Mrs. Abbett's Class

Syllabus

 

Geometry

 

Textbook:

            Glencoe:  Geometry – Integrated, Applications, and Connections

                           

General Description:

            Mrs. Abbett’s Geometry Class implements the shift from geometry as a course in proof to geometry as a representation of the world around us.  Then study of geometry also encompasses its close relationship with algebra by using coordinate and algebraic means to verity the synthetic representations.  In each chapter, students use algebraic tools to verify properties of figures presented on a coordinate plane.  Each lesson opener will motivate students to master the content they need to solve application, connection, or integration problems presented in the lesson.  Additional applications, connections and integration in the exercises enable students to apply   what they have learned.

 

 

Objectives: First Semester

 

1. Student will be able to solve problems involving angle measurement in polygons and circles. (7C2)
2. Student will be able to determine derived measurements. (7C5)
3. Student will be able to solve problems using indirect measurement by choosing appropriate technology, instruments and formulas. (7C9)
4. Student will be able to describe the general trends on how to change one measure affects other measures in the same figure. (7C11)
5. Student will be able to determine the ratio of similar figure perimeters, and area using ratio of similitude. (7C13)
6. Student will be able to represent and analyze the properties of geometric shapes using coordinate geometry. (9A1)
7. Student will be able to determine if a triangle is possible using side lengths and triangle inequality. (9A3)
8. Student will be able to solve pictorial or word problems that involve geometric relationships within a single geometric shape or figure, including the Pythagorean theorem. (9A5)
9. Student will be able to analyze the results of a combination or reflections, and translations of a figure, and determine alternate motions that could produce the results. (9A6)
10. Student will be able to combine simple construction techniques to construct squares, equilateral triangles or other simple combinations of equal sequent, angles, etc. (9A7)
11. Student will be able to describe and apply properties of a  polygon in a problem solving situation. (9A9)
12. Student will be able to classify angle relationship for two or more lines crossed by a transversal. (9A10)
13. Student will be able to analyze geometric situations using Cartesian coordinates. (9A11)
14. Student will be able to represent transformations of an object in the plane using sketches, coordinates and vectors. (9A12)
15. Student will be able to gain insights   into and answer questions in m other areas of mathematics using geometric models. (9A15)
16. Student will be able to calculate distance, midpoint coordinates, and slope using coordinate geometry. (9A16)
17. Student will be able to identify and apply properties of medians, altitudes, and angle bisector, perpendicular bisector, and midline of a triangle. (9A18)
18. Student will be able to analyze geometric situations using Cartesian coordinates and other coordinate systems such a s navigation, polar and spherical systems. (9A19)
19. Student will be able to represent and describe with the language of geometry real life objects, paths and regions in space. (9A 21)
20. Student will be able to examine the congruence of similarity of objects using transformations. (9B3)
21. Student will be able to solve problems using triangle congruence and similarity figures. (9B5)
22. Provide example and counter examples to either illustrate or disprove conjectures about geometric characteristics. (9C4)
23. Represent, solve ,and  explain numerical and algebraic relationships using geometric concepts. (9C3)
24. Develop a formal proof for a given geometric situation. (9C8
25. Develop conjectures about geometric situations with and without technology. (9C8)
26. Describe the difference between inductive argument and deductive argument. (9C 10)
27. Prove conjectures about geometric figures on the plane and in space. (9C11)
28. Explain the ideal of formal and informal proof to non-geometric situations. (9C12)
29. Recognize Pythagorean Triples (9D1)
30. Determine distances and angle measures using indirect measurement and properties of right triangle. (9D6)

 

 

Geometry Text Contents First Semester:

 

1. Discovering Points, Lines, and Angles.

-          The Coordinate Plane

-          Points, Lines and Planes

-          Using Formulas

-          Measuring Segments

-          Midpoints and Segment Congruence

-          Exploring Angles

-          Angle Relationships

 

 

1. Connecting Reasoning and Proof

-          Inductive Reasoning and Conjecturing

-          If-Then Statements and Postulates

-          Deductive Reasoning

-          Using Proof in Algebra

-          Verifying Segments Relationships

-          Verifying Angle Relationships

 

