geometry unit and lesson overviews 2014 - weebly

Geometry Unit & Lesson Overviews Mathematics

CRANSTON PUBLIC SCHOOLS Revised 9/17/2014

Unit: 1.1 Foundations of Geometry Days : 8

Essential Questions

• How do you use undefined terms as the basic elements of Geometry?

• What tools and methods can you use to construct and bisect segments and angles?

• How can you use angle pairs to solve problems?

• How can you find midpoints of segments and distances on the coordinate plane?

Content to be Learned Skills

• Students will learn the vocabulary and symbols for the

basic elements of geometry

• Students will construct and bisect segments and angles

• Students will use angle pairs to solve problems

• Students will find midpoints of segments and distances

on the coordinate plane

• Using a protractor to measure angles

• Using a compass to construct segments and angles

• Using angle relationships to write and solve equations to

find unknown angles

• Using midpoint and distance formulas

Assessments Standards

There are 2 assessments planned for this unit CC.9-12.G.CO.1

CO.5

CO.9

CO.12

CC.9-12.A.CED.4

CC.9-12.G.GPE.4

Sample Instructional Activities Resources

Sections 1-1, 1-2, 1-3, 1-4, 1-6 in the CoreMath Text Explorations in CoreMath Textbook



Unit 1.1 Understanding Points, Lines & Planes Lesson 1 of 5 Days 1

Lesson Focus

1. Standards Addressed 2. Content to be Learned 3. Mathematical Practices 4. Essential Questions

CC.9-12.G.CO.1 Students need to be able to

properly name geometric figures

so they can communicate ideas

precisely to others

CCSS.Math.Practice.MP8 How do you use undefined terms

as the basic elements of

geometry?

5. Prerequisite Knowledge 6. Essential Vocabulary 7. Possible Misconceptions 8. Teaching Materials

None Point Ray

Line Line Segment

Plane Endpoint

CoreMath Textbook

Instruction

9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) • Revisiting the basic geometric concepts of points, lines and

planes

• Working with students to understand how to use the basic undefined terms to help explain and communicate other geometric ideas

• Instructing students as to the proper method of naming/identifying these basic terms

• Identifying/Naming basic geometric terms

• Using undefined terms to help explain/define other geometric

terms (Ex. An angle is made up of two rays that have the same

endpoint, but go in different directions)



Unit 1.1 Segments Lesson 2 of 5 Days 1

Lesson Focus


CC.9-12.G.CO.1 CC.9-12.G.CO.12

Different methods of copying

and bisecting a given segment

CCSS.Math.Practice.MP6 What tools and methods can you

use to copy and bisect a segment?


Basic geometric terms CoreMath Textbook

Instruction

9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) • Showing students multiple methods of copying/bisecting a

given segment (ruler, straightedge & compass, software, paper folding, etc.)

• Discovering multiple ways of copying/bisecting a given segment

• Attending to the precision of every method, making sure to pay

attention to details



Unit 1.1 Measuring Angles Lesson 3 of 5 Days 1

Lesson Focus


CC.9-12.G.CO.1 CC.9-12.G.CO.12

Different methods and tools

to copy and bisect angles

Identifying the types of

angles

CCSS.Math.Practice.MP5 What tools and methods can you

use to copy and bisect an angle?


Basic geometric terms and constructions

Angle

Vertex

CoreMath Textbook

Instruction

9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) • Guiding students to use/discover multiple methods of

copying/bisecting angles (compass & straightedge, protractor, paper folding, etc.)

• Using multiple methods to copy/bisect angles (compass &

straightedge, protractor, paper folding, etc.)

• Classifying an angle based on its measure



Unit 1.1 Pairs of Angles Lesson 4 of 5 Days 1

Lesson Focus


CC.9-12.A.CED.1 CC.9-12.G.CO.9

• Identifying different

types of angle pairs

(complementary/supplem

entary, vertical angles,

adjacent angles)

• Using the properties of

these angle pairs to solve

for missing values and

measures

CCSS.Math.Practice.MP3 How can you use angle pairs to

solve problems?


Measuring Angles Congruent Adjacent

Complementary Vertical

Supplementary

CoreMath Textbook

Instruction

9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) • Introducing the different types of angle pairs

• Helping students discover the properties that go with each pair type (Complementary angles add up to 90 degrees, Vertical angles are congruent, etc.)

• Identifying the different types of angle pairs

• Using the properties of each pair type to solve problems for missing

measures and values



Unit 1.1 Midpoint & Distance Formulas Lesson 5 of 5 Days 2

Lesson Focus


CC.9-12.G.GPE.4 CC.9-12.G.GPE.6

• Midpoint Formula

• Distance Formula

CCSS.Math.Practice.MP8 How can you find midpoints of

segments and distances in the

coordinate plane?


Midpoint

Distance

CoreMath textbook

Instruction

9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) • Review quadrants on the coordinate plane

• Use the Pythagorean Theorem to prove that the Distance Formula does in fact work

• Show that the Midpoint Formula is really just finding the average of the x & y coordinates

• Using the Midpoint and Distance Formulas to solve problems



Unit: 1.2 Geometric Reasoning Days : 6

Essential Questions

• What kinds of justifications can you use in writing algebraic and geometric proofs?

• How can you organize the deductive reasoning of a geometric proof?

• What are some formats you can use to organize geometric proofs?


