DAYLIGHT HOURS – USING REAL DATA IN THE CLASSROOM
Ari Klein, Mathematics Teacher – Cleveland Heights High School
[email protected] online at http://chuh.org/staff/aklein/aklein.html
DISCUSSION: Do students realize that some days of the year have more daylight than others?
ACTIVITIES WITHOUT TECHNOLOGY:
Students can find and track sunrise and sunset time in the daily paper and determine the number of hours
of daylight for a particular day.
How much daylight is there if sunrise is at 5:49 and sunset is at 19:07
Prerequisite skills: military time and borrowing from hours for subtraction
This could lead to discussions of what students will happen for the rest of week. Will the daylight hours
increase or decrease and by how much? Is the change per day constant?
FOLLOW-UP ACTIVITY:
Determine “solar noon,” the middle of the sunlight for a particular day. If students are involved with the
GLOBE program (Global Learning and Observations to Benefit the Environment) they need to collect all
atmospheric data within one hour of solar noon. Can students come up with techniques for doing this?
-take half the number of daylight hours and add onto sunrise or subtract from sunset
-Will averaging sunrise and sunset times work?
ACTIVITIES WITH TECHNOLOGY:
Another way of analyzing sunrise and sunset times is to look online.
The US Naval Observatory site can create sun and moon rise/set table for 1 year, anywhere in the world
based on named locations or latitude and longitude combined with the number of hours the site differs
from the Greenwich (universal time)
http://aa.usno.navy.mil/data/docs/RS_OneYear.html
Created tables should be saved at TEXT files to allow easy import to a spreadsheet.
If the location desired is not available, you can find Latitude and Longitude from various sites on the
internet. For instance, help is available from the Bahá'í faithful helping worshipers worldwide who want to
point correctly to Bahá'í Qiblih
http://www.bcca.org/misc/qiblih/latlong_us.html
Import text files into Excel and move the columns so all the data is by date.
The next problem is how to understand the data.
Ari Klein – GCCTM January 2003 page 1
FIRST BY HAND:
Just as students might have explored the data through the newspaper, problems arise from the online
data. Times are given as 3 or 4 digits numbers instead of as in a time format.
For January 1 the sunrise and sunset times for Cleveland, Ohio are 754 to 1708.
1708 – 754 = 954. This is not correct. Discuss why
Using the concept of the borrowed hour:
16 hours and 68 minutes less 7 hours and 54 minutes
9 hours and 14 minutes is correct.
With 3 and 4 digit numbers I was not able to find a simple way of converting into time without serious
programming in visual basic (which I cannot help you with).
Work done on the computer is probably best done first by hand. Brainstorming methods and trying them
out together is probably a good idea. My methods were to determine the minutes since midnight for the
two times, then subtract the minutes.
Find the number of minutes since midnight given a 3 or 4 digit number representing time:
Method 1:
Subtract 100 (one hour) then add 60 (minutes) so that the borrowing problem is eliminated
Then convert that number into minutes or hours.
This seems like a good idea and works for most numbers. Why doesn’t it work all the time?
Method 2:
Separate the hour from the minute by using truncation and subtraction
HOURS : Truncate (754 / 100) = truncates 7.54 to 7
MINUTES: Take the hour value that was found, multiply by 100, then subtract from 754
754 – (7 * 100) = 54
Now convert to minutes since midnight by multiplying 7 by 60 and adding 54
Do the same with sunset.
Now subtract the two to get daylight minutes
Ari Klein – GCCTM January 2003 page 2
USING A SPREADSHEET:
With a year of sunrise and sunset in data two columns in excel method 2 can be implemented by copying
the functions down the whole year.
A
B
C
D
E
F
G
H
I
J
1
set min
Day
Sunrise
Sunrise rise min since Sunset
Sunset
since
Month
Day # Sunrise Sunset
hour
minute midnight
hours
Minutes midnight
2
1-Jan
1
754
1708
7
54
474
17
8
1028
Functions:
E2: =trunc(C2/100)
F2: =C2-E2*100
G2: =E2*60 + F2
And so forth with Sunset.
Column H would be daylight minutes subtracting J2 – G2
Copying all functions throughout the year will complete all of the data
Before graphing the data it makes more sense to shift the days to start with March 21, an equinox
(although discussing what an equinox or solstice is by itself an interesting topic).
Shifting the data to September 21 is a little tricky. Graphing the data; day # or date versus daylight
minutes produces a graph that appears to be sinusoidal.
OTHER INTERESTING THOUGHTS ABOUT DAYLIGHT HOURS:
In different parts of the world there will be different amounts of daylight hours.
