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Implementing the 6 th Grade GPS via Folding Geometric Shapes

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Implementing the 6th Grade GPS via Folding Geometric Shapes

Presented by Judy O’Neal

- 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?

Regular pyramid

Tetrahedron - All faces are triangles

Find the third net for a regular pyramid (tetrahedron)

Hint – Pattern block trapezoid and triangle

- 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)

- Construct a scale model from net to geometric solid (downloadable*)
- 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)

- Materials per student:

- 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
= 2rh

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

- Surface Area = 2*Base area + Rectangle area
- Open cylinder
- Surface Area = Area of rectangle
- Surface Area of Open Cylinder = 2rh 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)

- Materials per student:

- 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?

- Materials needed – Geometric solids, paper (plain or centimeter grid), tape or glue

- 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

- Materials:
- 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

- Equivalent Nets for Rectangular Prisms http://www.wrightgroup.com/download/cp/g7_geometry.pdf
- Nets http://www.eduplace.com/state/nc/hmm/tools/6.html
- ESOL On-Line Foil Fun
http://www.tki.org.nz/r/esol/esolonline/primary_mainstream/classroom/units/foil_fun/home_e.php