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# LAND COVER CHANGE PROBLEM: 

MATH LESSON ON SCALING (% change)

Intermediate-Advanced Level

## Purpose

To create, scale and read maps to help interpret various ∆NDVI images.
Part of this is to create a photo/image interpretation selective key to
help interpret the aerial photograph of the study area.

## Background Information 

The end product of this lesson is to create map overlays that can help
interpret the ∆NDVI image. In a satellite images only the large roads
are visible. When the students find an area of change shown in the ∆NDVI
image that is not on or near a major road, trying to accurately identify
this area can be very difficult. Even when a roadmap or a topographic
map are used side by side to help identify students still misidentify
areas due to differences in the scale of the two images. To help make
the identification process easier this lesson shows students how to
rescale a map to the same scale as the satellite image. A couple of
interesting things are possible with a map that is rescaled to the
satellite image. First and foremost the map can be transferred to a
transparency using a copier and then overlaid on the satellite image.
Additionally, the scale of the satellite image can be calculated using
the percent needed to change the map and the original map scale. This
last possibility is easily checked due to the fact that Landsat images
have a standard pixel size of 28.5 meters by 28.5 meters.

Aerial photographs are used to determine what areas of this image have
change and why without leaving the classroom. This is basically a set of
areas on the aerial photograph that the students confidently identify
using collateral material or metadata. The metadata can be maps, graphs,
or images. "Some authors say that as a general rule, keys are more
easily constructed and used for the identification of man-made objects
and features than for natural vegetation and landforms...Basically, an
interpretation key helps the interpreter organize the information
present in the image form and guides him/her to the correct
identification of unknown objects." ([Aids to and Techniques of Image
Interpretation](http://rsrunt.geog.ucsb.edu/rscc/voll/lec2/2-4.html) )
The students will need to construct a simple key so they can figure out
what type of change has occurred in what area of the ∆NDVI image. In
this case we will primarily use a U.S. Geologic Survey Topographic
Quadrangle mosaic that was last revised in 1976 for the collateral
material or metadata.

![](media/image2.png){width="2.016535433070866in"
height="4.193700787401575in"}![](media/image3.png){width="2.1133858267716534in"
height="4.192913385826771in"}![](media/image4.png){width="1.8771653543307087in"
height="4.198031496062992in"}

MLS Figure 1. The above set of images and maps illustrates how different
data sets can work together to aid in the interpretation of the ∆NDVI
image. In the selected area, a section of the Patuxent River, there are
three different features that can be easily identified using the aerial
photograph.

Before we can even begin this task a couple of things need to happen.
All the maps need to be scaled to the same scale so that they can be
compared and or overlaid over each other. This is a process that will
require the students to compute the scale for images, satellite images,
which do not have scales provided. This will test their map reading and
critical thinking skills. The below flow chart will graphically shown
why these lessons are grouped together.

## Time

-   1-2 class periods to create the needed images.

-   Time out of class can be used for the data analysis.

## Level

Intermediate and Advanced

## Key Concepts

> Communication skills
>
> Read and understand images produced by others
>
> Produce images for other to interpret
>
> Find and describe locations on maps and computer-generated images
>
> Technology/Software Mastery
>
> Use computer software to scale images
>
> Use computer software to interpret images
>
> Computation and Estimation
>
> Use, interpret, and compare numbers
>
> Estimate distances from maps and actual sizes of objects from scale
> drawings
>
> Decide what degree of precision is adequate
>
> Critical-Response
>
> Be aware that there may be more than one good way to interpret a given
> set of findings

**Skills/Students will:**

-   understand the different data that can be determined from each image

-   (understand that the significance of a change is based on the
    original size)\*

-   use proportions to change a ratio to a percent

-   use percents to describe increase/decrease in an image

**Prerequisites:**

Students should

-   understand the concept of a scale on a map

-   be familiar with the process of solving proportions to find equal
    ratios

**Materials:**

-   Landsat ∆NDVI images

-   Aerial photograph of the same area

-   Topographic map of the same area

-   Student worksheet

-   Calculator

-   Metric Ruler

-   Copier that can enlarge and reduce images

-   Computer with internet access and a color printer

-   Transparencies that are copier safe

-   Overhead Projector

-   Permanent Marker, ultra fine point

## Preparation

-   Download the appropriate images and print them on a color printer

-   Copy images and maps on to transparencies

-   Find a prominent intersection. Mark the intersection on all 3 images
    with a dot.

