Mathematics

7.G.A.2 Parts 1 and 2

This is Common Core State Standards support video in mathematics. The standard we’re addressing now is 7.G.A.2. This standard states: Draw (freehand with ruler and protractor and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.

First, let’s see what other standards are related to 7.G.A.2. Back in fourth grade, there was standard 4.MD.C.5a that gave students exposure to angles and angle measure. This fourth-grade standard states: 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 two rays intersect the circle. Also in fourth grade, a related standard was 4.MD.C.6 that talked about actually measuring angles in whole-number degrees using a protractor. Also in fourth grade, there is standard 4.G.A.1 that talks about drawing points, lines, line segments, rays, and angles. And notice that they do introduce the classifications of angles based on measure, those being right, acute, and obtuse angles. This should, in turn, help students identify triangles based on angle measures. So, that would give them some background and a little bit of experience with the terms that would lead to right triangle, acute triangle, and obtuse triangle.

Note that this is the only Common Core standard where the term obtuse is mentioned. Also, the term acute only appears again at the high school level with regard to trigonometry. Another standard in fourth grade—this one is 4.G.A.2. This one talks about classifying two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specific size. Notice the last statement: Recognize right triangles as a category, and identify right triangles. So, at the fourth-grade level, they’ve already been exposed to the idea of a right triangle.

Notice that when addressing standard 7.G.A.2, that seems like an appropriate time to classify triangles based on angle measure. In seventh grade, there’s a related standard 7.G.B that talks about supplementary angles, which leads to the idea of the sum of the angles of a triangle being 180. This seventh-grade standard will help set the foundation for standard 8.G.A.2, which talks about two-dimensional figures being congruent to one another. Notice that this eighth-grade standard is the first mention of the concept of congruency in the Common Core standards. In eighth grade, we have this other standard 8.G.A.5 that talks about using informal arguments to establish facts about the angle sum of triangles.

If we switch our focus a little bit to the idea of constructing triangles from measures of sides versus angles, there is this high school standard, HSG.CO.C.10, that talks about proving theorems about triangles and that includes measures of interior angles of a triangle sum being 180, and the base angles of isosceles triangles being congruent. Notice that this high school standard is the only standard where the term isosceles appears. Also, the term scalene is not mentioned anywhere in the standards, and the term equilateral does not occur until the high school Geometry standards. So, it seems like there’s not much of a focus as far as the terminology when it comes to the classification of triangles based on the sides. Since this isn’t really addressed in the standards, this standard, 7.G.A.2, seems like an appropriate time to classify triangles based on the lengths of the sides.

In looking at the standard and these last statements dealing with the conditions that determine different contexts as far as the number of triangles, if we look at the idea of having more than one triangle formed, there’s standard 8.G.A.4 that talks about the similarity of figures. So, this standard here, 7.G.A.2, can help lay the foundation for the idea of similar figures that will come up in eighth grade. Again, this standard in eighth grade is the first formal mention of the concept of similarity in the mathematics content standards. But again, we can use this seventh-grade standard to start laying the foundation for that whether or not we use the actual mathematical terminology.

Teachers should be able to find some type of software that can actually be used to measure angles, and that’s what we have here in this case. We have a protractor that we can use to measure angles, and so, this is a very useful tool that again, we can use. Let’s start by focusing on constructing triangles from three measures of angles. Let’s say our task was to construct a triangle with angle measures 34, 71, and 75 degrees. Well, let’s take our ruler, and let’s draw a segment; and let’s start off by measuring off a 34-degree angle. So, we take our protractor and we do that; and so we use our ruler to mark off that segment so that we have this ray. This is the beginning foundation for our triangle. So far, we have our one angle of 34 degrees. Now let’s construct a 71-degree angle from this other vertex. Well, right now it’s just the endpoint, but it will be the vertex of our triangle.

Okay, we have a little bit of a problem here. The way that the protractor is set up, I can measure the 71 degrees here, but I need the angle measure over here. So, what am I going to do? Well, here’s a little trick. Notice that we get the supplement of 71 degrees. That’s 109. So, that’s a little trick that we can use. We can use the measure of 109 degrees over here, and the supplement will be 71 degrees over here on the left-hand side. So, that’s what we can do. We can use this little trick of using the supplement of an angle to get the angle that we want. So, let’s put our protractor in the proper location and we’ll measure off 109 degrees instead to get the 71-degree angle that we want as part of our triangle. So, now we would use our ruler and draw our segment.

Let’s get that out of the way and let’s worry about the 75-degree angle. Well, we’ve already got our triangle formed, so let’s see what the angle measure is for this third angle up at the top. So, we take our protractor, put it in position, look at our measurement, and yes, it is 75 degrees. So, we’re in business. We’re set. So, now we have our third angle being 75 degrees. Let’s knock off the extra part of the segments, and there we have it. We’ve constructed our triangle with measures 34, 71, and 75 degrees. Since we didn’t have any directions as far as how long or short to make the sides of this triangle, then students are going to come up with different sized triangles. They’ll all have the correct measurements, 34, 71, and 75 degrees, but they will be different sizes.

