Rockdale County Public Schools: MSP Courses

1
Rockdale County Public Schools MSP Courses

• Day 2 6th Grade

2
Unit 6 Symmetry

• Students will be able to
• determine and use lines of symmetry
• investigate and use rotational symmetry
• identify objects, both man-made and natural, in
  their own environment that have symmetrical
  properties

3
Part 1 Lines of Symmetry

• Unit 6 Symmetry

4
Lines of Symmetry

• A shape has line symmetry when it can be divided
  in half by one or more lines of symmetry. A line
  of symmetry divides a symmetrical object into two
  congruent sides. Figures that match exactly when
  folded in half have line symmetry.

Non-examples
Examples
How many lines of symmetry do each of the regular
polygons have?
You can use a Mira or folding to find the lines
of symmetry
5
Symmetry

• A yellow hexagon has 6 lines of symmetry since it
  can be folded into identical halves along the 6
  different colors shown below (left).
• A green triangle has 3 lines of symmetry since it
  can be folded into identical halves along the 3
  different colors shown above (right).

6
More Symmetry

• How many lines of symmetry are in a blue rhombus?
• Explain why a red trapezoid has only one line of
  symmetry.

7
Activities

• Alphabet Symmetry
• Figure This Challenge 5 Upside Down
• http//www.figurethis.org/challenges/toc05-08.htm
  
• Visual Webquest http//www.adrianbruce.com/Symmet
  ry/
• Links Learning Line Symmetry Lesson
• http//www.linkslearning.org/Kids/1_Math/2_Illustr
  ated_Lessons/4_Line_Symmetry/index.html
• Creating Line Symmetry
• Feet
• NCTM Illuminations Mirror Tool
• http//illuminations.nctm.org/ActivityDetail.aspx?
  ID24

8
Part 2 Rotational Symmetry

• Unit 6 Symmetry

9
Rotational Symmetry

• A shape has rotational symmetry when it can be
  rotated by an angle of 180º or less about its
  center and produce the same shape.

What is the angle of rotation for each shape
below?
10
Rotational Symmetry

• A yellow hexagon has rotational symmetry since it
  can be reproduced exactly by rotating it about an
  axis through its center.
• A hexagon has 60º, 120º, 180º, 240º, and 300º
  rotational symmetry.

11
Activities

• Learning Math
• http//www.learner.org/channel/courses/learningmat
  h/geometry/session7/part_b/index.html
• Visual Webquest
• http//www.adrianbruce.com/Symmetry/10.htm
• Coordinate Plane
• http//www.mathsonline.co.uk/nonmembers/gamesroom/
  transform/rotation.html
• Creating rotation symmetry
• Quilt Square
• Figure This Challenge 73 A shard or Two
• http//www.figurethis.org/challenges/c73/challenge
  .htm

12
Unit 7 Scale Factor

• Students will be able to
• Select and use appropriate units to measure
  length, perimeter, area, and volume.
• Measure lengths to the nearest ½, ¼, ?, and
  1/16 inch.
• Convert one unit of measure to another in the
  same system of measurement
• Use ratio, proportion, and scale factor to
  describe relationships between similar figures.
• Interpret and create scale drawings
• Solve problems using scale factors, ratios, and
  proportions

13
Part 1 Units of Measure

• Unit 7 Scale Factor

14
Motivation

• Children frequently have a difficult time
  choosing appropriate units of measure. For
  example, they often try to measure area using
  linear units (centimeters), or volume using
  two-dimensional units (square centimeters).
  Reflect on your own knowledge of metric units of
  measure. Does your knowledge of and familiarity
  with metric units have anything to do with your
  ability to choose an appropriate unit?

