Lesson Plan

Midpoints, Distance, and Slope

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Objectives

In this lesson, students will use geometric figures in the coordinate plane to find slopes of lines, distances between two points, and the midpoints between two points. Students will:

  • learn how the slopes of parallel and perpendicular lines are related.
  • use the distance formula to determine the length of a side in a geometric shape in the coordinate plane.
  • use the midpoint formula to determine the midpoint of the segment, given its endpoints in the coordinate plane.


Essential Questions

  • How can you use coordinates and algebraic techniques to represent, interpret, and verify geometric relationships?


Vocabulary

  • Concave: Curving inward; a curve is concave toward a point if it bulges away from the point; a polygon is concave if it is not convex, i.e., if at least one of its interior angles is greater than 180 degrees.  [IS.1 - Preparation]  [IS.2 - Struggling Learners and ELL Students]     
  • Convex: Curving outward; a curve such that any straight line cutting the curve cuts it in just two points; a polygon is convex if it lies on one side of any one of its sides extended, i.e., if each interior angle is less than or equal to 180 degrees.
  • Coordinate Plane: A surface for which any set of numbers locate a point, line, or any geometry element in space; for Cartesian coordinates, the point, can be located by its distances from two intersecting straight lines, the distance from one line being measured along a parallel to the other line.
  • Distance Formula: The formula that represents the length of the line segment joining two points. In analytic geometry, it is found by taking the square root of the sum of the squares of the differences of the corresponding rectangular Cartesian coordinates of the two points, .
  • Equation: A statement of equality between two quantities, generally divided into two types, identities and conditional equations. A conditional equation is true only for certain values of the unknown; an identity is true for all values of the variables.
  • Formula: A general answer, rule, or principle stated in mathematical language.
  • Geometry: The science that treats the shape and size of things; the study of invariant properties of given elements under specified groups of transformations.
  • Midpoint: The point that divides a line segment into two equal parts; the point that bisects the line.
  • Parallel Lines: Equidistant, apart; if two lines are cut by a transversal, and the sum of the interior angles on one side of the transversal is less than a straight angle, the two lines will meet if produced, and will meet on that side of the transversal. Only one line can be drawn parallel to a given line through a given point not on the line.
  • Perpendicular Lines: Two lines are perpendicular to each other if, in a plane, the slope of one of the lines is the negative reciprocal of the other; two straight lines that intersect such that they form a pair of equal adjacent angles.
  • Polygon: A closed-plane figure consisting of points called vertices and lines called sides, which have no common point except for end points. A polygon is convex if each interior angle is less than or equal to 180 degrees. A polygon is concave if it is not convex.
  • Slope: The angle of inclination; for a straight line, the tangent of the angle that the line makes with the positive x-axis.


Duration

90–120 minutes/1–2 class periods [IS.3 - All Students]


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