DEFEATING "MANHOLE MONSTERS" BY SIMILARITY
Grades 9 - 12
In this lesson and its extension opportunities, students should
learn the effect of scaling on perimeter, area, and volume. They will discover
that if a figure is scaled by a factor of s then the perimeter of the scaled
figure is s times the original perimeter, the area is s^2 times the original
area, and the volume is s^3 times the original volume.
They will make predictions of the effect of scaling on the volume of "grow-able"
figures after measuring original and increased lengths of those figures.
They will check their predictions against the actual volume as read from
a graduated cylinder.
They should learn that though scaling can be done in theory, it cannot always
be done in reality ñ sometimes bigger is not better as in the case
of the manhole monsters. They will construct their own manhole monsters
to illustrate this concept.
"Project Mathematics: "Similarity" (#3)"
Students will be able to:
- find perimeter and area of a plane figure after being scaled by a
factor of s;
- find perimeter, surface area, and volume of a solid figure after being
scaled by a factor of s, and
- find the relationship between weight and pressure per unit of area
per group of two students
- 1 grow-able figure (small figures in the shape of dinosaurs, etc.
which expand in water, and are available in toy stores)
- rulers
- graduated cylinders
- cubes for constructing solids
- 1 C container of play dough or plasticene clay
- four 1/8" dowels cut in 2" lengths
- four 1/4" dowels cut in 4" lengths
The day before the lesson, pairs of students should be given
a grow-able figurine and asked to name it, to measure the length of their
figurine with a tape measure, and to find the volume of their figurine by
submersing it in a graduated cylinder half filled with water. Measurements
should be recorded for later use. The figurines should remain submerged
and will continue "growing" overnight.
Before viewing the video, the students should measure the lengths of their
enlarged figurines. They should then predict the increased volumes of their
figurines based on the original length and volume as compared to the new
length.
The focus for viewing is a specific responsibility or task(s)
students are responsible for during or after watching the video to focus
and engage students' attention. As the students watch this video, it will
be their responsibility to watch the growth of length and to predict how
that will affect the growth of perimeter, area, and volume.
BEGIN the video at the Tacoma Narrows Bridge model and
continue through the Manhole Monsters clip. STOP the video after
the narrator says , "We will see how mathematics defeats these monsters.''
FAST FORWARD to section 6 (perimeter) and RESUME the video.
PAUSE the video after the narrator says, "If the scaling factor
is s then the perimeter of the triangle will be..." Have students finish
the sentence. RESUME the video so that students can hear the narrator
finish the sentence. PAUSE the video and ask students what the perimeter
of the scaled triangle would be if the original had sides of length a, b,
and c. Ask them to express this in two ways. (The answers should be say
+ sb+ say and s(a+b+c).) RESUME the video so that students can check
their answers. PAUSE the video immediately after this and ask students
to give the perimeter of a triangle that is the result of increasing a triangle
of sides 5, 7, and 10 by a scale factor of 3.
START the video at section 7 ( area) and pause after the narrator
asks, "If the rectangle is stretched vertically and horizontally by
a scale factor of s then the area is stretched by?" Have students finish
the sentence. RESUME the video so that students can hear the narrator
finish the sentence. STOP the video after the narrator states, "Triangular
area increases by a scale factor of s^2." Ask students to find the
area of a triangle that is the result of scaling a triangle of base 5 and
height 8 by a scale factor of 10.
RESUME the video and show into section 8 (volume) until the narrator
says, "The cube is scaled by a factor of 2." STOP the video
here and ask students, "What do you think the effect of a scale factor
of 2 will have on the volume of the cube?" Give the pairs of students
unit cubes and ask them to record the length of a side, the surface area,
and the volume of a unit cube. Ask them to then scale their cube by a factor
of 2, and record the same data for the larger cube. Ask them to compare
the data from the two cubes and to write those comparisons in sentence form.
START the video and have them check their comparisons. PAUSE
right after the narrator states the results, and ask students to generalize
the effect of a scale factor of s on length, area, and volume. RESUME
the video so that students can check their answers. STOP the video.
START the video and view through the end of Applications to Biology.
Ask students to write an answer to this question: "Why is it impossible
for a Manhole Monster to exist?" Have students read their answers.
(This part of video may have to be repeated.)
1. Student pairs will now calculate the actual volume of their
enlarged figurines by using their graduated cylinders. They will submerge
their enlarged figurines in a half-filled graduated cylinder and calculate
the displacement of water. This displacement will give them the volume of
their enlarged figurine. They will compare this volume to that which they
predicted before viewing the video, and that which was calculated based
on what they learned in the video. They will be asked to explain the differences
between these values. They will also be asked to state the incorrect assumption
that they made when predicting the volume of their enlarged figurine ñ
namely, that if the length increases by a factor of k, then the volume will
also increase by a factor of k.
2. Student pairs will create their own manhole monster out of play dough
or clay that can be supported on four 1/8 inch dowel legs. Next, they will
double the linear dimensions of their monster, (length, width, and height),
and its dowel legs. They will need four 1/4 inch dowel legs since diameter
is a linear measure and it too has doubled. They should find that the volume
of their enlarged monster is such that it weighs too much to be supported
by its new legs. This is because the area of the cross-section of the legs
has increased four-fold, but the volume of the monster has increased eight-fold,
thus causing too much pressure on the legs.
Have someone who is a model-railroad enthusiast speak to the
class about scale models. Have a cartographer visit the class and explain
the significance of scale to map-making. Have someone who builds models
in order to manufacture larger products explain the significance of models
to manufacturing.
Science, and Technology: Have the students build model bridges and
test for strength. Give prizes to those whose are strongest, and most efficiently
built.
Art: Students can re-construct a famous painting by taking squares
of a reduced copy and enlarging those squares. The end product will not
be known to the students until they have each contributed their enlarged
pieces of the puzzle.
MASTER TEACHERS: Elizabeth Marquez and Joyce Dul-Jacunski
NOTE TO TEACHER
The students should have prior knowledge of similar figures having the same
shape, scaling factors, area, perimeter, volume, the distributive law, and
simple powers.
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