Mathematics

K.G.B.4 Transcript

This is Common Core State Standards support video in mathematics. The standard is K.G.B.4. This standard reads: Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts (for example, number of sides and vertices or corners) and other attributes (for example, having sides of equal length).

First, let’s look at some of the other standards that K.G.B.4 is connected to or has something to do with. Standard K.MD.A.1 is related to it in that students would need to describe measurable attributes of objects such as length or weight. Standard K.MD.A.2, talks about directly comparing two objects with a measurable attribute in common. Standard K.G.A.2 is about correctly naming shapes regardless of their orientations or overall size.

Now, on this idea of shapes, what’s involved because neither standard K.G.B.4 nor K.G.A.2 really specify what shapes are fair game here? Which ones should we use and which ones are out of bounds? From the kindergarten introduction, there is this statement. It says: They identify, name, and describe basic two-dimensional shapes such as squares, triangles, circles, rectangles, and hexagons, presented in a variety of ways. So, this gives us the two-dimensional shapes that students need to work with at this grade level.

The introduction also talks about which three-dimensional shapes would be involved. These would be cubes, cones, cylinders, and spheres. With the introduction, we do now have an idea of what shapes we can actually use. In this same sequence of standards we have K.G.B.5 that speaks to modeling shapes in the world by building shapes from components. We also have standard K.G.B.6 that talks about composing simple shapes to form larger shapes. Standard KG.B.4 sets the foundation for standard 1.MD.A.1 that talks about ordering three objects by length, or comparing the length of two objects indirectly by using a third object.

Let’s look at analyzing and comparing two-dimensional shapes. We need to pay attention to something that could happen here that might lead us astray. When we’re talking about, let’s say a square; we have that on the left-hand side. On the right-hand side, we have that same square, but we also have its interior. Now, there is a danger here. When you’re talking about a square, we’re only talking about those four line segments. We’re not also talking about the interior. So, be real careful that your idea of a square doesn’t start leaning to this idea of the square and its interior. Same thing with the trapezoid, the second figure here—again, it does not include the interior. It is strictly those four line segments. That is what constitutes that trapezoid.

Let’s say you gave your students these four line segments, and the task is to put them together to form a four-sided figure. We have a short segment, a longer one, and then two that are somewhere in between those that are equal to each other. Some students might do this. We have our shortest one. Connect that to one of those two middle ones. Now they connect the larger one and then the other middle one that has an equal segment the same size. This is the figure that they come up with.

Some students might do this. Here’s our short one. Connect that one to this one. Here’s our long side. Now connect it to this one. These are not the same figures. Notice that the difference between them is that in this example here, students connected the two equal segments to each other whereas over here, the two equal segments were not connected to each other. This resulted in two totally different figures. Again, it just depends on what the students connected to what.

Let’s say you gave the students four equal-sized segments. Some students would come up with something like this, a square. But, some students could take those same four line segments and do this. Notice what they’ve created. All four sides are equal, but you don’t have the right angles, so that’s really a rhombus. So, it’s possible for them to take four segments and create different quadrilaterals with them, and even if the four segments or equal, they could still come up with different figures, like here a square versus a rhombus.

Let’s say you gave them three segments. Okay, some students will put those three segments together to form this triangle. Taking those same three segments, some students would form this triangle. But something curious happens here. If I were to compare these two triangles, that these two different students made, from those same three sized segments, I rotate this around. Move it over. Notice that those two triangles were congruent. Students will see that when they’re given the same three segments to work with, they’re all going to come up with the same triangles.

Let’s look at this idea of informal language. There are some dangers that we have to look out for. For example, the term corners—the Standard English definition of the term corner has connotations that lead away from the idea of a point or a vertex. Corner often suggests an area or space, for example, the corner of a room, the corner of a boxing ring, the southeast corner of a state, or a street corner. If we look at this figure, well when we put it all together, informally this is what students will think of when they think of the corner of a room, and that’s totally different than that one point, the vertex that constitutes mathematically the corner of the room.