1. Using Perpendicular and Parallel Lines

-          Parallel Lines and Transversals

-          Angles and Parallel Lines

-          Slopes of Lines

-          Proving Lines Parallel

-          Parallels and Distance

-          Non-Euclidean Geometry

-    Spherical Geometry

 

1. Identifying Congruent Triangles

-          Classifying Triangles

-          Measuring Angles in Triangles

-          Exploring Congruent Triangles

-          Proving triangles Congruent

-          More congruent Triangles

-          Analyzing Isosceles triangles

 

1. Applying Congruent Triangles

-          Special Segments in Triangles

-          Right Triangles

-          Indirect Proof and Inequalities

-          Inequalities for Sides and Angles of a Triangle

-          The Triangle Inequality

-          Inequalities Involving Two Triangles

 

 

 

1. Exploring Quadrilaterals

-          Parallelograms

-          Testing for Parallelograms

-          Rectangles

-          Squares and Rhombi

-          Trapezoids

 

 

Objectives Second Semester

1                    All objectives from 1^st semester will be used as well as

2                    Develop and describe surface area and volume formulas for cones and cylinders by relating pyramids to cones and prisms to cylinders(7C3)

3                    Determine linear measures, perimeters, areas, , surface areas and volumes of similar figures using the ratio of similitude. (7c11)

4                    Identify relationships among circles, arcs, cords, Tangents and secants.((9B7)

5                    Solve problems in and gain insight into other disciplines and other areas of interest such as art and architecture using geometric ideas. (9B8)

6                    Analyze and describe the transformations that lead to successful tessellations of one of more figures. (9B9)

7                    Create and arguments concerning geometric ideas and relationships such as congruence similarity, Pythagorean relationships for areas and volume. (9 C 1)

8                     Identify the basic trigonometric ratios in terms of length of the sides of a right triangle and an acute angle. (9D2)

9                    Solve for missing sides length using the trigonometric ratios in right triangles. (9D3)

10                Determine and justify the side length relationship present in the 45-45 90 triangle and 30-60 -90 triangles. (9D4)

11                Determine the ratio of length of sides of a right triangle with given measurements for its acute angles. (9D5)

12                Solving problems using 45-45-90 and 30-40-90 triangles. (9D7)

13                Solve problems using the Law of Sine and The Law of Cosine. (9D8)

 

 

 

 

Geometry Text Contents:

 

1. Connecting Proportions and Similarity

-          Using proportions

-          Exploring Similar Polygons

-          Identifying Similar Triangles

-          Parallel Lines and Proportional Parts

-          Parts of Similar Triangles

-          Fractals and Self-Similarity

 

1. Applying Right Triangles and Trigonometry

-          Geometric Mean and the Pythagorean Theorem

-          Special Right Triangles

-          Ratios in Right Triangles

-          Angles of Elevation and Depression

-          Using the Law of Sines

-          Using the Law of Cosines

 

1. Analyzing Circles

-          Exploring Circles

-          Angles and Arcs

-          Arcs and Chords

-          Inscribed Angles

-          Tangents

-          Secants, Tangents and Angle Measures

-          Special Segments in a Circle

-          Equations of Circles

 

1. Exploring Polygons and Area

-          Polygons

-          Tessellations

-          Area of Parallelograms

-          Area of Triangles, Rhombi and Trapezoids

-          Area of Regular Polygons and Circles

-          Geometric Probability

-          Polygons as Networks

 

1. Investigating Surface Area and Volume

-          Exploring Three-Dimensional Figures

-          Nets and Surface Area

-          Surface Area of Prisms and Cylinders

-          Surface Area of Pyramids and Cones

-          Volume of Prisms and Cylinder

-          Volume of Pyramids and Cones

-          Surface Area and Volume of Spheres

-          Congruent and Similar Solids

 

 

1. Continuing Coordinate Geometry

-          Graphing Linear Equations

-          Writing Equations of Lines

-          Scatter Plots and Slopes

-          Coordinate Proof

-          Vectors

-          Coordinates in Space

 

1. Investigating Loci and Coordinate Transformations

-          What is Locus?

