Measurement Goal 2

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MeasurementGoal 2

• Grade 8
• NC SCOS objectives
• Sandra Davidson
• NBCT EA Math

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Measurement (4 weeks)

• 2.01 Determine the effect on perimeter, area, or
  volume when one or more of the dimensions of
  two- and three-dimensional figures are changed.
• 2.02 Apply and use concepts of indirect
  measurement.

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1. Work with a partner to measure the following
classroom items.

• How many paperclips long is your right little
  finger?
• How many math books wide is this room?
• How many arms long is the chalkboard?
• How many hands high is the door?
• How many pencils long is the width of your desk?
• How many paperclips long is the circumference of
  your calf?
• How many shoes long is the length of this room?

• How many paperclips high is the teachers desk?
• How many erasers long is your leg from you knee
  to your ankle?
• Would it be easier to measure the length of the
  room in paperclips, erasers, staples, or
  notebooks? Why?
• Is it a shortcut to cut through the 7th grade
  hall to get to the cafeteria? How did you
  determine this?

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Measure each item below in the indicated
measuring instruments.

• Distance from the floor to the top of your desk
• yardstick (in.) ______
• meter stick (cm.) ______
• tape measure (in.) ______
• Width of a locker
• yardstick (in.) ______
• meter stick (cm.) ______
• tape measure (in.) ______
• Height of the door
• yardstick (in.) ______
• meter stick (cm.) ______
• tape measure (in.) ______

• Length of a math book
• ruler (in.) ______
• meter stick (cm.) ______
• Circumference of your wrist
• ruler (in.) ______
• tape measure (in./cm.) _____
• Which measurements were made easier by using a
  tape measure than by using a wooden ruler?

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2. Use a geoboard to find the perimeter and area
of each figure.
Review perimeter and area of rectangles,
parallelograms, triangles, and trapezoids. (p.
283 and 287)
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3. Patterns in Perimeter If each side has a
length of one unit, what is the perimeter of the
other figures in each set? Write an equation for
each pattern.
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4. Minimizing PerimeterOn graph paper draw all
possible rectangles with an area of 36.

• How many different rectangles can you create?
• How can you tell when you have found all possible
  solutions?
• I want to fence in this area for my dog. The
  fencing costs 24.00 per meter. Which rectangle
  would be the least expensive to fence in?
• Compare this price to the most expensive
  rectangle to fence in.

• Repeat these steps for a garden with an area of
  24 square ft.
• Make a conjecture about the relationship between
  perimeter and area.
• Make a conjecture about the minimum fencing
  needed for an area of 100 sq. feet.
• Review Area of Compound Shapes (p.283-284)

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5. Area of Compound FiguresSeeing Is Believing
(NCTM Navigating Measurement)

• Imagine cutting out an 8x8 square, and a triangle
  the same size. Glue the triangle, overlapping
  part of the square, to create shape D.
• What is the area of shape D?
• Explain, in detail, how you found your answer.
• Review circumference and area of circles (p.296)

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6. Area of Circles

• Three students are considering the diagrams
  shown, in which circles are inscribed in a square
  with a side length of 8 inches.
• Carlos says that the shaded portion in diagram C
  is larger than the shaded portion in diagram B or
  A.
• Bill disagrees, saying that the shaded portion in
  diagram A is larger than the shaded portion in
  diagram B or C.
• Alicia disagrees with both boys. She says that
  all three diagrams have the same portion shaded.
  
• Who is correct? Justify your answer.
• Follow-up worksheet (p. 296)

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Circles- Grass for Goats(Balanced Assessment)

• The Jacobsens keep their goat on a chain that is
  3 meters long.
• 1. If they chain the goat to a metal hook in the
  center of their yard, what is the area of the
  grass that the goat can reach to eat?
• Justify your answer with a sketch.

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Grass for Goats (continued)

• 2. Sometimes the Jacobsens chain the goat to the
  corner of a shed that is 5 meters by 4 meters.
  What is the area of grass that the goat can
  reach? Justify your answer with a sketch.
• 3. Suppose the goat was chained to the center of
  the 4-meter shed wall. Would the amount of grass
  the goat can reach be greater than what he could
  reach when he was chained to the corner of the
  shed? Justify your answer with a sketch.

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Double the Dimensions
What happens to the perimeter and area of each
figure when the dimensions are doubled? When the
dimensions are tripled? Use graph paper to prove
your answers with sketches.
Figure C
Figure A
Figure B
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7. Ratios - Perimeters and Areas(NCTM Navigating
Measurement)

• 1. Find the scale factor of each pair of
  rectangles by computing the ratio
  (dilation/original) of the widths and lengths.

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The ratio of corresponding parts of similar
figures is called the scale factor.

