What is GeoGebra and How Does It Work?

GeoGebra is dynamic mathematics software for schools that joins geometry, algebra, and calculus. On the one hand, GeoGebra is an interactive geometry system. You can do constructions with points, vectors, segments, lines, and conic sections as well as functions while changing them dynamically afterwards. On the other hand, equations and coordinates can be entered directly. Thus, GeoGebra has the ability to deal with variables for numbers, vectors, and points. It finds derivatives and integrals of functions and offers commands like Root or Vertex. These two views are characteristic of GeoGebra: an expression in the algebra window corresponds to an object in the geometry window and vice versa.

GeoGebra’s User Interface

GeoGebra’s user interface consists of a graphics window and an algebra window. On the one hand you can operate the provided geometry tools with the mouse in order to create geometric constructions on the drawing pad of the graphics window. On the other hand, you can directly enter algebraic input, commands, and functions into the input field by using the keyboard. While the graphical representation of all objects is displayed in the graphics window, their algebraic numeric representation is shown in the algebra window.

The user interface of GeoGebra is flexible and can be adapted to the needs of your students.

• Open a toolbox by clicking on the lower part of a button and select another tool from this toolbox.

Hint: You don’t have to open the toolbox every time you want to select a tool. If the icon of the desired tool is already shown on the button it can be activated directly.

Hint: Toolboxes contain similar tools or tools that generate the same type of new object.

• Check the toolbar help in order to find out which tool is currently activated and how to operate it.

Definition of Geometry

Geometry is the investigation of “properties, measurements and relationships of points, lines, angles, surfaces and solids.” Geometry developed from a practical need to determine land boundaries (survey), figure the size (area) of a field, measure the volume of a silo (cylinder), and determine the relative positions of three-dimensional objects in a defined space. Man’s fascination with the stars and the heavens became the science of astronomy, which led to the development of trigonometry and its unique computational methods. Studying geometry helps students hone their spatial visualization skills, which helps them function better in the physical world. Points, lines, angles, surfaces and solids are all used in painting, sculpture and architecture. The artist must understand the relationship of these components in order to create in any medium. Various engineering disciplines use geometry to build bridges and dams, design freeway systems, mine for minerals and drill for oil. Geometry is used every day in many professions.

On the TExES 7-12 Mathematics 7–12 Standard III the following is required for Geometry and Measurement: The mathematics teacher understands and uses geometry, spatial reasoning, measurement concepts and principles and technology appropriate to teach the statewide curriculum (TEKS) to prepare students to use mathematics.

Domain III — Geometry and Measurement

Competency 012:

The teacher understands geometries, in particular Euclidian geometry, as axiomatic systems. The beginning teacher: A. Understands axiomatic systems and their components (e.g., undefined terms, defined terms, theorems, examples, counterexamples).

B. Uses properties of points, lines, planes, angles, lengths and distances to solve problems.

Assignments:

Project 1

Question 1: Use GeoGebra software to create the following objects:

Point
Line
Line segment
Ray
A 3-dimensional object

Question 2:

2.1 Use the GeoGebra software to create a triangle and measure the angles inside the triangle. Illustrate that the sum of any triangle is 180.

2.2 Create a quadrilateral with the GeoGebra software and measure the angles.

2.3 Create a regular pentagon with the GeoGebra software and measure the angles.

2.4 Prove by induction that the sum of the interior angles of an n-gon is equal to (n-2)180.

Question 3. 1. Use the GeoGebra software to create a triangle and one exterior angle. Measure the exterior angle and compute the sum of the measure of the opposite angles (opposite to the exterior angle). What do you notice?

Activity 1

Question 1:

Use the following items:

Piece of paper
Scissors
Ruler

Draw a triangle on a piece of paper and cut out the triangle. Now use the fact that the sum of the angles on a straight line add to 180 degrees to prove that the sum of the interior angles of a triangle (a 3-gon) equals 180 degrees.

Question 2:

What would be the sum of the interior angles of a quadrilateral (a 4-gon)?

Question 3:

Derive a formula for the sum of the interior angles of an n-gon

Discussion 1

Read through the material and then answer the questions.

