Developing an Understanding for Measurement Through Visual Models

1
Developing an Understanding for Measurement
Through Visual Models

• Jim Rahn
• LL Teach, Inc.
• www.jamesrahn.com
• James.rahn_at_verizon.net

2
Comparing and Contrasting Measurement Systems

• There are many ways to describe the can of soda.
• It could be described by
• its height
• its weight
• its capacity
• its temperature

3

• Estimate its height. Write your response on your
  communicator. Remember to use units on measure
  on your answer.

4 ½ inches
4

• Compare the weight of the can with a 1 pound of
  beans
• Guess the weight of the full soda can. Remember
  to put a unit of measure on your answer.

13 ounces
5

• Estimate the number of fluid ounces of liquid
  that the can holds.

12 fluid ounces
6

• About how cold is the can of soda if it has just
  been removed from the refrigerator?

About 35 degrees
7
Thinking about Length

• Lets think about the metric system of measuring
  length.
• About how large is each measurement?
• Millimeter
• Thickness of a dime
• Centimeter
• Width of your smallest finger
• Meter
• More than a yard
• Kilometer
• A little over ½ mile (.62)

8
Common Conversions

• 1000 millimeters (mm) 1 meter (m)
• 100 centimeters (cm) 1 meter (m)
• 1000 meters (m) 1 kilometer (km)

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(No Transcript)
10
Thinking about Weight

• Lets think about the metric system of measuring
  length (About how large is each measurement?)
• Gram
• Weight of a paper clip or a dollar bill
• Kilogram
• About the weight of 2 pounds
• A liter of water weighs 1 kilogram

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(No Transcript)
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Thinking About Capacity

• Lets think about the metric system of measuring
  length (About how large is each measurement?)
• Liter
• A little greater than 1 quart
• Milliliter
• 1 cubic centimeter

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(No Transcript)
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Thinking about Temperature

• Temperature of ice
• 32F
• Temperature of a freezer
• 15 to 25F
• Below freezing
• Below 32F
• Below 0 we use negative numbers
• 1, 2,

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Thinking about Temperature

• Freezing
• 32F (Fahrenheit)
• Body temperature
• 98.6F
• Hot weather
• 85 to 100F
• Comfortable weather
• 65 to 75F
• Temperature in refrigerator
• 38 to 42F

16
Change these Temperatures to Celsius

• Temperature of ice
• 32F
• Temperature of a freezer
• 15 to 25F
• Below freezing
• Below 32F
• Below 0 we use negative numbers
• 1, 2,

17
Change these temperatures to Celsius

• Freezing
• 32F (Fahrenheit)
• Body temperature
• 98.6F
• Hot weather
• 85 to 100F
• Comfortable weather
• 65 to 75F
• Temperature in refrigerator
• 38 to 42F

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Reviewing Converting Basic Units

• Work in pairs.
• Each pair will receive two standard die, two
  markers and the appropriate weight, capacity, or
  length template. Each player should also use
  his or her Communicator.
• Players alternately roll the dice and shade in
  that number of units on the appropriate template.
  The first player who cannot model the measure on
  the template because it is too full, wins the
  game.
• In addition to shading the template, use the
  Single Unit Customary Game Record sheet to keep
  a running total of the accumulated quantity.

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Sample Game

• Round 3
• Player A rolls a 6 and a 6, so Player A shades in
  12 additional ounces on the template.
• Player B rolls a 1 and a 3, so Player B shades in
  4 additional ounces on the template.
• Round 4
• Player A rolls a 2 and a 5, so Player A shades in
  7 additional ounces on the template.
• Player B rolls a 6 and a 4 so Player B shades in
  10 additional ounces on the template.
• Play continues until 4 or more quarts is reached.

• Round 1
• Player A rolls a 4 and a 2, so Player A shades in
  6 ounces on the template.
• Player B rolls a 6 and a 3 so Player B shades in
  9 ounces on the template.
• Round 2.
• Player A rolls a 3 and a 5, so Player A shades in
  8 additional ounces on the template.
• Player B rolls a 5 and a 2 so Player B shades in
  7 additional ounces on the template.

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Think About It

• Do you understand more about the relationship
  between the two systems?
• Do you have a deeper understanding for the size
  of certain units?
• Do you understand the systems of measurement
  better?

