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Jun 4, 2017Jun 4 at 12:09pm
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To evaluate each expression, I started out by writing down my
birthday, 6/15/94 and then I wrote down the problems keeping
in mind the variables a, b, and c; my problems looked like this,
6^3 - -15^3, (6- -15) [6^2+6(-15) + -15^2] and (-15 – 94)/ [2(-
15) – 6]. To start on the first problem, I broke down each
number and wrote it like this, 6 X 6 X 6 + 15 X 15 X 15, I did
this because the exponent of each number was 3. I then did the
multiplication and got this, 216 + 3,375. Lastly, I added the two
remaining numbers and got 3,591. For my second problem, I
started by changing (6 - - 15) into (6 +15) because a positive
minus a negative is a positive. I then made [6^2 +6(-15) + -
15^2] into [36(-90) + 225]. After doing this I added my first
two numbers and got 21. I then did 36(-90) and got -3,240;
bringing my problem to look like this, (21) (-3,240 + 225). To
finish off the problem I first added -3,240 and 225 resulting in –
3,015; I then multiplies (21) (-3.015) and got for my result -
63,315. For my last problem, I did not get an integer for my
final answer but I did get the lowest terms for the answer I got.
To start off I wrote the problem as I did with all the
expressions, I then did (-15 – 94) getting -109 and [2(-15) – 6]
resulting in -30 – 6. Now I have a problem that looks like this, -
109/ -30 – 6; I keep the -109 and I subtract the -30 and the 6,
getting -36. Now my problem looks like, -109/ -36. The – 36 is
the divisor in this problem. The answer I got was -3.027 and I
reduced it to -3.03.
The Reasons for the Seasons
Ask a fifth-grader why he or she believes Earth
has seasons, and the answer usually involves a
mistaken notion about Earth’s distance from

the Sun. Not only are elementary students often
stumped by the seasons, but adults also commonly
misunderstand this concept—even Harvard University
graduates (Schneps, Sadler, and Woll 1988).
Children understand that temperature usually fluctuates
depending upon one’s nearness to a heat source, which gives
rise to the false analogy of the Sun’s heat and its presumed
effect on Earth’s seasonal temperature fluctuations. Another
explanation for this widespread erroneous impression
may lie in the two-dimensional drawings that often depict
Earth’s orbit around the Sun. Most diagrams emphasize the
elliptical nature of Earth’s orbit. Although it is technically
elliptical, Earth’s orbit is a nearly perfect circle, with only a
2% difference between its apogee (the point in Earth’s orbit at
which it is farthest from the Sun’s center) during the month
of June and its perigee (the point in Earth’s orbit at which it
is closest to the Sun’s center) during the month of January.
Perigee occurs in January, corresponding with the Northern
Hemisphere’s winter, and the apogee occurs in June,
corresponding
with the Northern Hemisphere’s summer. The construction of a
three-dimensional model of the
changing seasons using simple materials has been successful
in correcting students’ misinterpretation of the cause
of the seasons (Lambert and Ariza 2008).
Like the other planets, the Earth rotates on its axis as
it revolves around the Sun. Earth is currently tilted 23.5º
on its axis and remains in the same alignment with respect
to the background stars throughout its orbit around the
Sun, which takes 365.2 days. The
North Pole always points toward
Polaris or the North Star. We know
that Earth is tilted 23.5º because of
the geometric relationship between
Earth and the Sun. The difference
between the angle of the midday Sun

