Stage Six: Interior and Exterior Angles

The last stage of angles I introduce to my students are interior and exterior angles. This has been the most difficult topic for my students to understand so practicing these formulas is important.

Students need to know that:
- The sum of the measure of the interior angles of a triangle always equals 180 degrees
- The sum of the measure of the interior angles of a quadrilateral is 360 degrees

Interior Angles

To find the sum of the interior angles you use the formula:

180n-360

[EXAMPLE]: The sum of the interior angles of a hexagon is:

 (180 x 6) - 360 = 

ANSWER: 720 degrees

To find out the measure of each interior angle you use the formula:

(180n-360)/n

[EXAMPLE]: The measure of each interior angle of a hexagon is:

(180 x 6) - 360 = 720

720/6 =

ANSWER: 120 degrees

The thing about math is that it is a subject that intertwines often. I have shown this to be true in previous blog posts and will now show it again.

Exterior Angles

The sum of the measure of exterior angles is always 360 degrees

SO, to find the measure of each exterior angle, supplementary and complementary angles need to be understood. First we found that the measure of each interior angle is 120 degrees in the previous examples. Supplementary angles are 180 degrees and complimentary angles are 90 degrees meaning that these interior angles cannot be complimentary but can be supplementary. 

To find the measure of each exterior angle:

We minus 120 from 180 and get 60. Since a hexagon has 6 sides, we would multiply 60 by 6. Is the final result 360? YES! This means our exterior angles are:

ANSWER: 60 degrees

Personally, while I was in school I struggled to understand interior and exterior angles. Be aware of students who are not understanding and provide ways for them to understand outside of the classroom. Here are some useful sources for interior and exterior angles:

WEBSITES:

http://www.regentsprep.org/regents/math/geometry/gg3/indexGG3.htm

http://en.wikipedia.org/wiki/Internal_and_external_angle

VIDEOS:

Stage Five: Angles, Angles, and More Angles! (Vertical, Supplementary, Complementary)

Geometry Review

When we discuss geometry, angles are going to be a popular topic. It is necessary for students to have a firm grasp on the topic because it appears in geometry so often. Before continuing with the next stage of angles and geometry topics, teachers should quiz students on what they already know. In my own classroom I like to group students together and assign each group certain questions. This is a quick way to get an idea of who is grasping the subject or not.

Some examples of questions I asked were:

• What are the fundamental building blocks of geometry?
• What does a line segment look like?
• What do angle measurement problems consist of?
• Which is an obtuse angle? 130 degrees, 185 degrees, 90 degrees?
• What are three examples of a polygon?
• What are three examples of a polyhedron?

Vertical, Supplementary, and Complementary Angles

After quizzing and making sure my students understand these first few geometry stages, we take a look at the next stage of angles: vertical, supplementary, and complementary. Vertical angles are created by intersecting lines which are pairs of angles whose sides are two pairs of opposite rays. Supplementary angles are two angles whose sum of measure is 180 degrees. Complementary angles are two angles whose measure is 90 degrees.

Here is a trick I learned in one of my college classes to remember these angles. This helped me to remember them and can be quite useful when learning and teaching these angles. With supplementary angles, turn the S and U in supplementary into 80 and put a 1 in front of the word. For complementary angles, take the "C" and the "O" and turn it into 90. Vertical angles are congruent meaning they are exactly the same. Here's a chart I made of these tricks:

WEBSITE:
http://www.algebralab.org/lessons/lesson.aspx?file=geometry_anglescomplementarysupplementaryvertical.xml

Stage Four: Angle Measurement

Measuring Angles & Angle Measurement Problems

This is one of the tougher concepts to understand when it comes to angles. With angle measurement problems we look at degrees, minutes, and seconds. I had a tough time teaching these problems to my students because it takes some patience to understand them.

A degree is 1/360 of a rotation about a point. They are subdivided into 60 equal parts which are minutes. Each minute is subdivided into 60 parts which are seconds. Thus a measurement of 29 degrees, 47 minutes, 13 seconds is written as:

Here is a video from the Khan Academy explaining angle measurements:

Here is also a video explaining how to add and subtract angle measurements:

And a website:
http://www.algebra.com/algebra/homework/Angles/Angles.faq.question.514507.html

Although these problems are not seen often, it is important for students to understand them so that they can better understand angles. For these it is all about comprehending the formula

Stage Three: Angles (Straight, Obtuse, Acute, and Right)

Angles Introduction

Angles are two rays with the same endpoint. They measure the rotation of these rays which are considered the sides of the angle. These rays have a common endpoint which is called the vertex. 

Angles can be a hard concept to understand when measuring, so it is important to know those fundamental building blocks of geometry: points, lines, and planes. When I was younger, I struggled a lot to understand angles and geometry in general for that matter. Without understanding the building blocks, you cannot understand geometry. 

Straight, Obtuse, Acute, and Right Angles

There are a few different types of angles to remember: straight, obtuse, acute, and right. A straight angle is exactly 180 degrees and an obtuse angle is greater than 90 degrees but less than 180 degrees. An acute angle is less than 90 degrees and a right angle is exactly 90 degrees.

A trick you can use to help your students remember the differences between these angles is to associate the pronunciation of the words with its meaning. OBtuse angles are big while acute angles are cute and small. Straight angles can only be 180 degrees while right angles can only be 90 degrees so they should be easier to comprehend.

WEBSITE: http://www.mathsisfun.com/angles.html

Stage Two: Polygons and Polyhedrons

Polygons

After teaching the building blocks of geometry, I like to focus on polygons and polyhedrons. A polygon is a simple, closed curve with sides that are line segments. What are simple and closed curves?

Simple curve = a curve that does not cross itself

Closed curve = a curve that starts and stops at the same point

Here are examples of polygons and the number of sides they have:

With polygons, we also look at convex and concave curves:

Convex curves are simple, closed curves such that the segment connecting any two points in the interior of the curve is wholly contained in the interior of the curve. Concave curves are simple, closed curves that are not convex meaning that it is possible for a line segment connecting two interior points to cross outside the interior of the curve.

Polyhedra

A polyhedron is a simple closed surface made up of polygonal regions, or faces. Faces are flat surfaces that forms part of the boundary of an object. Vertices are the points an object has and edges are particular line segments that join the vertices.

To find the relationship between these parts of a polyhedron, we use the formula:

VERTICES + FACES - EDGES = 2 or V + F - E = 2

WEBSITE: http://world.mathigon.org/Polygons_and_Polyhedra

Stage One: Geometry Building Blocks

Points, Lines, and Planes

To introduce students to the subject of geometry, I began with its fundamental building blocks: points, lines, and planes.

A point is the most fundamental of the building blocks and represents position with a dot and a letter. A line has no thickness and extends forever in two directions and is uniquely determined by two points. A plane is two-dimensional surface consisting of points and lines.

Here are some charts I created that helped my students begin to understand geometry:

These building blocks are key in being able to learn and teach geometry. My students got together in groups and described these building blocks to each other so that they could gain a better grasp on geometry.

Blogroll

Here are some of my personal favorite math blogs that I think everyone should check out!

thegeometryteacher

misscalcul8

Continuous Everywhere but Differentiable Nowhere

ThinkThankThunk

Math Teacher Mambo