So, this sets up this idea of these triangles being similar. So, in fact, if you look at this in terms of the actual sides, we will have more than one triangle formed. So, this might be an example here of what some of the different triangles that students come up with would look like. Now, we can use this and link it to standard 8.G.A.4, but what we can also do is link this to standard 8.G.A.5 and use it to discover or informally prove that the sum of the interior angles of the triangle is 180 degrees. So, this is what we can do to do that.Pick out three different sized triangles, but all of them have the same three degree measures. So, let’s place one triangle like this. Then, let’s take another one and invert it over. Fit it like this, and then we’ll take our third triangle and place it like this. Now, notice the three different angles here that come from three different triangles. We have our 71, our 75, and our 34, and if you add them up, that is 180 degrees and that is our straight angle here. If we double-check the sum, yes, it’s 180.

Now, the standard talks about constructing triangles from three measures of angles, but we can throw the students a little bit of a curve. What if we asked them to construct a triangle with angle measures 22 and 136? We’ve only given them two angle measures. Well, let’s see what happens. Let’s draw a segment as our starting point. We’ll take our protractor and let’s start off measuring the angle that is 136 degrees. So, we do that. Take our segment, and so we have our angle, and now let’s pick this endpoint over here. Well, we couldn’t measure the angle 22 directly, but what we can do is get the supplement 158, which would give us the 22 degrees here that we’re looking for.

So we do that. Let’s draw it, and so, we’re almost set except we need to extend this other segment a little bit. There you go. Now the question is, well, that third angle in this triangle, what is the measure for that one? If we take our protractor and measure it, look what happens. It’s 22 degrees. Let’s check our figures. If we combine 22 and 22 and 136, that is 180, and again, since students didn’t have any specific lengths for the sides, they will end up making more than one triangle out of it. They’re going to have similar triangles. So, again, this might be an example of what they come up with, but they’re all going to look exactly alike even though their lengths of the sides might be different sizes.

Let’s take yet another example. Let’s say you give them this task. Construct a triangle with angle measures 93, 64, and 43. Okay, just like before, we have a little bit of a problem. I want my 93-degree angle on the left, not on the right. So, let’s use our little trick where we can use the supplement of it, and let’s reverse it so we've got the 87- degree angle over here, which will give us the 93-degree angle that we’re looking for. So, let’s do that. So, we have our 93-degree angle. Let’s go ahead and go with the 43-degree angle that we need and use this endpoint on the left because it’ll be a lot easier to measure the angle that way. So here we go. Extend your segment. So, we have our 43-degree and our 93-degree angles.

So, now the question is the 64 degrees. Is that going to be that top angle? Well, let’s measure it. Okay, wait a minute. We’ve got a problem. That’s not 64 degrees. The protractor is saying 44 degrees Well, it turns out this is impossible. There’s no way that we can do this. There’s no triangle that can be formed with these three angles, and of course, what students can do is verify whether or not those three angles that were given add up to be 180 degrees. And of course, they’ll find out no. So, given similar tasks where the sum of the three angles is not 180 degrees, students will discover that no triangle is possible.

So far, students have constructed triangles based on information regarding angle measure. The results included acute, right, and obtuse triangles. Students should note that when only given information about angle measures, the constructed triangles were similar even though no information was provided regarding side lengths, and students should also realize that the sum of the three interior angles of a triangle must be 180 degrees.

So far we’ve focused on constructing triangles using three measures of angles, but now let’s switch our attention to constructing triangles based on measures of the sides. Let’s say this is our task. We need to construct a triangle with side lengths of 3 inches, 4 inches, and 5 inches. So, let’s take our ruler, and we’ll measure off those three lengths—3 inches, 4 inches, and 5 inches. Now we need to take those and combine them to make a triangle. So, we do that, and this is our result. When students compare their work, they will see that, hey, we all have the same triangle. So, the result will be that this does result in a unique triangle. Every single student will have a triangle that’s exactly the same as everyone else’s.

Something to pay attention to here—notice that no sides are of equal length. So, if we use side length as our criteria, this is a scalene triangle. Just out of curiosity, let’s use our protractor. One of those looks like a right angle, and when we measure it, sure enough, it is 90 degrees. So, the angle that we measured is 90 degrees. So, if we use angle measure as the criteria, then this is a right triangle in addition to being scalene based on the lengths of the sides.

Let’s take this task. Construct a triangle with side lengths of 3 inches, 4 inches, and 8 inches. So, we measured off those three lengths. Now let’s use our length of 8 inches. Let’s form our triangle. Oops, wait a minute. This isn’t working. We can’t make a triangle out of these three given lengths, so no triangle is possible. With repeated examples of this type, students should discover that the sum of the lengths of the two shorter sides of a triangle have to be greater than the length of the longest side. So here, the two shorter sides, 3 and 4, that’s a sum of 7, which is shorter than our length of 8 for the longer segment. So, this wasn’t possible because again, the sum of the two shorter sides of the triangle has got to be longer than the longest side.