So, again, there’s some danger with the informal language here. The term corner doesn’t exactly lead us to a single point. So, to be more correct we really should use the mathematical term vertex or vertices, rather than corner, because the informal use of corner doesn’t really get us to where we want to be. Again, there’s a need to reinforce the mathematical idea of a point where lines or edges meet, and that’s not going to happen using the term corner.

Let’s look at the term side. That has some dangers to it also. If we have a two-dimensional figure such as this one, we call those sides informally. Then we have a box, and we tend to call those segments sides also. But that’s a little bit confusing. So, mathematically, on a rectangular prism, a box, we don’t call those sides. We call them edges to distinguish from the two-dimensional idea of sides. But, wait a minute. We also tend to informally call the rectangle and its interior that constitute those parts of a rectangular prism sides, and that’s confusing.

You’ve got the term side, but they look totally different, if that’s how we’re going to interpret them using two-dimensional versus three-dimensional figures. So, mathematically, to avoid that confusion, we don’t call that part of a rectangular prism a side. We call that a face. Mathematically we should call, on a two-dimensional figure, this a segment. But on a three-dimensional figure, we call that an edge. And when we’re talking about the rectangle and its interior, on a rectangular prism, that is referred to as a face of the rectangular prism.

Let’s focus on analyzing and comparing two-dimensional shapes. Remember from the kindergarten introduction, that’s going to involve squares, triangles, circles, rectangles, and hexagons. Let’s say you gave the students all of these figures. Well, what the students should notice is that all of these are triangles because they all have three segments or three sides. They’re all composed again of three line segments. All have three vertices. Now, when students start comparing some of those triangles to each other, if we take those two, let’s move this one around to where they’re in the same orientation.

Those triangles, the three sides appear to be of equal length. If we look at this pair, we have a shorter side. We have a long side. So, the three sides of those triangles appear to all be different lengths. If we look at this pair, we have one pair of sides that definitely appears to be longer than the others. So, that’s one conclusion that we can reach there. If you change the orientation of a triangle, it will look different. If we started off with this triangle here and we move it around, for example there, that looks different. That one looks different. That looks different from the original also. So does this one.

Let’s look at some quadrilaterals, four-sided figures. The kindergarten standards only mention squares and rectangles, but that’s not the only types of four-sided figures that they’re going to encounter. Well, we have a square and a rectangle here, but then we also have all those others. What they have in common is that they all have four segments.They all have four vertices. If you compare some of these, this pair, it looks like the four segments are of equal length. If we compare these two, those appear to have two different pairs of segments of equal lengths.

If we look at this quadrilateral, that one appears to have one pair of segments of equal length. This quadrilateral over here doesn’t appear to have any segments that are of equal length. If you change the orientation of a four-sided figure, it’s going to look different. So, if we start off with this figure here and rotate it, that looks different than the original. So does this. If we start off with this four-sided figure and rotate it, that looks different, and so does this. However, what if we started off with a rectangle, and we mark off, we put an X on one of the segments to identify that specific one. Let’s say we rotate that to where now that segment is on the bottom instead of the top; but if we remove those Xs, you can’t tell one rectangle from the other. It looks like it never changed orientation.

Let’s do another example with a rectangle. Let’s start off with this one. We mark off that left-hand side to distinguish it from the other three segments. Let’s say we do a reflection to where now that side is on the right instead of the left. But, again, without those markings, it doesn’t look like the rectangle is different. What if we take a square and we mark off one segment to identify it from the others? We rotate it. Rotate it again. Rotate it again. But in each case, it still looked like we never did a rotation whatsoever. Let’s start off with a square, and rotate it clockwise to where then we’re in this position. Even though some call this a diamond, it’s still a square. It’s just in a different orientation. So, that is one danger of introducing new terminology like diamond. It’s still a square. The only difference is that it’s not in the typical orientation that you see most squares.

