Mathematics

3.G.A.2 Transcript

This is Common Core State Standards support video in mathematics. The standard is 3.G.A.2; this standard states partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into four parts with equal area, and describe the area of each part as one-fourth of the area the shape. There is a predecessor to this standard back in the second grade—Standard 2.G.A.3—that talks about partitioning circles and rectangles into two, three, or four equal shares. It also pushes on the idea of recognizing that equal shares of identical wholes need not have the same shape. What's interesting is that although these standards are in the Geometry Domain, they both lay the early foundation for fractions. In that Number and Operations for Fractions Domain, there's a footnote that states that grade three expectations are limited to fractions with denominators two, three, four, six, and eight. So we'll take that into consideration and apply just those numbers of equal parts to this standard. So we will just deal with partitioning these shapes into two, three, four, six, and eight equal parts. This standard talks about partitioning shapes into equal parts with equal areas, but what shapes should we use? Well back in first grade, we have Standard 1.G.A.2 that talks about composing two-dimensional shapes, and they specifically mention rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles.

And in second grade, we have Standard 2.G.A.1 that talks about recognizing and drawing shapes having specified attributes, and they specifically mentioned triangles, quadrilaterals, pentagons, hexagons, and cubes. The Standard 3.G.A.1, which is a predecessor to this standard, talks about understanding that shapes in different categories may share attributes and specifically mentioned are rhombuses, rectangles, and squares. So we look at the big picture and look at all three of these together to make our decision as far as well what shapes should be used per standard 3.G.A.2. Now keep in mind that we're talking about parts with equal areas, so that should automatically eliminate cubes because that's three dimensional not two-dimensional, then trapezoids and pentagons—those will be very difficult to divide up into equal parts. So we should go ahead and eliminate those, and even though they're the simplest of the polygons, triangles really won't work well either. Let's look at why. If you take say a scalene triangle, it'd be very difficult for students to even split this up into two equal parts.

Now again, how are students going to know at this level that these two are equal areas? Let's take a right triangle. The same situation; let's say I wanted to split this up into three equal areas. Again it would take a lot more knowledge and skills to be able to prove or determine that all three of these are equal parts. What about an equilateral triangle? It'll work to some extent; we could do something like this, and these would be equal areas, but again that would be it. We'd be limited to being able to partition this just into two equal parts and not any further. Based on our investigation, it appears that rectangles, squares, circles, and hexagons would be the best shapes to use at this level for this standard.

At the same grade level, there's Standard 3.MD.B4, and it talks about measuring lengths using rulers marked with halves and fourths of an inch, and it's very unlikely that you'll find commercially-made rulers that are marked off in just halves and fourths, so you might have to create your own but, the way that this standard ties into this one is that you can have students actually create, for example, rectangles and measure the lengths that you need. So for example here, they could create a rectangle that's say 4 inches long; then it wouldn't be that much trouble to mark off and figure out exactly how to divide this up into four equal parts. Now keep in mind that we're talking about areas; we're talking about all of the inside here of these rectangles. So again it's not just the rectangle, it’s the area in this rectangle that we need to be concerned about. And again we measured, and we know that each of these smaller rectangular areas would be one-fourth the area the original. Now what we can do is use the rulers and mark it off to where we would cut each one of those smaller rectangles in half, and so now we have eight equal parts.

So we would know that each of these smaller rectangular areas is one-eighth the area the original. Students could create a rectangle that is say 3 inches long and again be very easy to measure off and split this up into three equal areas. And again a reminder, it's not just the rectangle, it's the rectangular area. So each of these small rectangular areas again is one-third the area of the original. And just like we did before, we can measure off and cut each one of those smaller ones in half, so now we would know that each of those small rectangular areas is one-sixth the area of the original. Now there's other ways to do this. We don't have to do vertical lines; we can also do horizontals so this should be a slightly different approach to cutting this up to where each of the smaller rectangle areas is one-fourth the area the original. Our challenge here would be how to partition this to be shapes that are all one-eighth of the original area. You should have some students that would come up with this solution pretty quickly. Then, hey just, find the midpoint and just draw a vertical bar, and bingo, those four equal parts have become eight equal parts.

At this point, we've been creating the rectangles, but if the focus of the standard is moreon the act of partitioning, students can do this by folding rectangular strips to attain the desired fractional points for the regional area. So, for example, you could take a piece of paper, a rectangular strip, and fold it in half to get your two equal parts. Then of course you could fold it in half again to get four equal parts, and then in half again to get eight equal parts. They can also do a trifold to get three equal parts, and then of course fold that in half either horizontally or vertical to get six equal areas. Students can also fold circles to attain the desired fractional parts for the original area—halves, fourths, and eighths would be the best to work with here. So they could take this circle, fold it in half, and then when they unfold it, they'll have a crease where you have two equal parts. Then you can take the folded semicircle, and fold it again, and when you unfold it it'll have creases where you'll have four equal areas. And then you can take that and fold it in half again, and you'll have a circle where you have creases that will have divided this up into eight equal areas.

Now so far, we've been measuring and doing vertical and horizontal lines to do thepartitioning, but in the case of rectangles, you can also use a diagonal. So if I draw a diagonal like so, I've created two smaller triangular areas that are each one half the area of the original rectangle. Now I can take this a step further and draw a second diagonal, and I have divided this up to where each of the smaller triangular areas is one-fourth the area of the original, but this is third grade, and here's something that might be a problem. Even though it can be shown algebraically that each of these four triangles is equal in area, the fact that the two pairs have different shapes would be confusing to third-grade students. They might have a hard time seeing that this triangle here and this triangle here both have the same area.