15
Measurement

• What is measurement?Measurement is the process
  of quantifying the properties of an object by
  expressing them in terms of a standard unit.
  Measurements are made to answer such questions
  as, How heavy is my parcel? How tall is my
  daughter? How much chlorine is in this water?
• How do we measure?The process of measuring
  consists of three main steps. First, you need to
  select an attribute of the thing you wish to
  measure. Second, you need to choose an
  appropriate unit of measurement for that
  attribute. Third, you need to determine the
  number of units.
• What procedures are used to determine the number
  of units?Some measurements require only simple
  procedures and little equipment -- measuring the
  length of a table with a meter stick, for
  example. Others -- for example, scientific
  measurements -- can require elaborate equipment
  and complicated techniques.

16
Metric vs. Customary
The concept of a coherent system may be
confusing. If the customary system were coherent
(which it is not), then there would be a single
base unit for length area and volume would also
be based on this unit. For example, if the inch
were the base unit, then we would measure area in
square inches (not in acres or square yards) and
volume in cubic inches (not in pints or cubic
feet). In a coherent system, units can be
manipulated with simple algebra rather than by
remembering complex conversion factors.
http//www.visionlearning.com/library/module_vie
wer.php?mid47lc3 http//ts.nist.gov/WeightsAnd
Measures/Metric/lc1136a.cfm http//physics.nist.go
v/cuu/Units/background.html
17
Metric System

• The metric system was introduced in France in the
  1790s as a single, universally accepted system of
  measurement. But it wasn't until 1875 that the
  multinational Treaty of the Meter was signed (by
  the United States and other countries), creating
  two groups the International Bureau of Weights
  and Measures and the international General
  Conference on Weights and Measures. The purpose
  of these groups was to supervise the use of the
  metric system in accordance with the latest
  pertinent scientific developments.
• One of the strengths of the metric system is that
  it has only one unit for each type of
  measurement. Other units are defined as simple
  products or quotients of these base units.
• For example, the base unit for length (or
  distance) is the meter (m). Other units for
  length are described in terms of their
  relationship to a meter A kilometer (km) is
  1,000 m a centimeter (cm) is 0.01 of a meter
  and a millimeter (mm) is 0.001 of a meter.
• Prefixes in the metric system are short names or
  letter symbols for numbers that are attached to
  the front of the base unit as a multiplying
  factor. A unit with a prefix attached is called a
  multiple of the unit -- it is not a separate
  unit. For example, just as you would not consider
  1,000 in. a different unit from inches, a
  kilometer, which means 1,000 m, is not a
  different unit from meters.

18
Metric Units and Prefixes

• Estimate the sizes of these objects in metric
  units before you start the activity, and pay
  attention to the patterns that emerge as you
  change prefixes
• http//www.learner.org/channel/courses/learningmat
  h/measurement/session3/part_a/units.html

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Measuring Length

• As we've seen, the base unit for length (or
  distance) is the meter. Meter comes from the
  Greek word "metron," which means "measure."
• Many of us do not have a strong intuitive sense
  of metric lengths, which may be a result in part
  of our limited experience with metric measures
  and estimates. It is, however, important to have
  referents for measures, as referents make
  measurement tasks easier to interpret and provide
  us with benchmarks against which to test the
  reasonableness of our measures.

20
Estimating Length

• When measuring objects using the metric system,
  it is important to establish benchmarks for
  common lengths, such as meter, decimeter, and
  centimeter. In addition, you should actually make
  the measurements and compare your estimates to
  your measurement data. Reconciling the
  differences between your estimates and measures
  will help you improve your ability to make
  reasonable estimates using the metric system.

21
Measuring Liquids

• Measures of liquid volume, sometimes referred to
  as capacity, include the liter (L) and the
  milliliter (mL). These terms are holdovers from
  an older version of the metric system and,
  because they are so well known, are approved for
  use with the current SI. Volume, whether liquid
  or solid, is a measure of space. Solid volume is
  measured using cubic meters (m3) as the base
  unit. Liquid volume is most often measured using
  liters. In Session 8, we will explore measures of
  solid volume in detail, but we will begin to
  examine the relationships among measures of solid
  and liquid volume in this session.
• By definition, a liter is equivalent to 1,000 cm3
  (or one 1 dm3). This leads to the conclusion that
  1 mL is equivalent to 1 cm3. Large volumes may be
  stated in liters but are usually recorded in
  cubic meters