Geometry (Links to an external site.)Links to an external site. is a subject in mathematics that focuses on the study of shapes, sizes, relative configurations, and spatial properties. Derived from the Greek word meaning “earth measurement,” geometry is one of the oldest sciences. It was first formally organized by the Greek mathematician Euclid around 300 BC when he arranged 465 geometric propositions into 13 books, titled ‘Elements’. This, however, was not the first time geometry had been utilized. As a matter of fact, there exists evidence to believe that geometry dates all the way back to 3,000 BC in ancient Mesopotamia, Egypt!

Geometry has been the subject of countless developments. As a result, many types of geometry exist, including Euclidean geometry, non-Euclidean geometry, Riemannian geometry, algebraic geometry, and symplectic geometry.

This discussion primarily focuses on the properties of lines, points, and angles. We will also place emphasis on geometric measurements including lengths, areas, and volumes of various shapes. By the end of this section it won’t be hard to see that geometry is all around us!

Points

Lines and rays

Midpoints

Intersection

TransversalSpace

Question 1: (4 points)

Define the following terms:

1.1 Point

1.2 Line

1.3 Plane

1.4 Space

Question 2: (3 points)

Define a transversal line. What type of angles are formed by a transversal?

Question 3: (3 points)

3.1 What is Euclidean geometry?

3.2 Why should we study Euclidean geometry?

Week 3

Competency 012:

The teacher understands geometries, in particular Euclidian geometry, as axiomatic systems. The beginning teacher:

C. Applies the properties of parallel and perpendicular lines to solve problems.

E. Describes and justifies geometric constructions made using compass and straightedge, reflection devices and other appropriate technologies.

F. Demonstrates an understanding of the use of appropriate software to explore attributes of geometric figures and to make and evaluate conjectures about geometric relationships.

Competency 013:

The teacher understands the results, uses and applications of Euclidian geometry. The beginning teacher:

B. Analyzes the properties of circles and the lines that intersect them.

D. Computes the perimeter, area and volume of figures and shapes created by subdividing and combining other figures and shapes (e.g., arc length, area of sectors).

Reading

Axiomatic systems:

An Axiomatic system is a set of axioms from which some or all axioms can be used in conjunction to logically derive a system of Geometry. In an axiomatic system, all the axioms that are defined must be consistent where there are no contractions within the set of axioms. The first mathematician to design an axiomatic system was Euclid of Alexandria. Euclid of Alexandria was born around 325 BC. Most believe that he was a student of Plato. Euclid introduced the idea of an axiomatic geometry when he presented his 13 chapter book titled The Elements of Geometry. The Elements he introduced were simply fundamental geometric principles called axioms and postulates. The most notable are Euclid five postulates which are stated in the next passage.

Any two points can determine a straight line.
Any finite straight line can be extended in a straight line.
A circle can be determined from any center and any radius.
All right angles are equal.
If two straight lines in a plane are crossed by a transversal, and sum the interior angle of the same side of the transversal is less than two right angles, then the two lines extended will intersect.

According to Euclid, the rest of geometry could be deduced from these five postulates. Euclid’s fifth postulate, often referred to as the Parallel Postulate, is the basis for what are called Euclidean Geometries or geometries where parallel lines exist. There is an alternate version to Euclid fifth postulate which is usually stated as ”Given a line and a point not on the line, there is one and only one line that passed through the given point that is parallel to the given line.”

However, some mathematics believed that the Euclid Fifth Postulate was suspect or incomplete. As a result, mathematicians have written alternate postulates to the Parallel Postulate. These postulates have led the way to new geometries called Non-Euclidean Geometries.

A History of :

The ancient Babylonians calculated the area of a circle by taking 3 times the square of its radius, which gave a value of pi = 3. One Babylonian tablet (ca. 1900–1680 BC) indicates a value of 3.125 for pi, which is a closer approximation. The Rhind Papyrus (ca.1650 BC) gives us insight into the mathematics of ancient Egypt. The Egyptians calculated the area of a circle by a formula that gave the approximate value of 3.1605 for pi. The first calculation of pi was done by Archimedes of Syracuse (287–212 BC), one of the greatest mathematicians of the ancient world. Archimedes approximated the area of a circle by using the Pythagorean Theorem to find the areas of two regular polygons: the polygon inscribed within the circle and the polygon within which the circle was circumscribed. Since the actual area of the circle lies between the areas of the inscribed and circumscribed polygons, the areas of the polygons gave upper and lower bounds for the area of the circle. Archimedes knew that he had not found the value of pi but only an approximation within those limits. In this way, Archimedes showed that pi is between 3 1/7 and 3 10/71. A similar approach was used by Zu Chongzhi (429–501), a brilliant Chinese mathematician and astronomer. Zu Chongzhi would not have been familiar with Archimedes’ method—but because his book has been lost, little is known of his work. He calculated the value of the ratio of the circumference of a circle to its diameter to be 355/113. To compute this accuracy for pi, he must have started with an inscribed regular 24,576-gon and performed lengthy calculations involving hundreds of square roots carried out to 9 decimal places. Mathematicians began using the Greek letter π in the 1700s. Introduced by William Jones in 1706, use of the symbol was popularized by Leonhard Euler, who adopted it in 1737. Circles: A circle is a simple closed shape. It is the set of all points in a plane that are at a given distance from a given point, the centre; equivalently it is the curve traced out by a point that moves so that its distance from a given point is constant. The distance between any of the points and the centre is called the radius.