on an equinox (September or March) and a solstice (December
or June) is equal to 23.5º.
As Earth revolves around the Sun, its axis remains
tilted 23.5º in the same direction. However, the direction
of Earth’s tilt with respect to the Sun does change, causing
the seasons. When the Northern Hemisphere is tilted
toward the Sun, that half of the Earth receives more direct
sunlight and has summer. At the same time, the Southern
Hemisphere is tilted away from the Sun and has winter.
In this lesson, students employ a simple model to
learn how Earth’s tilt and revolution around the Sun
causes our seasons.
Julie Lee Lambert ([email protected]) is an associate
professor at Florida Atlantic University in Boca Raton,
Florida. Suzanne Smith Sundburg ([email protected]
verizon.net) is a freelance science writer and editor in
Arlington, Virginia.
References
Lambert, J. and E.N. Ariza. 2008. Improving achievement
for linguistically and culturally diverse learners
through an inquiry-based Earth systems curriculum.
Journal of Elementary Science Education 20 (4):
61–79.
Schneps, M.H., P.M. Sadler, and S. Woll. 1988. A private
universe: Misconceptions that block learning [Videorecording].
Santa Monica, CA: Pyramid Films.
Explaining Seasons With
Tilting Toothpicks
What causes the seasons?
Grade Level: Grades 4–6
Process Skills: Observing, modeling, inferring, and
communicating
Engage
To assess students’ prior knowledge, first each student
answered a brief preassessment (see NSTA Connection).

The preassessment helped determine whether
students thought Earth’s changing distance to the Sun
causes seasons or whether students thought that the tilt
physically changes during different seasons. Additionally,
it helped teachers determine if students knew that
the Northern and Southern Hemispheres are experiencing
opposite seasons when shown a diagram of the
Sun’s rays and a tilted Earth.
Teams of students were then asked to make a model of
the seasons using a small craft light, four Earth models
made of Styrofoam, four toothpicks, and a protractor.
Students were told that the toothpick represented Earth’s
axis and to push the toothpick into the ball through the
North Pole so that the end would go out at the South
Pole. They also were told that each Earth model should
represent one season.
Teams were asked to sketch their physical model and
to answer a series of questions (the summary of embedded
assessments is available online; see NSTA Connection).
Each team then presented its model. The initial models
revealed students’ naïve or alternative conceptions. Most
of the teams initially explained the seasons as being the
result of Earth’s distance to the Sun. Most teams had the
tilt of the summer and winter Earth models correct, but
they were not sure what to do with the tilt in the spring
and fall Earth models. Figure 1 shows a typical model in
which the students placed Earth closer to the Sun during
the summer season and farther in the winter with the correct
tilt, but then made the tilt more vertical for the spring
and fall season. Occasionally, a model did not match the
verbal explanation. For example, a team may have said
that it kept the tilt the same, but the model showed a
change in the direction of the tilt (Figure 2).
Explore and Explain
Teams were then asked to read a narrative describing
Earth’s orbit and its proximity to the Sun throughout

the seasons, its tilt on its axis in relation to the Sun,
and the amount and angle of direct rays of sunlight
that each hemisphere receives during a particular
season (see NSTA Connection). Based on the information
contained in the story, the teams were asked
to revise their models accordingly. Each team’s revised
model was then checked, and the previous
explanation was expanded on during a whole-class
discussion. Assessment was again embedded (see
NSTA Connection).
Each team eventually constructed a correct model of
the seasons. One student helped his team understand the
changing seasons by using a protractor to place each of the
four toothpicks (without the Styrofoam Earth spheres) on
the base, each at a 23.5º angle and all pointing the same
direction. Immediately, students on his team seemed to
understand the cause of the seasons. This simple explanation
seems to help most students construct a correct
model of the seasons.
The lesson highlighted one of the more difficult
concepts underlying the cause of the seasons—the idea
of direct and indirect light. Students sometimes asked
why the Arctic is not warmer when it receives almost 24
hours of daylight during the summer. To help students
understand why regions near the equator are warmer,
a teacher can hold a flashlight perpendicular to a line
drawn on a board. Using a marker, the bright area can
be circled. Then the light should be moved so that it
shines over the line at an angle, and the marker should
again be traced around the bright area. Students will
observe that the area was smaller when the light was
shone at a perpendicular angle, and therefore the Sun’s
rays would be spread over less surface area and the
area would be much warmer. When Sun’s rays strike
Earth’s surface nearer the equator, the Sun’s radiation
is spread over a smaller area than at higher latitudes.