An interesting investigation for all students could be in comparing daylight times from different locations,
especially if they are varying in latitude.
For younger students data collected from the Naval Observatory (or other site) can be averaged by month
or pick one date from the middle of each month as a representative of the month. Students can make bar
graphs by month to make comparisons.
Try to lead students to choose places that will show the greatest difference. For example:
Sao Paulo, BRAZIL
23° 33' S 46° 38' W (-3 hours)
Nome, Alaska
64° 30' N
165° 26' W
(-9 hours)
Cleveland OH
41° 24' N 81° 51' W (-5 hours)
These three places give interesting comparisons being far north of Cleveland and in the southern
hemisphere. Here is a glimpse.
Ari Klein – GCCTM January 2003 page 3
Average Monthly Temperatures
1400
1200
Minutes of sunlight
1000
800
600
400
200
0
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Month
Cleveland
Anchorage
Sao Paulo
SINE FUNCTION ACTIVITIES:
I have had students investigate the Sine function using a guided discovery with a graphing calculator
(enclosed in this packet). Discussions about amplitude, period, and shifts vertically and horizontally
helped students to know what was needed to create a function that fit the data for daylight hours in
Cleveland, Ohio. Using day number students should be able to determine how much daylight there will
be with some accuracy. Graphing the daylight hours from the data on the same chart as the derived
function is a way of seeing if the function is close to accurate.
I have no idea why, but a 5th degree polynomial regression in excel fits these curves with a lot of accuracy
as well. This is under “adding trendline” to a graph.
It is also possible to create a trendline manually by picking apart the sine function and matching the cells
for the days of the year with a new column. It is interesting to see how the created function comes close
to the one already graphed.
OTHER WAYS OF INVESTIGATING FUNCTIONS (not for everyone):
For those of you who have worked with Cabri Geometry (on a computer or on the TI-92 calculator) there
is a wonderfully powerful way to investigate functions by graphing them implicitly. That is, use the locus
function of Cabri to plot points so that you can manipulate various parts of the functions one at a time.
This includes using the “Show Axes” tool to put up a set of axes, put a point on the x-axis and find its
coordinate, place a vertical line through the point. Then use the calculate function to evaluate a function
at that point. After using “measurement transfer” to find a coordinate on the vertical line with the desired
value, finish by using the “locus tool” to find all such points as the x-axis point you started with moves
along the axis. The advantage is that the numbers you used to create the function can be changed to
see the effect on the graph. I have done these implicit functions for a few different generic types of
functions and it is powerful.
Ari Klein – GCCTM January 2003 page 4
TEMPERATURES:
Daylight hours and temperature are related. Temperature is another thing that students are aware
changes throughout the year. Students can be asked to make a simple graph of what they think
temperature does throughout the year. Many younger students keep track and graph this type of data. It
is possible to do a more thorough investigation of temperature, collecting information from the internet.
The National Weather Service, Cleveland station has been collecting temperatures in Cleveland since the
1870’s. The data is available online:
http://www.erh.noaa.gov/cle/climate/cle/climatecle.html
or linked through Cleveland State University, NWF
http://www.csuohio.edu/nws/climate/cle/climatecle.html
The only drawback to collecting the information is that it is textual by month. I have had students
assigned to collected different months, then pass them electronically over a network for that the data for a
whole year can all be put on one spreadsheet. Moreover, it makes sense to put all of the data on a sheet
that already has the daylight hours data on it. Since there is so much data, these graphs are pretty
smooth sinus curves.
Students can graph mean temperature, average high and average low temperatures together with
daylight hours. Regression curves can be determined so that students can come away with a formula
that can accurately determine average temperature for any day of the year. Graphing one year’s worth of
data can display how much variety there can be to the average and the strength of years of collected
data.
Students should be prompted to make the connection between daylight hours and temperature. The
delay in the high temperature with the day with the most daylight hours, etc. The warming and cooling of
the earth based on the amount of daylight it receives is interesting.
GLOBE (Global Learning and Observations to Benefit the Environment)
Other ideas for collected information could include Lake Erie water temperature and Soil Temperature.
The Globe web site is an interesting way to download data that students have collected throughout the
world. For instance, temperature data can be searched for by a particular day of the year for all of the
Globe schools that entered data for that day. This data can be sorted in a variety of ways. For instance,
sort by latitude to see what relationship there is between fronts moving west to east in the United States
or changes that occur North to South (if any). You do not have to be connected with the Globe program
to download data. Within the Globe site students can look at the visualizations that are interesting.
http://globe.fsl.noaa.gov/
Ari Klein – GCCTM January 2003 page 5