• 2. Find the perimeter of each rectangle.
• Rectangle A ____ Rectangle B ____
  Rectangle C____
• 3. Find the scale factor for the perimeters of
  each pair of rectangles.
• A and B ______ A and C ______ B and C
  ______
• 4. How does the scale factor of the sides compare
  to the scale factor of the perimeters?

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The ratio of corresponding parts of similar
figures is called the scale factor.

• 5. Compute the area of each rectangle.
• Rectangle A ______ Rectangle B ______
  Rectangle C ______
• 6. What is the scale factor of the areas of each
  pair of rectangles?
• A and B ______ A and C ______ B
  and C _______
• 7. How does the scale factor of the areas compare
  with the scale factor of the sides?
• 8. Find the scale factor of the sides and the
  scale factor of the areas of the two rectangles
  on the back of your worksheet.

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8. Determining the Effect on Perimeter and Area
when Changing One or Two Dimensions

• 1. Find the perimeter and area of a rectangle
  with a length of 9.5 inches and a width of 6.5
  inches.
• 2. Double the length of the rectangle above.
  What is the new perimeter and area? Write a
  comparison statement about the original and new
  perimeter and area.
• 3. Now double both the width and length of the
  rectangle in problem one. What is the new
  perimeter and area? Write a comparison statement
  about the original and new perimeter and area.
• 4. Mr. Green is preparing a new garden. His
  original plot was 13 ft. by 20 ft, but now he is
  doubling each side. How many yards of fencing
  are required for the new garden? By how much is
  Mr. Green increasing the area of his garden?

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Determining the Effect on Perimeter and Area
(continued)

• 5. The perimeter of Rexs dog pen is 28 ft. If
  the width of the pen is 5 ft, how long is the
  pen? Rex is a puppy, but he is growing fast and
  his owner wants to triple the dimensions of his
  pen. Find the new perimeter and area of Rexs
  pen. How does the new pen compare to the
  original puppy pen?
• 6. John and Mary each have a garden in the shape
  of a square. If the area of Johns garden is
  four times the area of Marys garden, how do the
  perimeters of the two gardens compare? Draw a
  sketch to justify your answer.
• 7. A square scarf is folded in half to form a
  rectangle. If the resulting rectangle has a
  perimeter of 15 inches, what are the area and
  perimeter of the original square scarf? Draw a
  sketch to justify your answer.

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9, 10. Volume of Prisms and Cylinders
(p.310) and Surface Area (p.318)

• Rectangular Prism Triangular Prism
• V B x h V B x h
• V l x w x h V (½bh) x h
• Cylinder
• V B x h
• V ? r2 x h

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Problem Solving with Volume

• 1. Joy has a 2 gallon tank that houses two
  goldfish. The base of the tank is 14 inches by 6
  inches. The height of the tank is 8 inches. For
  her birthday she was given a new fish tank that
  had a base of 28 inches by 12 inches and a height
  of 16 inches. Approximately how many gallons of
  water would the new tank hold?
• 2. Joys brother has a tank that holds x gallons
  of water. If each dimension were doubled, how
  many gallons would the new tank hold in terms of
  x ?

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11. Shodor Interactive ActivitySurface
Area/Volume http//www.shodor.org/interactivate/a
ctivities/SurfaceAreaAndVolume/
Create a rectangular prism that has a length
(depth) of 2 cm, a width of 3 cm, and a height of
4 cm. Record the volume and surface area.

• Describe the effect doubling one dimension has on
  the volume of a rectangular prism. Prove this
  with patterns of exponents.
• Describe the effect doubling one dimension has on
  the surface area of a rectangular prism.

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Interactive Activity (continued)

• Describe the effect doubling two dimensions has
  on the volume of a rectangular prism. Prove this
  with patterns of exponents.
• Describe the effect doubling two dimensions has
  on the surface area of a rectangular prism.

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Interactive Activity (continued)

• Describe the effect doubling all three dimensions
  has on the volume of a rectangular prism. Prove
  this with patterns of exponents.
• Describe the effect doubling all three dimensions
  has on the surface area of a rectangular prism.
  Can this be proven with patterns of exponents?

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Interactive Activity (continued)
Create a new rectangular prism with a length
(depth) of 4 cm, a width of 1 cm, and a height of
2 cm.

• Describe the effect tripling one, two, and all
  three dimensions has on the volume of a
  rectangular prism. Prove this with patterns of
  exponents.
• Describe the effect tripling one, two, and all
  three dimensions has on the surface area of a
  rectangular prism. Can any of these be proven
  with patterns of exponents?

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Interactive Activity (continued)

• Describe the effect quadrupling one, two, and all
  three dimensions has on the volume of a
  rectangular prism. Prove your statement with
  patterns of exponents.
• Describe the effect quadrupling one, two, and all
  three dimensions has on the surface area of a
  rectangular prism. Can any of these be proven
  with patterns of exponents?