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Creating a Ruler
22
Pick up one of the strips of paper. This strip
will be considered to be one unit.
23
Take the blue strip and fold it in half. This
line indicates that the unit has been folded into
two equal parts or 1 halves. Draw the line on
the fold.
24
Take the red strip. It also represents 1 unit,
but this time we will fold it into four equal
parts by folding it in half and then in half
again. When you open the strip you will have
four equal parts or the strip as been divided
into fourths. Draw the line on each fold.
25
Take the white strip. It also represents 1 unit,
but this time we will fold it into eight equal
parts by folding it in half, then in half again,
and then in half one more time. When you open the
strip you will have eight equal parts or the
strip as been divided into eighths. Draw the
line on each fold.
26
Take the yellow strip. It also represents 1
unit, but this time we will fold it into sixteen
equal parts by folding it in half, then in half
again, and then in half one more time.
This time open the fold paper once so it looks
like a book. Fold each of the edges of the
book to the binding. This will create 16 equal
parts or sixteenths. Draw the line on each
fold.
27
To make the unit strip each of the strips will be
placed so it overlaps about half the one before
it. Start with the yellow strip, then the white
strip, red strip and blue strip. Each time line
up the matching marks as best as possible.
28
Once they are aligned take the four pieces of
tape and place them vertically/diagonally to hold
the strips together. (You can also have students
glue the strips onto a 9 by 12 piece of
construction paper.)
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• How many equal parts has the yellow strip been
  divided into?
• What would one part of the yellow strip be? Why?
• What would 3 parts of the yellow strip be as a
  fraction? Why?

30

• How many equal parts has the white strip been
  divided into?
• What would one part of the white strip be? Why?
• What would 5 parts of the white strip be called
  as a fraction? Why?

31

• How many equal parts has the red strip been
  divided into?
• What would one part of the red strip be? Why?
• What would 3 parts of the red strip be called as
  a fraction? Why?

32

• How many equal parts has the blue strip been
  divided into?
• What would one part of the red strip be? Why?
• What would 1 part of the blue strip be called as
  a fraction? Why?

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• Place the unit fraction stick/ruler into your
  Communicators.
• Put a line through, shade, or put Xs on top of
  the part of the Unit Sticks I describe.

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• Use the white strip to show 3/8.
• Turn over the Communicator.
• Show your answer on the count of 3.

35

• Use the yellow strip to show 7/16.
• Turn over the Communicator.
• Show your answer on the count of 3.

36

• Use the red strip to show 1/4.
• Turn over the Communicator.
• Show your answer on the count of 3.

37

• Use the white strip to show 5/8.
• Turn over the Communicator.
• Show your answer on the count of 3.

38

• Use the blue strip to show 1/2.
• Turn over the Communicator.
• Show your answer on the count of 3.

39

• Use the red strip to show 3/4.
• Turn over the Communicator.
• Show your answer on the count of 3.

40

• Use the yellow strip to show 9/16.
• Turn over the Communicator.
• Show your answer on the count of 3.

41

• Use the yellow strip to show 12/16.
• Turn over the Communicator.
• Show your answer on the count of 3.

42

• Use the white strip to show 6/8.
• Turn over the Communicator.
• Show your answer on the count of 3.

43

• Use the red strip to show 2/4.
• Turn over the Communicator.
• Show your answer on the count of 3.

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• Use the yellow strip to show 4/16.
• Turn over the Communicator.
• Show your answer on the count of 3.

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• Use the yellow strip to show 15/16.
• Turn over the Communicator.
• Show your answer on the count of 3.

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• Use the quarter strip to show 3/4.
• Turn over the Communicator.
• Show your answer on the count of 3.

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• Use the sixteenths strip to show 11/16.
• Turn over the Communicator.
• Show your answer on the count of 3.

48

• Use the halve strip to show 2/2.
• Turn over the Communicator.
• Show your answer on the count of 3.

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• Use the eighths strip to show 7/8.
• Turn over the Communicator.
• Show your answer on the count of 3.

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• Use the sixteenths strip to show 7/16.
• Turn over the Communicator.
• Show your answer on the count of 3.

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• Use the quarter strip to show 1/4.
• Turn over the Communicator.
• Show your answer on the count of 3.

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• Use the sixteenths strip to show 10/16.
• Turn over the Communicator.
• Show your answer on the count of 3.

