.

Find probabilities of events given the occurrence of other events.

Find probabilities of events that are mutually exclusive and events that are not mutually exclusive.

Find probabilities of complements.



Create an organized list, table, and tree diagram to represent sample space.

Create a multi-stage tree diagram.

Use the Fundamental Counting Theorem to determine the number of possible class ring designs.

Find the probability that three people are voted president, vice president, and secretary.

Use permutations with repetition.

Find the probability of choosing three pairs of socks using combinations.

Find the probability choose a heart and then a red card from a standard deck of cards.





Addendum – Math Practices

1. CCSS.Math.Practice.MP1 Make sense of problems and persevere in solving

them.

Mathematically proficient students start by explaining to themselves the meaning

of a problem and looking for entry points to its solution. They analyze givens,

constraints, relationships, and goals. They make conjectures about the form and

meaning of the solution and plan a solution pathway rather than simply jumping

into a solution attempt. They consider analogous problems, and try special cases

and simpler forms of the original problem in order to gain insight into its solution.

They monitor and evaluate their progress and change course if necessary. Older

students might, depending on the context of the problem, transform algebraic

expressions or change the viewing window on their graphing calculator to get the

information they need. Mathematically proficient students can explain

correspondences between equations, verbal descriptions, tables, and graphs or

draw diagrams of important features and relationships, graph data, and search

for regularity or trends. Younger students might rely on using concrete objects or

pictures to help conceptualize and solve a problem. Mathematically proficient

students check their answers to problems using a different method, and they

continually ask themselves, “Does this make sense?” They can understand the

approaches of others to solving complex problems and identify correspondences

between different approaches.

2. CCSS.Math.Practice.MP2 Reason abstractly and quantitatively.

Mathematically proficient students make sense of quantities and their

relationships in problem situations. They bring two complementary abilities to

bear on problems involving quantitative relationships: the ability to

decontextualize—to abstract a given situation and represent it symbolically and

manipulate the representing symbols as if they have a life of their own, without

necessarily attending to their referents—and the ability to contextualize, to pause

as needed during the manipulation process in order to probe into the referents for

the symbols involved. Quantitative reasoning entails habits of creating a coherent

representation of the problem at hand; considering the units involved; attending

to the meaning of quantities, not just how to compute them; and knowing and

flexibly using different properties of operations and objects.

http://www.corestandards.org/Math/Practice/MP1






3. CCSS.Math.Practice.MP3 Construct viable arguments and critique the

reasoning of others.

Mathematically proficient students understand and use stated assumptions,

definitions, and previously established results in constructing arguments. They

make conjectures and build a logical progression of statements to explore the

truth of their conjectures. They are able to analyze situations by breaking them

into cases, and can recognize and use counterexamples. They justify their

conclusions, communicate them to others, and respond to the arguments of

others. They reason inductively about data, making plausible arguments that take

into account the context from which the data arose. Mathematically proficient

students are also able to compare the effectiveness of two plausible arguments,

distinguish correct logic or reasoning from that which is flawed, and—if there is a

flaw in an argument—explain what it is. Elementary students can construct

arguments using concrete referents such as objects, drawings, diagrams, and

actions. Such arguments can make sense and be correct, even though they are

not generalized or made formal until later grades. Later, students learn to

determine domains to which an argument applies. Students at all grades can

listen or read the arguments of others, decide whether they make sense, and ask

useful questions to clarify or improve the arguments.

4. CCSS.Math.Practice.MP4 Model with mathematics.

Mathematically proficient students can apply the mathematics they know to solve

problems arising in everyday life, society, and the workplace. In early grades, this

might be as simple as writing an addition equation to describe a situation. In

middle grades, a student might apply proportional reasoning to plan a school

event or analyze a problem in the community. By high school, a student might

use geometry to solve a design problem or use a function to describe how one

quantity of interest depends on another. Mathematically proficient students who

can apply what they know are comfortable making assumptions and

approximations to simplify a complicated situation, realizing that these may need

revision later. They are able to identify important quantities in a practical situation

and map their relationships using such tools as diagrams, two-way tables,

graphs, flowcharts and formulas. They can analyze those relationships

mathematically to draw conclusions. They routinely interpret their mathematical

results in the context of the situation and reflect on whether the results make

sense, possibly improving the model if it has not served its purpose.