Terminology:

• Annulus: the ring-shaped object, the region bounded by two concentric circles.

• Arc: any connected part of the circle.

• Centre: the point equidistant from the points on the circle.

• Chord: a line segment whose endpoints lie on the circle.

• Circumference: the length of one circuit along the circle, or the distance around the circle.

• Diameter: a line segment whose endpoints lie on the circle and which passes through the centre; or the length of such a line segment, which is the largest distance between any two points on the circle. It is a special case of a chord, namely the longest chord, and it is twice the radius.

• Disc: the region of the plane bounded by a circle.

• Lens: the intersection of two discs.

• Passant: a coplanar straight line that does not touch the circle.

• Radius: a line segment joining the centre of the circle to any point on the circle itself; or the length of such a segment, which is half a diameter.

• Sector: a region bounded by two radii and an arc lying between the radii.

• Segment: a region, not containing the centre, bounded by a chord and an arc lying between the chord’s endpoints.

• Secant: an extended chord, a coplanar straight line cutting the circle at two points.

• Semicircle: an arc that extends from one of a diameter’s endpoints to the other. In non-technical common usage it may mean the diameter, arc, and its interior, a two dimensional region, that is technically called a half-disc. A half-disc is a special case of a segment, namely the largest one.

• Tangent: a coplanar straight line that touches the circle at a single point.

Length of circumference

Further information: Circumference The ratio of a circle’s circumference to its diameter is π (pi), an irrational constant approximately equal to 3.141592654. Thus the length of the circumference C is related to the radius r and diameter d by: {C=2r=d.,}

C=2r

Area enclosed

Area enclosed by a circle = π × area of the shaded square Main article: Area of a circle As proved by Archimedes, in his Measurement of a Circle, the area enclosed by a circle is equal to that of a triangle whose base has the length of the circle’s circumference and whose height equals the circle’s radius,[7] which comes to π multiplied by the radius squared:

Equations

Cartesian coordinates

Circle of radius r = 1, centre (a, b) = (1.2, −0.5) In an x–y Cartesian coordinate system, the circle with centre coordinates (a, b) and radius r is the set of all points (x, y) such that {(x-a)^{2}+(y-b){2}=r^{2}.} This equation, known as the Equation of the Circle, follows from the Pythagorean theorem applied to any point on the circle: as shown in the adjacent diagram, the radius is the hypotenuse of a right-angled triangle whose other sides are of length |x − a| and |y − b|. If the circle is centred at the origin (0, 0), then the equation simplifies to {x^{2}+y{2}=r^{2}.! }

Properties

• The circle is the shape with the largest area for a given length of perimeter. (See Isoperimetric inequality.)

• The circle is a highly symmetric shape: every line through the centre forms a line of reflection symmetry and it has rotational symmetry around the centre for every angle. Its symmetry group is the orthogonal group O(2,R). The group of rotations alone is the circle group T.

• All circles are similar.

• A circle’s circumference and radius are proportional.

• The area enclosed and the square of its radius are proportional.

• The constants of proportionality are 2π and π, respectively.

• The circle which is centred at the origin with radius 1 is called the unit circle.

• Thought of as a great circle of the unit sphere, it becomes the Riemannian circle.

• Through any three points, not all on the same line, there lies a unique circle. In Cartesian coordinates, it is possible to give explicit formulae for the coordinates of the centre of the circle and the radius in terms of the coordinates of the three given points. See circumcircle.

Activity 2

Activity on Pi.

Use a compass to draw a circle on a piece of paper.
Find the length of the diameter of the circle.
Find the length of the circumference of the circle.
Calculate the ratio of the circumference to the diameter.
What is this ratio called?