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• Use the eighths strip to show 4/8.
• Turn over the Communicator.
• Show your answer on the count of 3.

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• Use the sixteenths strip to show 5/16.
• Turn over the Communicator.
• Show your answer on the count of 3.

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• Use the half strip to show 1/2.
• Turn over the Communicator.
• Show your answer on the count of 3.

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• Use the 8ths strip to show 2/8.
• Turn over the Communicator.
• Show your answer on the count of 3.

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• Use the sixteenths strip to show 2/16.
• Use the eighths strip to show 1/8.
• What do you notice about 2/16 and 1/8?

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• Use the sixteenths strip to show 12/16.
• Use the eighths strip to show 6/8.
• What do you notice about 12/16 and 6/8?

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• Use the eighths strip to show 6/8.
• Use the quarter strip to show 3/4.
• What do you notice about 6/8 and 3/4?

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• Show 2/8 on the Unit Stick
• Show 2/8 on the Giant Inch
• What are two other names for 2/8?

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• Use the eighths strip to show 2/8.
• Use the quarter strip to show 1/4.
• What do you notice about 2/8 and 1/4?

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Place the Giant Inch in the Communicator
63

• Shade in 2/16 on the unit stick.
• Shade in 1/8 on the unit stick.

Label 2/16 and 1/8 on the Giant Inch
64

• Shade in two more sixteenths on the Unit Stick.
• How many eighths is 4/16?
• What is another name for 4/16 in quarters?

Label 4/16, 2/8 and 1/4 on the Giant Inch
65

• Shade in two more sixteenths on the Unit Stick.
• How many eighths is 6/16?

Label 6/16, and 3/8 on the Giant Inch
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• Shade in two more sixteenths on the Unit Stick.
• How many eighths is 8/16?
• What is another name for 8/16 in quarters?
• What is another name for 8/16 in halves?

Label 8/16, 4/8, 2/4, and 1/2 on the Giant Inch
67

• Shade in two more sixteenths on the Unit Stick.
• How many eighths is 10/16?
• What is another name for 10/16 in quarters?

Label 10/16 and 5/8 on the Giant Inch
68

• Shade in two more sixteenths on the Unit Stick.
• How many eighths is 12/16?
• What is another name for 12/16 in quarters?

Label 12/16, 6/8, and 3/4, on the Giant Inch
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• Shade in two more sixteenths on the Unit Stick.
• How many eighths is 14/16?

Label 14/16 and 7/8 on the Giant Inch
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• Shade in two more sixteenths on the Unit Stick.
• How many eighths is 16/16?
• How many fourths are 16/16?
• How many halves are 16/16?

Label 16/16, 8/8, 4/4, 2/2 and 1 on the Giant Inch
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• Show 3/16 on the Unit Stick
• Show 3/16 on the Giant Inch

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• Show 9/16 on the Unit Stick
• Show 9/16 on the Giant Inch

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• Show 3/8 on the Unit Stick
• Show 3/8 on the Giant Inch
• What is another name for 3/8?

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• Show 7/8 on the Unit Stick
• Show 7/8 on the Giant Inch
• What is another name for 7/8?

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• Show 1/2 on the Unit Stick
• Show 1/2 on the Giant Inch
• What are three other names for 1/2?

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Think About It

• Was understanding of the ruler developed?
• Were equivalent fractions explored?
• Were pictures or visual of equivalent fractions
  presented?
• Were you just taught rules on how to read a
  ruler?
• Did the construction of the ruler help you
  understand the ruler better later?
• Is a ruler less mysterious?

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Understanding Area
78

• Use one geoband to make a rectangle whose area is
  15 square units.
• Determine the dimensions of a rectangle with the
  specified area of 15 square units.
• What other size rectangle could be made if the
  geoboard was larger?
• What if the dimensions were not counting number?
  

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• Which one or ones do not have an area of 15
  square units.

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How can you determine the area of a rectangle?
81

• Use the geoboard to make a square with an area of
  36 square units and one vertex at (3, 1).
• Lift the geoband from the vertex at (9, 7) so
  that a right triangle is formed.
• Determine the area of the triangle two ways.

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• Make a square with an area of 16 square units and
  a vertex at (2, 3).
• Lift the geoboard from the vertex at (2, 7) to
  create a right triangle.
• Find the area two ways.