22
Estimating Capacity

• Whereas the base unit for volume is the cubic
  meter, most practical day-to-day situations find
  us determining the capacity of smaller
  containers, and thus cubic centimeters or cubic
  millimeters might also be used. The relationship
  between cubic centimeters and milliliters (1 cm3
  1 mL) and between cubic decimeters and liters
  (1 dm3 1 L) is an important one to establish.
  Models can help people visualize these
  relationships. If you have metric base ten
  blocks, then the "small units cube" (1 cm3) is
  equivalent to 1 mL, and the "thousands cube" is
  equivalent to 1 dm3 this cube, if hollow, will
  hold 1 L. Compare a milliliter and a cubic
  centimeter as well as a liter and a cubic
  decimeter. If possible, pour 1 L of water into a
  hollow decimeter cube.

23
Measuring Mass

• Whereas weight measures the gravitational force
  that is exerted on an object, mass measures how
  much of something there is thus, mass is closely
  related to volume. The weight of an object can
  change depending on its location (e.g., on the
  Earth or on the Moon), but the mass of the object
  (how much of it there is) always stays the same.
• Mass and weight are often confused, because our
  two systems of measurement use different terms.
  In the metric system, kilograms and grams are
  measures of mass, but in the U.S. customary
  system, ounces and pounds are measures of weight.
  When using the metric system, we should really
  state that we are measuring mass, saying, for
  example "I have a mass of 60 kg" rather than "I
  weigh 60 kg," but this goes against convention.

24
Mass (continued)

• The base unit of mass is the kilogram (kg). In
  the 1790s, a kilogram was defined as the mass of
  1 L (cubic decimeter, or dm3) of water
• Though that definition has changed somewhat with
  time, here is a definition that is close enough
  for ordinary purposes There are 1,000 g in 1 kg,
  and 1,000 g occupy a volume of 1,000 cm3, or 1 L.
  Therefore, 1 g of water weighs the same as 1 cm3
  of water and occupies 1 mL of space. In other
  words, for water
• 1,000 g 1 kg 1,000 cm3 1 dm3 1 Land1 g
  1 cm3 1 mL

25
Mass (continued)

• Kilograms are used to weigh just about everything
  but very light objects (which are weighed in
  grams) and very heavy objects (which are weighed
  using metric tons). A gram is almost exactly the
  weight of a dollar bill. A metric ton is
  equivalent to 1,000 kg (so it can also be thought
  of as a megagram) and should not be confused with
  the common American ton in the U.S. customary
  system. In fact, the metric ton is often referred
  to by its French and German name, tonne, to
  distinguish it as a metric measure. Most cars
  have a mass of between 1 and 2 tonnes a large
  diesel freight locomotive has a mass of
  approximately 165 tonnes.
• As with metric lengths, it is useful to establish
  benchmarks for metric mass measures.

26
Part 2 Unit Conversions

• Unit 7 Scale Factor

27
Using the Identity Property
Conversion is changing the units of measure from
one measure to a different measure.
Example Convert 15 feet to inches.
15 feet
?
15 ? 12 inches
The feet divide out.
180 inches
NLVM Converting Units http//nlvm.usu.edu/en/na
v/frames_asid_272_g_3_t_4.html?openinstructions

28
Dimensional Analysis

• A method of problem-solving that focuses on the
  units used to describe matter.
• Also known as the bridge method
• A ratio of equivalent values used to express the
  same quantity in different units
• Always equal to 1
• Changes the units of a quantity without changing
  the values

29
Example
4 quarts are equivalent to 1 gallon and 1 gallon
is equivalent to 4 quarts
30
Metric Conversions

• Kilometers to meters to centimeters
• Liters to milliliters
• Kilograms to grams to milligrams

31
Step by step process

• First take what is given and change into a
  fraction
• Next find a conversion factor that you know and
  place next to what is given
• Make sure the unit in the first fraction is now
  on the bottom of the next fraction
• Repeat with more fractions and cancel units as
  you go
• Once the unit you want is on top of the last
  fraction, stop!
• Multiply across the top, multiply across the
  bottom and then divide.

