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A short class with Sarah and her students. The other ways are under More.
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Welcome everyone, today we will learn about mapping points in a 2D coordinate system. Can anyone tell me what defines a point in this system?
Isn't it the x and y coordinates?
That's right! Points are defined by their x and y coordinates. For example, a point could be (2,3). Now, consider what happens when we have 5 distinct points. What do you think?
There must be some relationships between the points?
Exactly! Today we will explore how many midpoints can have integer coordinates based on these points.
Now, let’s talk about the Pigeonhole Principle. Who remembers what this principle states?
If you have more items than containers, then at least one container must hold at least two items?
Perfect! Now let’s apply this principle to our 5 points. We can categorize these points based on whether their x and y coordinates are odd or even. How many categories do we have?
That would be four categories: even x & even y, even x & odd y, odd x & even y, and odd x & odd y.
Exactly, and since we have 5 points but only 4 categories, there must be at least one category with at least two points. Let’s investigate the significance of this!
Now that we've identified our points, let’s discuss how to calculate the midpoint of two points. Can anyone tell me the formula?
I think it's (x1 + x2)/2, (y1 + y2)/2.
You got it! Now, if both points fall into the same category, how does that affect the midpoint?
If they share the same category, then their x-coordinates will both be even or both will be odd, right? So the sum will always be even.
Very good! Therefore, their midpoint will be an integer. This shows that at least one midpoint must have integer coordinates.
Relates to categorizing points based on their coordinates.
Fundamental theorem for proving the existence of pairs of points.
Method to calculate the center point between two coordinates.
Essential for confirming midpoints result in whole numbers.
When points we pick and pairs we see, even or odd, they must agree!
Imagine a set of distinct friends, on a quest to find their matching midpoints. Each one represents either even or odd, and together they prove that pairs can be quite odd in their evenness!
E-O pairs: Every Odd pairs Odd, Every Even pairs Even - E.O.E.E.
MAP: Midpoint According to Pairs - helps to remember how to calculate midpoints.
Term
Midpoint Formula
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Pigeonhole Principle
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Integer Coordinates
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A pair of numbers used to identify a point in a two-dimensional space, represented as (x, y).
A principle stating that if more items are put into fewer containers, then at least one container must hold more than one item.
The point that is exactly halfway between two given points in a coordinate system.
Points that are not the same; each point has unique coordinates.
Coordinates on a coordinate plane where both the x and y values are whole numbers, without fractions or decimals.
This section explores the concept of mapping arbitrary distinct points in a 2D plane to ordered pairs, demonstrating the existence of midpoints with integer coordinates using the Pigeonhole Principle.
In this section, we examine 5 distinct points in a 2D coordinate system and apply the Pigeonhole Principle to establish that at least one pair of points exists such that their midpoint has integer coordinates. This is accomplished by mapping the points based on their even or odd properties and analyzing the midpoints mathematically.
In section 6.2, titled 'Mapping to Ordered Pairs', we delve into a foundational concept in discrete mathematics involving points in a two-dimensional plane. The section begins with the introduction of 5 arbitrary, distinct points, defined by their integer coordinates (x,y). The objective is to utilize the Pigeonhole Principle to show that among these points, there is guaranteed to be a pair whose midpoint results in integer coordinates. To do this, we categorize each point based on the parity (even or odd) of its coordinates into four possible combinations. Applying the Pigeonhole Principle, which states that if you have more items than containers, at least one container must hold more than one item, assures that at least two points will fall into the same category. Consequently, this leads us to prove that the coordinates of the calculated midpoints will always be integers. The conclusion reinforces the idea that discrete mathematics can effectively model and solve real mathematical problems.