-   Put the students into groups of 3 or 4.

**Procedure**

1.  Give each group a scaling packet, metric ruler, aerial photograph,
    ∆NDVI images and topographic map of the same area.

2.  Each student should have one image he/she is in charge of- the
    fourth student can be the checker.

3.  The following questions will help familiarize your students with
    their images as well as the purpose of this lesson. Each question
    should be discussed with the class. Be sure everyone knows how the
    images were taken/made.

4.  Ask everyone in the group to look at the NDVI image. "Can you tell
    what everything is? Why or why not?" (taken by a satellite thousands
    of miles above the Earth)

5.  Then ask them to look at the aerial photograph and answer the same
    questions. (taken by a camera on an airplane in the Earth's
    atmosphere)

6.  Finally, have them look at the topographic map and answer the same
    questions. (made by map makers who have visited the areas)

7.  The students should come to the conclusion that they can name
    everything on the topographic map, most things on the photograph,
    but very little on the NDVI image. So we would like to use the map
    and the photo to help us identify everything in the NDVI image.

8.  The only problem is that the images were taken from very different
    distances. Therefore each image covers a different amount of space
    and has a different scale. In other words, a mile in real life is
    much smaller on the image taken by a satellite than one taken by an
    airplane.

9.  If we enlarge or shrink our images so that the scales were the same
    size we could lay the transparencies on top of each other and label
    everything on the NDVI image.

10. The students should now be able to complete numbers one and two.
    Check each group's image to make sure they were able to find the
    same intersection on all three images.

11. The

12. You may need to help students set up and solve their proportions.

13. Unfortunately, the copying machine will not ask you for a distance.
    When you have a two-dimensional picture (length and width- square
    centimeters) a one-dimensional measurement (just length-
    centimeters) is not sufficient.

14. The other reason that a measurement is not very practical is because
    you cannot use this number to compare how much you had to enlarge or
    shrink an image. Just imagine if one person were able to jump one
    and a half feet up in the air and somebody else was able to jump
    three feet. At first, you might admire the second person's talent
    and think the first person didn't have much talent. But if I told
    you that the first person was only three feet tall and in second
    grade and the other person was seven feet tall, the situation might
    look a little different. The second-grader jumped half, or 50 %, of
    his height, while the taller person jumped about 43 percent. With
    percents you can compare the new amount with the original amount.

15. You can compare how much you had to enlarge two different size
    images. If you would like to increase an image by 100%, you would be
    doubling its size. If you wanted to increase it by 50 %, your final
    image would be one and a half times its original size (smaller than
    double). The percent, which you would like to increase an image, is
    called the percent of increase. If you were reducing an image, this
    percent would be called the percent of decrease.

16. Let's take an example.

+------------+-----------+------------+-----------+----------+--------+
| Images     | Distance  | Scale (in  | D         | Ratio    | P      |
|            | (in cm)   | cm per     | ifference |          | ercent |
|            | between   | mile)      | between   |          | of     |
|            | two dots  |            | two       |          | In     |
|            |           |            | scales    |          | crease |
+------------+-----------+------------+-----------+----------+--------+
| T          | 18        | [cm]{.     | X         | X        | X      |
| opographic |           | underline} |           |          |        |
| Map        |           | [18]{.     |           |          |        |
|            |           | underline} |           |          |        |
|            |           | =          |           |          |        |
|            |           | [36]{.     |           |          |        |
|            |           | underline} |           |          |        |
|            |           |            |           |          |        |
|            |           | mi 0.5 1   |           |          |        |
+------------+-----------+------------+-----------+----------+--------+
| Aerial     | 6         | [cm]{.     | 24        | [24]{.un | 200 %  |
| Photograph |           | underline} |           | derline} |        |
|            |           | [6]{.      |           | =        |        |
|            |           | underline} |           | [2]{.un  |        |
|            |           | =          |           | derline} |        |
|            |           | [12]{.     |           |          |        |
|            |           | underline} |           | 12 1     |        |
|            |           |            |           |          |        |
|            |           | mi 0.5 1   |           |          |        |
+------------+-----------+------------+-----------+----------+--------+
| ∆ NDVI     | 10        | [cm]{.     | 16        | [16]{.un | 80 %   |
| Image      |           | underline} |           | derline} |        |
|            |           | [10]{.     |           | =        |        |
|            |           | underline} |           | [4]{.un  |        |
|            |           | =          |           | derline} |        |
|            |           | [20]{.     |           |          |        |
|            |           | underline} |           | 20 5     |        |
|            |           |            |           |          |        |
|            |           | mi 0.5 1   |           |          |        |
+------------+-----------+------------+-----------+----------+--------+