32
Example

• How many meters are in 48km?

33
Part 3 Similar Figures

• Unit 7 Scale Factor

34
Similar Figures

• Similar figures are figures that have the same
  shape but may be of different sizes. In similar
  figures, corresponding angles are congruent and
  corresponding segments are in proportion.

35
Similarity

• Which figure is similar to this one?

Math.com (Similar Figures) http//www.math.com/sch
ool/subject1/lessons/S1U2L4GL.html
36
Scaling

• A scale of 11 implies that the drawing of the
  grasshopper is the same as the actual object. The
  scale 12 implies that the drawing is smaller
  (half the size) than the actual object (in other
  words, the dimensions are multiplied by a scale
  factor of 0.5). The scale 21 suggests that the
  drawing is larger than the actual grasshopper --
  twice as long and twice as high (we say the
  dimensions are multiplied by a scale factor of
  2).

37
Scale Drawings

• Copy the picture onto graph paper and label the
  coordinates. Multiply the coordinates of each
  point by 2, creating points A H. Plot these
  points on the same grid.
• What is the ratio of proportionality?

38
Activities

• Interactive Activity Quad Person
• http//www.learner.org/channel/courses/learningmat
  h/algebra/session4/part_c/index.html
• NLVM Transformation-Dilation
• Illuminations Shape Tool
• GSP Dilations
• Similar Figures
• Comic Expansions/Inca Birds

39
Scaling Factors Proportions

• Since, similar figures have equal angles and
  proportional sides. the sides of one figure can
  be obtained by multiplying the other by the
  scaling factor or by setting up proportions.

40
Finding Unknown Lengths of Sides in Similar
Triangles
EXAMPLE
Find the length of the side labeled n of the
following pair of similar triangles.
n
9
SOLUTION
14
8
Since the triangles are similar, corresponding
sides are in proportion. Thus, the ratio of 8 to
14 is the same as the ratio of 9 to n.
41
Find Missing Side Lengths

• Why are the two triangles similar? (how were the
  angles formed?) How can you find the height of
  the lamp post?

How tall is the lamp post if it has a shadow 5
meters long, your friend is 2 meters tall and
your friends shadow is 1 meter long?
42
Shadows

• Because the suns rays are parallel, the
  triangles are similar.

43
Scale Factor

• NCTM E-Ex. Side Length Area of Similar Figures
• http//standards.nctm.org/document/eexamples/chap6
  /6.3/index.htm
• NCTM E-Ex Side Length, Volume, Surface Area
• http//standards.nctm.org/document/eexamples/chap6
  /6.3/part2.htm
• NCTM E-Ex Ratios of Areas
• http//standards.nctm.org/document/eexamples/chap7
  /7.3/index.htm

44
Part 4 Scale Drawings

• Unit 7 Scale Factor

45
Similarity-Enlarging/Reducing

• Enlarging or reducing a figure produces two
  figures that are similar. Similar figures have
  the same shape but are not necessarily the same
  size. More formally, we state that two figures
  are similar if and only if two things are true
  (1) The corresponding angles have the same
  measure, and (2) the corresponding segments are
  in proportion (by a scale factor).
• What happens when we enlarge a figure by a scale
  factor of 2? Since in similar figures the
  corresponding sides are in proportion, each of
  the sides of the enlarged similar figure is twice
  as long as the corresponding side of the original
  figure. So, for example, in the enlargement of
  the trapezoid shown below on the left, the
  enlarged trapezoid is similar to the small
  trapezoid because the angles are congruent and
  each of the sides is proportionally larger (twice
  as long)
• Building similar figures, however, is not always
  so straightforward! For example, the trapezoid
  below is not similar to the original trapezoid.
  The angles are congruent, but the corresponding
  sides are not proportional -- some of the sides
  have been "stretched" more than others

