content area: mathematics geometry grade level: 10 & instruction/curriculum/math/9...5.3 kite...

Content Area: Mathematics

Course Title: Geometry Grade Level: 10

Introducing Geometry

Reasoning in Geometry

Using Tools of Geometry

Marking Period 1

Discovering and Proving Triangle Properties

Discovering and Proving Polygon Properties

Discovering and Proving Circle Properties

Marking Period 2

Transformations and Tessellations

Area

The Pythagorean Theorem

Marking Period 3

Volume

Similarity

Trigonometry

Marking Period 4

Date Created: May 2012

Board Approved on: August 27, 2012

Geometry

Unit Title Lessons to include Time Common Core Math Standards

Introducing

Geometry

1.1 Building Blocks of Geometry

3 wks

G-CO 1, G-MG 1

Alg Skills-Midpoint G-GPE 4,6

1.2 Poolroom Math G-CO 1, G-MG 1

1.3 What's a Widget G-CO 1, G-MG 1

1.4 Polygons G-CO 1, G-MG 1

1.5 Triangles G-CO 1, 10; G-MG 1

1.6 Special Quadrilaterals G-CO 1, G-MG 1

1.7 Circles G-CO 1, G-MG 1

1.8 Space Geometry G-CO 1, G-MG 1

1.9 A Picture is Worth a Thousand Words G-CO 1, G-MG 1

Reasoning in

Geometry

2.1 Inductive Reasoning

2 wks

G-CO 5, G-MG 1

2.2 Finding the nth Term G-CO 5, G-MG 1

2.3 Mathematical Modeling G-CO 5, G-MG 1

2.4 Deductive Reasoning G-CO 5, G-MG 1

Using Tools

of Geometry

3.1 Duplicating Segments and Angles

2 wks

G-CO 12

3.2 Constructing Perpendicular Bisectors G-CO 12

3.3 Constructing Perpendiculars to a Line G-CO 12

3.4 Constructing Angle Bisectors G-CO 12

3.7 Constucting Points of Concurrency G-CO 10,12

Discovering

and Proving

4.1 Triangle Sum Conjecture

3 wks

G-CO 10

4.2 Properties of Isosceles Triangles G-CO 10

4.3 Triangle Inequalites G-CO 10

4.4 Are There Congruence Shortcuts? G-CO 8,10

Triangle

Properties

4.5 Other Shortcuts?

G-CO 8,10

4.6 Corresponding Parts of Congruent Triangles G-CO 7, 10

4.8 Proving Special Conjectures G-CO 10

Discovering and

Proving Polygon

Properties

5.1 Polygon Sum Conjecture

3 wks

G-CO 10

5.2 Exterior Angles of a Polygon G-CO 10

5.3 Kite and Trapezoid Properties G-CO 10

5.4 Properties of Midsegments G-CO 10

5.5 Properties of Parallelograms G-CO 11

5.6 Properties of Special Parallelograms G-CO 11

5.7 Proving Quadrilateral Properties G-CO 11

Discovering and

Proving Circle

Properties

6.1 Tangent Properties

3 wks

G-C 2-4

6.2 Chord Properties G-C 2-4

6.3 Arcs and Angles G-C 2-4

6.5 The Circumference/Diameter Ratio G-C 2-4

6.6 Around the World G-C 2-4, G-MG 3

6.7 Arc Length G-C 2-5

Midterm Review Chapters 1-6 2 wks all of the above

& Exam

Transformations

and

Tessellations

7.1 Transformations and Symmetry

2 wks

G-CO 2-5

7.2 Properties of Isometries G-CO 2-5

7.3 Compositions of Transformations G-CO 2-5

8.1 Rectangles and Parallelograms

G-CO 11, G-GPE 7

8.2 Triangles, Trapezoids, and Kites G-CO 10, G-GPE 7

8.3 Area Problems G-CO 10-11, G-GPE 7

8.4 Regular Polygons G-CO 10-11, G-GPE 7, G-CO 13

Area

8.5 Circles 3 wks

G-C 5

8.6 Any Way You Slice It G-CO 10-11, G-GPE 7, G-CO 13, G-C 5

8.7 Surface Area G-CO 10-11, G-GPE 7, G-CO 13, G-C 5

The Pythagorean

Theorem

9.1 The Theorem of Pythagoras

3 wks

G-SRT 4-5

9.2 Converse of Pythagorean Theorem G-SRT 4-5

9.3 Two Special Right Triangles G-SRT 4-5

9.4 Story Problems G-SRT 4-5, 8

9.5 Distance in Coordinate Geometry G-SRT 4-5,G-GPE 4,7

Volume

10.2 Prisms and Cylinders

3 wks

G-GMD 1-4, 2

10.3 Pyramids and Cones G-GMD 1-4, 2

10.4 Volume Problems G-GMD 1-4, 2

10.6 Spheres G-GMD 1-4, 2

10.7 Surface Area of Sphere G-GMD 1-4, 2

Similarity

11.1 Similar Polygons

3 wks

G-SRT 2

11.2 Similar Triangles G-SRT 2-5

11.3 Indirect Measurement/Similar Triangles G-SRT 2-5

11.4 Corresponding Parts of Similar Triangles G-SRT 2-5

11.5 Proportions with Area G-SRT 5