83

• Make a square with an area of 25 square units and
  a vertex at (8, 3).
• Lift the geoboard from the vertex at (3, 8) to
  make a right triangle.
• Find the area two ways.

84

• Make a rectangle whose area is 20 square units
  anywhere on the geoboard.
• What will be the area of the related right
  triangle? Why?
• Lift the geoband from one of the vertices of the
  rectangle creating a right triangle that has the
  same base and height as the original.
• By counting full and partial squares, estimate
  the area of the triangle formed.

85

• Will all the rectangles that are 20 square units
  will be congruent? Why or why not?
• Will all related right triangles have the same
  area?
• Are there other rectangles that are 20 square
  units that cant be built on the geoboard?
• Will all related right triangles for these new
  rectangles have the same area?
• What can you conclude?

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Area of Non-right Triangles

• Make a square with an area of 36 square units and
  one vertex at the ordered pair (0, 0).
• Take another geoband and make a right triangle
  whose base and height both measure 6 units, and
  whose right angle is at the ordered pair (0, 0).
  What is the area of this triangle?

87

• Move the geoband forming the right triangle so
  that the top vertex is at the ordered pair (3,
  6).
• What makes this triangle isosceles?
• Why isnt it also a right triangle?
• What would be the name of this triangle if it
  were classified by its angles instead of its
  sides?

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• What is the area of this isosceles triangle? Why?
  
• How do you know for sure?
• What is the length of the base of this triangle?
  What is its height? What is its area?

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• Now move the top vertex of the isosceles triangle
  to the ordered pair (4, 6).
• What kind of triangle is now formed?
• Why is this a scalene triangle?
• If this triangle were classified by its angles,
  what would it be called?
• What do you think the area of this triangle is?
  Is it still half of the square?
• Show that it is 18 square units.

90

• Make a right triangle with an area of 24 square
  units whose right angle vertex is at the ordered
  pair (4, 0).
• Make the related rectangle. What is the area of
  the rectangle?
• What is the area of the triangle?

91

• Move the geoband so that the top vertex is at the
  ordered pair (6, 8) instead of (4, 8).
• What kind of triangle did the right triangle
  become?
• Why is it scalene?
• What is the area of this scalene triangle?
• Support your answer.

92

• If you make different triangles by moving the
  geoband along the top base of the rectangle, the
  triangle changes shape. But does it change its
  area in relationship to the rectangle?
• What is the area of each triangle that is formed?
  Why is this?

93

• All triangles have a related rectangle.
• Which of the triangles above can be derived from
  a square whose area is 36 square inches?
• What is the base and height of the related
  rectangle for Triangle B?

94

• Make a triangle on your geoboard that has a
  height of 8 units and a base with a length of 3
  units. Show the triangles.
• Use another geoband to make the rectangle from
  which your triangle could be derived. Hold up the
  answers. Are all the rectangles you made
  congruent?
• Are all the triangles the same area? Why?

95

• Make a rectangle that has the same height and
  base as a scalene triangle whose area is 16
  square units.

96

• Draw on a Communicator a scalene acute triangle
  similar to the one shown.
• Draw the height and base of this triangle as
  shown. What must be true?
• Draw in the relative rectangle.
• Turn the Communicator several times and draw a
  new height from this direction.
• Is the height and base the same?

97

• Make a parallelogram on your geoboard by
  connecting the coordinates (1, 5) (3, 7) (10,
  7) and (8, 5)
• Find the area of the parallelogram. Support your
  reasoning.

98

• Make a parallelogram on your geoboard by
  connecting the coordinates (0, 8) (6, 8) (10,
  4) and (4, 4)
• Find the area of the parallelogram. Support your
  reasoning.

99

• Graph both parallelograms on the geoboard.
• (0,10), (3,10), (7,6), and (4,6)
• (0,5), (4,5), (10,0) and (6,0)
• Find their area.

100
Area for Figures Made from a Combination of
Polygons

• Use a geoboard and a geoband to create a square
  with an area of 16 square units and a vertex at
  the ordered pair (0, 0).
• Use a second geoband to create another square of
  16 square units that DOES NOT overlap the first
  one and is anchored at the ordered pair (4, 0).
• Form a triangle from the second square by lifting
  the geoband from the vertex at (8, 4).
• Determine the area of the figure formed by the
  square and the triangle.
• What is this figure called?