Week 4

Competency 014:

The teacher understands coordinate, transformational and vector geometry and their connections. The beginning teacher: A. Identifies transformations (i.e., reflections, translations, glide-reflections, rotations, dilations) and explores their properties.

B. Uses the properties of transformations and their compositions to solve problems.

C. Uses transformations to explore and describe reflectional, rotational and translational symmetry.

D. Applies transformations in the coordinate plane.

E. Applies concepts and properties of slope, midpoint, parallelism, perpendicularity and distance to explore properties of geometric figures and solve problems in the coordinate plane.

F. Uses coordinate geometry to derive and explore the equations, properties and applications of conic sections (i.e., lines, circles, hyperbolas, ellipses, parabolas).

G. Relates geometry and algebra by representing transformations as matrices and uses this relationship to solve problems.

H. Explores the relationship between geometric and algebraic representations of vectors and uses this relationship to solve problems.

Competency 009:

The teacher understands trigonometric and circular functions, analyzes their algebraic and graphical properties and uses them to model and solve problems. The beginning teacher: A. Analyzes the relationships among the unit circle in the coordinate plane, circular functions and the trigonometric functions.

B. Recognizes and translates among various representations (e.g., written, numerical, tabular, graphical, algebraic) of trigonometric functions and their inverses.

C. Recognizes and uses connections among significant properties (e.g., zeros, axes of symmetry, local extrema) and characteristics (e.g., amplitude, frequency, phase shift) of a trigonometric function, the graph of the function and the function’s symbolic representation.

D. Understands the relationships between trigonometric functions and their inverses and uses these relationships to solve problems.

E. Uses trigonometric identities to simplify expressions and solve equations.

F. Models and solves a variety of problems (e.g., analyzing periodic phenomena) using trigonometric functions.

G. Uses graphing calculators to analyze and solve problems involving trigonometric functions.

Competency 019:

The teacher understands mathematical connections both within and outside of mathematics and how to communicate mathematical ideas and concepts. The beginning teacher: A. Recognizes and uses multiple representations of a mathematical concept (e.g., a point and its coordinates, the area of a circle as a quadratic function of the radius, probability as the ratio of two areas, area of a plane region as a definite integral).

B. Understands how mathematics is used to model and solve problems in other disciplines (e.g., art, music, science, social science, business).

Competency 020:

The teacher understands how children learn mathematics and plans, organizes and implements instruction using knowledge of students, subject matter and statewide curriculum (Texas Essential Knowledge and Skills [TEKS]). The beginning teacher:

A. Applies research-based theories of learning mathematics to plan appropriate instructional activities for all students.

B. Understands how students differ in their approaches to learning mathematics.

C. Uses students’ prior mathematical knowledge to build conceptual links to new knowledge and plans instruction that builds on students’ strengths and addresses students’ needs.

D. Understands how learning may be enhanced through the use of manipulatives, technology and other tools (e.g., stop watches, rulers).

E. Understands how to provide instruction along a continuum from concrete to abstract.

F. Understands a variety of instructional strategies and tasks that promote students’ abilities to do the mathematics described in the TEKS.

Competency 011: The teacher understands measurement as a process.

The beginning teacher: A. Applies dimensional analysis to derive units and formulas in a variety of situations (e.g., rates of change of one variable with respect to another) and to find and evaluate solutions to problems.

B. Applies formulas for perimeter, area, surface area and volume of geometric figures and shapes (e.g., polygons, pyramids, prisms, cylinders, cones, spheres) to solve problems.

C. Recognizes the effects on length, area or volume when the linear dimensions of plane figures or solids are changed.

D. Applies the Pythagorean theorem, proportional reasoning and right triangle trigonometry to solve measurement problems.

E. Relates the concept of area under a curve to the limit of a Riemann sum.

F. Uses integral calculus to compute various measurements associated with curves and regions (e.g., area, arc length) in the plane, and measurements associated with curves, surfaces and regions in three-space.

Competency 012:

D. Uses properties of congruence and similarity to explore geometric relationships, justify conjectures and prove theorems.

G. Compares and contrasts the axioms of Euclidean geometry with those of non-Euclidean geometry (i.e., hyperbolic and elliptic geometry).

