Geometry - The Basics & Beyond

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  • Lectures 37
  • Contents Video: 4.5 hours
    Other: 0 mins
  • Skill Level Beginner Level
  • Languages English
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About This Course

Published 11/2014 English

Course Description

The demand for my basics course was so high, I decided to expand it and charge a small fee for the extra time and effort for pulling these lectures and external resources together. The first expansion includes explanation and practice for about a quarter of the Common Core standards. Specifically it includes all the “Congruence” standards which encompasses the new focus on transformations as well as a lot of traditional two column proofs. Eventually I hope to make it to make it through all of the Common Core standards - either here or in other courses.

THE BASICS - The original Course:

Having taught Geometry for almost a decade, I've learned that one of the biggest challenges is simply the vocabulary. This first (and original) section is a very broad overview of Geometry and the language that it uses. If you've never heard this vocabulary, this will introduce it. If you have, this will reinforce it and put it into context.

Said another way there are a few different kinds of students who would benefit from this section:

  • Those who are about to take Geometry
  • Those who are taking Geometry and want a review or another perspective
  • Those who are curious as to the overall nature of Geometry but not wanting to take time for an entire course

Some of the vocabulary and concepts discussed are used in every day conversations, whereas others are specialized and not usually found in everyday conversation (I'm looking at you, hypotenuse!). Some of these terms SHOULD be familiar while others will not be. The course also includes a lot outside resources to practice as well as Udemy-style quizzes.

There's an old joke about what you remember from a college course five years after you've taken it. For Economics, it's supply and demand. For Chemistry, it's the periodic table is the organization of the elements. For English Composition, it's always start with an outline. This limited course is along those same lines: it will give you the general feeling for topics in Geometry without many of the details.

The first section could be watched in one sitting if you don't do any of the exercises. A better way to take it would be to target one day for a week to watch one video (less than 10 minutes) and then spending 20 minutes after each lecture to explore the additional material that's included.

SECTION 2 - Common Core Congruence Standards

First, a word about Common Core. IF you live in a state that has rigorous Mathematical Standards, Common Core is simply a nationalized version of a variation of what has been taught over the last 20 years. It stresses some new topics and ignores some traditional stuff, but it’s simply a road map for a GIANT TOPIC. In 180 school days, only so much can be learned about any given field - these standards attempt to point out what’s important and what’s not: Both in general and related to what will be on State/National assessment tests.

There are 13 standards related to “Congruence” in the New Common Core and this section explains (at varying levels of depth in it’s first iteration) all of them. I attacked this first because it had transformations in it and it was the one topic I knew I would have to do a fair amount of research on before I could teach it.

This section is primarily for students:

  • Interested in learning more about transformations and why they have become a new item of importance
  • Need more and different examples of two column proofs and strategies for solving these.

What are the requirements?

  • This course only assumes basic english fluency and basic numerical fluency.
  • A notebook for taking notes is always a good idea
  • Many of the supplemental exercises are on the internet and will require internet access but not particular browser or software is required.
  • A positive mental attitude (and a full house) beats a bad attitude (and a straight) any day of the week

What am I going to get from this course?

  • SECTION 1
  • Master the vocabulary needed for success in a Geometry class
  • Describe a polygon, give examples and non-examples
  • List the basic parts of a circle
  • Calculate the area and perimeter of rectangles and triangles
  • Describe the concepts of perimeter, area, and volume
  • Calculate the volume of a right prism (given the area of the base)
  • Categorize angles, triangles, and undefined terms by sight
  • Name angles, rays, lines, and segments using symbolic shorthand
  • SECTION 2
  • Apply the concepts of precise definitions and counter-examples
  • Construct items such as perpendicular segment bisectors using both electronic tools and paper, compass, straight-edge and-pencil
  • Define rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
  • Determine the image produce by a series of transformations
  • Use transformation to prove SSS, SAS, ASA, and AAS
  • Determine the series of transformations required to create a particular image (given a pre-image)
  • Use transformations to prove congruency of shapes
  • Use two-column proofs to prove theorems about triangles, quadrilaterals, and parallel lines

What is the target audience?