Partitioning rectangles with diagonals beyond cutting it up into two equal areas should be done with squares so that each of the resulting triangles is the same shape and size. Third-grade students will then be able to see that they are equal in area. Then they could take it a step further and do something like this. You created eight smaller right triangles that are all going to be equal areas. It makes sense that hexagons would be the best shape to partition in illustrating parts that are one-sixth the area of the original. Of course, all students need to do is to draw diagonals to connect the nonconsecutive vertices. And so here, we have our six equal parts—six equal areas. Students can then use this partitioning to design a more accurate partition for parts that are all one-third the area of the original. All they have to do is knock out part of those diagonals, and they'll create something that looks like this. So now you have your hexagon pretty actually partitioned to three parts that are all one-third the area the original. Although when you look at this your mind plays tricks; your eyesight kind of makes this look like it's a cube.

Let's go back and revisit what we've done. We did this particular example where each of the smaller rectangular areas was one-third the area of the original, but how would a third-grader take this up a notch and prove that each of those three smaller shapes have equal areas? Well logic dictates that they have the same area if the figures are exactly the same shape and the same size, so all the students would need to do to prove this informally at this level would be to simply take one of these smaller rectangles, and slide it over and you see that it fits exactly over another one; so I know, hey, these have to have the exact same area because they're pretty much carbon copies of each other. Now let's say we have a rectangle and we've used a diagonal to cut it up into two equal areas, so now the question is we're saying that each of these two smaller triangle areas is one-half the area the original. But as before, how would a third-grader prove that these two triangular shapes do in fact have the same area?

Well let's go ahead and change the color of one of them so we can tell one from the other. Then we need to kinda slide one on the rectangles over, and so now whatwe need to do is we're going to have to rotate this around, and then slide it over, and so there we have it. One fits exactly over the other. So again it's pretty obvious that these two have to have the same area, because they're exactly the same shape and exactly the same size. Now although we're using an area context, we're really laying a foundation for the concept of congruency. What's interesting is that if you do a search of the Common Core Standards in math, and you're looking for the word congruent, it doesn't appear until eighth grade.

So the Common Core Standards don't even address congruency until that point, eighth grade, and that seems a little bit late. And hopefully you'll see from these types of activities that you can lay the foundation for congruency at a much earlier level. Again, even though this standard deals with area, we can use this context to lay the foundation for the concept of congruency. The main adjustment that we need to make is to switch our focus to the polygons themselves instead of area, so in this case, we transition from the rectangular areas to the rectangles themselves. These two rectangles are congruent if and only if they're the exact same shape and size. Now at this grade level, it would be sufficient to prove by just sliding one rectangle over the other to see if it fits exactly. So we can do that. So now we see that, okay, one rectangle fits exactly over the other, so they are in fact congruent. If we concentrate on that simple idea of congruency as figures being the exact shape and the exact same size, you can see how we can teach this concept by expanding on what we do for the standard. And using translations and rotations for the most part would be sufficient to be able to do this proof at this level. As you can see the activities and the context, so we use for this standard really lend themselves and connect to the idea of congruency. So we can't and we shouldn't wait until the eighth grade to teach this fundamental idea. Again, even though we're talking about area here, we're still laying the foundation for congruency, because all you have to do is instead of thinking of the areas here, just think in this case of just the rectangles themselves. Then again, these two triangles are congruent if they're exactly the same shape and size, and again that's very easy to prove by just being able to slide one and put it exactly over the other. And that's all pretty much what the idea congruency would involve at this level. So again not impossible to do—you can lay this foundation and don't wait until eighth grade.

Here's another interesting observation, it's important to note that the Common Core State Standards in math do not specifically address fractions as part of a set rather than as part of the whole. It doesn't appear. So the question is, well, when do we do this? When do we address fractions as part of a set? We can actually do this as part of this standard part of these activities. So let's say, okay, go back to this example where we had our rectangle, and we partitioned this into four equal parts so each in the smaller rectangular areas is one-fourth the area of the original, and the original we can switch over and call the whole, which in this case would be our original rectangular area.

Again, the activities that we're doing for this standard can easily be adapted to establish a foundation for again interpreting fraction as part of a set. So we took, let's say, these four rectangles; each one of these rectangles would be one-fourth of this set or group of four rectangles. So pretty easy transition. And in fact, what you can do is transition to where instead of just these polygons, these rectangular figures, whatever, and use other objects, other manipulatives. For example, maybe you have some miniature cars. So now if I transition this over instead of having four rectangles, I have four cars. Well each of these cars is one-fourth of this set or group of four cars. Or let's say you have some little model trucks, again each of these trucks is one-third of this set of three trucks.

So again, the transitioning over to showing fractions as part of a set really would fit perfectly with the activities involved for this standard. If you look at the Standards for Mathematical Practice, here's the first four, and I think we could reasonably conclude that we would be addressing the second one. Students would be reasoning abstractly and quantitatively. They would be constructing viable arguments and critiquing the reasoning of others, and they would be modeling the mathematics. If you look at the last four of the Standards for Mathematical Practice, we could pretty much assert that students addressing this standard would be using appropriate tools strategically, and they would be attending to precision.