This is sample project based learning Mathematics . The students learnt "Linear Equation taking Arctic sea ice amount "Through this project the students developed the awareness that we should be very conscious about the global warming under facilitation of teacher Pratima Nayak

1. Topic :Arctic Sea Ice and Linear Equations
(Mathematics for Planet Earth)
Name of the author: Pratima Nayak
Genesis of the Project:
Glaciers are melting, sea levels are rising, cloud forests are drying, and wildlife is
scrambling to keep pace. It's becoming clear that humans have caused most of the
past century's warming by releasing heat-trapping gases as we power our modern
lives. Called greenhouse gases, their levels are higher now than in the last 650,000
years.
We call the result global warming. Scientists say it is climate change.
Are our students aware of it?
Arctic sea ice has declined by more than 86,000 square kilometres. This will affect
both terrestrial and marine animals. Summer ice will be completely melted in year
2060.
How do Mathematicians predict?
Some Mathematical model predicts that extent of ice in Arctic sea is a function of
time.
Introduction:
The project is not original work of the teacher. On the basis of this article Pratima
Nayak Mathematics Teacher of Kendriya Vidyalaya, Fort William,Kolkta took the
interdisciplinary project to teach her students “Equation of straight line “ in Class
XI.The project aims at learning of Mathematics, Science, Geography and History
together.The article Arctic Sea Ice and Linear Equations published through the link
http://www.mathaware.org/mam/2013/sustainability/Artic%20Sea%20Ice%20and%2
0Linear%20Equations%20Teacher%20Notes%20Final%203.31.13.pdf
She sincerely acknowledges Teacher Guide William C. Bauldry, Appalachian State
University, Victor J. Donnay, Bryn Mawr College, Thomas J. Pfaff, Ithaca College.
She did not change even the title of the project.
The writer gave the linear equation.
y = - 0.0921 t + 190.12. When y is amount of ice (square Kilometre) and t is year.
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2. The linear equation was framed taking average September Arctic Sea Ice Extent,
from 1979 to 2012. This model predicted that there will be no ice in Arctic sea in
summer 2064.
Results/ Conclusion given by the Author:
1. The horizontal axis measures time in years. The vertical axis gives the average
extent of Arctic sea ice in September. The units are square kilometres measured in
millions (million km2).
2. Finding the line of best fit.
Discussion with students explaining why they drew the line they did and seeing the
variation among the results. This would provide a teachable moment for the need to
have a precise definition of line of best fit.
3. Slope of line. The students will need to calculate the slope from the drawing by
calculating y = m x + c
4. In Excel, we used the Chart – Add Trend line feature and found that the line of
best fit is given by y = - 0.0921 t + 190.12.
This function is a mathematical model of the extent of sea ice. One can use a
mathematical model to predict the future – in this case the future extent of the Arctic
sea ice.
With help of this article/ concept of linear equation the teacher Pratima Nayak aims
to create the awareness about the global warming and its consequences as well as
about how human are responsible for the grave situation of earth among the
students. Here project title even remains same to acknowledge the original author.
For students she changed the project title to create brainstorming in the students.
Thus the project......
” Only one degree of difference....”
The project begins with the brain storming question to motivate the students for
learning.
“So, the Earth's average temperature has increased about 1 degree Fahrenheit
during the 20th century.
What's the big deal?”
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3. Specific Objectives:
1. The students will understand that the earth‟s temperature has increased by 1
but it creates a lot of changes to our mother earth. At 32 F ice is in solid form
and at 33 it changes to water.
2. The students will be aware about the grave problem “Global warming” or
„Climate Change” .They will study details of green house effects both natural
and manmade.
3. They will know the causes and consequences of the climate change.
4. They will study how the problem can be minimised.
5. They will be involved to make Planet Earth beautiful.
6. The students will comprehend that predictions about future of the Earth is
nothing but Mathematical modelling.
7. The students will apply Mathematical modelling to find linear equation which
can be applied to find the ice amount on Arctic sea in successive years.
Mathematical Content: (using ICT)
1. By the graph the students will construct knowledge of linear equation. The
students will comprehend that average extent of the Arctic sea ice in September
(measured in millions of square kilometers) as a function of time.( in years).
The time in year will be taken on horizontal axis and Arctic sea ice in September
will be taken on y axis.
2. Line of best fit. From the collected data they will mark a set of data points, and
find a line that fits the data as well as mathematically possible.
3. Mathematical modelling. The line of best fit is described by a formula or function. It
is the linear equation.
4. Slope and rate of change. The slope of a line can be interpreted as the rate of
change of the function. From the slope one can determine the approximate ice count
for successive year can be predicted.
5. Units. The units of rate of change in this problem are million km 2/ per year =
million km2/year which has a crucial meaning in the context of the problem. It tells
how much less ice there will be in each successive year.
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4. Method:
This lesson will be taught by project based learning using ICT tools.

