Implementing the 6th Grade Mathematics GPS via Centimeter

Cubes

Presented by Judy O’Neal (joneal@ngcsu.edu)

Topics Addressed

• Views of solid figures (polyhedra)• Volumes of right rectangular prisms

(polyhedra)• Surface area of right rectangular prisms

(polyhedra)• Proportional relationships (scale factors)• Connections among mathematical topics

A View from the Top 1• Use the numbers on the mat and your

centimeter cubes to construct the building whose top (footprint) view is shown below.

right

front

left

21

1

1

3

2

back

A View from the Top 2• Which of the architectural views below

represent the front, back, left, and right of your building?

I.H.G.

F.E.D.

C.B.A.

right

front

left

21

1

1

3

2

back

Architectural Plans 3A

Front Front Left Right Back

QUESTIONS FOR STUDENTS:• What is the relationship between the front and back

views? • What is the relationship between the left and right views?

1 3 1

3

2 11

A View from the Top 3• Use your cubes to construct the building

represented by the following mats.A. B. C.

FRONT FRONT

FRONT

• On centimeter grid paper (downloadable), draw the architectural plans for each building and label the front, back, left, and right view for each.

1 3 1

3

2 11

4 2 1

2

4

3

3 2

1

34 2

1

Architectural Plans 3B

QUESTIONS FOR STUDENTS:• What is the relationship between the front and back

views? • What is the relationship between the left and right views?

4 2 1

2

4

3

Front Front Left Right Back

Architectural Plans 3C

QUESTIONS FOR STUDENTS:• What is the relationship between the front and back

views? • What is the relationship between the left and right views?

3 2

1

34 2

1

Front Front Left Right Back

A View from the Top 4

• Use the plans below to construct a building. Record the height of each section of the building on the mat.

Top Front Right MAT

A View from the Top 5

• Use the plans below and centimeter cubes to construct a building. Record the height of each section of the building on the mat.

Top Front Right MAT

Keep this model intact for use later in this webcast.

Isometric Views from a Footprint/Mat

• Which isometric drawing shows the view from the left front corner of the building represented by the footprint below?

* Excerpt from student worksheet (downloadable) “Isometric Explorations”, pp. 113-114 (Navigating through Geometry in Grades 6-8, NCTM, 2002)

Sketching an Isometric Drawing

• Isometric dot paper has dots placed so that isosceles triangles can be drawn easily.

• Sketching a cube is much like drawing a pattern block yellow hexagon with three blue rhombi on top.

Isometric Drawings Practice

• Using isometric dot paper (downloadable), sketch each of these structures.

Volume of Building 5

Top Front Right MAT

• How many cubes are there in each layer of the solid (saved from earlier in the webcast)?

• What is the total number of cubes in this building (volume)?

Volume of a Rectangular Solid• Use centimeter cubes to construct solids made

up of the following stack of cubes.

• How many cubes are there in each layer of the solid?

____ ____ ____ ____ ____• What is the volume of this solid (total number of

cubes)? ____ cm3

4 1

1

2

52

1 2

3

What is a polyhedron?

• A polyhedron is a three-dimensional solid whose faces are polygons joined at their edges (no curved edges or surfaces). – The word polyhedron

is derived from the Greek poly (many) and the Indo-European hedron (seat).

Regular Polyhedron

• A polyhedron is said to be regular if its faces are made up of regular polygons (sides of equal length placed symmetrically around a common center).

Dodecahedron-12 Pentagonal Faces

Octahedron – 8 Triangular Faces

Cube – 6 Square Faces

Irregular Polyhedra

Faces are a combination of different polygons.

Non-Polyhedra

• Cylinder

• Cone

• Sphere

• Why aren’t each of these solids a polyhedron?

Polyhedra in our World

• Crystals are real-world examples of polyhedra.

• The salt you sprinkle on your food is a crystal in the shape of a cube.

What is a Prism?

• A prism is a polyhedron (three-dimensional solid) with two congruent, parallel bases that are polygons, and all remaining (lateral) faces are parallelograms.

What is a Right Prism?

• A right prism is a prism in which the top and bottom polygons lie on top of (parallel to) each other so that the vertical polygons connecting their sides are perpendicular to the top and bottom and are not only parallelograms, but rectangles.

• A prism that is not a right prism is known as an oblique prism.

What is a Right Rectangular Prism?