-          Locus and Systems of Linear Equations

-          Intersection of Loci

-          Mappings

-          Reflections

-          Translations

-          Rotations

-          Dilations

 

Homework:

 

            Overnight assignments are due at the beginning of the next day class period.  Absents students will turn in work the day after their return.  Each day’s assignments will post on the board in class.

 

 

Class rules:

 

Everyone receives classroom rules on the first day of school to keep in their journals.  Journals will be collected periodically and checked for rules, assignments, and notes.  The class will follow the school wide final exam policy.

 

Materials:

 

            Textbook

            Notebook

            Pencil

            Calculator – TI 30

 

Grading Policy:

 

            The district grading scale will be used in this class.  The grade will be determined in the following manner:

 

                        Homework                              20%

                        Journal                                     10%

                        Test and Quizzes                     50%

                        Finals                                       20%

 

 

Prerequisite Course:                 1^st Semester Geometry

 

 

Follow up Courses:                Integrated Algebra and Geometry, Algebra II, Trigonometry, Pre-calculus, and Calculus

 

 

 

Unit 8 – Right Angles

Geometry

9th grade

Mrs. Abbett

Chapter 8 – Right Triangles

21 days            Jan. 26 – Mar. 6

State Standards:

9A1 – Solve pictorial or word problems that involve geometric relationships within a single geometric shape or figure, including the Pythagorean Theorem.

9D1 – Recognize Pythagorean Triples.

9D2-Identify the basic trigonometric ratios in terms of lengths of the sides of a right triangle and an acute angle.

9D3 – Solve for missing side lengths using the trigonometric ratios in right triangles.

9D4  – Determine and justify the side length relationships present in 45° - 45° - 90° and 30° - 60° - 90° triangles.

9D7 – Solve problems using 45° - 45° - 90° and 30° - 60° - 90° triangles.

9D8 – Solve problems using the Law of Sines and the Law of Cosines.

Common Core Standards:

Standard (G-SRT.) – Understand by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

Standard (G-SRT.8) – Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Standard (G-SRT.11)Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles.

Daily Plans:

Day 1  Simplifying Radical Expressions

            Put students in pairs and have them come up with as many perfect squares as they can in three minutes.  Compile a list of perfect squares by having each pair report out.  If any of the first 25 are missing after they have all reported, fill them in.  Recall how the perfect squares are used to simplify radical expressions.  Give each student a radical to simplify and do Pippen activity where they visit other students and do each other’s problems.

Assign:            Radical expression practice sheet

Day 2  Pythagorean Theorem

Show video on how the Pythagorean Theorem is defined.  Show several different right triangles and have students identify the hypotenuse in each one.  Do examples with simplified radicals and with approximate decimal answers.  Compare the answers to each other.

Assign:            Worksheet

Day 3  Converse of the Pythagorean Theorem

Recall the meanings of inverse, converse, and contrapositive.  Find each one relating to the Pythagorean.  Brainstorm about ways you could find out if the converse was true for different triangles.  Do examples on smartboard.

Assign:            page 400-402 #3,6,12-14,22,24

Day 4  Pythagorean Theorem Practice

Use the following programs or sites and email your results to madmathletes@gmail.com

Sketchpad activity

http://www.shodor.org/interactivate/activities/PythagoreanExplorer/ and http://www.mathwarehouse.com/geometry/triangles/right-triangle.html

There will be bonus points if you find other sites on right triangles and share them with the class.

Day 5  Isosceles Right Triangles

Use Sketchpad to measure the legs and hypotenuse of several 45 – 45- 90 triangles.  Have students take the ratio of each leg to the hypotenuse and give their results.  Use theses results to come up with a rule for all 45-45-90 triangles.  Use the rule to solve problems of missing sides on the smartboard.

