Activity
Day 1
Time: 
1
class period 
Preparation
Time: 
5
minutes copying Activity sheet if you are providing
it to individual students 
Materials: 
Activity sheet (may be run off and used
as an overhead) 
When the students walk into
the room, ask them, “How many seconds are there in
a regular year?” Allow a few minutes for students to
respond. Do not judge answers.
Put the digit 1 on the board and tell them, “I’m
going to start putting zeros behind the 1, and you have to
tell me when to stop so we’ll know your estimate of
the number of seconds in a year.” It might be a good
idea for you and the class to reach an agreement on how many
answers you will accept. Don’t take a guess from each
student in the class or else you will be adding zeros for
the whole period.
Begin and keep adding zeros until someone
tells you to stop. Ask the class if they want to stop there
or keep going.
If they want to go on, mark the first zero in a different
color and add more zeros until someone else says, “Stop.” Have
the students put commas in the appropriate places in the
number and ask the students to read the numeral they have
chosen.
Tell the students, “As a class we will now calculate
the number. Anybody have any suggestions?” Allow a
few minutes for students to respond and have them write their
answers on the board. Hopefully someone will come up with
the correct procedure, but they may not include the units
of measure with the numbers. As you go over the procedure,
make sure to write in the units in the appropriate spots
and show how the units cancel out to get the answer, explaining
that this technique is called unit analysis.
If nobody suggests the correct procedure, point out that
unit analysis can be used to compute the answer to this question.
Hand out Activity sheet or put it on the overhead and have
the students do the work in their notebooks.
Activity sheet
Unit analysis means that you include the units of measurement
when you are doing calculations and cancel out the common
units in the numerators and denominators until you end up
with the units of measure you desire.
(To determine the number of seconds in a year, it might
be easier for the students to understand the process if you
split the calculation into smaller steps.)
To find
the number of seconds in a year, let’s start
by converting one year to days, then days to hours, then
hours to minutes, and finally minutes to seconds. The process
would look something like this:
2. How do you put this all together into one equation? Remember
to include the units so each of the numbers is in its correct
position in the expression.
(Give
the students the following scenario: A friend is visiting
from England and asks how
far it is from Tucson to Phoenix. You say that it’s
about 128 miles from your house. She then asks,” How
many kilometers is that?” Allow a few minutes for students
to respond. Do not judge answers.)
3. To convert 128 miles into kilometers, do the following
calculation:
The “miles” in the numerator of the first factor
is cancelled by the “miles” in the denominator
of the second factor leaving kilometers as the final unit
of measure for your computation.
Ask which problem was more difficult and why. (Point out
that the first problem is more difficult because there are
so many factors involved, but by canceling out the units
they can put the numbers in their correct place in the expression.
The second calculation is much easier because there are only
two factors involved.)
(The rest of the lesson is devoted to changing from empirical
units into metric units by using unit analysis. The conversion
factors needed are the following?
1.6 km = 1 mile
.625 mile = 1 km (this is calculated on the activity sheet)
1 sq mile = 2.56 sq km (this is calculated on the activity
sheet)
1 sq km = .39 sq miles (this is calculated on the activity
sheet)
4. How do you use the fact that approximately 1.6 km = 1
mile to determine how many miles are in one kilometer?
5a. Use a conversion factor from problem 4 to determine
how many square kilometers are in one square mile?
To
picture this, draw a square that is 1 mile on each side
and then
change the 1 mile into 1.6 km and find
the area of the square or
5b. Use
a conversion factor from problem 4 to determine how many
square miles are in one square kilometer.
To picture
this, draw a square that is 1 km on each side and then
change the 1 km into .6 mile
and find the area of the square or
5c.What is another way to determine
how many square miles are in one square kilometer? (Hint:
this uses your answer to problem 5a)
Take the reciprocal of your answer from question 5a:
Multiple
choice: For the following problems, calculate the answer
that best approximates the given measure. (The correct
answers are underlined.)
6. 4
miles = a.
3 km b. 5 km c. 6.4 km d. 8 km
7. 80 km = a. 40 miles b. 50 miles c. 60 miles d. 70miles
8. 120 sq miles = a.
307 sq km b. 250 sq km c. 192 sq km
d. 120 sq km
9. A friend of yours is confused and thinks that 12 km
is about 19 miles.
a. How do you know that he is wrong?
You know that 1 km is smaller than 1 mile, so the answer
must be smaller than 12 instead of being larger than 12
b. What should he do to get the correct answer?