• Students will learn the different types of justifications

that can be used when writing proofs

• Students will learn how to organize a proof

• Students will discover the different formats that can be

used to organize proofs

• Differentiating between theorems and postulates

• Justifying geometric statements


There is one assessment planned for this unit CC.9-12.G.CO.9


Sections 2-5, 2-6, and 2-7 in CoreMath Text Explorations in CoreMath Textbook



Unit 1.2 Algebraic Proofs Lesson 1 of 3 Days 1

Lesson Focus


CC.9-12.G.CO.9 Students will learn the different

types of justifications that can be

used to write an algebraic proof

CCSS.Math.Practice.MP7 What kinds of justifications can

you use in writing algebraic and

geometric proofs?


Properties of Equality Proof

theorem

postulate

Students need to be aware that

they have to justify EVERY step

in a mathematical proof

2-5 in CoreMath textbook

Instruction

9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) • Helping students translate/compare their verbal responses to

the question “Why?”, to actual mathematical properties and definitions. For example, “Subtract 5 from both sides” is really the Subtraction Property of Equality, and so on.

• Discuss the difference between a postulate and a theorem

• Help students “slow down” and analyze each step of solving problem

• Practicing/Repeating the concept of explaining why they're

performing each step.

• Understanding that it's not good enough any more just to do the

work, but the need to explain why they're doing what they're doing.

For many students, this is a struggle.



Unit 1.2 Geometric Proofs Lesson 2 of 3 Days 2

Lesson Focus


CC.9-12.G.CO.9 Students will work on

determining the best way to

organize the steps of a geometric

proof

CCSS.Math.Practice.MP3 How can you organize the

deductive reasoning for a proof?


Justifications for proofs 2-6 of CoreMath textbook

Instruction

9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) • Continue to work w/students on translating their answers to

the question “Why?” to mathematical reasons

• Begin to demonstrate effective ways to organize those answers into a geometric proof

• Applying the concepts and practices learned while doing Algebraic

proofs when introduced w/Geometric proofs

• Begin learning effective ways of organizing a proof



Unit 1.2 Flowchart and Paragraph Proofs Lesson 3 of 3 Days 2

Lesson Focus


CC.9-12.G.CO.9 Students will discover the

different ways available to

organize and present a proof

CCSS.Math.Practice.MP7 What are some formats you can

use to organize geometric proofs?


Writing two-column proofs 2-7 in CoreMath textbook

Instruction

9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) Introducing students to different formats for proofs (paragraph

and flowchart styles) Continue working on correct justifications of ALL steps in a proof.

Practicing writing their justifications in different formats (flowchart

and/or paragraph proofs)



Unit: 1.3 Parallel and Perpendicular Lines Days : 9

Essential Questions

• How many distinct angle measures are formed when three lines in a plane intersect in different ways?

• How can you construct perpendicular lines and prove theorems about perpendicular bisectors?

• How do you find the point on a line segment that partitions the segment in a given ratio?

• How can you use slope to write equations of lines that are parallel or perpendicular?


• Students will learn the types of angles that are formed by

a parallel line and its transversals

• Students will learn the properties of the angles formed

by parallel lines and its transversals

• Students will learn the relationship of the slopes of

parallel and perpendicular lines

• Determining the slope of a line parallel or perpendicular

to a given line

• Determining missing measures of angles formed by

parallel lines and their transversals


There are two assessments scheduled for this unit CC.9-12.G.CO.9

CC.9-12.G.CO.12

CC.9-12.G.GPE.5

CC.9-12.G.GPE.6


Sections 3-1, 3-2, 3-4, 3-5 and 3-6 in the CoreMath Text Textbook



Unit 1.3 Lines and Angles Lesson 1 of 4 Days 1

Lesson Focus



Instruction

9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing)



Unit 1.3 Parallel Lines and Transversals Lesson 2 of 4 Days 3

Lesson Focus



Instruction




Unit 1.3 Perpendicular Lines Lesson 3 of 4 Days 2

Lesson Focus



Instruction




Unit 1.3 Slopes of Parallel/Perpendicular Lines Lesson 4 of 4 Days 2

Lesson Focus



Instruction




Unit: 1.4 Transformations Days : 9

Essential Questions

• How do you draw the image of a figure under a translation, reflection, rotation and/or dilation?

• How can you use more than one transformation to map one figure onto another?

• How do you identify transformations that are rigid motions?


• Students will be able to draw the image of a figure under

a translation, reflection, rotation and/or dilation

• Students will be able to identify which transformations

are rigid motions

• Students will use more than one transformation to map

one figure onto another

• Draw figures using translations, reflections, rotations

and/or dilations

• Identifying rigid motion transformations

• Predict effects of a given rigid motion on a figure


One assessment to be given at the end of the unit CC.9-12.G.CO.2, CC.9-12.G.CO.4, CC.9-12.G.CO.5, CC.9-

12.G.CO.6


Ch. 1.7, 9.2, 9.1, 9.3, 9.4 and 9.7 in the Explorations in

CoreMath Textbook

Textbook



Unit 1.4 Transformations Lesson 1 of 6 Days 1

Lesson Focus


CC.9-12.G.CO.2

CC.9-12.G.CO.5

CC.9-12.G.CO.6

What is/isn't a rigid motion CCSS.Math.Practice.MP5 How do you identify

transformations that are rigid

motions?


None Transformation

Pre-image

Image

Rigid Motion

Ch.1.7 in Text

Instruction


Read first paragraph on page 337 to students to introduce the terms

transformation, pre-image and image.

Guide students through Explore Example 2 on page 38. Students should

work in pairs with focus on the Reflection Questions 2a and 2b.

Read definition of rigid motions on page 39 and have students complete

Example 3.

Students are reading along on page 37 and taking additional notes

when necessary.

Students complete Explore Example 2 on page 38

Students will read Example 3 and compare definition to their

answer to 2a and 2b.