See the “What Causes the Seasons?” Science 101 column
(Robertson 2007) for a detailed explanation of
this phenomenon.
Extend
Students next applied their understanding of the real
world by constructing a working sundial to measure the
time of day (find directions online; see NSTA Connection).
As the Sun shines on the sundial, the shadow of
the gnomon’s point will cover the current time on the
time dial (Figure 3).
Next, we made an astrolabe, an instrument used to
measure the angle of an object in the sky, such as the Sun
or Moon, above the horizon (see NSTA Connection). In
Greek, the word astro means “star,” and labe means “to
find.” Both the sundial and the astrolabe can be used to
track the Sun’s path across the sky throughout the day
or year.
Finally, students compared the number of daylighthours and the
path of the Sun for each season in cities
at different latitudes. Sunrise and sunset times
for most cities can be found on the U.S. Naval Observatory’s
Astronomical Applications website (see
Internet Resource).
Reference
Robertson, W. 2007. What Causes the Seasons? (Science
101) Science and Children 44(5): 54–57.
Internet Resource
U.S. Naval Observatory’s Astronomical Applications
http://aa.usno.navy.mil/data
A
sk a fifth

-
grader why he or she believes Earth
mistaken notion about Earth
’
s distance from
the Sun. Not only are elementary students often
stumped by the seasons, but adults also commonly
misunderstand this concept
—
even Harvard University
graduates (Schneps, Sadler, and Woll 1988).
Children understand that temperature usually fluctuates
depending upon one
’
s nearness to a heat source, which gi
ves
rise to the false analogy of the Sun
’
s heat and its presumed
effect on Earth
’
s seasonal temperature fluctuations. Another
explanation for this widespread erroneous impression

may lie in the two
-
dimensional drawings that often depict
Earth
’
s orbit aroun
d the Sun. Most diagrams emphasize the
elliptical nature of Earth
’
s orbit. Although it is technically
elliptical, Earth
’
s orbit is a nearly perfect circle, with only a
2% difference between its
apogee
(the point in Earth
’
s orbit at
which it is farthest fro
m the Sun
’
s center) during the month
of June and its
perigee
(the point in Earth
’
s orbit at which it
is closest to the Sun
’

s center) during the month of January.
Perigee occurs in January, corresponding with the Northern
Hemisphere
’
s winter, and the apogee
occurs in June, corresponding
with the Northern Hemisphere
’
s summer.
The construction of a three
-
dimensional model of the
changing seasons using simple materials has been successful
in correcting students
’
misinterpretation of the cause
of the seasons (Lam
bert and Ariza 2008).
Like the other planets, the Earth rotates on its axis as
it revolves around the Sun. Earth is currently tilted 23.5º
on its axis and remains in the same alignment with respect
to the background stars throughout its orbit around the

Sun, which takes 365.2 days. The
North Pole always points toward
Polaris or the North Star. We know
that Earth is tilted 23.5º because of
the geometric relationship between
Earth and the Sun. The difference
between the angle of the midday Sun
on an equinox
(September or March)
and a solstice (December or June) is equal to 23.5º.
tilted 23.5º in the same direction. However, the direc
tion
of Earth
’
s tilt with respect to the Sun does change, causing
Ask a fifth-grader why he or she believes Earth
mistaken notion about Earth’s distance from

on an equinox (September or March) and a solstice (December
or June) is equal to 23.5º.
tilted 23.5º in the same direction. However, the direction
of Earth’s tilt with respect to the Sun does change, causing
Reproduced with permission of the copyright owner. Further
reproduction prohibited without permission.
Science 101
Nelson, George
Science and Children; Summer 2005; 42, 8; ProQuest Central
pg. 44
Reproduced with permission of the copyright owner. Further
reproduction prohibited without permission.
EARTH SEASONS DON'T FIT OTHER PLANETS
Abstract
Translate [unavailable for this document]
[Venus] on the other hand is much closer to the sun.
Temperatures are much higher. Venus' tilt is less than three
degrees from vertical, but since Venus was found to rotate in a
direction opposite to most other planets, scientists list its tilt as
177.4 degrees, almost exactly upside down. This combined with
Venus' thick carbon dioxide atmosphere means there are
essentially no seasonal changes. The mean surface temperature
is about twice as hot as your oven's maximum setting. On top of