  • This course is meant for those wanting to explore introductory concepts at a high level. This would include a student who is about to take a geometry class, is feeling behind in their geometry class, or is wanting review before an end of semester exam. This class covers most of the vocabulary and concepts related to shapes, area, perimeter, volume, and angles.
  • Students looking for a more in-depth course should NOT take this course. This basic review does NOT cover proof, trig, pythagorean theorem, transformations, or extensive calcuations.

What you get with this course?

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30 day money back guarantee.

Forever yours.
Lifetime access.

Learn on the go.
Desktop, iOS and Android.

Get rewarded.
Certificate of completion.

Curriculum

Section 1: Geometry - The Basics
01:04

A goofy welcome video filmed long after most of the content.

05:49

What you'll learn:

What is and isn't a polygon?

How to draw a 13-gon.

Name the shapes for these clues:

-The Military Building in Washington DC

-The shape of most nuts (of nuts and bolts, not nuts and squirrels)

-The 1980 movie where Chuck Norris kicks butt (of course, he always kicks butt)

Rambling Teacher Thoughts:

Having taught Geometry for almost a decade, I've learned that one of the biggest challenges is simply the vocabulary. Some of the vocabulary is used daily, whereas others are specialized and not usually found in everyday conversation (I'm looking at you, hypotenuse!). This lectur introduces one of common intro lessons found in most Geometry books. Some of these terms SHOULD be familiar while others will not be. I'm including some outside resources to practice. If a term is confusing... Well, you're on the internet, look it up - someone else probably has a slightly different explanation that will probably fit into your existing knowledge base!

3 questions

Just making sure you were paying attention to the first lecture.

09:13

What you'll learn:

What is perimeter?

How is perimeter different from area?

What's the perimeter and area of a 4 cm by 10 cm rectangle?

What's the perimeter of a shape that has sides that are length of 3, 6, 4, 7, and 2?

What's the area of a triangle that has a height of 3 and a base of 8?

Rambling Teacher Thoughts:

Measuring length, calculating perimeter, and calculating area are practical skills that everyone should have.

There's no substitute for practice here and I'll include a bunch of practice links for the problems above and much harder ones (like kites and trapezoids).

Area and Perimeter: Are you spreading over or walking around?
3 questions
06:36

What you'll learn:

How are radius and diameter related?

What is pi?

What's the formula for area of a circle?

Rambling Teacher Thoughts:

The circle is a magical figure. Every point along the circle is exactly the same distance from the center. All circles are similar - that is, they are all resized variations of every other circle. If you take any segment and draw an identical size circle using each endpoint as a center, the resulting intersections are endpoints for a second segment which bisects the original segment. I know that sounds likely gobbledy-gook, but it's true!

There's an external link the practices some circle vocabulary coming up later but you need to learn how to name angles first.

Circles!
4 questions
06:06

This link offers some practice calculating the area and perimeter of squares, rectangles, triangles, kites, trapezoids, and parallelograms. The video explains the feedback interface as well as talks about a couple of the shapes that have not yet been discussed such as kites, trapezoids, and parallelograms.

If you want even FURTHER explanation about perimeter and area as well as some unusual quizzes. See the link to some other videos I did for my classes.

07:08

What you'll learn:

Defining the undefinable: Points, lines, and planes.

The inhabitants of Flatland: Segments and three classes of triangles

Hard to spell words like Scalene, Isosceles, and Equilateral

Symbols for naming segments, lines, and rays.


Rambling Teacher Thoughts:

Points, Lines, and Planes are the undefined terms that are the building blocks of geometry.

Flatland by Edwin Abbott is a fun little novella that explores this in detail as well as sets up a really interesting analogy for talking about the fourth dimension.

Trying to drive home a Point...
3 questions
03:09

A short motivational speech about how to raise your game to the next level.

The top 5 RESEARCH PROVEN EFFECTIVE** strategies for teaching (and thus learning) are as follows. Which ones work best for you? Which ones might you want to try more?

1) Identifying similarities and differences

2) Summarizing and Note Taking

3) Reinforcing Effort and Providing Recognition*

4) Homework and Practice

5) Nonlinguistic Representations (PICTURES!)