Students complete Example 3 and page 40-41 #’s 1-10, 15




Lesson Focus


CC.9-12.G.CO.2

CC.9-12.G.CO.4

CC.9-12.G.CO.5

CC.9-12.G.CO.6

Drawing the image of a figure

under a translation

CCSS.Math.Practice.MP2 How do you draw the image of a

figure under a translation?


Transformations and Rigid

Motions

Translation Ch 9.2 in Text

Instruction

9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) Teacher will define a translation using coordinates and guide students

through Example 3 on page 397

Students will take notes on translations and complete Example 3. Students

should now be able to do page 380 #’s 10-12 and page 381 #’s 1-10




Lesson Focus


CC.9-12.G.CO.2

CC.9-12.G.CO.4

CC.9-12.G.CO.5

CC.9-12.G.CO.6

Drawing the image of a figure

under a reflection

CCSS.Math.Practice.MP3 How do you draw the image of a

figure under a reflection?


Transformations and rigid

motions

Reflection Ch 9.1 in Text

Instruction

9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) Teacher will instruct students to read introductory paragraph on page 369

then guide students through Explore 1. If time and materials are available

use MIRA’s to introduce reflections. Teacher will then show students the

table on page 371 and guide them through Example 3.

Students will highlight important information in the initial paragraph on

page 369 and complete Explore 1. Students will highlight table on page 371

and complete Example 3. Students should then be able to complete pages

373-374 #’s 9-19 and memorize the table from page 371.




Lesson Focus


CC.9-12.G.CO.2

CC.9-12.G.CO.4

CC.9-12.G.CO.5

CC.9-12.G.CO.6

Drawing an image under a

rotation

CCSS.Math.Practice.MP5 How do you draw an image

under a rotation?


Transformations and rigid

motions

Rotation Remind students that 270

counterclockwise is the same as

90 clockwise.

Ch 9.3 in Text

Ruler

Protractor

Instruction

9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) Day 1:

Teacher will instruct students to read introductory paragraph on page 383

and the paragraph defining rotations on page 384. Teacher will then guide

students through Example 2 on page 384. Have students work in groups and

complete page 386 #’s 1-3

Day 2:

Teacher will instruct students to read and highlight the box on page 385 and

guide them through Example 3. Teacher will then have students try Extra

Example from 385T.

Day 1:

Students will read and take notes on page 383 and 384 or highlight

important material. They will then complete Example 2 and work in groups

or pairs on page 386 #’s 1-3

Day 2:

Students will highlight the box on page 385 and complete Example 3 and

the Extra Example. Students can now complete pages 386-387 #’s 4-9 and

memorize the table from page 385.



Unit: 2.1 Properties of Triangles Days : 15

Essential Questions

• How can you classify triangles in the coordinate plane?

• What are some theorems about angle measures in triangles?

• What special relationships exist among the sides and angles of isosceles triangles?

• What can you conclude about the medians of a triangle?

• What must be true about the segment that connects the midpoints of two sides of a triangle?

• How are side lengths and angle measures of triangles related?


• Students will classify triangles on the coordinate plane

• Students will identify special relationships related to

isosceles triangles

• Students will make conclusions using the medians of a

triangle

• Students will explore the segment that connects the

midpoints of two sides of a triangle

• Students will determine how side lengths and angle

measures of triangles are related

• Calculating distance and midpoint on the coordinate

plane

• Identifying relationships in isosceles triangles

• Making conclusions about the medians of a triangle

• Determining how side lengths and angle measures in

triangles are related


Two assessments – one midway through the unit, one at the end

of the unit

CC.9-12.G.GPE.4, CC.9-12.G.GPE.7, CC.9-12.G.CO.10


Ch. 4-2, 4-3, 4-9, 5-3, 5-1, 5-4 and 5-5 in the Explorations in

COREMath textbook

Textbook



Unit 2.1 Properties of Triangles Lesson 1 of 7 Days 1

Lesson Focus


CC.9-12.G.GPE.7 Using distance formula to

classify triangles

CCSS.Math.Practice.MP1 How can you classify triangles in

the coordinate plane?


The Distance Formula and perimeter

Ch. 4.2 in Textbook

Instruction

9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) • Asking students what information is needed to classify a triangle by

its sides

• Reviewing how to use the Distance Formula on the coordinate plane

• Explaining how to classify a triangle by its angles using only the side lengths

• Creating a summary table of the different types of triangles and their

characteristics

• Classifying triangles by their sides

• Classifying triangles by their angles, using only their side lengths




Lesson Focus


CC.9-12.G.CO.10 • What the sum of the

interior angles of a

triangle is

• How to find the measure

of an exterior angle of a

triangle

CCSS.Math.Practice.MP3 What are some theorems about

angle measures in triangles?


Parallel lines and transversals Interior angle, exterior angle,

remote exterior angle, corollary

Ch. 4.3 in Textbook

Instruction

9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) Using geometry software to demonstrate many different triangles and

how the sum of the interior angles are always equivalent Use triangle cutouts and protractors to demonstrate the same concept

as above

• Using the Triangle Sum Theorem to find missing interior angles

• Using the Exterior Angle Theorem to find missing exterior angles

• Use counterexamples to prove/disprove certain statements




Lesson Focus


CC.9-12.G.CO.10 How to use the relationships of

sides/angles of isosceles triangles

to solve for missing sides/angles

CCSS.Math.Practice.MP1 What special relationships exist

among the sides and angles of

isosceles triangles?


