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Implementing the 6 th Grade GPS via Folding Geometric Shapes - PowerPoint PPT Presentation

Implementing the 6 th Grade GPS via Folding Geometric Shapes. Presented by Judy O’Neal ([email protected]). Topics Addressed. Nets Prisms Pyramids Cylinders Cones Surface Area of Cylinders. Nets. A net is a two-dimensional figure that, when folded, forms a three-dimensional figure.

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Implementing the 6th Grade GPS via Folding Geometric Shapes

Presented by Judy O’Neal

([email protected])

• Nets

• Prisms

• Pyramids

• Cylinders

• Cones

• Surface Area of Cylinders

• A net is a two-dimensional figure that, when folded, forms a three-dimensional figure.

• Two 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.

• A 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.

• There are a total of 11 distinct (different) nets for a cube.

• Cut 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.

• 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.

• Are there others?

Tetrahedron - All faces are triangles

Find the third net for a regular pyramid (tetrahedron)

Hint – Pattern block trapezoid and triangle

Nets for a Regular Pyramid

• Square pyramid

• Pentahedron - Base is a square and faces are triangles

• Which of the following are nets of a square pyramid?

• Are these nets distinct?

• Are there other distinct nets? (No)

• Materials per student:

• 8.5” by 11” sheet of paper

• Scissors

• Ruler (inches)

• Black, red, and blue markers

• Tape

*http://www.mathforum.com/alejandre/mathfair/pyramid2.html(Spanish version available)

• 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”

• 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

• Print your name along the based of one of the sides of the pyramid.

• Fold along the lines and tape edges together.

• Closed cylinder (top and bottom included)

• Rectangle and two congruent circles

• What relationship must exist between the rectangle and the circles?

• Are other nets possible?

• Open cylinder - Any rectangular piece of paper

• Closed cylinder

• Surface Area = 2*Base area + Rectangle area

• 2*Area of base (circle) = 2*r2

• Area of rectangle = Circle circumference * height

= 2rh

• Surface Area of Closed Cylinder = (2r2 + 2rh) sq units

• Open cylinder

• Surface Area = Area of rectangle

• Surface Area of Open Cylinder = 2rh sq units

• Construct a net for a cylinder and form a geometric solid

• Materials per student:

• 3 pieces of 8½” by 11” paper

• Scissors

• Tape

• Compass

• Ruler (inches)

• 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.

• 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?

• Closed cone (top or bottom included)

• Circle and a sector of a larger but related circle

• Circumference 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

• Cut 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?

• 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.

• 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.

• 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 glue

Questions 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?

• Explain how cones and cylinders are alike and different.

• In what ways are right prisms and regular pyramids alike? different?

• Individually or in pairs, students build three similar cubes and create nets

• Materials:

• Centimeter cubes

• Centimeter grid paper

Questions for Students

• What is the surface area of each cube?

• How does the scale factor affect the surface area?

• M6M4

• Find the surface area of cylinders using manipulatives and constructing nets

• Compute the surface area of cylinders using formulae

• Solve application problems involving surface area of cylinders

• M6A2

• Use manipulatives or draw pictures to solve problems involving proportional relationships

• M6G2

• Compare and contrast right prisms and pyramids

• Compare and contrast cylinders and cones

• Construct nets for prisms, cylinders, pyramids, and cones

• M6P3

• Organize and consolidate their mathematical thinking through communication

• Use the language of mathematics to express mathematical ideas precisely