This is Common Core State Standards Support Video in mathematics. The standard is 4.MB.5ab, so there’s actually two standards here. The standard reads: Recognize angles in geometric shapes that are formed whenever two rays share a common endpoint, and understand concepts of angle measurement—Part A. An angle is measured with reference to a circle with the center at the common endpoint of the rays by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a "one-degree angle," and can be used to measure angles. Part B: An angle that turns through n one-degree angles is said to have an angle measure of n degrees.
Let’s look at the main portion, the introductory portion of this standard, and let’s concentrate on the definition of an angle. Angles are formed whenever two rays share a common endpoint. So if we had two rays such as we have here, and we put them together to where they have a common endpoint, now we have an angle. That common endpoint is called a vertex, and it’s usually designated by a single letter. It probably won’t apply at this grade level, but in some cases where there are several angles with a common vertex, it’s necessary to name the angles with three letters. So for example, here this angle could be named angle EAJ or angle JAE, depending on the order of rotation. Notice that the vertex is always going to be the middle point.
It’s important that students at this grade level get experience with angles with different orientations and sizes, so don’t just concentrate on angles like angle A. They need to see angles like angle B and angle C so again, they experience angles that have different sizes and different orientations. Students are going to experience angles as part of polygons. Now here, the phrase angle B is slightly misleading because by definition, an angle is the union of two rays with a common endpoint. But polygons have line segments for their sides, not rays. So in the case of polygons, it’s generally accepted that the angle is formed by the union of two line segments with a common endpoint like we have here.
Now there’s a lot of little details involved with angles. There is a distinction between angles and angle measurement. So here, if we say angle B and the measure of angle B, they’re not synonymous. Saying something like angle B is 100 degrees is actually not correct. It’s really the measure of angle B is 100 degrees. This would be synonymous to, let’s say a 3 foot board. Well, we have the board itself, and then the length of it is 3 feet.
Let’s focus on Part A. The first statement is really the meat and potatoes here. An angle is measured with reference to a circle with its center at the common endpoint of the rays by considering the fraction of the circular arc between the points where the rays intersect the circle. So let’s look at a visual representation to get a handle on what this says. So when we are measuring an angle, it’s going to be with regard to a circle with its center at the common endpoint of the rays. So in other words, the center of the circle and the vertex of the angle are one and the same point. Now the angle is measured by considering the fraction of the circular arc between the points where the rays intersect the circle. So the rotation is this here, this arc.
Note that the measure of an angle deals with rotation in a counterclockwise direction along a circular path until the arc intersects the second ray, and the end result is usually designated in this manner where you’re going to have your angle and you’re going to have an arc that’s marked off. But it’s important that the kids realize that this is actually what that previous abbreviated diagram represents. It’s not just this. It’s actually this amount of rotation along this circle. Students are probably going to see diagrams where the interior of the angle is shaded, but again, keep in mind the measure of an angle deals with rotation, not area. So again, make sure the students don’t get confused with this.
Also, the measure of an angle deals with rotation, not the actual length of the arc. So notice here we have this arc. Here’s another circle, and then here’s a third circle. Notice that the whole time the angle never changed. Now the lengths of these arcs did because of the size of the circles, but again, we’re dealing with rotation. So again, very important that the kids see the distinction that, again, the length of the arc is not the measure. The rotation around the circle is what the degree measure is.
The second statement in Part A states that an angle that turns through 1/360 of a circle is called a one-degree angle and can be used to measure angles. Now it’s important to address the multiple meanings of the term degree because it does have several meanings outside of mathematics and even within mathematics. So for example, in medicine you could have a second-degree burn. In academics, a degree can refer to achievement such as a college degree. And even within mathematics degree is a unit of measure for temperature, and now we’re using degree as a unit of measure, but for rotation.
So let’s look at this idea of a 1 degree angle. If I were to mark off this distance that’s 1/360 of the distance around the circle and keep doing that until such time as I get all the way around a circle, we would have 360 of those. And just one of those little arcs would be 1 degree. So that would be a 1 degree angle and again, that’s 1/360 of all the way around a circle.
Let’s look at Part B: an angle that turns through "n" 1 degree angles is said to have an angle measure of n degrees. It might be easier for students to relate to something that they know, such as linear measure. So, 1 degree is the unit of measure for rotation on any circle just like 1 inch is a unit of measure for length on a ruler. So they’re synonymous except one is dealing with circular rotation and the other is dealing with straight-line linear distance. So 4 inches for example, would be four iterations of 1 inch each laid end to end. So that’s what 4 inches really is. In a similar manner, a 4-degree angle would be four iterations of a 1-degree angle of rotation counterclockwise along a circle. If we look at the statement and substitute numbers for the variables, it makes it a lot easier to understand. So if we take the example that we just did, an angle that turns through 4 one-degree angles is said to have an angle measure of 4 degrees.
Now as far as measuring angles, there’s probably all kinds of things that you can find, either on the internet or different types of software, that can be used to measure angles. Here’s one example here. So notice that I can measure any angle, and in fact, we can even go past 180 degrees. And notice that this particular software measures both the angle and the remaining part of the rotation around this circle. Notice that by eliminating the bottom half of the circle we have just the top semi-circle, which many of you will recognize as a protractor from back in your high school days. So this is pretty much what we have, a protractor. We can use this to measure angles. But it’s important that kids realize that that protractor is actually an incomplete picture because again, keep in mind that we’re actually measuring angles, and remember what angles are. And again, we’re dealing with rotation around a circle.
At this point it’s important to note that standards do not address the symbolic representation of degree measure. So without any examples as far as test items, we’re really not sure how that’s going to be represented. At this point, go ahead and show students what the degree representation is, which of course, is a superscript zero or "O" if you’d rather the think of it as an "O." So again, it’s important that students know what the degree measurement is as far as the representation, and of course, be careful. Make sure the students do not confuse the representation for degree with a zero.
Students are expected to use protractors and measure with them as noted in standard 4.MD.6 which states: Measure angles in whole-number degrees using a protractor. Sketch angles of a specified measure. So let’s say the task was to find the measure of this angle. So students will need to take their protractor, line up the vertex and then just rotate and measure. So this angle looks like it’s about 50 degrees. The standard also calls for sketching angles of a specified measure. So let’s say the task is to sketch an angle whose measure is 114 degrees. So we’re going to take our protractor, place it to where the vertices align, and then we’re going to measure on here 114 degrees. And then what need to do is go ahead and mark where we are, which would be about here. And then what students need to do is take that mark and just complete the ray through there, starting at the vertex, and we’ve sketched an angle that’s 114 degrees.
The use of the protractor will also be important in learning and applying standard 4.MD.7 which focuses on the composing and decomposing of angles. So for example, this 70 degree angle can be split up into other smaller angles. So for example, we could draw this second ray here starting at the vertex. And then if we measure this, it looks like it’s about 20 degrees. So we have split this up to where we have a 20 degree angle here and a 50 degree angle here. And then of course, together they would constitute the original 70 degree angle. So again, you can compose and decompose angles just like you can compose and decompose numbers.
There’s a lot to standard 4.MD.5. Parts A and B are critical. It’s a very important standard because it establishes a foundation for what angles are and the idea of the angle measurement.