• A right rectangular prism is a right prism in which the upper and lower bases are rectangles.

• A rectangular prism has six rectangular faces.

• How many edges?

What is a Cube?

• A cube is a right rectangular prism with square upper and lower bases and square vertical faces.

• How many faces? edges?

Cubes in our World

• The world's largest cube is the Atomium, a structure built for the 1958 Brussels World's Fair. The Atomium is 334.6 feet high, and the nine spheres at the vertices and center have diameters of 59.0 feet. The distance between the spheres along the edge of the cube is 95.1 feet, and the diameter of the tubes connecting the spheres is 9.8 feet.

Caroline the Cube

• On Caroline’s first birthday, she looks like one centimeter cube.

• Help Caroline finish building herself on her 2nd birthday.(Hint: Build a cube whose length, width, and height are 2 cm.)

• How many blocks define Caroline on her 2nd birthday? (What is her volume?)

Caroline’s Surface Area

The area of the exposed surfaces of a solid object is its surface area.

• What is Caroline’s surface area on her 1st birthday?

• On her 2nd birthday?• On her 3rd birthday?• On her 5th birthday?• On her nth birthday?

Volume of a CubeConsider the 3-cube and the 5-cube on the left.

• How long is the front bottom edge?right bottom edge?

• What is the area of the base (number of cubes in the bottom layer)?

Recall the area of a square is (side length)2

• How many layers are there (height)?• How many total cubes (volume)?• Volume is area of the base * height.

• Since all dimensions of a cube are equal, the volume of a cube is (side length)3 or V=s3.

Surface Area of a Cube• Suppose the length, width, and height

of the given cube is 2 cm. What is the surface area?

• What happens to the surface area of a cube when all of the dimensions are tripled?

• What happens to the edge length of a cube when the surface area is doubled?

• What can be said about the number of edges in each of these cubes?

Building Right Rectangular Prisms• Using 12 centimeter cubes, build all possible

rectangular prisms.

• Which model has the largest surface area for the given volume of 12 cubic centimeters (cm3)?

Excerpt from Student Activity Sheets (downloadable) “To the Surface and Beyond”, pp. 112-113, Navigating through Measurement in Grades 6-8, NCTM, 2002.

Volume of Right Rectangular Prism

• Using centimeter cubes, build a right rectangular prism with front edge length of 3 cubes, right edge width of 2 cubes, and height of 2 cubes.

• How many cubes are contained in the prism?• What is the area of the base (front edge length * right

edge width)? • What is the height?• What is

(front edge length) * (right edge width)*(height)?• How does this compare to the total number of cubes in

the prism?

• In general, the volume of a right rectangular prism is V = length * width * height or V = lwh.

Scale Factor, Volume, and Surface Area of a Rectangular Prism

• Excerpt from Student Activity Sheet (downloadable) p. 119-120 from Navigating through Measurements in Grades 6-8, NCTM, 2002.

• Two rectangular prisms have similar shapes. The front and back faces are the same shape, the top and bottom faces are the same shape, and the two remaining faces are the same shape.

• What is the scale factor (ratio)of the edges of the prisms?

• What is the scale factor of the surface areas of the prisms?

• How does the scale factor of the two volumes compare with the scale factor of the edges?

GPS Addressed

M6M2 Select and use units of appropriate size and type to measure volume

M6M3 Determine the formula for finding the volume of fundamental solid figures Compute the volumes of fundamental solid figures, using appropriate units

of measure Estimate the volumes of simple geometric solids

M6M4 Find the surface area of right rectangular prisms using manipulatives Compute the surface area of right rectangular prisms using formulae

M6A2 Describe proportional relationships mathematically using y = kx, where k is

the constant of proportionalityM6G2

Interpret and sketch front, back, top, bottom, and side views of solid figures M6P4

Understand how mathematical ideas interconnect and build on one another to produce a coherent whole

Websites for Additional Exploration

• Learning about Length, Perimeter, Area, and Volume of Similar Objects Using Interactive Figures: Side Length and Area of Similar Figures

http://standards.nctm.org/document/eexamples/chap6/6.3/index.htm

• InterMathhttp://www.intermath-uga.gatech.edu/homepg.htm

• Linking Length, Perimeter, Area, and Volume

http://illuminations.nctm.org/LessonDetail.aspx?ID=L261

• Eric Weisstein’s World of Mathematics http://mathworld.wolfram.com/topics/Polyhedra.html