As a self-motivated life-long learner, you need to have some "go-to" strategies as well as might want to mix things up from time to time.

*Does this seem odd to do for yourself? Positive self-talk is one of the best things you can do - why wait for someone else to tell you that you worked hard and got better at something?

**Kudos to R. Marzano et al for their awesome book "Classroom Instruction that Works"

07:53

What you'll learn:

Cubic measure is how much space something takes up

Analogy - Perimeter:area :: Surface Area : volume

Analogy - Right Prisms:Rectangles :: Pyramids:Triangles

Metaphor: A cone is when a circle and a pyramid have a baby.

Hard Cold Fact: Circles and Spheres have formulas - memorize circle, find out if sphere is on the test

Rambling Teacher Thoughts:

This briefly talks about 3-D and Volume and then encourages you to look at these external links which have more tutorials and videos that I created.

More Videos and Mini Quizzes

Visual Volume Practice

Visual Volume and Surface Area

Surface Area and Volume Practice

A real challenge - Only if you dare

More about Lateral Area and calculating surface area

08:27

What you'll learn:

What makes an angle?

What are degrees?

How many degrees are in a straight line and a right angle?

What are the classifications of angles?

How does one use a protractor?

Rambling Teacher Thoughts:

Angles and degrees are used a lot in every day life. 45 and 90 degree angles come up all the time. They are the 1/4 and 1/2 of the angle world. Protractors are useful tools for certain situations but, not being an architect, rarely use one.

For more fun interactive practice see this protractor practice link. After watching the intro, try the "Up to 180 degrees in 10s". With this exercise, I'd suggest using the protractor a few times and then just estimating. You should be able to get most angles with only two guesses.

08:22

What you'll learn:

Exactly how to line up a protractor on the paper.

The classifications of angles.

How to figure out an angle in a linear pair or a complementary pair.


Rambling Teacher Thoughts:

Blah blah blah blah. I actual rambled enough in the last description.

The promised circle vocabulary quiz.

Hints for link.

A Tangent is a line that crosses a circle at exactly one point. (Get two close, touch two points, Your a secant!)

A Major Arc is more than 180 degrees (more than half a circle)

A hypotenuse is the long side of a RIGHT triangle

Congruent means same size and shape

WOW - that's literally about a days lecture in 4 lines!!

Article

Just a link for practice.

A lot of this was vocabulary.

Here's some more practice

There's a few words we haven't covered. You can look them up or make an intelligent guess based on context (it is matching!).

Section 2: Common Core Congruence: Transformations and Proofs
10:11

What does it mean to be congruent?

What is a transformation?

What is the symbol for congruence?

What are different ways that two figures might be marked congruent?

3 questions

Most of my quizzes are fill in the blank. Since spelling and word choice matter, you may often need to decide if you got it right or wrong. Did you understand the concept but just pick the wrong word? Not sure? e-mail me!

10:20

CCSS.Math.Content.HSG.CO.D.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.

CCSS.Math.Content.HSG.CO.D.13
Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

ADDITIONAL RESOURCES HERE:

EACH OF THE WORKSHEETS, list the pages in the book where instruction can be found. In addition you can check out these video examples:

creating a congruent segment

creating an equilateral triangle

making congruent angles (safety compass) making congruent angles(old style)

bisecting a segment

bisecting angles (safety compass) bisecting angles (old style)

creating a perpendicular from a point ON the line

creating a perpendicular from a point NOT on the line

creating parallel lines

3 questions

Most of my quizzes are fill in the blank. Since spelling and word choice matter, you may often need to decide if you got it right or wrong. Did you understand the concept but just pick the wrong word? Not sure? e-mail me!

10:51

CCSS.Math.Content.HSG.CO.A.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.

CCSS.Math.Practice.MP3 Construct viable arguments and critique the reasoning of others. (Counter examples)

Definitions Quiz
7 questions
08:27

CCSS.Math.Content.HSG.CO.A.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).

Transformations Quiz 1
3 questions
05:01

CCSS.Math.Content.HSG.CO.A.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).