Assign:            Practice page

Day 6  30-60-90 right triangles

Put students in groups of 3and give each group a 30-60-90 triangle.  Define short leg, long leg, and hypotenuse and have each group label their triangle with these definitions of the sides.  Discuss the different relationships of each of the sides.  Give each group a measurement for either their short leg, long leg, or hypotenuse.  Have them used the discussed relationships to find the measures of the other two sides.  Have one group member report out their results while another draws it on the smart board.  Do all examples this way.

Assign:            408-409 #2,3,5,7,9,10,11, and 13

Day 7  Right Triangle Problems

Discuss things that you see in real life that form right triangles.  Pull up some ACT problems that use the Pythagorean Theorem.  Discuss different procedures for solving the problems.  Show how to draw and label a diagram.  Review how to put the answer in both simplified and decimal form.  Do examples as a whole class on the Smart board.

Assign:            Finish Example problems

Day 8  Geometric Mean

Draw a diagram of two right triangles making up a larger right triangle.  Discuss what relationships occur between the right triangles.  Set up proportions to represent the relationships.  Use these proportions to find missing parts of the triangles.  Do examples on the smart board.

Assign:            page 400-402 #1,2,4,5 and 7-11

Day9   Mid-Chapter Review

Day10             Mid-Chapter Test

Day 11            Trigonometric Ratios

Define sin, cos, and tan as ratios in a right triangle.  Demonstrate how to label the sides adjacent, opposite, and hypotenuse depending on the angle used.  Practice setting up ratios with no numbers involved in pairs.  Then randomly assign numbers to the sides and have students rotate around the room setting up ratios for sin, cos, and tan.

Assign:            Worksheet

Day 12            Evaluating Sin, Cos, and Tan on the calculator

Recall the ratios from yesterday.  Discuss why sin and cos can never be less than zero or greater than one.  Show students a trig table and talk about how you would find the values using the table.  Now have the students use the trig function buttons on their calculators and compare the results.  Discuss how to go backwards to find the angle.  Do examples in a whip around.

Assign:            Finish Example Problems

Day 13and 14:                        Finding Missing Sides and Angles of Right Triangles

Put the students into small groups.  Give each person in the group an angle and a side measure.  Have the students make three different right triangles using those measures.  Discuss how to find the missing sides using the given side and angle measure.  Have each group compare their answers.  Talk about setting up an equation to find a missing side with one side and one angle.  On the second day, give the students two sides and have them find the angles and the missing side.  Discuss all of the different procedures that you could use to find each of the missing parts.  Have three students take different approaches and then compare their results.  Do examples as a whole class.

Assign:   Day 13 – missing sides worksheet, Day 14 – finish examples.

Day 15            Solving Right Triangle Problems with Trig Functions

Follow the same strategy as on day 7 with the first type of right triangle problems.  Compare the two types of problems and decide how you would know which concept to use to solve the problems.  Do examples on smart board.

Assign:  Worksheet

Day 16                        Law of Cosines for a Missing Side

Ask how we could find a missing side of a triangle if it is not a right triangle.  Derive the law of cosines.  Demonstrate how to label the parts of the triangle and then put them into the formula.  Do examples as a whole class on the smart board.

Assign:            Law of Cosines puzzle sheet

Day 17                        Law of Cosines for a Missing Angle

Review yesterday’s lesson.  Extend the concept to using the Law of Cosines to find a missing angle.   Review how to use the calculator to find cos^-1x.  Practice problems in pairs.  If the pairs disagree on an answer, help them determine the correct solution.

Assign:  Worksheet

Day 18                        Law of Sines for a Missing Side

Recall proportions from chapter 7.  Discuss how a proportion could be set up using .  Discuss when it would be appropriate to use the Law of Sines instead of the law of cosines.

Assign:  Page 429, #10 - 15

Day 19            Law of Sines for a Missing Angle

Recall yesterday’s lesson.  Extend lesson to finding a missing side.  Then give each pair of students a triangle with some given information and have each pair find the missing sides and angles.  Have them present their solutions on the smart board, discussing how they found each part and why they used that particular formula.

Assign:  Page 429, #16 - 24   

Day 20                        Unit 8 Review

Day 21            Unit 8 Test