11.6 Proportions with Volume G-SRT 5

11.7 Proportional Segments Between Parallel Sides G-SRT 4

Trigonometry

12.1 Trigonometric Ratios

2 wks

G-SRT 6-7

12.2 Problem Solving G-SRT 6-8

Final Review Chapters 7-12 2 wks all of the above

& Exam

Total # of Weeks

36

wks

Geometry

Course Title: Geometry Grade Level: 9 and 10

Overarching Essential Questions

What are the building blocks of geometry?

How is reasoning used in geometry?

What types of angle relationships occur in geometry?

How can you show triangle congruence and how is it used to help solve other problems?

What are the properties of polygons in regards to angles and sides?

What are the properties of circles?

What types of transformations occur in geometry and how do we create them?

What are the different properties of right triangles including the Pythagorean Theorem?

What are similar figures and how to we use them in calculations?

Overarching Enduring Understanding

Course Description

Geometry is an academic course that incorporates both plane and solid geometry. It emphasizes

logical reasoning, connecting ideas, and applying knowledge of concepts to solve real-world

problems. This course introduces segments, lines, angles, bisectors, The Pythagorean Theorem

and its Converse, triangles, quadrilaterals, other special polygons, circles, perimeter, area,

volume, transformations, congruency, similarity, etc. Students will explore concepts through

investigation and learn to verify conclusions using logical reasoning. Algebra skills are reviewed

and correlated to problems solving. Proofs are introduced, but are not the major focus.

Introducing Geometry(Chapter 1)

Essential Questions

What are building blocks of geometry?

What makes a good definition?

What types of polygons are there?

What are circles?

What are the different 3-D shapes?

Key Terms

Point, line, plane, collinear, coplanar, segment, congruent, midpoint, bisect, ray, angle, side,

vertex, reflex measure, protractor, counterexample, skew, right angle, acute angle, obtuse angle,

complementary, supplementary, vertical angles, linear pair, polygon, diagonal, convex, concave,

perimeter, equilateral, equiangular, regular, scalene, isosceles, trapezoid, kite, parallelogram,

rhombus, rectangle, square, circle, radius, chord, diameter, tangent, concentric, minor arc, major

arc, semicircle, central angle, isometric drawing, cylinder, cone, pyramid, prism, sphere,

hemisphere, net

Objectives

Students will be able to:

Learn and use the terminology of terms, including: points, lines, planes, segments, rays,

angles, collinear, coplanar, and congruence.

Use a protractor to create or measure an angle and ruler to measure a segment.

Create good definitions using the following technique: classify, differentiate, and test by

looking for a counterexample.

Define and classify polygons including triangles and special quadrilaterals.

Define a circle and identify its parts.

Draw 3-D figure as isometric drawings or nets.

Standards associate with objectives

MA.G.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. MA.G.CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. MA.G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle;

prove or disprove that the point (1, 3) lies on the circle centered at the origin and containing the point (0, 2). MA.G.GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. MA.G.MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).★

Suggested Lesson Activities

Essential problems for each section covered in the textbook.

Performance assessment problems.

Explorations.

Differentiation /Customizing learning (strategies)

Allow students to work in peer pairs or groups.

Allow for multiple representations of solutions.

