How Far Between Cities? Part II
by Suzanne Alejandre
Teacher Lesson Plan
Feedback Form || Student Page
Open the Java Applet
Note: It will open in a separate window. If you are displaying the page for students, arrange your browser windows so that the applet and the directions can be easily viewed. If students are working individually they should be encouraged to do this.
Ask the question:
How can you find the distance between Rialto and Loma Linda using this map?
Assuming that care has been taken to have horizontal and vertical line segments, prompt students to notice the:
- display of x (in pixels): How can you calculate how long a line segment is in pixels if you know the x coordinate of both endpoints of that line segment?
- display of y (in pixels): How can you calculate how long a line segment is in pixels if you know the y coordinate of both endpoints of that line segment?
- Pythagorean Theorem: Can you find the hypotenuse of a right triangle given the measure of the other two sides of the triangle?
- scale of pixels to miles: If you know there are 25 pixels in 1 mile, how do you calculate the distance of something in miles given the number of pixels?
NOTE: A pixel is the basic unit of the composition of an image on a computer monitor (usually a single-colored dot).
Students or groups of students working together should complete the following calculations:
Note: The numbers given below are given only as a guideline since there isn't really a place where the students know exactly that they are on the mark for the city location. Students might figure that as long as they're touching the dot somehow, they're on it. Reasonably close answers to what is given below should be accepted.
Rialto is located on the given map: (196, 180)
Loma Linda is located on the given map: (402, 57)
The difference between the two locations on the "x" side of the triangle is 206.
The difference between the two locations on the "y" side of the triangle is 123.
206^2 + 123^2 = 42436 + 15129 = 57565
To find the length of the hypotenuse of the right triangle that was formed, find the square root of 57565. Rounding to the nearest unit, it is about 240.
Using the ratio: 25 pixels/240 pixels = 1 mile/9.6 miles
Rialto is about 9 and a half miles from Loma Linda.
Practice: Ask students to find the distances between other cities in the Inland Empire including:
Fontana and San Bernardino
Muscoy and Loma Linda
Ask students to generalize this method and write an explanation of the steps they would take to calculate the distance between any two cities on the map.
Back to Part I
Math Forum Resources
[Problems of the Week]
Selections from Math Forum Problems of the Week:
- AlgPoW: A Tale of Two Intercepts - posted February 7, 2000
- Find the distance between two y-intercepts.
- GeoPoW: Building a Bridge Across a Chasm - March 20-24
- So they're standing on the edge of this chasm. It isn't really that far across, but they need to find a log or tie together enough rope to reach a tree on the other side.
[Math Forum Resources]
Selections from Teacher2Teacher Discussions:
- Pythagorean Theorem
- One way to teach the Pythagorean Theorem
[Math Forum Resources]
Selections from Ask Dr. Math Archives:
- 30-60-90 and 45-45-90 Triangles
- If I have a triangle that is 30-60-90 or 45-45-90, how do I find all the sides when given only one side?
- Defining Distance Mathematically
- What is wrong with D' = sqrt(X^2 - X'2)?
- FAQ: The Pythagorean Theorem
- What is the Pythagorean theorem? When would you use it? How can we prove it?
- Figuring Ratios
- How do you figure the ratio of something?
- Finding a Point Equidistant From Two Other Points
- Point A is (-5,-3), and point B is (-1,-5); to be equidistant from A and B, what should the value of k be for the point (3,k)?
- First Math Teacher
- Who was the first math teacher? How did Pythagoreas come up with a^2 + b^2 = c^2?
- How Long is the Hypotenuse?
- In a right triangle, the lengths of the segments connecting the points of trisection of the hypotenuse to the vertex of the right angle are 7 and 9...
- Midpoint of a Straight Line Segment
- What is the midpoint of this: (-3,4) (5,-4)? Use the distance formula.
- Ratio and Proportion: Beaches and Hawks
- On a map, the scale states that 3 inches represent 125 miles. Two beaches are 5.2 inches apart. How far apart are they in miles?
- What does the second number in a ratio stand for?
- Teaching about Bearings
- Can you help us learn how to locate a point using two bearings?
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