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Math that’s oUt
oF this World
1. What does the last frame of the
cartoon suggest about how some
people feel about math?
2. Astronomers measure distances
in the solar system in astronomical
units (AUs). By definition,
1 AU = the average distance from
the sun to Earth. This distance is
approximately 150,000,000 km,
or about 93,000,000 mi.
a. The distance from the sun to
Mercury is about 58,000,000 km.
Describe two ways to convert that
distance to an AU.
b. Describe a shortcut you could
use to convert a distance in millions of kilometers to an AU.
3. Complete the table at right to find
the mean distance from the sun in
AUs for each planet listed.
4. The speed of light is calculated to
be about 299,792,458 m/sec.
a. About how far, in kilometers,
does light travel per second?
b. How long will it take light to
travel 1 AU? Round to the nearest tenth of a minute. (Recall:
1 AU = 150,000,000 km.)
c. How long will it take light to
travel from the sun to each
planet? Round each result to the
nearest tenth of a minute.
Planet
Mercury
challenge
5. Draw a scale model of the solar
system, in AUs, along a number
line. This will show the relative
distance of each planet from the
sun. Draw the sun on the left, and
space out the planets to the right.
If you use the scale 1 cm = 1 AU,
you should be able to list the solar
system along a number line drawn
diagonally on 8 1/2 in. × 11 in.
paper.
Mean distance from the
sun (in millions of km)
conversion
calculation
aU (rounded to
nearest tenth)
58
58 ÷ 150 ≈ 0.387
0.4
Venus
108
Earth
150
Mars
228
Jupiter
778
Saturn
1427
Uranus
2871
Neptune
4497
from the August 2013 issue of
Copyright © 2013 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.
This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
Edited by David B. Spangler, McGraw-Hill Education, and Katie A. Hendrickson, Ohio University, Athens. Classroom teachers interested
in field-testing or submitting a cartoon should contact David Spangler, [email protected]. The cartoons must include
the date and the newspaper syndicate that holds the copyright.
SOLUTIONS
1. Answers will vary. The cartoon
suggests that some people will
literally run away from anything
requiring mathematics.
3. See the table below.
2.a. Solution methods will vary.
Method 1: Divide the number of kilometers as follows:
58,000,000 ÷ 150,000,000
Method 2: Solve the following proportion, where x is the
distance in AUs:
58, 000, 000 km 150, 000 km
=
x AU
1 AU
b. B
efore converting, divide each
number by 1,000,000 (to remove
the final 6 zeros of each number). Then divide the number of
kilometers by 150.
Planet
Mean Distance from
Sun (in millions of km)
Conversion
Calculation
AU (rounded to
nearest tenth)
Mercury
58
58 ÷ 150 ≈ 0.387
0.4
Venus
108
108 ÷ 150 = 0.72
0.7
Earth
150
150 ÷ 150 = 1
1.0
Mars
228
228 ÷ 150 = 1.52
1.5
Jupiter
778
778 ÷ 150 ≈ 5.187
5.2
Saturn
1427
1427 ÷ 150 ≈ 9.513
9.5
Uranus
2871
2871 ÷ 150 = 19.14
19.1
Neptune
4497
4497 ÷ 150 = 29.98
30.0
4. a. D
ivide 299,792,458 by 1,000.
The speed of light is approximately 299,792 km/sec.
b. 1 50,000,000 km ÷ 299,792 km/sec.
≈ 500 sec., or about 500/60 =
8.3 min. It will take light
about 8.3 min. to travel 1 AU.
c. To find the approximate amount
of time it will take light to travel
from the sun to each planet,
multiply the number of AUs
for each planet by 8.3. (See the
calculations on page 18.)
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Vol. 19, No. 1, August 2013
callspkrs_0513
●
NCTM journals 0813
Mathematics Teaching in the Middle School
17
Mercury: 8.3 min. × 0.4, or
about 3.3 min.
Venus: 8.3 min. × 0.7, or about
5.8 min.
Earth: 8.3 min. × 1, or about
8.3 min.
Mars: 8.3 min. × 1.5, or about
12.5 min.
Jupiter: 8.3 min. × 5.2, or about
43.2 min.
Saturn: 8.3 min. × 9.5, or about
78.9 min.
FIELD-TEST COMMENTS
The first challenge in my sixth-grade
prealgebra class was defining astrology!
