Work in the following math link and leave a comment explaining one of your discoveries about convex and concave polygons. If you didn't make any learning discoveries, keep digging (asking question and searching for answers), until you do.
http://www.mathopenref.com/polygonconcave.html
Monday, November 29, 2010
Monday, November 22, 2010
Building Blocks of Geometry
Points, lines, segments, rays and angles are the building blocks of geometry. Understanding these concepts is essential, since they will be used in a wide variety of applications.
Points and lines are basic ideas (also known as abstractions) in geometry that do not have definitions - they are undefined terms. Points and lines are not things; they can not be seen or touched. However, our understanding of these ideas evolves out of our experiences with physical objects and situations that can be seen, touched or manipulated. For example, the notion of a point can be suggested by the tip of a pencil or the smallest dot you can make; it has no dimension. The notion of a line can be suggested by railroad tracks, a string held tight, or the path of a light eam. These physical models help cultivate the notion of these ideas in our minds.
Ancient Greek mathematician Euclid described a line by using two critieria or postualtes:
(1) through any two points there is always a line, and (2) every line contains at least two points. These postulates, which detail facts that Euclid believed were self-evident appear in every geometry book that has since been ever written.
We can verify the first postulate by drawing a pair of points. No matter how we draw the two points, a line can always be drawn through them. The second postulate uses the phrase at least two to describe the number of points on a line. Euclid knew that a line contains an infinite number of points. Why not simply state that in the postulate? One reason is that these postulates present facts that are the foundation for further geometry facts. Euclid wanted his postulates to be as simple and as brief as possible. At least two points allows for but does not stipulate an infinite number of points.
A segment is a part of a line that contains two points (called endpoints) and all the points between them. A ray is a subset of a line that contains a point (called the endpoint) and all the points of the line on one side of the point.
Think about what you have just read. Think about Eucalid's postulates. Think about how it changes or deepens (or confuses) your understanding of points and lines. Do you need to study more to get it? See what else you can find out to deepen your understanding.
Leave a comment on your thinking.
Points and lines are basic ideas (also known as abstractions) in geometry that do not have definitions - they are undefined terms. Points and lines are not things; they can not be seen or touched. However, our understanding of these ideas evolves out of our experiences with physical objects and situations that can be seen, touched or manipulated. For example, the notion of a point can be suggested by the tip of a pencil or the smallest dot you can make; it has no dimension. The notion of a line can be suggested by railroad tracks, a string held tight, or the path of a light eam. These physical models help cultivate the notion of these ideas in our minds.
Ancient Greek mathematician Euclid described a line by using two critieria or postualtes:
(1) through any two points there is always a line, and (2) every line contains at least two points. These postulates, which detail facts that Euclid believed were self-evident appear in every geometry book that has since been ever written.
We can verify the first postulate by drawing a pair of points. No matter how we draw the two points, a line can always be drawn through them. The second postulate uses the phrase at least two to describe the number of points on a line. Euclid knew that a line contains an infinite number of points. Why not simply state that in the postulate? One reason is that these postulates present facts that are the foundation for further geometry facts. Euclid wanted his postulates to be as simple and as brief as possible. At least two points allows for but does not stipulate an infinite number of points.
A segment is a part of a line that contains two points (called endpoints) and all the points between them. A ray is a subset of a line that contains a point (called the endpoint) and all the points of the line on one side of the point.
Think about what you have just read. Think about Eucalid's postulates. Think about how it changes or deepens (or confuses) your understanding of points and lines. Do you need to study more to get it? See what else you can find out to deepen your understanding.
Leave a comment on your thinking.
Geometer's Sketchpad
Click this link to find today's challenges with angles. You will look for your grade level and then you will click on the first activity. You will be working to accurately predict angle measurements and to draw given angle measurements.
Complete the work in the given amount of time and leave a comment about your discoveries and learning. (One mark for each valid statement up to a maximum of 3 points.)
Complete the work in the given amount of time and leave a comment about your discoveries and learning. (One mark for each valid statement up to a maximum of 3 points.)
Monday, November 15, 2010
Saturday, November 13, 2010
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