VOLUME 2013 Judo Math Inc.
7 th grade Geometry Discipline: Black Belt Training Order of Mastery: Surface Area/Volume 1. 2D vs. 3D: slicing 3D shapes to get 2D shapes (7G3) 2. Surface Area 1 (7G6) 3. Volume or prisms (7G6) 4. Volume of pyramids (7G6) 5. Volume and Surface Area applications (7G6) Welcome to the Black Belt Surface Area & Volume Last discipline we spent a good amount of time contemplating the 2 dimensional world. Now you have advanced to the 3 dimensional world. You might be thinking Hey, I have always been in the 3 dimensional world and, well, you would be correct because the whole world we live in is in 3 dimensions. So that can of soda you re drinking, or that sand castle bucket that you use at the beach, or even the pyramids in ancient Egypt all have a volume and a surface area and it can be handy to know how to calculate that. Not only will you get to learn how to find the volume of some pretty awesome shapes, but you will even get to do a little bit of slicing and dicing in this discipline! When you move to high school and take a class like calculus, you will move onto finding the volume of super crazy shapes like: We have to start with some simpler shapes first so buckle your seatbelt because it s going to be an awesome ride! Good Luck Grasshopper. Standards Included: 7.G.3 Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. 7.G.6 Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. 2013 Judo Math Inc.
1. 2D vs. 3D: slicing 3D shapes to get 2D shapes (7G3) Cube Rectangular Prism Triangular Prism Sphere Square based Pyramid Triangle Based pyramid Cylinder Cone Above is a table of the most common 3-D shapes. In this belt it will come in handy to be able to draw these. In the space below, practice drawing each of the shapes above. Use a ruler to make your drawings as neat as possible. If you often have trouble classifying which shape is which, put your drawings onto notecards with the name on front and drawing on the back and practice whenever you have spare time! Space for your sketches 1
Flowing Water Challenge: Below are drawings of 3 dimensional vessels. Imagine they are full of water that is flowing through a pipe at the bottom of the bessel. Sketch diagrams for each vessel to show how the shape of the surface of the waterchanges as the water flows out of each vessel. For each drawing, write what the formed shape is. 2
Now assume that there is one vessel attached to another vessel and water is flowing out of the top vessel and into the bottom vessel. Draw the shapes of the surface of the water as it flows OUT of the top vessel and the shape of the bottom vessel as the water flows into it! 1 (Example) Vessels Top shape surface Bottom shape water surface 2 3 3
Vessels Top shape surface Bottom shape water surface 1 2 3 4
Draw a plane to show where the cube on the left would need to be sliced could be sliced to make the following cross sections: There is one other cross section of a cube not mentioned here. Can you figure out what it is?! 5
1. Below is a diagram showing a side view and a top view of a vase. Both views are drawn accurately and are half their real size. Your job is to decide which is bigger: the height or the circumference of the real vase! Have fun! 6
2. Surface Area (7G6) Surface area is the total area of the surface of a three-dimensional object. Last year you learned about nets. Write your own definition for a net here: Nets are: 1. Use the following net to find the surface area of this rectangular prism: 2. Draw a net to find the surface area of this figure: 7
3. James covered the box to the right with sticky backed decorating paper. The paper costs 3 cents per square inch. How much money will James need to spend on paper? 4. The diagram shows a prism whose base is a trapezoid. The surface area of the prism is 72cm. Find the value of x. 5. Alisa s camping tent is not waterproof, so she wants to put a tarp on it, in case it rains. The tarp will cover two of the faces. Which size tarp should they buy, 50ft 2, 100ft 2, or 150ft 2. 8
6. Calculate the surface area of the following square pyramid. The net drawn below might help! 7. Some teachers like to tell their students that the formula for the surface area of a pyramid with a square bottom is SA=s 2 +2sl. Explain where this formula comes from using the image here and WHY you don t actually need to memorize this formula! 9
8. Draw and label a triangular prism, a square based pyramid, and a cube that have a surface area of 48 cm 2. 10
3. Volume prisms (review) (7G6) Volume is how much three-dimensional space a shape occupies or contains. Often teachers will make their students memorize lots of different formulas for the volume of various prisms, but you don t actually have to! In fact, there is only one thing you really need to be able to see in prisms: Volume of Prism = Area of base x height The only tricky part is identifying the base of the prism. Check out these the base is the shaded part and their height is identified. H As you can see, the base isn t ALWAYS on the bottom. No matter what, the two faces that are parallel and congruent are the bases of the prism. Also if you take any cross section of a prism parallel to those bases by making a cut through it parallel to the bases, the cross section will look just like the bases. In each of the figures below, shade the base and label the height with an H: 11
Shade each base, label the height, then find the volume. 12
9. Jennie purchased a box of crackers from the deli. The box is in the shape of a triangular prism (see diagram below. If the volume of the vox is 3,240 cubic centimeters, what is the height of the triangular face of the box. How much packaging material was used to construct the cracker box? Explain how you got your answer? 10. In the office, there is a filing cabinet that is 1 foot wide, 3 feet deep and 3 feet tall. You have hundreds of file folders to put into this cabinet. Assuming that each folder is 12 inches wide, 1 inch thick, and 9 inches tall, how many folders can you fit into this cabinet? 13
11. As you know very well by now, the volume for a prism is Volume of Prism = Area of base x height Using pictures and words, explain why this might be the formula that works? Words like cross section, area, height, base, and lengths might come in handy! 14
4. Volume pyramids (7G6) A pyramid is a polyhedron with a single base and lateral faces that are all triangular. All lateral edges of a pyramid meet at a single point, or vertex. In order to find the volume of a pyramid, we are going to attempt to see how a pyramid and prism are related to eachother. In the pictures above, you can see that the differences between a pyramid and a prism that have the same base. How many pyramids do you think would fit inside a prism with the same base? 1. Now get a piece of cardstock. Your goal for today is to create a net for a prism and a pyramid that have the same exact base shape and the same height. The two pictures below are examples but are not exact representations of what you could create. 2. Now fold up your nets. Get your hands on some sand, dirt, or rice and fill up the pyramid. Dump the contents into the prism. How many pyramids does it take to fill up the prism? Record your answer and any notes here: 15
If you properly constructed and built your nets in the last activity, hopefully you found out that it takes 3 pyramids to fill up a prism if they have the same base shape and same height. So the volume of a pyramid is exactly 1 of the volume of a prism. This leads us to the 3 Pyramid Volume Theorem The volume V of any pyramid with height h and a base with area B is equal to one-third the product of the height and the area of the base. Prism Formula: V = 1 3 Bh Where B is the area of the base. 16
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8. You are an archeologist studying two ancient pyramids. What factors would affect how long it took to build each pyramid? Given similar conditions, which pyramid took longer to build? Explain your reasoning. 19. A pyramid has a volume of 40 cubic feet and a height of 6 feet. Find one possible set of dimensions of the rectangular base 18
5. Volume and Surface Area mini project (7G6) You have been asked to design a sports bag for the middle school basketball team. The length of the bag will be 20 inches The bag will have circular ends of diameter 11 inches. The main body of the bag will be made from 3 pieces of fabric: a piece for the curved body and the two end pieces. When cutting out pieces of fabric for the bag, each piece will need an extra ½ inch all the way around so that they can be stitched together. 1. Make a sketch of the pieces you will need to cut out for the body of the bag. On your sketch show all the measurements you will need. 2. Suppose you are going to make one of these bags from a roll of fabric 1 yard wide. What is the shortest length of fabric you can cut from the roll? 3. Finally, design another shape for the bag that would have a larger volume but would uuse a similar amount of fabric. 19