Measure the distance of the dots. Set up a ratio to find the scale of cm
per mile. Figure out how many cm you will need to add to each scale to
make it equal the largest scale. This will tell you how much you need to
increase each image by. Set up another ratio of the amount of cm needed
for each image over the original number of cm equivalent to one mile.
Finally turn this ratio into a percent for the percent of increase.

17\. Note: when entering these percents on a copying machine, be sure to
add 100 % to the student's figures since their figure is the percent of
increase.

18\. Answers to student's practice chart.

+------------+-----------+------------+-----------+----------+--------+
| Images     | Distance  | Scale (in  | D         | Ratio    | P      |
|            | (in cm)   | cm per     | ifference |          | ercent |
|            | between   | mile)      | between   |          |        |
|            | two dots  |            | two       |          |        |
|            |           |            | images    |          |        |
+------------+-----------+------------+-----------+----------+--------+
| T          | 13        | [13]{.     | 1.5       | [        | 20 %   |
| opographic |           | underline} |           | 1.5]{.un |        |
| Map        |           | =          |           | derline} |        |
|            |           | [7.5]{.    |           | =        |        |
|            |           | underline} |           | [1]{.un  |        |
|            |           |            |           | derline} |        |
|            |           | 2 1        |           |          |        |
|            |           |            |           | 7.5 5    |        |
+------------+-----------+------------+-----------+----------+--------+
| Aerial     | 9         | [9]{.      | 4.5       | [        | 100 %  |
| Photograph |           | underline} |           | 4.5]{.un |        |
|            |           | =          |           | derline} |        |
|            |           | [4.5]{.    |           | =        |        |
|            |           | underline} |           | [1]{.un  |        |
|            |           |            |           | derline} |        |
|            |           | 2 1        |           |          |        |
|            |           |            |           | 4.5 1    |        |
+------------+-----------+------------+-----------+----------+--------+
| ∆ NDVI     | 18        | [18]{.     | X         | X        | X      |
| Image      |           | underline} |           |          |        |
|            |           | =          |           |          |        |
|            |           | [9]{.      |           |          |        |
|            |           | underline} |           |          |        |
|            |           |            |           |          |        |
|            |           | 2 1        |           |          |        |
+------------+-----------+------------+-----------+----------+--------+

Name \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

**[Student Packet on Scaling]{.underline}**

*Materials*: You will need an aerial photograph, a topographic map,
∆NDVI image (the ∆NDVI image that is from the oldest date to the most
recent, most likely the 1984 to 1997) and metric ruler. Make sure you
can identify each of these images and maps.

Lay out your three images. Find the dot at the intersection of two roads
on each image.

1.  Look at your topographic map to find another intersection. Mark this
    intersection with a pen.

2.  Find the same intersection on the other two images. Mark these as
    well.

3.  Have your teacher check your three images and then initial here.
    \_\_\_\_\_\_\_

4.  Measure the distance (in cm) between the two dots on the topographic
    map. Fill in the distance in the 1^st^ column of the chart below.

5.  Measure the distances (in cm) between the two dots on the aerial
    photograph and the NDVI image. Record them in the 1^st^ column on
    the chart below.

  ------------- ----------- ------------- ------------ ----------- ---------
  Images        Distance    Scale (in cm  Difference   Ratio       Percent
                (in cm)     per mile)     between two              
                between two               images                   
                dots                                               

  Topographic                                                      
  Map                                                              

  Aerial                                                           
  Photograph                                                       

  ∆ NDVI Image                                                     
  ------------- ----------- ------------- ------------ ----------- ---------

Here is an example of a completed chart to help you as you answer the
following questions.