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12. RatiosSurface Areas and Volumes

• 1. What is the scale factor of the edges of the
  prisms, if E is the original and D the dilation?
• 2. Compute the surface area of each rectangular
  prism.
• Surface area of prism D ______ units2
• Surface area of prism E ______ units2
• 3. What is the scale factor of the surface areas
  of the prisms? ______

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Ratios-Surface Areas and Volumes

• 4. How does the scale factor of the surface areas
  compare to the scale factor of the edges?
• 5. Find the scale factors of the volumes.
• volume prism D ____ volume prism E ____
• Scale factor ______

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Ratios-Surface Areas and Volumes

• 6. How does the scale factor of the volumes
  compare with the scale factor of the edges?
• 7. Summarize your conclusions, including your
  discoveries about the relationships between the
  scale factors of the edges, surface areas, and
  volumes.

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Problem Solving - Volume and Surface Area

• 1. The Yummy Tastin Company is exploring
  packaging options for their oatmeal. One option
  is a cylindrical package with a diameter of 5
  inches and a height of 10 inches. The other
  option is a rectangular box with a length of 7
  in., a width of 2 in., and a height of 14 in. If
  the packaging material costs ¼ cents per square
  inch, what option would cost the company the
  least?
• 2. A company manufactures two different size
  cylindrical containers for storage of pool
  chemicals. If the radius and height of the
  larger container are twice the radius and height
  of the smaller container, by what factor does the
  volume increase?

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13. Snap Cube Activity NC SCOS 2.01

• Students work in groups of three or four to build
  the situation they are given using snap cubes.
• There are seven situations.
• Each group should be prepared to present and
  explain their situation and the cubes they have
  built to the class.

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14. Class discussion Review for Test What
happens to the area and volume of a figure when
the dimensions double? Triple?

• 1. When the length of a rectangle doubles, how
  does the area change?
• 2. When the length and width of a rectangle
  doubles, how does the area change?
• 3. When the length and width both triple, how
  does the area change?
• 4. What happens to the area if the length and
  width become half as long?

• 5. When the length of the edges of a cube
  doubles, how does the volume change?
• 7. What happens to the volume of a cube when the
  edges becomes three times as long? Four times as
  long? Half as long?
• 8. What happens to the area of a circle when the
  radius doubles? Triples?

Test on Changing Dimensions
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15. Indirect Measurement(NCTM Navigating through
Geometry)

• Use indirect measurement to find the distance
  across the pond. Describe your method of finding
  the distance.

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Indirect Measurement (continued)

• A billboard is 18 feet high and casts a 24 foot
  shadow. A flagpole next to the billboard casts a
  60 ft. shadow. Find the height of the flagpole.

• Jesse wanted to determine the height of the oak
  tree located in the school yard. His shadow
  measured 2 ft. 6 in. If Jesse is 6 ft. 2 in.
  tall, what is the height of the tree if the
  shadow of the tree is approximately 35 feet at
  the same time of day.

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Indirect Measurement (continued)

• Tonis dad works in construction and knows that
  safety is of utmost concern when placing a ladder
  against a house. If he is using a 24 ft. ladder,
  he makes sure that he has the foot of the ladder
  on flat ground 8 ft. from the house. In order to
  keep this same ratio, how far from the house
  should he place an 8 ft. ladder? If he is using
  the 24 ft. ladder as described above, what is the
  vertical distance from the top of the ladder to
  the ground?

Similar Figures Worksheet (NCDPI Strategies)
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16. Finding Heights (NCTM Navigating Measurement)

In ancient times (600 B.C.), the Greek
mathematician Thales used shadows to determine
the height of pyramids in Egypt.

• Thales measured the height of a vertical rod (4
  ft.) and the length of its shadow (5ft.). He
  knew that the right triangle formed by the rod
  and its shadow was similar to the triangle formed
  by the height of the pyramid and the distance
  from the center of the pyramids base to the tip
  of its shadow (535 ft.). To find the pyramids
  height, he compared the ratio of the length of
  the rod to the length of its shadow with the
  ratio of the pyramids height to the distance
  from the center of its base to the tip of its
  shadow.
• What is the height (h) of the pyramid?

d 535 ft.
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Use Thales shadow method to find the height of
the flagpole outside your school.

• Work in groups of three.
• 1. Did everyone in your group find the same ratio
  for the length of their height to the length of
  their shadow? Explain.
• 2. Did everyone in your group find the same
  approximate height for the object?
• 3. Did Thales shadow method work? Explain.

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References

• Balanced Assessment Middle Grades
• Connected Mathematics
• Exemplary Mathematics Assessments Tasks for
  Middle grades
• Mathscapes
• NCDPI Strategies
• NCTM Navigating Through Geometry Grades 6-8
• NCTM Navigating Through Measurement Grades 6-8