Brain storming

Data collection from mostly www.nasaclimate.org

Analysis

Interpretation

Developing Mathematical concept

Presentation of Report

Evaluation
Lesson launch.
Short videos about the issue. http://www.youtube.com/watch?v=_m-M37vc-m0
http://www.youtube.com/watch?v=HHxHxI4VuAM
SSM/I Arctic Sea Ice (Dec 1991 - Nov 2011)
http://www.youtube.com/watch?v=AztEry44A9A&feature=youtu.be
Arctic Sea Ice Extent, 1979-2012: From NSIDC was shown to the students.
They were asked essential questions
1. “What are its consequences?”
2. “How many years it will continue?”When Arctic sea will be free from summer
ice? Will it affect us?
Certainly this brain storming questions will arouse an interest to do a research work
on this topic.
Project plan
Grouping: The students will be grouped with a group leader.
Calendar of the activities will be prepared.
Time line for each activity will be set.
Evaluation criteria will be planned.
Project implementation:
Sstudents of Class XI of Kendriya Vidyalaya ,Fort William,Kolkata participated in
construction of knowledge under the facilitation of teacher Mrs.Pratima Nayak.
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5. The students will be assigned activities: They have to present the research work on

Green house effects

Melting of sea ice.

Use of Mathematics in predicting the ice extent from previous and present
data.
1. They students collected data. (http://climate.nasa.gov/) and other websites.
2. They developed average extent of the Arctic sea ice in September (measured
in millions of square kilometres) as a function of time.( in years).
3. Using MS Excel they found line that fits the data as well as mathematically
possible.
4. They found the slope of the line from Graph using software “ Geo gebra”.
Data:
5. They were able to predict the amount of ice content from the linear equation y
=mx+c
Students work:
1. The students will make a PowerPoint presentation on Green House effect
covering following concepts.
Humans have increased the amount of carbon dioxide in the atmosphere by more
than a third since the industrial revolution. Changes this large have historically taken
thousands of years, but are now happening over the course of decades.
The rapid rise in greenhouse gases is a problem because it is changing the climate
faster than some living things may be able to adapt. Also, a new and more
unpredictable climate poses unique challenges to all life.
Historically, Earth's climate has regularly shifted back and forth between
temperatures like those we see today and temperatures cold enough that large
sheets of ice covered much of North America and Europe. The difference between
average global temperatures today and during those ice ages is only about 5
degrees Celsius (9 degrees Fahrenheit), and these swings happen slowly, over
hundreds of thousands of years.
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6. Now, with concentrations of greenhouse gases rising, Earth's remaining ice sheets
(such as Greenland and Antarctica) are starting to melt too. The extra water could
potentially raise sea levels significantly.
As the mercury rises, the climate can change in unexpected ways. In addition to sea
levels rising, weather can become more extreme. This means more intense major
storms, more rain followed by longer and drier droughts (a challenge for growing
crops), changes in the ranges in which plants and animals can live, and loss of water
supplies that have historically come from glaciers.
2. Sea ice volume is an important climate indicator. It depends on both ice
thickness and extent and therefore more directly tied to climate forcing than