+------------+-----------+------------+-----------+----------+--------+
| Images     | Distance  | Scale (in  | D         | Ratio    | P      |
|            | (in cm)   | cm per     | ifference |          | ercent |
|            | between   | mile)      | between   |          | of     |
|            | two dots  |            | two       |          | In     |
|            |           |            | scales    |          | crease |
+------------+-----------+------------+-----------+----------+--------+
| T          | 18        | [cm]{.     | X         | X        | X      |
| opographic |           | underline} |           |          |        |
| Map        |           | [18]{.     |           |          |        |
|            |           | underline} |           |          |        |
|            |           | =          |           |          |        |
|            |           | [36]{.     |           |          |        |
|            |           | underline} |           |          |        |
|            |           |            |           |          |        |
|            |           | mi 0.5 1   |           |          |        |
+------------+-----------+------------+-----------+----------+--------+
| Aerial     | 6         | [cm]{.     | 24        | [24]{.un | 200 %  |
| Photograph |           | underline} |           | derline} |        |
|            |           | [6]{.      |           | =        |        |
|            |           | underline} |           | [2]{.un  |        |
|            |           | =          |           | derline} |        |
|            |           | [12]{.     |           |          |        |
|            |           | underline} |           | 12 1     |        |
|            |           |            |           |          |        |
|            |           | mi 0.5 1   |           |          |        |
+------------+-----------+------------+-----------+----------+--------+
| ∆ NDVI     | 10        | [cm]{.     | 16        | [16]{.un | 80 %   |
| Image      |           | underline} |           | derline} |        |
|            |           | [10]{.     |           | =        |        |
|            |           | underline} |           | [4]{.un  |        |
|            |           | =          |           | derline} |        |
|            |           | [20]{.     |           |          |        |
|            |           | underline} |           | 20 5     |        |
|            |           |            |           |          |        |
|            |           | mi 0.5 1   |           |          |        |
+------------+-----------+------------+-----------+----------+--------+

Now we know that the distance between these two points is different on
each image/map. However, the distance in real life between these two
points is the same. Our next step is to find out the scale for each
image. In other words, how many centimeters on the map equals one mile
in real life.

5.  One of these images already has a printed scale. Which image? Write
    the scale in centimeters per mile next to the image in column 2. (If
    the scale is not in cm per mi, write and solve a proportion to find
    the answer.)

6.  Now we need to figure out the scale for each of the other images.
    Solve the following proportion for each image and write the scale
    ratio in the proper box in column 2.

[Distance on map in cm\_\_]{.underline} \_\_\_\_\_\_\_ =
\_\_\_[?]{.underline}\_\_\_ \_\_\_\_\_\_\_ = \_\_\_[?]{.underline}\_\_\_

Distance in real life in mi 1 1

7.  Put a star on the chart by the image that has the largest scale. You
    can tell which image has the largest scale by looking at how far
    apart your two dots are. The image that has the most centimeters for
    each actual mile has the largest scale. We will call this image your
    target image. Put an x in each empty box across from your target
    image.

Now, we need to enlarge the other two images on a copying machine until
all three scales are the same size. The question is how much do we need
to enlarge it. Our answer will need to be in a percent.

8.  Subtract the scale (centimeters equivalent to one mile) from one
    image (recorded in the 2^nd^ column) from the scale on your target
    image (also recorded in the 2^nd^ column). Record your answer in the
    third column. Do the same for the last empty box in column three.

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9.  Fill in column four with the ratios of the difference (column 3) to
    the scale (number of centimeters equivalent to one mile). Be sure to
    simplify.

10. In the space below, change these two ratios to percents using
    proportions. Record your answers in the last column of your chart.

These percents are the percent you would like your images to be
increased. Write the percent that goes with each of your two images and
your name on a piece of sticky paper. Attach each percent to the proper
image. Give these images to your teacher so that she can enlarge them
the proper amount. While you are waiting, finish the following chart. By
checking these answers, you should have a good idea if you filled in
your chart correctly.

  ------------- ----------- ----------- ------------ ------------ -----------
  Images        Distance    Scale (in   Difference   Ratio        Percent
                (in cm)     cm per      between two               
                between two mile)       images                    
                dots                                              

  Topographic   13                                                
  Map                                                             

  Aerial        9                                                 
  Photograph                                                      

  ∆ NDVI Image  18                                                
  ------------- ----------- ----------- ------------ ------------ -----------