5.1+Extending+the+Coordinate+Grid

MT 3.05.09 Big Idea:Negative numbers help us to model many real world situations. Essential Question:How do I multiply and divide integers? Notes from class: Quadrants: The four sections into which the coordinate plane is divided by the x and y-axes. (3,4) the first number is the x-coordinate and the second number is the y-coordinate.


 * __5.1 Extending the Coordinate grid__**


 * The //x-//axis on a coordinate grid is a horizontal number line, and the //y//-axis is a vertical number line. On the coordinate grids you have worked with so far, all the values on the //x//- and //y//-axes have been greater than or equal to 0.



Just as we extended the number line in in Investigation 1 to include negtive numbers, we can extend the //x//- and //y-// axes of the coordinate grid to include negative numbers.



When the axes are extended, they divide the grid into four regions called //quadrants//. We can number these quadrants, starting with the region at the upper right and continuing counterclockwise. The quadrants are usually numbered by using roman numerals as shown below. Melina made a coordinate grid and plotted the points (4,3), (-3,1), (-4,-5) and (3,-3). Study her work, and see if you can figure out what she did. Recall that the two numbers that describe a point are called the //coordinates// of the point. The first number is the //x-coordinate// and the second number is the //y-coordinate//. For example, the first point Melina plotted has coordinates (4,3); the //x-coordinate// is 4, and the //y-coordinate// is 3.**

__**Problem 5.1**__

First she went along the x-axis to find the first number (the x-coordinate) and then from that point she went up to the second number (y-coordinate) and then plotted the point. She did this for all four points.
 * A. Describe how Melina located each of the four points on the coordinate grid.**

Example:(-3,1) First go along the x-axis and find -3 then from -3 go up to 1 now plot your point.

You could make a parallelogram because there are four points which means the shape can easily be a quadrilateral and the points are diagonal to each other which makes it a parrelogram if you join the points together in order of their quadrants(start at quadrant 1 then go to 2 then 3 then 4 and finish in 1).
 * B. What polygon could you make by connecting the four points? Justify your answer.**
 * C. On a coordinate grid, plot four points that are the vertices of a square, such that both coordinates of each point are positive integers.**


 * D. On a coordinate grid, plot four points that are the vertices of a square such that both coordinates of each point are negative integers.** [[image:MELANIE_2_squares.jpg]]


 * E. On a coordinate grid, plot four points that are the vertices of a square, such that one point has two negative-integer coordinates, one point has two positive-integer coordinates, and each of the other points has one positive-integer coordinate and one negative-integer coordinate.**

The pair of points could be (-1,5) and (3,5) or (-1,-3) and (3,-3)
 * F. Two vertices of a square are (3, 1) and (-1, 1). Find the coordinates for every pair of points that could be the other two vertices. **

__**Problem 5.1 Follow-up**__


 * Imagine that you can walk on a coordinate grid. Each integer unit is one step, and you must stay on the grid lines. Suppose you want to walk from point (4,2) to point (2,1), taking the least number of steps possible**


 * You could go to steps to the left and then 1 step down or you could go 1 step down and then 2 steps tp the left. Paths between two points that require the least possible number are called minimal paths.**

a. (-4,-2) to (5,3)** 9 steps to the right and 5 steps up or 5 steps up and nine steps right. 9 steps to the right and 1 step down or 1 step down and 9 steps to the right.
 * 1. For each pairs of points ,describe two minimal paths from the first point to the second point.
 * b.(-4,3) to (5,2)**

3 steps left and 2 steps up or 2 steps up and 3 steps left.
 * c. (2,-4) to (-1,-2)**

a. Locate two points on the coordinate grid such that it will take 12 steps to travel from one of the points to the other on a minimal path. No I don't think so because there are different options and not all graphs are the same for example the graph in the book and the one above are slightly different.
 * 2.**
 * b. Will everyone name the same two points for part a? Why do you think this is so? **