Student Actions

Teacher Actions

• Develop Accurate and Appropriate Procedural Fluency by correctly using efficient procedures with formulas and/or mathematical models to solve for distance, midpoint, and slope in a variety of tasks.

• Develop A Deep and Flexible Conceptual Understanding by determining how and when to apply concepts of distance, midpoint, and slope to solve mathematical problems involving polygons.

• Develop Mathematical Reasoning when using knowledge of distance, midpoint, and slope to develop arguments, and counterarguments, for appropriate methods in solving polygon problems. 

• Use and connect mathematical representations by guiding students to determine between coordinate geometry and algebraic formulas when calculating distance, midpoint, and slope.

• Build procedural fluency from conceptual understanding by modeling distance, midpoint, and slope with properties of polygons

• Encourage productive struggle by providing a variety of tasks that encourage students to use multiple representations of the slope, distance, and midpoint formulas.

• Encourage the use of technology when available. 

Key Understandings

Misconceptions 

• The endpoints of a line segment determine the side length, midpoint, and slope of the line segment.

• Slope is used to determine if lines or line segments are parallel, which can also be used to identify properties of polygons.

• The distance formula and midpoint formulas are used to determine if line segments are congruent, which can also be used to identify properties of polygons.

• The distance formula is an application of the Pythagorean Theorem. 

• Students confuse positive and negative slope values.
• Students will incorrectly identify the x-values and y-values when calculating the slope.
• Students use an incorrect process for slope, distance, or midpoint.
• Students may not recognize how to use the coordinate plane to calculate midpoint and distance.
• Students may not correctly determine an endpoint of a segment when given a midpoint and the other endpoint. 
• Students confuse the midpoint and distance formulas and incorrectly identify when it is most appropriate to use either process. 

  Knowledge Connections

Prior Knowledge

Leads to 

• Use the Pythagorean theorem to find the distance between any two points in a coordinate plane. (PA.GM.1.2)

• Write square roots of constants and monomial algebraic expressions in simplest radical form. (A1.N.1.1)

• Add, subtract, multiply, divide, and simplify square roots of constants, rationalizing the denominator when necessary. (A1.N.1.2) 

• Use knowledge of solving equations with rational values to represent, use and apply mathematical models (e.g., angle measures, geometric formulas, dimensional analysis, Pythagorean theorem, science, statistics) and interpret the solutions in the original context. (A1.A.1.1)  

• Analyze and interpret mathematical models involving lines that are parallel, perpendicular, horizontal, and vertical. (A1.A.4.2)

• Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) to solve problems involving angle measures and segment lengths using mathematical models, algebraic reasoning, and proofs. (G.2D.1.5) 

• Apply the properties of congruent or similar polygons to solve problems using mathematical models and algebraic and logical reasoning. (G.2D.1.8)

• Use numeric, graphic, and algebraic representations of transformations in two dimensions (e.g., reflections, translations, dilations, rotations about the origin by multiples of 90 ) to solve problems involving figures on a coordinate plane and identify types of symmetry. (G.2D.1.11) 

• Use the distance and midpoint formula, where appropriate, to recognize and write the radius r, center (h,k), and standard form of the equation of a circle (x - h)²+(y - k)²=r² with and without graphs. (G.C.1.2)

• Apply the distance formula, the Pythagorean theorem, and the Pythagorean theorem converse (approximate and exact values, including Pythagorean triples) to solve problems, using algebraic and logical reasoning and mathematical models. (G.RT.1.1)