extent alone. The melting of once-permanent ice is already affecting native
people, wildlife and plants. Polar bears, whales, walrus and seals are
changing their feeding and migration patterns. And along Arctic coastlines,
entire villages will be uprooted because they're in danger of being swamped.
The native people of the Arctic view global warming as a threat to their
cultural identity and their very survival.
The year 2013 marks the first October with an extent above 8 million square
kilometres (3.09 million square miles) since 2009 and only the second since 2006.
From 1979 to 2006, average October extent was never below 8 million square
kilometres, and several years had October extents above 9 million square kilometres
(3.47 million square miles). The lowest October extent, less than 7 million square
kilometres (2.7 million square miles), was observed in 2007. The linear trend in
October ice extent is –7.1 % per decade relative to the 1981 to 2010 mean, or –
63,400 square kilometres per year (–24,500 square miles per year).
It is observed from satellite record that the extent of sea ice this September , 2013 is
substantially greater than last year‟s record low.
Specific activity/ Mathematical concept
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7. Extent of Ice (Million per
year
square Kilometer)
2003
6.15
2004
6.04
2005
5.57
2006
5.89
2007
4.28
2008
4.67
2009
5.36
2010
4.9
2011
4.61
2012
3.61
2013
5.35
The current project is done with average September Arctic Sea Ice Extent,
from 2003 to 2013. Is there a change in slope of the linear equation? How to
predict for next year?
Results/ Conclusion ( Excel file)
7
Amount of Million per square
Kilometers
6
5
4
3
2
1
0
2000
2002
2004
2006
Year
2008
2010
2012
2014
Applying linear equation, Y = m x + c
They took the year 2003 to 2013.
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8. 6.15 = 2003 X m + c
5.35 = 2013 X m + c
Solving two equations, m = – 0.08 and c = 166.4
So the final linear equation
Y =  0.08 x + 166.4
The Arctic sea will be Ice free in 2070 summer.
So , increased ice of summer 2013 leaves a little hope for world.
Conclusion:
While a 1-degree temperature rise may sound puny, global warming has set in
motion lots of changes: glaciers are melting, many birds are beginning their
migrations earlier, some islands are becoming submerged by rising seas, melting ice
is causing polar bears to starve and so on.
If nothing is done, an international consortium of 2,000 scientists (called the
Intergovernmental Panel on Climate Change) predicts temperatures could rise 11
degrees by century's end.
To avoid the most dangerous effects of warming, scientists say an 80 percent
reduction in greenhouse-gas emissions is needed by 2050.
Evaluation:
Evaluation was done with help of Power point Rubric and research report rubric and
test on slope of linear equations.
Teacher’s reflection:
This is not original idea of the teacher. Once the students know to predict ice amount
in successive year they will curious to verify the result and ultimately they will be
involved the happenings in the environment
Mrs.Pratima Nayak,Post Graduate Teacher,
Kendriya Vidyalya,Fort William,Kolkata-700021
27,Joy Krishna Paul Road,Kolkata-700023
Email address: pnpratima@gmail.com
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9. Students’ activity
Global Warming( students assignment)

Today carbon dioxide is at an "unprecedented" level not seen for at least the last
800,000 years.

Sea level is set to continue to rise at a faster rate than over the past 40 years.
Waters are expected to rise by between 26 cm (10 inches) at the low end and 82
cm (30 inches) at the high end. Arctic sea will be ice free in summer in 2070.