that, there is no place on the planet to escape the heat, day or
night, equator or pole.
Full Text
As spring arrives I am reminded of a question I'm often asked,
that goes something like this: "What season is Venus (or
another planet) visible?" Planet visibility relies on the earth's
position and the planet's position relative to the sun. But
because planets move at different rates, they don't follow our
seasons from year to year.
I suspect this is one way that we earthlings demonstrate our self
centeredness by expecting all our experiences to be affected by
the seasons. Of course we have four seasons, determined by our
location and the tilt of the earth. In school we learn that people
in the southern hemisphere have seasons opposite ours. So
seasons are not even consistent over the whole planet. How then
can we expect the rest of the solar system to follow our earthly
cycles? Yet we do, falling into an easy set of expectations.
Are there seasons on other planets? Yes and no. Mars has
seasons due to its tilt of less than one degree different from
earth. But the red planet's orbit is almost twice as large as
earth's, so seasons last about twice as long. Also due to its
greater distance from the sun, Mars has temperatures that rarely
reach above freezing. Seasonal warming has been known to
cause months-long dust storms across the red planet.
Venus on the other hand is much closer to the sun.
direction opposite to most other planets, scientists list its tilt as
177.4 degrees, almost exactly upside down. This combined with
Venus' thick carbon dioxide atmosphere means there are
essentially no seasonal changes. The mean surface temperature
is about twice as hot as your oven's maximum setting. On top of
that, there is no place on the planet to escape the heat, day or
night, equator or pole.

I'm just happy to enjoy the spring of our northern hemisphere on
good old planet earth. Spring officially began at 1:26 p.m.
March 20. At the same time we can admire Venus and Mars
from afar. Find Mars located between the horns of Taurus the
Bull in the west as soon as it gets dark. Venus is our brilliant
morning star for a while, low in the east before sunrise. One of
the marvels of the universe is that the more we examine it the
more we confront the unexpected.
Maness is the director of astronomy at the Virginia Living
Museum in Newport News. Nature Notes is a bi-weekly column.
You can access the museum's Web site at
www.valivingmuseum.org. *
Illustration
Photos (b&w) courtesy of The Virginia Living Museum;
Caption: Right now, Venus is our bright morning star. Admire
Mars this spring. Logo (b&w) Virginia Living Museum
Word count: 467
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Copyright Chicago Tribune Co. Mar 21, 2006
Abstract
Temperatures are much higher. Venus' tilt is less than
three degrees from vertical, but since Venus was found to rotate
in a direction opposite to most other planets,
scientists list its tilt as 177.4 de
grees, almost exactly upside down. This combined with Venus'
thick carbon
dioxide atmosphere means there are essentially no seasonal
changes. The mean surface temperature is about
twice as hot as your oven's maximum setting. On top of that,
there is no pla
ce on the planet to escape the heat,
day or night, equator or pole.
Full Text
that goes something like this: "What
season is
Venus (or another planet) visible?" Planet visibility relies on

the earth's position and the planet's position
relative to the sun. But because planets move at different rates,
they don't follow our seasons from year to year.
I suspect this is o
ne way that we earthlings demonstrate our self centeredness by
expecting all our experiences
to be affected by the seasons. Of course we have four seasons,
determined by our location and the tilt of the
earth. In school we learn that people in the southern
hemisphere have seasons opposite ours. So seasons are not
even consistent over the whole planet. How then can we expect
the rest of the solar system to follow our
earthly cycles? Yet we do, falling into an easy set of
expectations.
Are there seasons on o
ther planets? Yes and no. Mars has seasons due to its tilt of less
than one degree
different from earth. But the red planet's orbit is almost twice as
large as earth's, so seasons last about twice as
long. Also due to its greater distance from the sun, Mar
s has temperatures that rarely reach above freezing.
Seasonal warming has been known to cause months
-
long dust storms across the red planet.
de
grees from vertical, but since Venus was found to rotate in a
direction opposite to most other planets,
scientists list its tilt as 177.4 degrees, almost exactly upside
down. This combined with Venus' thick carbon
dioxide atmosphere means there are essenti