Triangle Sum Theorem Legs, vertex angle, base, base

angles

Ch. 4.9 in Textbook

Instruction

9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) • Defining what an isosceles triangle is and its properties

• Showing different methods of constructing an isosceles triangle • Identifying what information is needed to determine if a triangle is

isosceles

• Using reasoning and/or algebra to find unknown angle measures and

sides

• Applying the Isosceles Triangle Theorem




Lesson Focus


CC.9-12.G.GPE.4 What the median/altitude of a

triangle is and their properties

CCSS.Math.Practice.MP8 What can you conclude about the

medians of a triangle?


Midpoint Formula Coordinate Proofs Equation of a line

Median, concurrent, centroid,

altitude, orthocenter

Altitudes connect to the

midpoints of sides

Ch. 5.3 in Textbook

Instruction

9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) Some Possible Approaches: Teachers can provide students with cut-out triangles as manipulatives to fold and measure as a way of examining and introducing the properties in the section

Manipulating the medians of triangles either through geometry software or

actual hands-on manipulatives to discover their properties




Lesson Focus


CC.9-12.G.GPE.2 What the Angle and

Perpendicular Bisectors of a

triangle are and their properties

CCSS.Math.Practice.MP6 How do you write the equation of

a parabola given its focus and

directrix?


Distance Formula Incenter, Circumcenter Ch. 5.1 in Textbook

Instruction

9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) Some Possible Approaches: Modelling how the properties of Angle and Perpendicular Bisectors of triangles can come into play in other areas of mathematics (more specifically, conics in these examples).

Discovering how seemingly unrelated topics in mathematics can and do

overlap




Lesson Focus


CC.9-12.G.GPE.4 The definition of the

midsegment of a triangle and

how to find its measure

CCSS.Math.Practice.MP5 What must be true about the

segment that connects the

midpoints of two sides of a

triangle?


Distance & Midpoint Formulas Coordinate proofs Slope of Parallel Lines

Midsegment Ch 5.4 in Textbook

Instruction

9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) Some Possible Approaches:

• Using a variety of questioning strategies to elicit constructive and meaningful dialogue regarding triangle midsegments

• Use geometry software to allow students to explore midsegment properties

Discovering the properties of midsegments of triangles and how they apply

to all types of triangles34




Lesson Focus


CC.9-12.G.CO.10 How the size of the sides and

angles of a triangle relate to each

other

CCSS.Math.Practice.MP3


How can you use inequalities

related to triangle side lengths and

angle measures in proofs?


Triangle Sum Theorem Ch. 5.5 in Textbook

Instruction

9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) Some Possible Approaches: Review the Transitive Property of Inequality and encouraging the students to discover the relationships of sides/angles within a triangle by exploring/manipulating a variety of triangle types

• Using diagrams to illustrate angle-side relationships in triangles

• Applying the inequality properties to previously learned concepts

involving writing proofs



Unit: 2.2 Triangle Congruence Days : 9

Essential Questions

• How can you use properties of rigid motions to draw conclusions about corresponding sides and angles in congruent

triangles?

• How can you establish the SSS, SAS, ASA, AAS, and HL triangle congruence criteria?

• How can CPCTC be used to in proofs?


• Students wil use properties of rigid motions to draw

conclusions about corresponding sides and angles in

congruent triangles

• Students will use SSS, SAS, ASA, AAS, and HL triangle

congruence criteria to prove triangles are congruent

Students will use CPCTC to establish proofs

• Using SSS, SAS, ASA, AAS, HL triangle congruence

criteria and CPCTC in proofs


One assessment given at the end of the unit CC.9-12.G.CO.7

CC.9-12.G.CO.8

CC.9-12.G.CO.10

CC.9-12.G.SRT.5


Ch 4-4, 4-5, 4-6 and 4-7 in the Explorations in COREMath

Textbook

Textbook



Unit 2.2 Triangle Congruence Lesson 1 of 4 Days 1

Lesson Focus


CC.9-12.G.CO.7 How to write congruence

statements

CCSS.Math.Practice.MP6 How can you use properties of

rigid motions to draw conclusions

about corresponding sides and

angles in congruent triangles?


Congruence Corresponding parts Ch 4.4

Instruction

9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) • Teacher will guide students through Example 1 on page 147 and the

definition of CPCTC and then have students try Example 2 on page 148 • Students will highlight important information from Example 1 and the

definition of CPCTC and complete Example 2. Students will then work

on pages 149-150 #’s 1-8 and page 151 #’s 1-8




Lesson Focus


CC.9-12.G.CO.7

CC.9-12.G.CO.8

CC.9-12.G.SRT.5

How to use SSS and SAS to

show that two triangles are

congruent.

CCSS.Math.Practice.MP3 How can you establish the SSS

and SAS triangle congruence

criteria using properties of rigid

motions?


Congruence and triangles

Ch 4.5

Instruction


Teacher will read description of SSS and SAS by Proof 1 and Proof 4.

Remind students of previous knowledge such as the reflexive property,

definition of midpoint, vertical angles and angles formed by parallel lines.

Show multiple examples of congruent triangles using SSS and SAS.

Day 2:

Show students how to find missing congruent parts when given the

necessary criteria and one or two congruent parts.

Day 1:

Students will highlight and take notes on SSS and SAS congruence criteria

and complete examples using the criteria in different ways (separate

triangles, using the reflexive property, vertical angles)

Day 2:

Students will find missing congruent parts when given the necessary

congruence criteria.




Lesson Focus


CC.9-12.G.CO.8

CC.9-12.G.SRT.5

How to use ASA, AAS and HL

to prove that two triangles are

congruent.

CCSS.Math.Practice.MP4 How can you establish and use

the ASA, AAS and HL triangle

congruence criteria?