Although the concept of an angle is not introduced in the standards until Grade 2, the foundational knowledge can be explored at this level. If we take these two segments and connect them together at one vertex, we have a certain size angle here, and we can relate that to that type of angle in a figure such as a quadrilateral like this one. We can make that angle a little bit larger, in fact, a right angle, which would correspond to the right angles of this rectangle. We could also widen it some more to where this larger angle could be part of a quadrilateral such as this one.

Let’s say you gave students this instruction: draw a square. So they would do that. This other task is to draw a rectangle, so the students did that. If you had given the instruction “draw a rectangle” and students drew these two different figures, they would both be correct. However, if the instructions were to draw a square, well, the figure on the left is correct, but not the one on the right because not all four segments are of the same length. It’s important that we start instilling the idea that a square is a special type of rectangle, and that’s a key idea. It is a special type of rectangle. They’re not totally disjoint sets that have nothing to do with each other. Again, a square is a special type of rectangle. Students need to realize that. If I start off with a rectangle, and just change the lengths of the sides, again notice the only thing that changed was two pairs of sides were changing lengths to the point where all four are the same; again, very important that students realize this.

Let’s look at some other figures, in this case, circles. All of these are circles. Students should see that circles have no vertices. They’re not composed of straight line segments. If students start comparing some of the circles to each other, these two circles appear to be the same size. If you change the orientation of a circle, it still looks the same. So, for example, if I mark off this point A then I rotate it around, although I’ve done rotations, the circle still looks exactly the same. It doesn’t look any different.

The introduction for kindergarten mentions hexagons, so let’s look at them very quickly. Students should note that a hexagon has six segments. A hexagon has six vertices. The segments of a hexagon do not have to be the same length. If you change the orientation of the hexagon, it will look different. If I start off with these two hexagons and I rotate them both, the results do not look the same.

From the kindergarten introduction, the three-dimensional shapes included cubes, cones, cylinders, and spheres. There will be some parallel observations to two-dimensional shapes such as appearance changing because of different orientations. So, some of the things that we’ve talked about with two-dimensional figures are going to apply to three-dimensional figures as well.

Let’s look at a cube. A real-life example would be a box. When students count them up, they’ll see that there are 12 edges. They should note that in a cube, the edges are all the same length. A cube has six faces. The six faces appear to be the same size, and in fact, they should be because the sides are made up of squares and their interiors. Students should also notice that there are eight vertices. When you change the orientation of a cube, you will change the appearance. So we take this cube and we move it around, change what it looks like.

Now let’s look at a cone. A cone has only one vertex. Students should also notice that a cone has no segments. The bottom of a cone, if you will, is a circle. Changing the orientation will change the appearance of a cone. So we start off with a cone in this orientation. Rotate it. Looks different again. When it ends up in this position, it does again look different than what it did at the beginning.

Let’s look at a cylinder. A real-life example would be a can. Students will notice that the bottom and the top of a cylinder are circles. There are no vertices. Changing the orientation of a cylinder will change the appearance. Now let’s look at a sphere. A real- life example would be a ball such as a basketball or a baseball. There are no vertices. There are no faces. Although, in pure mathematics, you do have lateral surface area, and some mathematicians do consider that a face. But of course at this level, we’re not going to worry about that. There are no edges in a sphere. What’s different here is that changing the orientation of a sphere will not change the appearance. If I take this sphere and I rotate it around, spin it, and so forth, it still looks the same. In summary, your three-dimensional shapes included the cube, cone, cylinder, and sphere and these are the ones that students should be accountable for at this level.

Let’s look at our eight standards of mathematical practice. If we do the activities that were modeled in this video, students would reason abstractly and quantitatively. They would construct viable arguments and critique the reasoning of others. Looking at the last four of those standards, in this standard they should attend to precision, and they would look for and express regularity in repeating reasoning.