Plot rational coordinates, reflect points and find axis-parallel distances

Learning goal: Locate rational points in all four quadrants, explain reflections caused by sign changes and find distances between points sharing an x- or y-coordinate. Draw a polygon from given vertices and use coordinates to determine its horizontal and vertical side lengths.

An ordered pair (x,y) gives horizontal position first and vertical position second. Start at the origin, move according to x, then according to y. For (−3,2), move three units left and two units up. The negative x does not make both coordinates negative. Axis labels and the ordered pair determine each movement separately.

Quadrant I has positive x and positive y; Quadrant II has negative x and positive y; Quadrant III has both negative; Quadrant IV has positive x and negative y. Points on an axis have a zero coordinate and do not belong to one of the four quadrants. The origin has both coordinates zero.

Compare (−3,2) with (3,2). Their y-coordinates match and their x-coordinates are opposites, so the points are reflections across the y-axis. Compare (−3,2) with (−3,−2): x stays fixed while y changes sign, so the reflection is across the x-axis. Changing both signs places the point at (3,−2), reached by reflecting across both axes in succession. Reflecting across the same axis twice returns the original point, because the opposite of the opposite of a number is the number itself: −(−3) = 3. A point on the y-axis, such as (0,2), is its own reflection across that axis because 0 is its own opposite.

The same reasoning applies to nonintegers. A point at (−1.5,0.5) lies one and one half units left and one half unit up. Reflecting across the y-axis gives (1.5,0.5). Use a scale that allows halves to be placed accurately; do not round a coordinate to the nearest integer merely because the grid has bold whole-number lines.

In a fictional garden map, plot A(−3,2), B(3,2), C(3,−2) and D(−3,−2), then connect A to B to C to D to A. The vertices form a rectangle crossing both axes. The axis units represent meters. Connecting points in a different order could create crossed segments instead of the intended boundary, so follow the stated vertex order.

A and B share y = 2, so their connecting side is horizontal. From x = −3 to zero is three meters, then zero to x = 3 is another three. The length is |−3| + |3| = 6 meters. B and C share x = 3, so the vertical side is |2| + |−2| = 4 meters. The rectangle's perimeter is 6 + 4 + 6 + 4 = 20 meters. In the real-world problem of fencing this garden, twenty meters is the length of fence needed, found entirely from coordinate distances.

Adding absolute values works for opposite-side coordinates because the segment passes through zero. When both varying coordinates are on the same side of zero, subtract their distances from zero. Points (−5,1) and (−2,1) are three units apart: five minus two. Adding five and two would count a path to zero and back, not the direct segment between the points.

A general horizontal distance is the absolute value of the difference between x-coordinates when the y-coordinates match. For −5 and −2, |−2 − (−5)| = |3| = 3. You can justify the result with the number line even before becoming fluent with signed subtraction. A corresponding vertical distance uses the y-coordinates when x matches.

The diagram shows the rectangle and a separate marked half-unit point. All units on both axes use the same physical scale. The horizontal six-meter side is longer than the vertical four-meter side in the same 6:4 ratio. This makes the picture consistent with the computed lengths.

Do not apply an axis-parallel distance rule to a diagonal. A and C share neither coordinate. Adding horizontal and vertical travel gives a route along perpendicular segments, not the straight diagonal distance. This lesson asks for horizontal and vertical sides and routes; a diagonal requires additional reasoning.

Worked vertical segment crossing an axis: E(−4,−1.5) and F(−4,2.5) share x = −4, so the segment is vertical. It crosses zero in the varying coordinate, giving length 1.5 + 2.5 = 4 units. The fixed negative x-coordinate does not contribute to the segment's length.

Worked reflection correction: “Reflect (−3,2) across the x-axis to get (3,2).” That changes horizontal position, so it is a y-axis reflection. Across the x-axis, retain x and reverse y: (−3,−2).

Guided graphing: Plot the four garden vertices, label the quadrants and connect the boundary in order. Identify reflection relationships between vertices. Use number-line reasoning to find each side length and check the perimeter.

Independent performance: Draw a new polygon from coordinates in all four quadrants with horizontal and vertical sides. Include at least one fractional coordinate on a separate point, show reflections across each axis and identify an axis point correctly. Calculate several same-coordinate distances, including points on the same side of zero and on opposite sides. State units and explain why your method does not directly give a diagonal length.