46
Activities

• Comic Expansions
• Inca birds
• Dinosaur
• Figure This Challenge 61 Statue of Liberty
• http//www.figurethis.org/challenges/c61/challenge
  .htm

47
Unit 8 Solids

• Students will be able to
• develop their understanding of solid figures
  (including perspective drawings, nets,
  compare/contrast).
• determine the surface area of solid figures
  (right rectangular prisms and cylinders.)
• determine the volume of fundamental solid
  figures (right rectangular prisms, cylinders,
  pyramids, and cones.)

48
The Geometry of Solid Figures

• The concept of volume should include visualizing
  layers of unit cubes filling solid figures
  including right rectangular prisms, cylinders,
  pyramids and cones. In contrast, surface area
  should be conceptualized as tiling solid figures
  including right rectangular prisms and right
  circular cylinders. Instruction and activities
  should incorporate a variety of strategies
  including, but not limited to, using
  manipulatives, constructing nets, developing
  patterns, applying formulas and using appropriate
  technology. Estimation, algebra, ratio,
  proportion and scale factors will be an integral
  part of Unit 8 along with all of the Process
  Standards. Including opportunities for students
  to write, develop high-level questions, and hold
  discussions/debates with their peers will not
  only enhance their understanding and learning,
  but will foster improved critical thinking
  skills.

49
Space Figures

• Space figures are figures whose points do not all
  lie in the same plane.
• Prisms, Cylinders, Pyramids, Cones, Spheres

50
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).

51
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).

Octahedron 8 Triangular Faces
Cube 6 Square Faces
Dodecahedron-12 Pentagonal Faces
52
Irregular Polyhedra
Faces are a combination of different polygons.
53
Non-Polyhedra

• Cylinder
• Cone
• Sphere
• Why arent each of these solids a polyhedron?

54
Prisms

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

A prism is named by the shape of its base.
55
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.

56
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?

57
Creating Prisms

• If you move a vertical rectangle horizontally
  through space, you will create a rectangular or
  square prism.
• If you move a vertical triangle horizontally, you
  generate a triangular prism. When made out of
  glass, this type of prism splits sunlight into
  the colors of the rainbow.

58
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?
• How do the number of faces, number of vertices,
  and number of edges relate in a prism? in a
  pyramid?

59
Investigating Vertices, Edges, and Faces

• For each of the Power Solids listed in the table,
  count and record the number of vertices, edges,
  and faces.
• Describe any patterns you observe.

60
Investigating Vertices, Edges, and Faces Teacher
Notes

• One pattern that may emerge from the table is
  Eulers Formula
• V F E 2
• where V number of vertices, F number of
  faces, and E number of edges.

61
Visualization Practice

• Suppose you see the footprint of a prism whose
  base is shown below.
• Without actually making the prism, explain how
  could you tell how many vertices, edges, and
  faces it has.

62
Whats My Solid?

• I have one circular face. I also have a curved
  surface. What geometric solid am I?
• Answer cone
• I have six faces. All of my edges are the same
  length. Which geometric solid am I?
• Answer cube
• I have an odd number of vertices. I have the same
  number of faces and vertices. Which geometric
  solid am I?
• Answer square pyramid
• I have two triangular faces. I have three
  rectangular faces. Which geometric solid am I?
• Answer triangular prism
• I have two circular faces. I have a curved
  surface. Which geometric solid am I?
• Answer cylinder
• I have an even number of vertices. I have the
  same number of faces and vertices. Which
  geometric solid am I?
  
• Answer triangular pyramid

63
Part 1 Perspective Drawings

• Unit 8 Solids

64
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.

65
A View from the Top 2

• Which of the architectural views below represent
  the front, back, left, and right of your
  building?

66
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.

4
2
1
1
3
1
3
2
2
3
3
3
4
2
2
1
1
4
1
1
67
Architectural Plans 3A
1
3
1

• 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?