There is "high confidence" that over the last two decades the Greenland and
Antarctic ice sheets have been melting, glaciers have receded in most parts of the
world, and Arctic sea ice has continued to shrink in terms of extent.
Research Report presented by students
Evidences of global warming or climate change.
1. Global Surface Temperature
This graph illustrates the change in global surface temperature relative to
1951-1980 average temperatures. Global surface temperatures in 2012 were the
ninth warmest on record. (Source: NASA/GISS) The gray error bars represent the
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10. uncertainty on measurements. This research is broadly consistent with similar
constructions prepared by the Climatic Research Unit and the National Atmospheric
and Oceanic Administration.
The time series at right shows the five-year average variation of global surface
temperatures from 1884 to 2012. Dark blue indicates areas cooler than average.
Dark red indicates areas warmer than average.
2. Carbon dioxide Concentration
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11. Carbon dioxide (CO2) is an important heat-trapping (greenhouse) gas, which is
released through human activities such as deforestation and burning fossil fuels, as
well as natural processes such as respiration and volcanic eruptions. The chart on
the left shows the CO2 levels in the Earth's atmosphere during the last three glacial
cycles, as reconstructed from ice cores. The chart on the right shows CO2 levels in
recent years, corrected for average seasonal cycles.
The time series at right shows global distribution and variation of the concentration of
mid-troposphere carbon dioxide in parts per million (ppmv) at an altitude range of 313 kilometres (1.9 to 8 miles).
3. Arctic Sea Ice
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12. September Arctic sea ice is now declining at a rate of 11.5 percent per decade,
relative to the 1979 to 2000 average 1. Arctic sea ice reaches its minimum each
September. The graph above shows the average monthly Arctic sea ice extent in
September from 1979 to 2012, derived from satellite observations. The September
2012 extent was the lowest in the satellite record.
The time series at right shows the annual Arctic sea ice minimum since 1979, based
on satellite observations.
4. Land Ice
Data from NASA's Grace Satellites show that the land ice sheets in both Antarctica
and Greenland are losing mass. The continent of Antarctica (left chart) has been
losing more than 100 cubic kilometres (24 cubic miles) of ice per year since 2002.
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13. The time series at right shows average ice mass changes in Greenland each year for
the month of September. Purple and blue colours indicate the areas and amount of
ice loss, and white and red indicates areas of ice gain. The measurements are
calculated in terms of centimetres of equivalent water height change per year.
5. Sea Level
Sea level rise is caused by two factors related to global warming: the added water
coming from the melting of land ice, and the expansion of sea water as it warms up.
The above graphs show how much sea level has changed since 1993 (right, satellite
data record) and about 1880 (left, coastal tide gauge data). Source data files can be
found here and here.
The time series at right shows average annual sea-surface height variations. Red
and yellow are regions where sea level is higher than normal; purple and dark blue
show where sea level is lower.
Increase in Temperature
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14. Causes:
Most climate scientists agree the main cause of the current global warming trend is
human expansion of the "greenhouse effect" -- warming those results when the
atmosphere traps heat radiating from Earth toward space.
Certain gases in the atmosphere block heat from escaping. Long-lived gases,
remaining semi-permanently in the atmosphere, which do not respond physically or
chemically to changes in temperature are described as "forcing" climate change
whereas gases, such as water, which respond physically or chemically to changes in
temperature are seen as "feedbacks."
Gases that contribute to the greenhouse effect include:

Water vapour. The most abundant greenhouse gas, but importantly, it acts as a
feedback to the climate. Water vapour increases as the Earth's atmosphere
warms, but so does the possibility of clouds and precipitation, making these some
of the most important feedback mechanisms to the greenhouse effect.

Carbon dioxide (CO2). A minor but very important component of the atmosphere,
carbon dioxide is released through natural processes such as respiration and
volcano eruptions and through human activities such as deforestation, land use
changes, and burning fossil fuels. Humans have increased atmospheric
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15. CO2 concentration by a third since the Industrial Revolution began. This is the
most important long-lived "forcing" of climate change.

Methane. A hydrocarbon gas produced both through natural sources and human
activities, including the decomposition of wastes in landfills, agriculture, and
especially rice cultivation, as well as ruminant digestion and manure management
associated with domestic livestock.