Multiply 12 km by .6 mile/ km to get 7.2 miles
Day
2
Time: 
1
class period 
Preparation
Time: 
5
minutes reserving computer lab 
Materials: 
Access to computers with geographic data 
Activity
Ask students. “Do you think the size of a country
or of the number of neighbors it has affects its status or
economy? Why?” Allow a few minutes for students to
respond. Do not judge answers.
Show
students the data on the United States from the Infrared
Analyst's Guide to Worldwide Environments web page or the
World Fact Book:
http://iac.dtic.mil/iria/
http://www.cia.gov/cia/publications/factbook/
If you use the first website, the distances will be given
in miles and area in square miles. The second website uses
a different format and uses metric units. Both provide useful
information.
Activity Sheet
When
studying a country it’s important to know many
of its geographic features. As a class, we will find the
following information for Brazil (The information on Brazil
is on another sheet)
1. What is the total length of the land boundary (including
coastline and neighboring countries)?
a. in miles b. in kilometers
2. What is the length of the land boundary that is coastline?
a. in miles b. in kilometers
3. Calculate what percentage of the total land boundary
is coastline.
4. Find the length of the land boundary that is shared with
the each of the neighboring countries
a. in miles b. in kilometers
5. Calculate what percentage of the total land boundary
is shared with each individual neighboring country.
6. Find the total length of the land boundary that is shared
with neighboring countries.
a. in miles b. in kilometers
7. Calculate what percentage of the total land boundary
is the total length of the land boundary that is shared with
the neighboring counties.
8. What is the total area, including land and water?
a. in square miles b. in square kilometers
9. Find the land area of the country:
a. in square miles b. in square kilometers
10. Calculate
what percentage of the total area is land.
11. Find the area of the country that is water:
a. in square miles b. in square kilometers
12. Calculate what percentage of the total area is water.
13. Find the area of the land that is arable.
a. in square miles b. in square kilometers
14. Calculate what percentage of the land area is arable.
15. Find the area of the land that is used for other purposes.
a.
in square miles b. in square kilometers
16. Calculate what percentage of the land area is used for
other purposes.
Have students share how these measurements might affect the
status and economy of Brazil (or whatever country you choose
to use as an example). What other data might be useful?
Homework
Have students calculate the same information for their country.
Remind them that this data will be a part of the country
display so that it should be presented accurately and neatly.
(This might be information they wish to graph for their country
displays.)
Day 3
Time: 
1
class period 
Preparation
Time: 
5 minutes copying blank piegraphs 
Materials: 
Blank piegraphs sheets 
Activity:
Tell
the students, “Being able to present data in
an appropriate manner is one of the standards in mathematics.
Simply having the data is not enough; you need to be able
to communicate it to others. Given the data that you found
yesterday about your country, what forms of representation
would you choose to make the data clear? Why would you choose
this form?” Allow a few minutes for students to respond.
Do not judge answers.
Using the data we have for Brazil, we are going to make
a piegraph to represent each of the following:
a. The percentage of the total land boundary that is coastline
with the percentage of the total land boundary that is on
land. The total percentage that is on land may be divided
into the percentage of the total land boundary that is shared
with each of the neighboring countries.
You might recommend to the students that they do the calculations
in their notebooks and then draw the appropriate angles on
their piegraphs. Things will be much neater this way.
To calculate
the “piece” of the piegraph that
represents the percentage of the total land boundary that
is coastline, multiply the percentage you have for the coastline
by to get the number of degrees in the central angle of the
piece. For example, to get the piece of the piegraph for
coastline,
You then draw a section of the circle that subtends a central
angle. The other part of the circle represents the percentage
of the total boundary that borders neighboring countries.
The central angle would be
However,
this section may be split up into the appropriately sized
sections for each of the neighboring countries. For example,
Argentina would be represented by a section with a central
angle of
You
would do the same calculations for the other neighboring
countries (see separate sheet).
b. The percentage of the total land area that is covered
by water (lakes and rivers) compared to the percentage of
the total land area that is land. Again, the section that
represents land can then be split up into smaller sections
that represent arable land and land used for other purposes.
Homework
Have students make piegraphs of the same information for
their country. Remind them that this data will be a part
of the country display so that it should be presented accurately
and neatly. (This might be information they wish to graph
for their country displays.)
Closure
The students reach closure as they work on the graphs for
their country displays.
