Often when they are investigating geometry, students are shown angles of different sizes and given various tools to draw them and measure them without really developing an understanding of what angles are.
Understanding what each individual angle represents can help students apply reasoning skills when classifying and comparing angles.
By the end of this lesson, your children will be able to understand what angles represent, and how they are measured in degrees.
Many students make it all the way through their geometry studies without every really making a connection to what angles represent. This leads to confusion and frustration. By understanding the bigger picture, the whole subject becomes much clearer. An angle is actually a section taken out of a circle.
By understanding the measure of the whole circle, it is easy to understand and recognize what parts of that circle represent. In this lesson, you will be investigating angles as parts of a circle.
Have your children take the Pre-Test below to see if they are ready for this lesson. If they get 7 or less correct, review fractions of a circle with them before continuing on to the lesson.
A circle is a round figure made by points that are all the same distance from the figure's center point. A circle contains 360 equal sections. Using the unit of measurement one "degree" for each section, we say that the circle has 360 degrees. The symbol ° is often used in place of the word "degrees".
A typical angle that you see when you are studying geometry is nothing more than a slice taken out of a complete circle. The endpoint of the angle, the point where the two rays come together to form the angle, matches the center point of the circle. If you imagine filling in the rest of the circle around the angle you are given, it will give you a better understanding of the size of your angle.
All angles measure 360° or less, since that is the maximum number of degrees in a full circle.
Your children should become familiar with circles and pie charts and make the connection to the fact that an angle is a piece of a circle just like a slice of pie is a piece of a whole pie.
This full circle has 360°. It can also be called a full rotation.
Angles that look like one quarter of a circle have approximately one quarter of the degrees of the full circle, so, 360 ÷ 4 = 90°. A 90° angle forms a square corner at the endpoint. It is called a right angle.
An angle that goes half way around the circle measures 180° and forms a straight line. In fact, it is called a straight angle.
An angle that goes three fourths of the way around the circle measures 270°. It is called a reflex angle, because it is more than 180° but less than 360°.
It doesn't matter which direction the angle is turned, because you are measuring how much of a circle is between the two rays that form the angle. In other words,and
are both right angles, with both measuring 90°, even though they are facing in different directions.
Remind your children to picture a circle in their minds when they see an angle. They can think of the angle they are considering as a slice or wedge taken out of the full circle. Since the full circle measures 360°, any slice taken from it has to measure less than 360°.
Angles are classified, or sorted, by their size. Some of the names used for classifying angles have been shown above, but some are new. Let's look at all of them together:
Acute Angles: angles that are smaller than 90°.
Right Angles: angles that measure exactly 90°.
Obtuse Angles: angles that measure more than 90°.
Straight Angles: angles that measure exactly 180°.
Reflex Angles: angles that measure more than 180°.
Full Rotation: angles that measure exactly 360°.
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