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Country Borders and Area

By: Kirstin Bittel, Brink Harrison, Sally Rusk, and Rachel Hughes


Time: 2-3 class periods
Preparation Time: 10 minutes to copy Activity sheets
Materials: Activity sheet 1
Activity sheet 2

Abstract
A math lesson applying unit analysis to deal with metric units when analyzing the percentage of land use in a specific country.

Purpose: To determine the perimeter and area of a country and find out what percentage of each is used for specific purposes. (explore)


Objectives

First class period students will be able to:
i. Use unit analysis to set up ratios to compute measurements
ii. Convert empirical measurements to metric
iii. Judge the reasonableness of results when converting to metric

Second class period students will be able to:
iv. Calculate the perimeter and area of a country
v. Calculate the percentage of perimeter that borders other features (neighboring countries, oceans, etc.)
vi. Calculate the percentage of area in the country that is water (lakes and rivers.)
vii. Calculate what percentage of land is used for agriculture and other uses.

Third class period students will be able to:
ix. Use proportions to calculate the portion of a pie-graph that represents a specific percentage of land use
x. Construct pie-graphs to represent data on borders and land use


National Science Education Standards
Measurement:
Apply appropriate techniques, tools, and formulas to determine measurement
Use unit analysis to check measurement computations
Understand measurable attributes of objects and the units, systems, and processes of measurement.

Number and Operations
:
Judge the reasonableness of numerical computations and their results

Representation
:
Use representations to model and interpret physical, social, and mathematical phenomena


Related and Resource Websites
http://iac.dtic.mil/iria/
http://www.cia.gov/cia/publications/factbook/

 

 

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 pie-graphs
Materials: Blank pie-graphs 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 pie-graph 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 pie-graphs. Things will be much neater this way.

To calculate the “piece” of the pie-graph 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 pie-graph 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 pie-graphs 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.


Embedded Assessment

The first day of this lesson will be assessed through informal discussion of ways to apply unit analysis and guided practice as the students do the Activity sheet. The second day of the lesson will be assessed by guided practice as the class as a whole finds the information for Brazil. Individual or small group assessment can be done by informal observation as the students do the calculations for their country displays. The third day will be assessed by guided practice as the class as a whole makes the pie-graphs for Brazil. Individual or small group assessment on the third day can be done by informal observation as the students do the calculations for their country displays.

Things to check with embedded assessments:
Can students convent measurements from empirical to metric?
Can students calculate percentages?
Can students calculate ratios?
Can the students measure angles in the pie-graphs?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


 

 

 

 

 

 

 

 



PULSE is a project of the Community Outreach and Education Program of the Southwest Environmental Health Sciences Center and is funded by:


an
NIH/NCRR award #16260-01A1
The Community Outreach and Education Program is part of the Southwest Environmental Health Sciences Center: an NIEHS Award

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Supported by NIEHS grant # ES06694


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Last update: March 7, 2007
  Page Content: Rachel Hughes
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