3
2
1
1
68
Architectural Plans 3B
4
2
1
2
3
4
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?

69
Architectural Plans 3C
3
2
3
4
2
1
1
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?

70
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

71
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 on today.

72
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)

73
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.

74
Isometric Drawings Practice

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

75
Activities

• Building Viewpoints http//www.learner.org/catalo
  g/resources/activities/middle/mid_math.html
• Illuminations Isometric Draw Tool
• http//illuminations.nctm.org/ActivityDetail.aspx?
  ID125

76
Part 2 Nets

• Unit 8 Solids

77
Nets

• A net is a 2-dimensional representation
• of a 3- dimensional geometric figure.
• Nets can be drawn on a sheet of paper,
• cut out, and taped together to form a
• 3-dimensional shape. There are often
• many ways to draw a net for a
• 3-dimensional figure.
• Polydrons are useful for constructing nets.

78
Identical Nets

• 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.

79
Nets for a Cube

• A net for a cube can be drawn by tracing faces of
  a cube as it is rolled forward, backward, and
  sideways. Which of the following are nets for a
  cube?
• How many different nets are there for a cube?
  Using centimeter grid paper (downloadable), draw
  all possible nets for a cube.

80
Nets for a Cube

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

Figure This Challenge 55 Decorating
Boxes http//www.figurethis.org/challenges/c55/cha
llenge.htm
81
Nets 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.
• Are there others?

82
Pyramids

• A pyramid has a base and triangular faces that
  meet at a vertex.
• Triangular Square
• Pyramid Pyramid
• Here are two of the nets for a triangular
  pyramid. Find the third.

83
Nets for a Square Pyramid

• Which of the following are nets of a square
  pyramid?
• Are these nets distinct?
• Are there other distinct nets? (No)

84
Cylinders Cones

• A cylinder is 3D object with two
• parallel, circular bases.
• Sketch a net for a cylinder and describe
• how the various edges must be related.
• A cone is a 3D object that has a circular
• or elliptical base and a vertex.
• Sketch a net for a cone and describe each part.

85
Creating Cylinders Cones

• If you move the center of a circle on a
• straight line perpendicular to the circle,
• you will generate a cylinder.
• A cone can be generated by twirling a
• right triangle around one of its legs.

86
Nets for a Cylinder

• 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

87
Building a Cylinder

• 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)

88
Building 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.

89
Nets for a Cone

• 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

90
Cone Investigation

• 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?

91
Cone 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.

92
Creating 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 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?

93
Nets for Similar Cubes Using Centimeter Cubes

• 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?
• What is the volume of each cube?
• How does the scale factor affect the volume?

94
Scaling Laws

• Lengths always scale with the scale factor.
• Areas always scale with the square of the scale
  factor.
• Volumes always scale with the cube of the scale
  factor.

95
Part 3 Surface Area of right rectangular prisms
and cylinders

• Unit 8 Solids

96
(No Transcript)
97
Surface Area of Building 5

• Top Front Right MAT
• How many square faces are there in each view
  (top, front, back, right, left, and MAT) of the
  solid created earlier today?
• What is the total number of square faces in this
  building (surface area)?

98
Surface Area of a Rectangular Solid

• Use centimeter cubes to construct solids made up
  of the following stack of cubes.
• How many square faces are there in each layer of
  the solid?
• _____ _____ _____ _____ _____
  _____
• What is the surface area of this solid? ______
  cm2

4
1
3
2
1
5
2
1
2
99
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?

100
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.

101
Surface Area of a Cylinder

• Closed cylinder
• Surface Area 2Base area Rectangle area
• 2Area 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

102
Surface Area Formulas
Surface Area of a Cube 6 a2
(a is the length of the side of each edge of the
cube)
Surface Area of a Right Rectangular Prism 2ab
2bc 2ac
(a, b, and c are the lengths of the 3 sides)
Surface Area of a Cylinder 2 pi r 2 2 pi r h
(h is the height of the cylinder, r is the radius
of the top)
103
F-103 of 25
F-103 of 24
F-103 of 24
104
Part 6 Volume

• Unit 8 Solids

105
Volume of Building 5

• Top Front Right MAT
• How many cubes are there in each layer of the
  solid (saved from earlier today)?
• What is the total number of cubes in this
  building (volume)?