We had a discussion about horoscopes,
and then the cartoon made more sense
to my students. They readily agreed
that many people avoid math.
Most students had little trouble
computing the conversion of kilometers to AUs; some students set up
a division problem, whereas others set up a ratio. A few had difficulty understanding that it was not a
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18
Uranus: 8.3 min. × 19.1, or about
158.5 min.
Neptune: 8.3 min. × 30, or about
249 min.
5.
multiplication problem. One student
insisted that you should multiply,
so I asked her, “How many AUs are
in 300,000,000 km?” She promptly
answered, “Two,” and said she multiplied. I then asked, “Does 300,000,000
× 150,000,000 = 2?” She said, “No;
I meant I added.” So I asked, “Does
300,000,000 + 150,000,000 = 2?” A
very puzzled expression was followed
by the “aha!” moment when she realized that division was required.
I allowed students to use calculators to complete the table at this point
in the lesson. Students understood
the meaning behind the conversions,
so the rest of the lesson was very
smooth—including calculating the
amount of time it will take for light to
travel from the sun to each planet.
The Challenge problem was assigned as homework. Students demonstrated the ability to transfer the data
to a number line. Some tried to incorporate relative sizes of the planets and
discovered that they really did not have
those data. Others wanted to show the
solar system in motion and drew the
planets in their orbit rather than on a
number line. Overall, they enjoyed the
activities and developed a better grasp
of larger distances and scale.
Judy Kraus
Hyde Park Middle School
Las Vegas, Nevada
Mathematics Teaching in the Middle School
●
Vol. 19, No. 1, August 2013
My eighth-grade geometry students
completed this activity. I particularly
enjoyed the cartoon itself because
it caused some of my students to
do Internet searches for astronomy
and astrology soon after reading it,
and it opened discussions in their
groups about these terms. Being top
students, they are not used to being
scared of math, so the humor was a
bit lost on them.
The Challenge problem brought
a very nice conclusion to their work,
making them carefully discuss the best
strategy for drawing the scale model.
In the context of the course, we had
already spoken about distance (from a
point to a line, a line to a line, and so
on) and also discussed distances from
points of concurrency of a triangle
to sides and vertices. The cartoon allowed a discussion of a different type
of distance.
My students raised questions as
to how the value of 1 AU was computed and consequently the average
distance from each planet (including
Earth) to the sun. My students were
familiar with and have visited Cape
Canaveral’s Kennedy Space Center, so
“space” talk was a natural extension to
the cartoon discussion.
Students in geometry need to get a
glimpse of non-Euclidean geometry.
Discussions of space can be used as a
doorway to this content, and at the
beginning of the year, this cartoon
activity can be used as a good introduction. In addition, other classes,
like prealgebra, can benefit from these
questions, especially the Challenge
problem. I have seen more than one
middle school student draw a time
line for their social studies class in
which the divisions were equidistant,
regardless of the year expanse between events. With these issues and
classes in mind, this cartoon would be
an excellent activity for an interdisciplinary discussion.
Sandra Argüelles Daire
Ada Merritt K–8 Center
Miami, Florida
I tried this cartoon with my seventhgrade prealgebra class. Many students
did not understand the point of the
cartoon. Question 4 needs more information to help lead students in the
conversion of the speed of light from
meters per second to minutes per AU.
I do like the level of challenge that
this question presented.
Machele Lynch
St. Patrick School
Carlisle, Pennsylvania
Ed. note: As a result of the feedback
above, we adjusted the final wording
of question 1 from “What is the point
of the cartoon?” to “What does the
last frame of the cartoon suggest about
how some people feel about math?”
We suggest that you begin the activity
by asking students, “What is the point
of the cartoon?” If more guidance is
needed, ask the current question. We
also added some steps to question 4 to
help students with the conversions.
Vol. 19, No. 1, August 2013
●
OTHER IDEAS
• Have students find the mean
distance from the sun in AUs
for each planet when the mean
distances are given in millions of
miles. Students should conclude
that the AUs are the same,
whether the distances are given
in kilometers or in miles.
• Have students calculate the
amount of time it will take to
travel 1 AU using miles instead of
kilometers. Given the conversion
factor for kilometers to miles, they
could calculate the speed of light
in mi./sec. They should realize
that it will take the same amount
of time, whether they start with
miles or kilometers.
• Have students use scientific notation to write planetary distances given in miles or kilometers.
Mathematics Teaching in the Middle School
19