Transformations Quiz 2
2 questions
10:16

CCSS.Math.Content.HSG.CO.A.3
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

Mapping Quiz
4 questions
09:33

CCSS.Math.Content.HSG.CO.A.4
Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

CCSS.Math.Content.HSG.CO.A.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.

Transformations Quiz 3
2 questions
10:27

"Tragedy is when I cut my finger. Comedy is when you walk into an open sewer and die." - Mel Brookes

CCSS.Math.Content.HSG.CO.A.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.

Transformations Quiz 4
3 questions
03:42

CCSS.Math.Content.HSG.CO.B.6
Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

14:51

CCSS.Math.Content.HSG.CO.B.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.

CCSS.Math.Content.HSG.CO.B.8

Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

Congruence Quiz 2
7 questions
05:36

CCSS.Math.Content.HSG.CO.C.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.

03:39

CCSS.Math.Content.HSG.CO.C.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.

08:33

CCSS.Math.Content.HSG.CO.C.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.

2 questions

Just a quick check-in.

12:43

CCSS.Math.Content.HSG.CO.C.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.

07:27

CCSS.Math.Content.HSG.CO.C.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.

Angle Values and you-know-who!
2 questions
07:01

CCSS.Math.Content.HSG.CO.C.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.

05:26

CCSS.Math.Content.HSG.CO.C.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.

10:20

CCSS.Math.Content.HSG.CO.C.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.

ADDITIONAL RESOURCES:

A Great Site if it's up and running: http://feromax.com/cgi-bin/ProveIt.pl?task=getproofslist

Proof shown: http://www.teach4ever.net/Geometry/MedProof8-1.htm

Four Simple Proofs: http://www.teach4ever.net/Geometry/MedProof6-1.htm

Intro to proofs using Algebra: http://www.teach4ever.net/Geometry/MedProof0Notes.html

Last two proofs
2 questions
07:48

CCSS.Math.Content.HSG.CO.C.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.

Section 3: Same Bat-Shape, Different Bat-Size
08:11

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

3 questions

Most of my quizzes are fill in the blank. Since spelling and word choice matter, you may often need to decide if you got it right or wrong. Did you understand the concept but just pick the wrong word? Not sure? e-mail me!

06:26

G.SRT.A.1b Verify experimentally the properties of dilations given by a center and a scale factor: the dilation of a line segment is longer or shorter in the ratio given by the scale factor.

4 questions

Most of my quizzes are fill in the blank. Since spelling and word choice matter, you may often need to decide if you got it right or wrong. Did you understand the concept but just pick the wrong word? Not sure? e-mail me!

07:14

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.

09:30

G.SRT.3 - Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

The external resource attached to this lesson has some proportion practice. Only some of the questions are specifically about similar objects but all of the questions are helpful in thinking about setting up a proportion and then solving using the cross-multiplication method. Ignore the request to submit your name and student number: that's only useful for my formal classes.

Similarity Quiz 3 - Angle Angle
5 questions
10:40

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. (unbolded part covered briefly in previous lesson)

Pythagorean Proof Quiz
2 questions
07:31

G.SRT.5 - Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Similar Problems Quiz
5 questions
07:43

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.

45-45-90 and 30-60-90 Triangle Quiz
4 questions

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Instructor Biography

Michael Taylor, Teacher of Mathematics and Computer Science

Teaching is something I came to a little later in life. My first love was technology and that's what my Duke degree is in. I've kept one foot in the programming world most of my life. But several years after graduating, I realized something else was calling me and strangely discovered it was the desire to teach. I went back to school and got a Masters in Education and a Teacher Certificate - and that's what I've been doing for most of my adult life.

I've worked in the public and private sector. I've worked with students of all races and socio-economic backgrounds. My primary subjects have been Math and Computer Science although I like to think I teach a lot about how to be a good student [We're all life-long learners!!]. I did a year stint in Hungary as English as a Second Language teacher and an even shorter volunteer job teaching fractions to 3rd graders using my limited Bahasa in Jakarta.

I recently returned to the programming world for a change of pace but there's that tug to teach again (and all this Math knowledge slipping away!). The courses I'm providing are an attempt to feed my teaching desire, share the knowledge I've gained, and let me explore the snowballing phenomenon of MOOCs.

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