5. CCSS.Math.Practice.MP5 Use appropriate tools strategically.

Mathematically proficient students consider the available tools when solving a

mathematical problem. These tools might include pencil and paper, concrete

models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra

system, a statistical package, or dynamic geometry software. Proficient students

are sufficiently familiar with tools appropriate for their grade or course to make

sound decisions about when each of these tools might be helpful, recognizing

both the insight to be gained and their limitations. For example, mathematically

proficient high school students analyze graphs of functions and solutions

generated using a graphing calculator. They detect possible errors by

strategically using estimation and other mathematical knowledge. When making

mathematical models, they know that technology can enable them to visualize

the results of varying assumptions, explore consequences, and compare

predictions with data. Mathematically proficient students at various grade levels

are able to identify relevant external mathematical resources, such as digital

content located on a website, and use them to pose or solve problems. They are

able to use technological tools to explore and deepen their understanding of

concepts.

6. CCSS.Math.Practice.MP6 Attend to precision.

Mathematically proficient students try to communicate precisely to others. They

try to use clear definitions in discussion with others and in their own reasoning.

They state the meaning of the symbols they choose, including using the equal

sign consistently and appropriately. They are careful about specifying units of

measure, and labeling axes to clarify the correspondence with quantities in a

problem. They calculate accurately and efficiently, express numerical answers

with a degree of precision appropriate for the problem context. In the elementary

grades, students give carefully formulated explanations to each other. By the

time they reach high school they have learned to examine claims and make

explicit use of definitions.







7. CCSS.Math.Practice.MP7 Look for and make use of structure.

Mathematically proficient students look closely to discern a pattern or structure.

Young students, for example, might notice that three and seven more is the

same amount as seven and three more, or they may sort a collection of shapes

according to how many sides the shapes have. Later, students will see 7 × 8

equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the

distributive property. In the expression x2 + 9x + 14, older students can see the

14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line

in a geometric figure and can use the strategy of drawing an auxiliary line for

solving problems. They also can step back for an overview and shift perspective.

They can see complicated things, such as some algebraic expressions, as single

objects or as being composed of several objects. For example, they can see 5 –

3(x – y)2 as 5 minus a positive number times a square and use that to realize that

its value cannot be more than 5 for any real numbers x and y.

8. CCSS.Math.Practice.MP8 Look for and express regularity in repeated

reasoning.

Mathematically proficient students notice if calculations are repeated, and look

both for general methods and for shortcuts. Upper elementary students might

notice when dividing 25 by 11 that they are repeating the same calculations over

and over again, and conclude they have a repeating decimal. By paying attention

to the calculation of slope as they repeatedly check whether points are on the

line through (1, 2) with slope 3, middle school students might abstract the

equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when

expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might

lead them to the general formula for the sum of a geometric series. As they work

to solve a problem, mathematically proficient students maintain oversight of the

process, while attending to the details. They continually evaluate the

reasonableness of their intermediate results.







Connecting the Standards for Mathematical Practice to the Standards for Mathematical Content

The Standards for Mathematical Practice describe ways in which developing

student practitioners of the discipline of mathematics increasingly ought to

engage with the subject matter as they grow in mathematical maturity and

expertise throughout the elementary, middle and high school years. Designers of

curricula, assessments, and professional development should all attend to the

need to connect the mathematical practices to mathematical content in

mathematics instruction.

The Standards for Mathematical Content are a balanced combination of

procedure and understanding. Expectations that begin with the word

“understand” are often especially good opportunities to connect the practices to

the content. Students who lack understanding of a topic may rely on procedures

too heavily. Without a flexible base from which to work, they may be less likely to

consider analogous problems, represent problems coherently, justify

conclusions, apply the mathematics to practical situations, use technology

mindfully to work with the mathematics, explain the mathematics accurately to

other students, step back for an overview, or deviate from a known procedure to

find a shortcut. In short, a lack of understanding effectively prevents a student

from engaging in the mathematical practices.

In this respect, those content standards which set an expectation of

understanding are potential “points of intersection” between the Standards for

Mathematical Content and the Standards for Mathematical Practice. These

points of intersection are intended to be weighted toward central and generative

concepts in the school mathematics curriculum that most merit the time,

resources, innovative energies, and focus necessary to qualitatively improve the

curriculum, instruction, assessment, professional development, and student

achievement in mathematics.

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