Use varying degrees of difficulty.

Reasoning in Geometry (Chapter 2) Essential Questions

What is inductive and deductive reasoning?

How do we find a function rule?

What types of relationships occur within angle relationships: such as linear pairs, vertical

angles, and parallel lines cut by a transversal?

Key Terms

Inductive reasoning, conjecture, concurrent lines, deductive reasoning, corresponding angles,

alternate interior angles, alternate exterior angles, transversal.

Objectives


Generalize basic number patterns using inductive reasoning

Apply mathematical models to problem solving.

Discover relationships between special pairs of angles


MA.G.CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. MA.G.CO.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. MA.G.MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).★


Essential problems for each section as listed in the book.


Explorations Differentiation /Customizing learning (strategies)




Using Tools of Geometry (Chapter 3)

Essential Questions

How do you duplicate segments and angles using a compass and straightedge?

How do you construct perpendicular bisectors and angle bisectors using a compass and

straightedge?

What are points of concurrency?

Key Terms

Perpendicular bisector, median, midsegment, altitude, concurrent, circumcenter, incenter,

circumscribed, inscribed.

Objectives


Duplicate a segment and an angle

Make conjectures about perpendicular bisectors.

Construct a perpendicular from a point to a line.

Construct an angle bisector.

Discover properties of points of concurrency.


MA.G. CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

MA.G.CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.




Explorations.





Discovering and Proving Triangle Properties (Chapter 4)

Essential Questions

What is the sum of the three angles of any triangle?

What are the special properties of isosceles triangles?

What triangle inequalities exist and how can they be applied in problem solving?

Which shortcuts exist for establishing triangle congruence?

How is deductive reasoning implemented in establishing triangle congruence?

Key Terms

Legs, vertex, base, base angles, remote int. angles, exterior angle, included side/angle, CPCTC

Objectives


Determine the three angles of any triangle (total 180 degrees).

Determine that base angles of an isosceles triangle are congruent.

Determine that the sum of any two sides of a triangle must be greater than the third side.

Determine that the largest angle will be opposite the largest side.

Determine that an exterior angle of a triangle is equal to the sum of the two remote

interior angles.

Discover which shortcuts guarantee triangle congruence.

Provide reasons for proofs involving congruent triangles


MA.G.CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. MA.G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

MA.G.CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.








Discovering and Proving Polygon Properties (Chapter 5)

Essential Questions

What is the total of the interior and exterior angles of a polygon?

What are some special properties of kites, trapezoids, parallelograms, rhombuses,

rectangles, and squares?

What are the properties of midsegments of triangles and trapezoids?

Key Terms

n-gon, vertex angles, isosceles trapezoid, opposite angles, opposite sides.

Objectives

Students will be able to

Discover and use a formula for finding the sum of the interior angles of a polygon.

Discover that the sum of the exterior angles of any polygon is 360 degrees.

Discover properties of special quadrilaterals and use those properties to solve problems.

Determine properties of midsegments by investigation.

Use properties discovered involving triangle congruence and CPCTC.


MA.G.CO.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. MA.G.CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.




Explorations





Discovering and Proving Circle Properties. (Chapter 6)

Essential Questions

What is the relationship between a tangent and a radius at the point of tangency?

What is the relationship between tangent segments from the same exterior point?

What are some properties of chords?

How is an inscribed angle related to its intercepted arc?

What is the relationship between circumference and diameter?

How is arc length different than arc measure and how is it calculated?

Key Terms

Tangent segments, tangent circles, intercepted arc, inscribed angle, cyclic quadrilateral, secant,

circumference, arc length

Objectives


Discover and use properties of tangents of circles.

Discover and use properties of chords to a circle.

Discover the relationship between an inscribed angle and the intercepted arc measure.

Use circle properties.

Use the formula for circumference.

Calculate arc length

Standards associate with objectives.

MA.G.C.2 Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. MA.G.C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. MA.G.C.4 (+) Construct a tangent line from a point outside a given circle to the circle MA.G.MG.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).★





Allow students to work in groups.



Transformations and Tessellations (Chapter 7) Essential Questions

What types of transformations are there?

What is a reflection and reflectional symmetry?

What are the properties of isometries?

How do you create a tessellation with regular polygons?

How do you create a tessellation using only translations?