Competency 013:

A. Analyzes the properties of polygons and their components.

C. Uses geometric patterns and properties (e.g., similarity, congruence) to make generalizations about two- and three-dimensional figures and shapes (e.g., relationships of sides, angles).

D. Computes the perimeter, area and volume of figures and shapes created by subdividing and combining other figures and shapes (e.g., arc length, area of sectors).

E. Analyzes cross-sections and nets of three-dimensional shapes.

F. Uses top, front, side and corner views of three-dimensional shapes to create complete representations and solve problems.

Week 9:

3D shapes:

TEXES Standards and Competencies:

The teacher understands the results, uses and applications of Euclidean Geometry.

The beginning teacher:

A. Analyzes the properties of polygons and their components.

B. Analyzes the properties of circles and the lines that intersect them.

C. Uses geometric patterns and properties (e.g. similarity and congruence) to make generalizations about two-and-three dimensional figures and shapes (e.g. relationships of sides and angles)

D. Computes the perimeter, area and volume of figures and shapes created by subdividing and combining other figures and shapes (e.g. arc-length and area of sectors)

3D Shapes

There are many types of three-dimensional shapes. You’ve surely seen spheres and cubes before. In this lesson, you’ll learn about polyhedra — three-dimensional shapes whose faces are polygons — and you’ll also learn about two special types of polyhedra: prisms and pyramids.

Polyhedra

A die is in the shape of a cube. A portable DVD player is in the shape of a rectangular prism. A soccer ball is in the shape of a truncated icosahedron. These shapes are all examples of polyhedra.

A three-dimensional shape whose faces are polygons is known as a polyhedron. This term comes from the Greek words poly, which means “many,” and hedron, which means “face.” So, quite literally, a polyhedron is a three-dimensional object with many faces.

The faces of a cube are squares. The faces of a rectangular prism are rectangles. And the faces of a truncated icosahedron are pentagons and hexagons — there are some of each.

The other parts of a polyhedron are its edges, the line segments along which two faces intersect, and its vertices, the points at which three or more faces meet.

Surface Area Formulas

In general, the surface area is the sum of all the areas of all the shapes that cover the surface of the object.

Cube:

Surface Area of a Cube \(= 6\cdot a^2\), where \(a\) is the length of the side of each edge of the cube.

In words, the surface area of a cube is the area of the six squares that cover it. The area of one of them is \(a*a\), or \(a^2\). Since these are all the same, you can multiply one of them by six, so the surface area of a cube is 6 times one of the sides squared.

Volume of a cube: \(V= a^3\).

Cube

Example 9.1

Find the surface area of a cube with a side of length 3 cm.

Solution:

Given that a = 3, we have that the

Surface area of a cube \(= 6a^2 = 6(3)^2 = 54 cm^2\).

Example 9.2

Find the volume of a cube with a side length of 3 cm.

Solution:

Volume of a cube: \(V= a^3 = 3^3=27 cm^3\).

Sphere:

A sphere is a 3-dimensional object shaped like a ball.

Every point on the surface is the same distance from the center.

Surface area of a sphere:

The surface area of a sphere is the area covering the outside of the sphere - think of it as the rubber covering a kickball or the surface of the earth. Because it is curved, it is much harder to measure the surface area of a sphere than a box, so we need an equation to determine the area.

\(A=4\pi\cdot r^2\).

Example 9.3

Find the surface area of the sphere with diameter 11.9 inches.

Figure 9.5

Solution:

If the diameter is 11.9 inches, then the radius is \(\frac{11.9}{2}=5.95\) inches. Therefore, the surface area is:

Area = \(4\cdot \pi \cdot 5.95^2 = 141.61 \cdot \pi \ inches^2\).

Sphere

Volume of a sphere:

In three dimensions, the volume inside a sphere (that is the volume of a ball) is:

\(V=\frac{4}{3}\cdot\pi\cdot r^3\),

where \(r\) is the radius of the circle.

Example 9.4

Find the volume of a sphere with diameter 11.9 inches.

Solution:

The volume of a sphere is:

\(V=\frac{4}{3}\cdot\pi\cdot r^3=\frac{4}{3}\cdot\pi\cdot 5.95^3=280.86\pi \ inches^3\).

Week 11

The Pythagorean Theorem:

The Pythagorean Theorem states that if a right triangle has side lengths \(a\), \(b\) and \(c\), where \(c\) is the hypotenuse, then the sum of the squares of the two shorter lengths is equal to the square of the length of the hypotenuse.