Let’s take on this task of constructing a triangle with side lengths of 4 inches, 4 inches, and 6 inches. Okay, so when students do that, this is what we come up with. When students compare, they’ll find out that, hey, this is a unique triangle. We all came up with exactly the same thing. In this case, we had two sides that were of equal length. If we use side length as a criteria, then this is an isosceles triangle, again because we had two sides being of equal length.

Just out of curiosity, one of those angles looks like it’s more than 90 degrees. So, let’s measure it. Yes, it looks like it’s 98. One of the angles is an obtuse angle, so if we use angle measure as a criteria, then this is an obtuse triangle. I wonder what the other angles have for their measures. Well, let’s see. If I measure this one, that’s 41 degrees. If we switch over to the other one, we have to use our supplement though. Yes, that’s 41 degrees also. Letting students do some discovery and some work on their own, they should be able to figure out that when you have an isosceles triangle, you can have two angles of equal measure as well as two sides of equal length.

Let’s do this task. Construct a triangle with all three side lengths of 5 inches. So, we have our three equal lengths of five each. Put them together and again, students are going to come up with a unique triangle. In this case again, all the students are going to get the exact same thing. What we have here are triangles that have all three sides being equal lengths. If we use side length as the criteria, then this is an equilateral triangle. Okay, we’re curious again about the angle measures. Well, let’s see. Let’s measure the angle on the right-most side. Well, let’s see. We have to use our supplement again—120 degrees, the supplement is 60, so that one is 60 degrees. Let’s measure this one. It’s an easier task. This one is 60 degrees. Let’s measure the top one. That one is 60 degrees. . .interesting. This time all of our angles are acute angles. If we use angle measure as the criteria, then this is an acute triangle. Now, all the angle measures were equal. So, using angle measure as the criteria, then this is also an equiangular triangle.

So, now students have constructed triangles based on information regarding side lengths. Results included scalene, isosceles, and equilateral triangles. Students should also note that when only given information about side lengths, the constructed triangles were congruent despite the fact that no information was given about the angles. Another interesting tidbit—if we do things right—students should discover that the sum of the lengths of the two shorter sides of the triangle had to be greater than the length of the longest side.

The standard really doesn’t specifically address this, but we’ve already seen that there seems to be some relationships between the angles and the sides. Remember we had this triangle. We had this task. We were given angle measures of 34, 71, and 75. Well, let’s measure the lengths of the sides. That appears to be 4. This one appears to be a little less than 7, about 6 7/8. This one looks to be about 7. Let’s see what kind of relationship there might be if we look at the angles and the sides together. If we look at the smallest angle, 34 degrees, notice the measure of the opposite side is four, which is the shortest of the three side lengths. The 75-degree angle was the largest angle, and notice that seven is the longest side that’s opposite of it. If we look at our 71-degree angle, that was the in-between angle measure and notice that the 6 7/8 side length lies in between the 4 and the 7. If students do this over and over, they should reach the conclusion that in the same given triangle, the larger the angle, the longer the opposite side is going to be.

Is there anything else? Well, let’s do some more digging. Let’s take the example that we did where we were given two equal side lengths, 4, and 4, and then also the 6. We measured the angles to be 98, 41, and 41. Well, look—98 is the biggest angle and the side opposite of it is the longest side. Well, the other two, 41 degrees, are equal angles and notice that, huh, the sides are equal, the ones opposite them. Then we had that equilateral triangle where all the sides were five and we measured the angles to all be 60. Notice what happens here. Given some experience, the students here should be able to determine that in the same given triangle, angles with equal measures will have opposite sides that are equal lengths.

So, with regard to triangles, there are a lot of facts and definitions that are not specifically addressed in the standards. This standard, 7.G.A.2, should be used as the foundation to introduce and teach all of these important items regarding triangles. So, this is a very important foundational standard. Again, there are a lot of things that it doesn’t specifically mention, but this is a good time to really address a lot of those other terms and other facts. Again, there are a lot of things that happen here. There are some things that happen with constructing the triangles based on the measures of the angles. It seemed like a lot of times that resulted with triangles that were similar as opposed to when we were given the task of constructing triangles given the lengths of the sides, students came up with the same triangles. So, they seemed to come up with congruent triangles. So, again, all kinds of little patterns like that, all kinds of facts and relationships that should be addressed at this point.

If we look at the standards for mathematical practice, by addressing some of the issues and doing some of the activities in this standard, students would reason abstractly and quantitatively. They would construct viable arguments and critique the reasoning of others. Yes, in this standard there’s a lot of opportunity for discovery and for discussion. If we look at the remaining standards of mathematical practice, students would use appropriate tools strategically, like your protractors especially. They would attend to precision. They would look for and make use of structure, and they would definitely look for and express regularity in repeated reasoning.