ASA and AAS confusion Ch 4.6

Instruction


Teacher will read ASA and AAS congruence criteria from Proof 1 on page

161 and Proof 3 on page 163. Teacher will explain on their own the HL

criteria (and which criteria does not work!)

Day 2:

Show students how to find missing congruent parts when given the

necessary criteria and one or two congruent parts.

Day 1:

Students will highlight and take notes on ASA and AAS congruence

criteria, take notes on HL and complete examples using the criteria in

different ways (separate triangles, using the reflexive property, vertical

angles)

Day 2:

Students will find missing congruent parts when given the necessary

congruence criteria.




Lesson Focus


CC.9-12.G.GPE.5 How to construct geometric proof

using triangle congruence criteria

and CPCTC.

CCSS.Math.Practice.MP6 How can CPCTC be used in

proofs?


Triangle congruence criteria Ch 4.7

Instruction


Teachers will remind students about properties, definitions, theorems and

congruence criteria that have been previously introduced (reflexive, SSS,

CPCTC, definition of midpoint) and also refer back to proofs done in

previous sections (algebraic, parallel line).

Day 2:

Teacher will guide students through sample proofs and fill-in-the-blank

proofs such as Example 5 on page 157 and Proof 3 on page 163.

Day 3:

Teacher will facilitate group work completing triangle congruence proofs.

Day 1:

Students will take notes on how to appropriately use properties, definitions,

theorems and congruence criteria in a proof.

Day 2:

Students will follow along completing Example 5 on page 157 and Proof 3

on page 163. They will then complete page 158 #’s 1-2 and page 164 #’s 1-

2.

Day 3:

Students will work in groups and complete triangle congruence proofs such

as page 171 #’s 1-3



Unit: 2.3 Triangle Similarity Days : 10

Essential Questions

• How can you use ratios of corresponding side lengths to solve problems involving similar polygons?

• How can dilations be used to show figures are similar?


• Students will use scale factor and proportions to solve for

missing side lengths of similar polygons

• Students will use ratios and angle measures to identify

whether figures are similar

Students will solve problems by applying the Triangle

Proportionality Theorem

• Write and use proportions to solve problems with similar

figures.


Two assessments throughout the unit G.SRT.2

G.SRT.5

MG.1


Sections 7-6, 7-3, 7-1, and 7-4 in the CoreMath text Textbook



Unit 2.3 Triangle Similarity Lesson 1 of 5 Days 1

Lesson Focus


CC.9-12.G.CO.2 Using scale factor to draw

dilations

CCSS.Math.Practice.MP3 How can you represent dilations

in the coordinate plane?


Properties of Dilations Scale Factor

Dilation

7-6 in CoreMath Text

Instruction

9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) • Remind students about dilations (9-7)

• Introduce scale factor • Use scale factor and ordered pairs to find the image under each dilation




Lesson Focus


CC.9-12.G.SRT.2 Using scale factors and angle

measures to determine whether

two figures are similar

CCSS.Math.Practice.MP8 How can you use ratios of

corresponding side lengths to

solve problems involving similar

polygons?


Simplifying ratios

Solving Proportions

Similarity

Ratio

Proportion

Corresponding parts

Confusing similarity and

congruence

7-1 in CoreMath Textbook

Instruction


• Introduce similarity by showing how the angles are congruent and the

corresponding sides will all simplify to the same scale factor.

• Have students determine if figures are similar or not

Day 2:

Remind students how to solve proportions Teach students how to find missing side lengths of similar figures

using proportions




Lesson Focus


CC.9-12.G.SRT.3 Using AA, SSS and SAS

triangle similarity criteria

CCSS.Math.Practice.MP6 What can you conclude about

similar triangles and how can you

prove triangles are similar?


Similarity Confusing SSS and SAS

congruence with similarity


Instruction

9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) • Remind students that in similar figures, the sides are proportional and

the angles are congruent

• Remind students about the Triangle Sum Theorem (4-3)

• Explain that depending which information is given in the problem, they can use ratios of corresponding sides and/or congruent angles to prove that the triangles are similar by SSS, AA or SAS




Lesson Focus


CC.9-12.G.SRT.5 Writing and solving proportions

involving a line parallel to one

side of a triangle

CCSS.Math.Practice.MP7 How does a line that is parallel to

one side of a triangle divide the

two sides it intersects?


Similarity and triangles Angles formed by parallel lines

Using incorrect proportions to

find the length of the base of the

triangle


Instruction


• Remind students that parallel lines make corresponding and same-side

interior angles

• Show students how to make and solve proportions made by parallel

lines

Day 2:

• Show students that 3 or more parallel lines intersected by two transversals make proportional segments




Lesson Focus


CC.9-12.G.SRT.5

CC.9-12.G.MG.3

Solving real-world problems

involving similar triangles and

proportions

CCSS.Math.Practice.MP3 How can you use similar

triangles and similar rectangles to

solve problems?


Similar polygons 7-5 in CoreMath textbook

Instruction


• Show students how to set up and solve triangle proportions involving

real-world problems (shadow, cliff)

Day 2:

• Show students how to set up and solve rectangle proportions involving

real-world problems (scale models, graphic design)

Possible extension: Take students outside on a sunny day and have them measure their own height and shadow and the shadow of an available pole and calculate the actual height of the pole



Unit: 3.1 Right Triangles Days : 15

Essential Questions

• How can you apply the Pythagorean Theorem?

• How do you find the tangent, sine and cosine ratios for the acute angles in a right triangle?

• How can you use trigonometric ratios to solve problems involving angles of elevation and depression?

• What can you say about the side lengths associated with special right triangles?