Key Terms

Image, isometry, rigid/nonrigid transformation, translation, rotation, reflection, reflectional

symmetry, rotational symmetry, composition, tessellation.

Objectives


Discover some basic properties of transformations and symmetry

Learn more about symmetry in art & nature.

Determine symmetries of plane figures.

Create transformations on a coordinate plane.

Create tessellations with regular and nonregular polygons.

Create tessellations using translations, and glide reflections.


MA.G.CO.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). MA.G.CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. MA.G.CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. MA.G.CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.




Explorations





Area (Chapter 8) Essential Questions

What formulas are used to calculate areas of figures?

How do you calculate the area of irregular figures?

What techniques can be used to calculate the area of portions of a circle?

How is surface area calculated?

Key Terms

Altitude, apothem, sector, segment, annulus, surface area, bases, lateral faces.

Objectives


Apply formulas to calculate the area of rectangles and parallelograms.

Apply formulas to calculate the area of triangles, trapezoids, and kites.

Solve real-world problems involving area.

Apply a formula for the area of regular polygons.

Apply a formula for the area of circles.

Calculate surface area of different figures.


MA.G.CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. MA.G.CO.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. MA.G.CO.12 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle MA.G.GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.★ MA.G.GMD.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. MA.G.C.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.




Explorations. Differentiation /Customizing learning (strategies)

Allow students to work in peer pairs and groups.



The Pythagorean Theorem (Chapter 9)

Essential Questions

What is the Pythagorean Theorem?

To what types of triangles does it apply?

How can we use the Pythagorean Theorem to show a right angle?

What are the relationships among the sides of a 45-45-90 and 30-60-90 triangle?

What is the distance formula and how is related to the Pythagorean Theorem?

Key Terms

Pythagorean Theorem, legs, hypotenuse, Pythagorean triples, 45-45-90 triangle, 30-60-90

triangle, distance formula.

Objectives


Apply the Pythagorean Theorem to solve problems.

Use the Converse of the Pythagorean Theorem.

Use special right triangles to solve problems.

Apply knowledge to solve real-world problems.

Use the distance formula.


MA.G.SRT.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. MA.G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. MA.G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.★

MA.G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle;

prove or disprove that the point (1, 3) lies on the circle centered at the origin and containing the point (0, 2). MA.G.GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.★




Explorations. Differentiation /Customizing learning (strategies)




Volume (Chapter 10)

Essential Questions

What does volume measure?

What are the formulas for volume of different solids?

How do we apply volume formulas to solve problems?

How do we determine the surface area of a sphere or hemisphere?

Key Terms

Right cylinder/prism, oblique cylinder/prism, great cylinder, volume, density

Objectives


Name prisms, cylinders, pyramids, and cones.

Determine the surface area & volume of prisms and cylinders.

Determine the surface area & volume of pyramids and cones.

Determine the volume of a sphere.

Solve for volume of composite figures.


MA.G.GMD.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. MA.G.GMD.2 (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. MA.G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.★

MA.G.GMD.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects MA.G.MG.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).★




Explorations.





Similarity (Chapter 11)

Essential Questions

What makes a figure similar to another figure?

How are proportions used to solve similarity problems?

What is the minimum criteria needed to show that two triangles are similar?

How do you use similar figures to solve real-world problems?

How do you determine the relationship between sides of a triangle cut by a parallel lines.

Key Terms

Proportion, similar figure, scale factor, indirect measurement.

Objectives


Set-up and solve proportions.

Calculate length of sides of similar polygons.

Determine triangle similarity shortcuts.

Solve problems using indirect measurement.

Calculate areas, surface areas, and volumes of similar figures


MA.G.SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor:

a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

MA.G.SRT.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. MA.G.SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. MA.G.SRT.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. MA.G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.




Explorations.





Trigonometry (Chapter 12) Essential Questions

What are the trig ratios?

How do you use trig ratios to solve problems?

Key Terms

Sine, cosine, tangent, opposite leg, adjacent leg, angle of elevation, angle of depression

Objectives


Write ratios to represent trig functions.

Solve for missing sides of triangles using trig functions.


MA.G.SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. MA.G.SRT.7 Explain and use the relationship between the sine and cosine of complementary angles. MA.G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in

applied problems.★




Explorations.





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