106
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
3
2
1
5
2
1
2
107
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.

108
Volume of a Cube

• Consider 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
• Vs3.

109
Cylinder Volume Exploration

• Materials Required
• Three sheets of 8.5 x 11 paper
• Tape
• Rice cereal, bird seed, popcorn, or other
  small pourable material
• Procedure
• Roll and tape one piece of paper along longest
  side (hotdog style). Label this cylinder A.
• Roll and tape other piece of paper along shortest
  side (burger style). Label this cylinder B
• Place shorter cylinder B on top of third piece of
  paper (or in a shallow box) and insert the taller
  cylinder A inside the shorter cylinder.
• Fill tall cylinder A with cereal.

110
Cylinder Volume Exploration (continued)

• Making Predictions
• Will the cylinders hold the same amount?
• If not, which cylinder A or B will hold more?
  Why?
• Verifying Predictions
• Slowly raise the taller cylinder A allowing the
  contents to fall into the shorter cylinder B.
• Do the cylinders have the same volume? Why or why
  not?

A
B
111
Cylinder Volume Exploration Teacher Notes

• Recall that the volume of a cylinder is found by
  multiplying the area of the base by the height.
  Thus
• V ?r2h
• Since r is squared, it has a bigger effect on V
  than does h. Therefore, the cylinder with the
  greater radius will have the greater volume.

112
Exploring Volumes of Other Cylinders

• Materials Required
• Two sheets of 8.5 x 11 paper
• Tape and scissors
• Rice cereal, bird seed, popcorn, or other small
  pourable material
• Procedure
• Fold a sheet of paper lengthwise and cut it in
  half forming two pieces each measuring 4 ¼ by
  11.
• Tape the two pieces together to form a rectangle
  4 ¼ by 22.
• Repeat this process to create another 4 ¼ by
  22 rectangle.

113
Exploring Volumes of Other Cylinders continued

• Procedure continued
• Roll and tape the two long rectangles into two
  different cylinders as shown.
• Label the cylinders C (4 ¼ high) and D (22
  high).
• Making Predictions
• Which cylinder C or D will hold more? Why?

C
D
114
Exploring Volumes of Other Cylinders continued

• Arrange the four cylinders in order from least
  volume to greatest volume.
• Write down your predictions.
• Test your predictions by filling.
• Drawing Conclusions
• Is there a pattern that relates the size of the
  cylinder and the volume?

115
Exploring Volume of Other Cylinders Teacher Notes

• As cylinders get taller and narrower, the
  cylinders hold less.
• As cylinders get shorter and wider, they hold
  more.
• You may wish to have students complete the table
  below to confirm their predictions.

116
Volumes of Related Cones and Cylinders
Investigation

• Materials Required
• Rice cereal or other pourable materials
• Power Solids cylinder and related cone (NOTE A
  cylinders related cone has the same base area
  and the same height as the cylinder.) Nets for
  paper models are downloadable.
• Procedure
• Examine the cylinder and its related cone and
  estimate how many cones it will take to fill the
  cylinder.
• Fill the cone and pour into the cylinder. Repeat
  the process until the cylinder is full.

117
Volumes of Related Cones and Cylinders
Investigation continued

• Drawing Conclusions
• Write a ratio that compares the volume of the
  cone to the volume of the cylinder.
• Write a rule to determine the volume of a cone
  based on the formula for the volume of a
  cylinder.

118
Volume of Cones and Cylinders

• Vcone (1/3)?r2h
• Vcylinder ?r2h

119
Volumes of Related Pyramids and Prisms
Investigation

• What is true about the areas of the bases of the
  pyramid and prism?
• What is true about the heights of the pyramid and
  prism?
• Estimate how many pyramids of cereal it will take
  to fill the prism.