ally no seasonal changes. The mean surface temperature is about
there is no place on the planet to escape the heat,
Abstract
Temperatures are much higher. Venus' tilt is less than
three degrees from vertical, but since Venus was found to rotate
in a direction opposite to most other planets,
Full Text
that goes something like this: "What season is
Venus (or another planet) visible?" Planet visibility relies on
the earth's position and the planet's position
relative to the sun. But because planets move at different rates,
they don't follow our seasons from year to year.
I suspect this is one way that we earthlings demonstrate our self
centeredness by expecting all our experiences
to be affected by the seasons. Of course we have four seasons,
determined by our location and the tilt of the
earth. In school we learn that people in the southern hemisphere
have seasons opposite ours. So seasons are not
even consistent over the whole planet. How then can we expect

the rest of the solar system to follow our
earthly cycles? Yet we do, falling into an easy set of
expectations.
Are there seasons on other planets? Yes and no. Mars has
seasons due to its tilt of less than one degree
different from earth. But the red planet's orbit is almost twice as
large as earth's, so seasons last about twice as
long. Also due to its greater distance from the sun, Mars has
temperatures that rarely reach above freezing.
Seasonal warming has been known to cause months-long dust
storms across the red planet.
direction opposite to most other planets,
Cori Berry
Jun 5, 2017Jun 5 at 9:21pm
Using my birth date 11/9/88 to solve the following equations
using the following variables A, B, and C. A=11 B=9=88.
My problems ended up looking like this: 1. (11^3) – (-9^3) 2.
(11- -9) +(11^2 + 11(-9) + -9^2 and the final problem, 3. (-9 –
88) / (2(-9) – 11). When solving my first problem I knew that
with the exponent number 3 it would look like this (11x11x11)
= 1,331 and (-9x -9x -9) = -729. So, then it is laid out like this
(1,331) – (-729) but, we are subtracting a negative integer it
changes to positive so, now it looks like this 1,331 + 729 =
2060. The second problem as shown above is would be written
like this (11- -9) + (11^2 +11(-9) +(-9^2). Then it would look

like this (11+9) (121 – 99 + 81) then (20) (103) =2060. For my
last problem (-9 – 88) / 2 (-9) – 11) next my answer came to this
(-97) / (-29). Now I am going to take the dividend by the divisor
number and reduce my answer to lowest terms. My answer to (-
97) / (-29) = 3.34. Since a negative and a negative is divided
together my answer becomes positive. This was a very big deal
for me considering I struggle very badly when it comes to math
so any positive criticism would be greatly appreciated. Thanks
in advance.
Cindi Gallegos
Jun 6, 2017Jun 6 at 6:46pm
Here are the equations based on my birthday: 6/20/77.
1. (a x a x a) - (b x b x b)
(6x6x6) – (-20 x -20 x -20x)
(216) – (-8000)
216 + 8000 = 8216
I changed the variables a, b, and c for my birthday. They are a =
6, b = -20, and c = 77.
1. (a-b)(a^2 +ab + b^2)
(6 - -20)(6^2 + 6(-20) + -20^2)
(6 + 20)(36 – 120 + 400)
26(312) = 8216
Since the integer is negative and we are subtracting, I changed
the sign from a – to a +. I noticed that the total was the same as
the first problem. I noticed that the exponents were different
from 3 in the first problem and 2 in the second problem.
1. (b – c) / (2b – a)
-20 -77 / 2(-20) -6
-97 / -46
= 2.11
I need to simplify the answer to the lowest terms. The divisor is
a negative number like the dividend.

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