HW # 4

In, On, or Out?

Do you know what a Lattice Point is?

 

Materials needed for this homework:
Compass
Graph paper (Don't have any? Get some here)
The definition of Boundary and Lattice Point
The Pythagorean Theorem
Your Book

 

 

 

 

Notice that:
  1. All the points (location of the trees in the orchard) are located in the first quadrant.
  2. This is a radius 10 orchard.
  3. The center is located at (0, 0)
Your job is to locate every point on the list (plot them) and decide  which ones are inside the boundary, outside the boundary or exactly on the boundary, and explain how you decided. (Boundary means the border; or using the language of circle it means the circumference).
 
Locate the center of the orchard (0,0) on the lowest  left corner of the graph paper.
Locate the points (0, 10) and (10, 0)
Why (0,10) and (10, 0)? Because this is a radius 10 orchard.
Use your compass to draw a quarter of a circle with the center in (0, 0)  and passing through the points (0, 10) and (10, 0).
Look at the diagram. It shows you the size of the orchard and its boundary.
 
Now, plot the location of the 13 trees given in the problem.
 
Make the following table
Point Inside
the boundary
Points on the boundary
Points outside the boundary
Undecided
d < 10 d = 10 d >10
to close to call
       
       
       
       
       
 
Notice that deciding whether a given point is within the orchard is fairly easy when the given point is on one of the axis, but for other points it can be more difficult. We called those points "undecided points."
 
How can we prove or be certain that a tree is within the orchard?
           That is that the tree is planted in your property.
 
Well, the problem is telling us that the radius of the orchard is 10 units.
 
Based on that fact we can conclude that:
If the distance between the center of the orchard and the center of a given tree is less than 10 units, the tree is inside the boundary.
 
If the distance between the center of the orchard and the center of a given tree is equal to10 units, the tree is on the boundary.
 
If the distance between the center of the orchard and the center of a given tree is greater  than 10 units, the tree is outside the boundary.
How can we find the distance between two points in a coordinate system?
 
Choose one of the undecided points.
Starting in the center of the orchard (0, 0) draw a line to the x-coordinate of the point in question.
From the x value (on the x-axis) draw a line to the point.
From the point draw a line to the center of the orchard.
 
Do you see a right triangle?
Can you tell the length of both legs of the right triangle?
 
The missing side, the hypotenuse, is the distance between the center of the orchard and the center of the tree.
 
Now you can use the Pythagorean Theorem to find the distance.
 
Remember:
If the distance between the center of the orchard and the center of a given tree is less than 10 units, the tree is inside the boundary.
 
If the distance between the center of the orchard and the center of a given tree is equal to10 units, the tree is on the boundary.
 
If the distance between the center of the orchard and the center of a given tree is greater  than 10 units, the tree is outside the boundary.
 
Repeat the process with all undecided points. Make sure to show all your work.
One of the mistake that some students make is doing all the calculations on the calculator and showing only the answer.
As students always show every step of their process.
That's why they are As students!
 
Here is your chance to become an A or B student.
 
Note: A Lattice point is a point whose coordinates are both integers.
 
Now that you are done, go ahead and answer this questionnaire

Mr. Lora

 

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