• Students will be able to find missing side lengths of right

triangles using the Pythagorean Theorem and Special Right

triangles

• Students will be able to apply trigonometric ratios to find

missing sides and/or angles

• Use different right triangle properties to solve problems


Two assessments and a Common Task (Cranston Marina) G.SRT.6

G.SRT.7

G.SRT.8


Explorations in COREMath Textbook:

5-7

8-1

8-2

8-3

8-4

5-8

Textbook



Unit 3.1 Right Triangles Lesson 1 of 6 Days 1

Lesson Focus 1. Standards Addressed 2. Content to be Learned 3. Mathematical Practices 4. Essential Questions

CC.9-12.G.SRT.8 Using the Pythagorean Theorem CCSS.Math.Practice.MP4 How can you apply the

Pythagorean Theorem?


Pythagorean Theorem

Triangle Sun Theorem

5-7

Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) Remind students how to use the Pythagorean Theorem and Triangle Sum

Theorem to find the missing sides and angles in a right triangle Practice using the Pythagorean Theorem to solve for missing sides of right

triangles





CC.9-12.G.SRT.5 Use geometric mean to find

lengths of sides of right triangles

CCSS.Math.Practice.MP3 How can you use triangle

similarity to prove the

Pythagorean Theorem?


Similar Triangles

Pythagorean Theorem

8-1

Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing)

• Show students that the altitude drawn to the hypotenuse of a right

triangle makes similar triangles

• Introduce and use geometric mean to solve proportions involving

similar right triangles

• Solve proportions using geometric mean





CC.9-12.G.SRT.6

CC.9-12.G.SRT.7

Finding sine, cosine and tangent

ratios in right triangles

CCSS.Math.Practice.MP3 How do you find the tangent,

sine and cosine ratios for acute

angles in a right triangle?


Adjacent

Opposite

Tangent

Sine

Cosine

Trigonometric ratio

8-2

Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) Day 1:

• Introduce the trig ratios and SOHCAHTOA and how to use each Day 2:

• Explain how to use trig ratios to solve real-world problems

Day 1:

• Apply trig ratios correctly Day 2:

• Solving real-world problems with trig ratios





CC.9-12.G.SRT.8 Students will use trigonometric

ratios to find acute angle

measures in right triangles

CCSS.Math.Practice.MP5 How do you find an unknown

angle measure in a right triangle?


Trigonometric ratios Inverse trigonometric ratios 8-3

Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) Day 1: Introduce how to use an inverse function to find an angle measure,

including necessary calculator keystrokes Day 2: Teach how to find acute angle measures in right triangles involving real-

world problems

Day 1: Use inverse trig functions to find acute angle measures in right triangles Day 2: Use inverse trig functions to solve real-world problems





CC.9-12.G.SRT.8 Solving problems involving

elevation and depression

CCSS.Math.Practice.4 How can you use trigonometric

ratios to solve problems

involving angles of elevation and

depression?


Angles formed by parallel lines

and a transversal

Trigonometric ratios

8-4

Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) Show students how to use alternate interior angles and trigonometric ratios

to solve real-world problems Solve problems using alternate interior angles and trig ratios





CC.9-12.G.SRT.6 Using properties of special right

triangles to find side lengths

CCSS.Math.Practice.MP7 What can you say about the side

lengths associated with special

right triangles?


Pythagorean Theorem Confusing 45-45-90 and 30-60-

90 triangles

5-8

Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) Day 1: Show students that the length of the hypotenuse of an isosceles right

triangle is 2 times the length of its leg Day 2: Show students that the hypotenuse of a 30-60-90 triangle is twice the length

of its short leg and that the long leg is 3 times the length of its short leg

Day 1: Solve problems involving isosceles right triangles and diagonals of squares

by applying 45-45-90 triangle properties Day 2: Solve problems involving 30-60-90 triangles and equilateral triangles with

an altitude drawn.



Unit: 3.2 Quadrilaterals Days : 8

Essential Questions

• What can you conclude about the sides, angles, and diagonals of a parallelogram?

• What are the properties of rectangles and rhombi?

• How can you use slope in coordinate proofs?


• Students will learn the properties of different types of

quadrilaterals (ie, parallelograms, rectangles, rhombi,

etc.)

• Students will use the properties of quadrilaterals as well

as slope, distance, and midpoint in coordinate proofs

• Students will solve problems using the properties of

quadrilaterals

• Using the slope, distance and midpoint formulas

• Using properties of parallel lines and transversals


There is one assessment planned for this unit G.CO.9

G.CO.11

G.GPE.4

G.SRT.5


Sections 6-2, 6-3, 6-4, 6-5, and 6-6 in the CoreMath Text Explorations in Core Math Textbook



Unit 3.2 Properties of Parallelograms Lesson 1 of 5 Days 1

Lesson Focus


CC.9-12.G.CO.11 CC.9-12.G.SRT.5

Properties of sides, angles, and

diagonals of parallelograms



What can you conclude about the sides,

angles, and diagonals of a

parallelogram?


• Theorems about parallel lines cut by a transversal

• Triangle congruence criteria

Diagonal

parallelogram

6-2 in Textbook

LCD Projector/Laptop/ELMO

Instruction

9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) • Remind students of difference between “opposite” and

“consecutive”

• Revisit parallel lines, and how they and their properties fit when dealing with a parallelogram

• Introduce the concept of a diagonal, and how the triangles that are formed by diagonals can help prove and support the various theorems and properties

• Use geometry software to construct parallelograms

• Identifying the diagonals of a parallelogram

• Identifying the “opposite” and “consecutive” sides and angles in a

parallelogram

• Using reasoning to extend what they already know about parallelograms

• Using reasoning and/or algebra to find unknown angle and side measures



Unit 3.2 Conditions for Parallelograms Lesson 2 of 5 Days 1

Lesson Focus


CC.9-12.G.CO.11 CC.9-12.G.SRT.5

How to prove that a quadrilateral

is a parallelogram based on given

information



What criteria can you use to prove that a

quadrilateral is a parallelogram?