Nitrous oxide. A powerful greenhouse gas produced by soil cultivation practices,
especially the use of commercial and organic fertilizers, fossil fuel combustion,
nitric acid production, and biomass burning.

Chlorofluorocarbons (CFCs). Synthetic compounds of entirely of industrial origin
used in a number of applications, but now largely regulated in production and
release to the atmosphere by international agreement for their ability to contribute
to destruction of the ozone layer. They are also greenhouse gases.
The role of human activity
In its recently released Fourth Assessment Report, the Intergovernmental Panel
on Climate Change, a group of 1,300 independent scientific experts from
countries all over the world under the auspices of the United Nations, concluded
there's a more than 90 percent probability that human activities over the past 250
years have warmed our planet.
The industrial activities that our modern civilization depends upon have raised
atmospheric carbon dioxide levels from 280 parts per million to 379 parts per
million in the last 150 years. The panel also concluded there's a better than 90
percent probability that human-produced greenhouse gases such as carbon
dioxide, methane and nitrous oxide have caused much of the observed increase
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16. Effects: ( Slides developed by students)
16

17. Activity -2 Melting of Ice ( Slides prepared by students)
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18. 18

19. Poster by students to create awareness
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20. Arctic Sea ice and linear equation:
In 2013,, August extent was the sixth lowest in the 1979 to 2013 satellite record.
August 2013 ice extent was 1.38 million square kilometres (533,000 square miles)
above the record low August extent in 2012. The monthly trend is –10.6% per
decade relative to the 1981 to 2010 average.
Overall, 10.03 million square kilometers (3.87 million square miles) of ice were lost
between the 2013 maximum and minimum extents
Many different computer models have been developed to predict when summer sea
ice will disappear from the Arctic.
Fifty per cent of these models say it will have gone by 2060.
March Average
September/Marc
September Average
h(minimum/max
Extent(millions of
imum)
square kilometers)
1979–2000 mean
7.0
15.7
1999/2000
6.2
15.3
2000/2001
6.3
15.6
2001/2002
6.8
15.4
2002/2003
6.0
15.5
2003/2004
6.2
15.1
2004/2005
6.1
14.7
2005/2006
5.6
14.4
2006/2007
5.9
14.7
Extent(millions
of square
kilometers)
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21. 2007/2008
4.3
15.2
2008/2009
4.7
15.2
2009/2010
5.4
15.1
2010/2011
4.9
14.6
2011/2012
4.6
15.2
2012/2013
3.6
15.0
http://nsidc.org/arcticseaicenews/
Results/ Conclusion (Excel file)
7
Amount of Million per square
Kilometers
6
5
4
3
2
1
0
2000
2002
2004
2006
Year
2008
2010
2012
2014
Applying linear equation, Y = m x + c
They took the year 2003 to 2013.
6.15 = 2003 X m + c
5.35 = 2013 X m + c
Solving two equations, m = – 0.08 and c = 166.4
So the final linear equation
Y =  0.08 x + 166.4
The Arctic sea will be Ice free in 2070 summer.
So , increased ice of summer 2013 leaves a little hope for world.
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22. Bibliography:
1. The 17-year pause in global warming is likely to last into the 2030s and the
Arctic sea ice has already started to recover, according to new research.
A paper in the peer-reviewed journal Climate Dynamics – by Professor Judith
Curry of the Georgia
Institute of Technology and Dr Marcia Wyatt – amounts to a stunning challenge to
climate science orthodoxy.
References:/ Credits
1. http://www.bbc.co.uk/climate/adaptation/life_at_home2.shtml
2. http://worldwildlife.org/species/galapagos-penguin
3. http://www.sciencedaily.com/releases/2013/08/130801142317.htm
4. http://climate.nasa.gov/
5. http://www.bbc.co.uk/climate/adaptation/life_at_home2.shtml
6. http://worldwildlife.org/species/galapagos-penguin
7. http://psc.apl.washington.edu/wordpress/research/projects/projectionsof-an-ice-diminished-arctic-ocean/
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