Presentation on theme: "Implementing the 6th Grade GPS via Folding Geometric Shapes"— Presentation transcript:
1Implementing the 6th Grade GPS via Folding Geometric Shapes Presented by Judy O’Neal
2Topics Addressed Nets Surface Area of Cylinders Prisms Pyramids ConesSurface Area of Cylinders
3NetsA net is a two-dimensional figure that, when folded, forms a three-dimensional figure.
4Identical NetsTwo nets are identical if they are congruent; that is, they are the same if you can rotate or flip one of them and it looks just like the other.
5Nets for a CubeA net for a cube can be drawn by tracing faces of a cube as it is rolled forward, backward, and sideways.Using centimeter grid paper (downloadable), draw all possible nets for a cube.
6Nets for a CubeThere are a total of 11 distinct (different) nets for a cube.
7Nets for a CubeCut out a copy of the net below from centimeter grid paper (downloadable).Write the letters M,A,T,H,I, and E on the net so that when you fold it, you can read the words MATH around its side in one direction and TIME around its side in the other direction.You will be able to orient all of the letters except one to be right-side up.
8Nets for a Rectangular Prism One net for the yellow rectangular prism is illustrated below. Roll a rectangular prism on a piece of paper or on centimeter grid paper and trace to create another net.
10Nets for a Regular Pyramid Tetrahedron - All faces are trianglesFind the third net for a regular pyramid (tetrahedron)Hint – Pattern block trapezoid and triangle
11Nets for a Square Pyramid Pentahedron - Base is a square and faces are triangles
12Nets for a Square Pyramid Which of the following are nets of a square pyramid?Are these nets distinct?Are there other distinct nets? (No)
13Great Pyramid at GizaConstruct a scale model from net to geometric solid (downloadable*)Materials per student:8.5” by 11” sheet of paperScissorsRuler (inches)Black, red, and blue markersTape*http://www.mathforum.com/alejandre/mathfair/pyramid2.html (Spanish version available)
14Great Pyramid at Giza Directions Fold one corner of the paper to the opposite side. Cut off the extra rectangle. The result is an 8½" square sheet of paper.Fold the paper in half and in half again. Open the paper and mark the midpoint of each side. Draw a line connecting opposite midpoints.4 ¼”8 ½”
15More Great Pyramid Directions Measure 3¼ inches out from the center on each of the four lines. Draw a red line from each corner of the paper to each point you just marked. Cut along these red lines to see what to throw away.Draw the blue lines as shown
16Great Pyramid at Giza Scale Model Print your name along the based of one of the sides of the pyramid.Fold along the lines and tape edges together.
17Nets for a Cylinder Closed cylinder (top and bottom included) Rectangle and two congruent circlesWhat relationship must exist between the rectangle and the circles?Are other nets possible?Open cylinder - Any rectangular piece of paper
18Surface Area of a Cylinder Closed cylinderSurface Area = 2*Base area + Rectangle area2*Area of base (circle) = 2*r2Area of rectangle = Circle circumference * height= 2rhSurface Area of Closed Cylinder = (2r2 + 2rh) sq unitsOpen cylinderSurface Area = Area of rectangleSurface Area of Open Cylinder = 2rh sq units
19Building a CylinderConstruct a net for a cylinder and form a geometric solidMaterials per student:3 pieces of 8½” by 11” paperScissorsTapeCompassRuler (inches)
20Building a Cylinder Directions Roll one piece of paper to form an open cylinder.Questions for students:What size circles are needed for the top and bottom?How long should the diameter or radius of each circle be?Using your compass and ruler, draw two circles to fit the top and bottom of the open cylinder. Cut out both circles.Tape the circles to the opened cylinder.
21Can Label Investigation An intern at a manufacturing plant is given the job of estimating how much could be saved by only covering part of a can with a label. The can is 5.5 inches tall with diameter of 3 inches. The management suggests that 1 inch at the top and bottom be left uncovered. If the label costs 4 cents/in2, how much would be saved?
22Nets for a Cone Closed cone (top or bottom included) Circle and a sector of a larger but related circleCircumference of the (smaller) circle must equal the length of the arc of the given sector (from the larger circle).Open cone (party hat or ice cream sugar cone)Circular sector
23Cone InvestigationCut 3 identical sectors from 3 congruent circles or use 3 identical party hats with 2 of them slit open.Cut a slice from the center of one of the opened cones to its base.Cut a different size slice from another cone.Roll the 3 different sectors into a cone and secure with tape.Questions for Students:If you take a larger sector of the same circle, how is the cone changed? What if you take a smaller sector?What can be said about the radii of each of the 3 circles?
24Cone Investigation continued A larger sector would increase the area of the base and decrease the height of the cone.A smaller sector would decrease the area of the base and increase the height.All the radii of the same circle are the same length.
25Making Your Own Cone Investigation When making a cone from an 8.5” by 11” piece of paper, what is the maximum height? Explain your thinking and illustrate with a drawing.
26Creating Nets from Shapes In small groups students create nets for triangular (regular) pyramids (downloadable isometric dot paper), square pyramids, rectangular prisms, cylinders, cones, and triangular prisms.Materials needed – Geometric solids, paper (plain or centimeter grid), tape or glueQuestions for students:How many vertices does your net need?How many edges does your net need?How many faces does your net need?Is more than one net possible?
27Alike or Different?Explain how cones and cylinders are alike and different.In what ways are right prisms and regular pyramids alike? different?
28Nets for Similar Cubes Using Centimeter Cubes Individually or in pairs, students build three similar cubes and create netsMaterials:Centimeter cubesCentimeter grid paperQuestions for StudentsWhat is the surface area of each cube?How does the scale factor affect the surface area?
29GPS AddressedM6M4Find the surface area of cylinders using manipulatives and constructing netsCompute the surface area of cylinders using formulaeSolve application problems involving surface area of cylindersM6A2Use manipulatives or draw pictures to solve problems involving proportional relationshipsM6G2Compare and contrast right prisms and pyramidsCompare and contrast cylinders and conesConstruct nets for prisms, cylinders, pyramids, and conesM6P3Organize and consolidate their mathematical thinking through communicationUse the language of mathematics to express mathematical ideas precisely
30Websites for Additional Exploration Equivalent Nets for Rectangular PrismsNetsESOL On-Line Foil Fun