Triangle Congruence criteria Properties of Parallelograms

6-3 in Textbook


Instruction

9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) • Revisit the definition of Parallelogram and what properties make a

parallelogram a parallelogram

• Give examples of problems where students have various pieces of information and explore whether or not that information is enough to prove that the quadrilateral is a parallelogram

• Revisit the different ways that a proof can be set up and presented

• Making arguments to prove/disprove whether or not the given information

supports the given conclusion

• Determining which criteria is necessary to prove that a quadrilateral is a

parallelogram



Unit 3.2 Properties of Special Parallelograms Lesson 3 of 5 Days 2

Lesson Focus


CC.9-12.G.CO.11 CC.9-12.G.SRT.5

Specific properties for

rectangles, rhombi and squares



What are the properties of rectangles

and rhombi?


• Triangle Congruence criteria

• Properties of Parallelograms Rectangle

Rhombus

Square

Mixing up the properties that are

specific to each special type of

parallelogram

6-4 in Textbook


Instruction

9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) • Present the specific properties that apply only to rectangles and

rhombi

• Discuss how the previous properties of parallelograms they've already learned, also apply to rectangles and rhombi

• Showing how special parallelograms fall into multiple categories (ie, a square is also a rectangle, and a rhombus, and a parallelogram)



rectangle, rhombus, or square



Unit 3.2 Quadrilaterals on the Coordinate Plane Lesson 4 of 5 Days 1

Lesson Focus


CC.9-12.G.GPE.4 Using slope, distance and

midpoint in quadrilateral proofs

on the coordinate plane

CCSS.Math.Practice.MP3 How can you use slope in coordinate

proofs?


• Writing coordinate proofs

• Slopes of parallel and perpendicular lines

6-5 in Textbook


Instruction

9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) • Asking students to list what information is needed to prove that a

quadrilateral is a rectangle, rhombus, etc.

• Asking students to state what coordinate plane formulas can be used to determine the required information

• Remind students of the slope criteria for parallel and perpendicular lines

• Students use reasoning to find the coordinates of a point that makes a

quadrilateral a parallelogram, rectangle, etc.

• Students prove that the diagonals of a rhombus are perpendicular by using

slopes

• Students use slope, distance, midpoint formulas to determine what type of

quadrilateral is presented on the coordinate plane



Unit 3.2 Other Quadrilaterals Lesson 5 of 5 Days 2

Lesson Focus


CC.9-12.G.CO.9 Properties of non-parallelogram

quadrilaterals (ie, kites,

trapezoids, etc.)



How can auxiliary segments be used in

proofs?


• Angles formed by parallel lines and transversals

• Properties of Parallelograms

Kite

Trapezoid

Applying the properties of

parallelograms to non-

parallelogram quadrilaterals

6-6 in Textbook


Instruction

9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) • Introduce the differences between kites and trapezoids and those

quadrilaterals that fall into the parallelogram family

• Use geometry software to introduce the Trapezoid Midsegment Theorem

• Students apply what they've learned to prove that a quadrilateral is a

trapezoid



trapezoid



Unit: 4.1 Circles Days : 13

Essential Questions

• How are arcs and chords of circles associated with central angles?

• What is the relationship between central angles and inscribed angles in a circle?

• What is the relationship between a tangent line to a circle and the radius drawn from the center to the point of tangency?

• When two tangents are drawn to a circle, how do you find the measure of the angle formed at their intersection?

• How can you write and use equations of circles in the coordinate plane?


Students will be able to name and find the measures of arcs, angles and

segments on and in a circle.

• Students will be able to write the equation of a circle and graph a

circle on the coordinate plane.

• Finding angle, segment and arc measure in a circle.

• Writing equations of a circle

• Graphing circles on the coordinate plane


Two assessments, one mid-unit and one end of unit G.C.2

G.C.3

G.CO.9

G.GPE.1


Exploration in CORE Math:

12-2, 12-4, 12-1, 12-5, 12-6, 12-7 Textbook



Unit 4.1 Circles Lesson 1 of 3 Days 2


G.C.2 Recognizing parts of a circle and

finding the measures of central

angles.

CCSS.Math.Practice.MP7 How are arcs and chords of

circles associated with central

angles?


Basic terms Chord

Central Angle

Inscribed Angle

Arc

Minor Arc

Major Arc

Semicircle

12-2


Parts of a circle, including: chords, arcs, central and inscribed angles

Day 2:

Find the measure of a central angle in a circle

Day 1:

Students will be able to name and recognize parts of a circle including

chords, arcs, central and inscribed angles.

Day 2:

Students will be able to find the measure of a central angle of a circle.





G.C.2

G.C.3

Finding the measure of inscribed

angles

CCSS.Math.Practice.MP5 What is the relationship between

central angles and inscribed

angles in a circle?


Inscribed and Central Angles 12-4


Find measure of an inscribed angle in a circle

Day 2:

Problem solving involving inscribed and central angles in a circle

Day 1:

Students will be able to find the measure of an inscribed angle in a circle.

Day 2:

Students will solve problems involving central and inscribed angles in a

circle.





G.C.2 Right triangles formed by a

tangent line to a circle and a

radius drawn to the point of

tangency

CCSS.Math.Practice.MP7 What is the relationship between

a tangent line to a circle and the

radius drawn from the center to

the point of tangency?