Fig. 11.1

Example 11.1

Use the Pythagorean theorem to determine the length of \(X\).

Fig 11.2

Step 1:

Identify the legs and the hypotenuse of the right triangle.

The legs have length ‘6 and ’8’ . \(X\) is the hypotenuse because it is opposite the right angle.

Step 2:

Since \(a^2+b^2=c^2\), we have that \(6^2+8^2=36+64=c^2\).

Step 3:

Solve for the unknown:

\(36+64=c^2\), so

\(100 = c^2\), and therefore we have that

\(c=10\).

Week 12

Translations

An isometry does not change shape or size.

Definition of Translation:

A geometry translation is an isometric transformation, meaning that the original figure and the image are congruent.

That is: “Sliding” or moving a shape without rotating or flipping it.

The shape still looks exactly the same, just in a different place.

Translation

Example 1:

Click on the link above and unselect:

“Show translated triangle” and

“Show arrows to translation”

Move the vector to \((0,0)\)

Create \(\triangle{ABC}\) with the following coordinates:

\({A(0,0), B(0,1), C(1,0)}\)

Perform the following translation:

\((x,y) \rightarrow (x+2, y-3)\)

This means that we will move the figure 2 units to the right and 3 units down.

Move the translation vector (purple arrow) two units to the right and 3 units down.

Now select:

“Show arrows to translation”

Then select:

“Show translated triangle”

It is clear that we translated

\(\triangle{ABC}\)

to \(\triangle{A'B'C'}\) with

\(A'(2,-3), B'(2,-2), C'(3,-2)\)

We also notice that

Every point of the shape must move:

The same distance
In the same direction.

Example 2:

Perform the following translation:

\((x,y) \rightarrow (x-4, y+\frac{1}{2})\)

We also have that \(\triangle{ABC}\) is congruent to \(\triangle{A'B'C'}\).

Example 3:

Make a vector \(u\) between two points \(A\) and \(B\) (use the tool “Vector between Two Points”“).
Make a polygon.
Use the tool “Translate Object by Vector image”.
Click on the polygon and then on the vector \(u\). A new polygon is created.
Drag the points \(A\) and \(B\) to change the direction and the length of the vector, the position of the new polygon changes.
Drag the vertices of the first polygon, the new polygon changes. Note that you can’t drag the vertices of the new polygon, these points are dependent objects.

Rotation

Definition:

A transformation in which a plane figure turns around a fixed center point. In other words, one point on the plane, the center of rotation, is fixed and everything else on the plane rotates about that point by a given angle.

Example 1:

Click on the following link:

Rotation

Rotate the image \(90^0\) clockwise.
Rotate the image \(180^0\) counter clockwise.

Example 2:

Click on the following link:

GeoGebra

Steps to rotate an object around a point:

Make a polygon and a point.
Use the tool Rotate Object around Point by Angle image.
Click on the polygon, then on the point, choose an angle in the pop-up window.

Reflection

Definition:

A transformation in which a geometric figure is reflected across a line, creating a mirror image. That line is called the axis of reflection.

So, it is an image or shape as it would be seen in a mirror.

Example 1

Download the following image:

cutie

Reflect the image about the y-axis and the x-axis.

Geometry: Basic Concepts

Mienie Roberts

1/17/2018

Week 1 and 2

Introduction

What is GeoGebra and How Does It Work?

GeoGebra’s User Interface

Definition of Geometry

Domain III — Geometry and Measurement

Competency 012:

Assignments:

Project 1

Activity 1

Discussion 1

Week 3

Competency 012:

Competency 013:

Reading

Terminology:

Length of circumference

Equations

Cartesian coordinates

Properties

Activity 2

Week 4

Competency 014:

Competency 009:

Competency 019:

Competency 020:

Competency 011: The teacher understands measurement as a process.

Competency 012:

Competency 013:

Week 9:

3D shapes:

TEXES Standards and Competencies:

3D Shapes

Polyhedra

Surface Area Formulas

Cube:

Example 9.1

Solution:

Example 9.2

Solution:

Sphere:

Surface area of a sphere:

Example 9.3

Solution:

Example 9.4

Solution:

Week 11

The Pythagorean Theorem:

Example 11.1

Week 12

Translations

Definition of Translation:

Example 1:

Example 2:

Example 3:

Rotation

Definition:

Example 1:

Example 2:

Reflection

Definition:

Example 1