120
Volumes of Related Pyramids and Prisms
Investigation continued

• Procedure
• Fill the Power Solids pyramid with cereal and
  pour into the Power Solids prism. Nets for paper
  models are downloadable.
• Repeat the process until the prism is full.

121
Volumes of Related Pyramids and Prisms
Investigation continued

• Drawing Conclusions
• Write a ratio that compares the volume of a
  pyramid to the volume of a prism.
• Write a rule to determine the volume of a pyramid
  based on the formula for the volume of a prism.

122
Volumes of Pyramids and Prisms

• Vpyramid (1/3)Base area height
• Vprism Base area height

123
Volume of Your Square Pyramid and Cube

• Using an inch-ruler and measuring to the nearest
  sixteenth of an inch, find the volume of your
  Power Solid square pyramid.
• What units are associated with the volume of your
  square pyramid?
• Based on the volume of the Power Solid square
  pyramid, find the volume of the Power Solid cube.

124
Getting Acquainted with the Square Pyramid

• How many lateral faces are in a square pyramid?
• What shape are the lateral faces?
• How many vertices are there?
• How many edges are there?
• Describe where the lateral edges meet.

125
Cross Section Method

• One way to determine the volume of rectangular
  prisms is by multiplying the dimensions (length
  width height). Another way to determine the
  volume is to find the area of the base of the
  prism and multiply the area of the base by the
  height. This second method is sometimes referred
  to as the cross-section method and is a useful
  approach to finding the volume of other figures
  with parallel and congruent cross sections, such
  as triangular prisms and cylinders. Notice in the
  figures below that each cross section is
  congruent to a base.

126
Compute Volume using the CSM

• The formula for the volume of prisms is V B
  h, where B is the area of the base of the prism,
  and h is the height of the prism. Does it matter
  which face is the base in each of the following
  solids? Explain.
• Find the volume of the figures above using the
  cross-section method.

127
Summary of Volume Formulas

• Volume is a measure of how much it would take to
  fill up a shape.
• If B is the area of the base figure, and h is the
  height from the base to the vertex.
• The formula for the volume of prisms and
  cylinders is
• V Bh
• The formula for the volume of pyramids and cones
  is

128
Prisms
129
Pyramid
130
Cylinder Cone
131
Activities

• Figure This Challenge 3 Popcorn
• http//www.figurethis.org/challenges/c03/challenge
  .htm
• NLVM Fill and Pour
• http//nlvm.usu.edu/en/nav/frames_asid_273_g_3_t_4
  .html
• NLVM How High?
• http//nlvm.usu.edu/en/nav/frames_asid_275_g_3_t_4
  .html
• Figure This Challenge 15 Big Trees
• http//www.figurethis.org/challenges/c15/challenge
  .htm

132
Surface Area Volume

• How about the relationship between surface area
  and volume? Do prisms with the same volume have
  the same surface area? Let's explore this
  relationship.
• Take 24 multilink cubes or building blocks and
  imagine that each cube represents a chocolate
  truffle. For shipping purposes, these truffles
  need to be packaged into boxes in the shape of
  rectangular prisms. Knowing that you must always
  package 24 truffles (i.e., your volume is set at
  24 cubic units), what are the possible dimensions
  for the boxes? Compute the Surface Areas.
• http//intermath.coe.uga.edu/dictnary/challeng.asp
  ?termid352

133
Scale Factor, Volume, and Surface Area of a
Rectangular Prism

• 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?

Excerpt from Student Activity Sheet
(downloadable) p. 119-120 from Navigating
through Measurements in Grades 6-8, NCTM, 2002.
134
References

• The information and activities in this
  presentation was primarily from the following
  sources
• http//www.georgiastandards.org/mathframework.aspx
  
• http//www.learner.org/channel/courses/learningmat
  h/index.html
• http//intermath.coe.uga.edu/