If you imagine the plane just like one of those huge blades Let me shade in the stuff, if you kind of view this, So it might look like this, and once again That I'm cutting with, and my cube is going to beĪ rectangle. That I'm actually cutting with, so the intersection of the plane That you are cutting with so, the intersection, let's see: This could be the plane Right over there, then the intersection of the plane I think you'd see where this is going, this side like that, and then you cut the bottom, You can actually cut like this: So, if you cut this side like this, and then cut that side like that, and then you cut this side like that That I'm talking about? Well for a rectangle On your own: How can you get these shapes How can you get that? And at any point, I encourage you: Pause the video and try to think about it So a square isĪ pretty straightforward thing to get, if you're doing a planar slice of a cube. it could look something like this.Īnd I can even color in the part of the plane that you couldĪctually see it the cube were opaque, if you couldn't see through it.īut if you could see through it you would see this dotted line,Īnd the plane would look like that. Let me see if I can do a decent,Īs you see you'd say a part of the plane, that is cutting this cube. So you can imagine a plane that did this, and if I wanna draw the broader plane, Of the glass cube, where you will see it, right over there, dotted line, and then it cuts this, right over here. We are to cut, just like this, the square is maybe the most obvious one, so it cuts the top right over there, it cuts the side, right over here, it cuts the side, I guess on the back To get the intersection of this cube, and that plane to be a square? We'll imagine that that plane So what am I talking about? Let's say we wanna deconstruct a square: How can we slice a cube with a plane Tennyson’s In Memoriam A.H.Is explore the types of 2D shapes we can construct,īy taking planar slices of cubes.It worked great! Now each cross-section stands nicely spaced and vertical. Then I carefully slid each cross-section into its appropriate slit. Update: I used spray adhesive to glue the base area to some foamboard and cut slits in it with an Xacto knife. You can download the templates provided by Nina Otterson here. That way, they will stand up straighter and stay evenly spaced. When I do this activity next year, I think I’ll glue the base area to foamboard, and have students insert the cross-sections into slits cut into the foamboard. Building a model using actual cross-sections made all the difference! I’ve never had students grasp the idea behind this type of volume as quickly and as easily as this group did. Once they understood that the thickness of the paper was dx, it was very easy to set up the integrals to calculate the volumes of their models. Here they are, taping the cross-sections onto the base area: Here are my students in action, cutting out the cross-sections: Students use the templates to cut out a cross-section that fits down the middle of the base area, and six others on each side. I had my students cut the square templates diagonally for isosceles right triangles, and horizontally for rectangles. In her session, Nina Otterson provided templates that fit the given base area for different shapes: semicircles, squares, and equilateral triangles. Here’s what the base area looks like, courtesy of ’s online function grapher: She has her students cut cross-sections of different shapes and apply them to a base area enclosed by two parabolas, y = x^2 – 3 and y = 3 – x^2. Nina Chung Otterson was the presenter, and she teaches at The Hotchkiss School in Connecticut. Last year, I went to the regional NCTM conference here in Nashville, TN, and one of the sessions I attended addressed this exact issue. It’s hard, because they have difficulty visualizing it. One of the hardest type of problem for calculus students to understand is calculating the volume of solids of known cross-sections.
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The earliest recorded discussion of the basic structure of matter comes from ancient Greek philosophers. Thus, a fixed mass of hydrogen, 2 g, combines with different masses of oxygen, 32 or 16 g, in a small whole-number ratio of 2 to 1, to form two different compounds - hydrogen peroxide, and water. According to this law, when two elements combine to form more than one compound, different masses of one element combine with the fixed mass of the other element in a small whole-number ratio. Thus, a sample of pure water will always have a 1 to 8 mass ratio of hydrogen to oxygen irrespective of the total mass of the water.ĭalton then proposed the law of multiple proportions based on his own theory. The law of definite proportions states that independent of the source, in a given chemical compound, the mass ratio of constituent elements always remains fixed. So, 18 g of water breaks down into 2 g of hydrogen and 16 g of oxygen. 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