Central and Inscribed Angles Tangent

Point of Tangency

Circumscribed Angle

12-1

Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) • •



Unit: 4.2 Extending Perimeter, Circumference and Area Days : 6

Essential Questions

• How can you find areas of irregular shapes?

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

• How do you calculate arc length in a circle?


Students will remember how to find the area and perimeter

(circumference) of polygons and circles.

Students will be able to find the areas of composite figures.

Students will be able to find the area of a sector of a circle.

• Students will be able to find arc length of a circle.

• Finding area of triangles, parallelograms, trapezoids and circles.

• Using areas of polygons and circles to find area of composite figures.

• Using proportional reasoning to fins the area of a sector and arc

length on a circle.


One assessment at the end of the unit G.MG.1

G.MG.3

G.CO.1

G.C.5

G.GMD.1


Explorations in CoreMath text: 10-2 and 12-3 Textbook



Unit 4.2 Extending Perimeter, Circumference and Area Lesson 1 of 2 Days 3


G.MG.1

G.MG.3

Finding the area of a region by

separating into simpler shapes

CCSS.Math.Practice.MP8 How can you find the area of

irregular shapes?


Basic area formulas 10-3

Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) Day 1: Reinforce basic area formulas (triangle, rectangle, circle, trapezoid) Day 2: Begin composite figures such as two triangles forming a quadrilateral or a

rectangle with two semicircles Day 3: More complex composite figures involving adding and subtracting multiple

figures

Day 1: Students will be able to apply basic area formulas Day 2: Students will be able to find the area of basic composite figures Day 3: Students will be able to find the area of complex composite figures



Unit 4.2 Extending Perimeter, Circumference and Area Lesson 2 of 2 Days 2


G.CO.1

G.C.5

G.MGD.1

Finding the area of a sector by

considering the fraction of a full

circle that is represented by a

given circle and also using a

similar process to find arc

lengths.



How do you find the area of a

sector of a circle and how do you

calculate arc length in a circle?


Central angles

Inscribed angles

Circumference

Sector

Arc length

Radian measure

Confusing arc length with area of

a sector

12-3

Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) Day 1: Use area of a circle to find area of a sector using proportional reasoning Day 2: Use circumference of a circle to find arc length using proportional

reasoning

Day 1: Students will be able to use their understanding of area of circles and

proportional reasoning to find area of a sector Day 2: Students will be able to us the their understanding of circumference of a

circle and proportional reasoning to find arc length



Unit: 4.3 Volume Days : 11

Essential Questions

• How do you calculate the volume of a prism or cylinder?

• How do you calculate the volume of a pyramid or cone and use volume formulas to solve problems?

• How do you calculate the volume of a sphere?

• How do you calculate the volume of composite figures?


• Students will be able to find the volume of prisms, cylinders,

pyramids, cones and spheres.

• Students will solve problems involving volume.

Students will be able to find the volume of composite figures.

• Finding volume of different figures


Two assessments, one mid-unit and one end of unit G.GMD.1

G.GMD.3

G.MG.1

G.MG.2


Explorations in CoreMath: 11-2, 11-3, and 11-4 Textbook



Unit 4.3 Volume Lesson 1 of 3 Days 3


G.GMD.1

G.GMD.2

G.GMD.3

Finding volume of figures CCSS.Math.Practice.MP1 How do you calculate the volume

of a prism or cylinder and use

volume formulas to solve design

problems?


Visualizing three dimensional

figures

Oblique cylinder 11-2

Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) Day 1: Introduce formulas and examples involving volume of prisms Day 2: Introduce formulas and examples involving volume of cylinders Day 3: Find volume of oblique prisms and cylinders

Day 1: Students will be able to find the volume of prisms Day 2: Students will be able to find the volume of cylinders Day 3: Students will be able to find the volume of oblique prisms and cylinders





G.GMD.1

G.GMD.3


of a pyramid or cone and use

volume formulas to solve

problems?


Volume of prisms and cylinders 11-3

Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) Introduce formulas and examples involving volume of pyramids and

cones

Students will be able to find the volume of pyramids and cones





G.GMD.2

G.GMD.3


of a sphere and use the volume

formula to solve problems?


Volume of cylinders and cones 11-4

Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) Day 1: Introduce formula and examples involving volume of a sphere Day 2 and 3: Using the volumes of prisms, cylinders, pyramids, cones and spheres, find

the volume of composite figures by adding and subtracting volumes

Day 1: Students should be able to find the volume of a sphere Day 2 and 3: Students should be able to find the volume of composite figures



Unit: 4.4 Probability Days : 9

Essential Questions

• What are permutations and combinations and how can you use them to calculate probabilities?

• How can you use probabilities to help you make fair decisions?

• How can you use geometric probability to solve problems?

• How do you find the probability of independent and dependent events?

• How do you find the probability of mutually exclusive events and overlapping events?


• Students will be able to use permutations and combinations to

calculate probability.

• Students will make educated decisions based on probability.

• Students will be able to calculate theoretical probabilities.

• Students will be able to use the multiplication rule and find

the probability of independent and dependent events.

Students will find the probability of mutually exclusive events and

overlapping events and apply the addition rule.

• Calculating factorials

• Applying the formulas for permutations and

combinations appropriately

• Decision making problems

• Probability of an event and its complement

• Independent and dependent events

• Conditional probability

• Mutually exclusive and overlapping events

Addition rule


One assessment at the end of the unit S.CP.9

S.MD.6

S.CP.1-5,7


Exploration in CORE Math:

13-1, 13-2, 10-6, 13-3, 13-5

Textbook

(Also see Glencoe Geometry 2005 section 11-5 for geometric

probability)

geometry unit and lesson overviews 2014 - weebly

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