Student Actions

Teacher Actions

• Develop Accurate and Appropriate Procedural Fluency when using accurate procedures to represent slanted line segments in the coordinate plane and developing efficient processes for solving various mathematical situations involving lengths of slanted segments.

• Develop Strategies for Problem-Solving by exploring a variety of problem-solving strategies for different mathematical situations to discover that the Pythagorean Theorem can be used to find the shortest distance between two points on a coordinate plane. For example, determining the shortest distance between two locations on a map. 

• Implement tasks that promote reasoning and problem-solving by allowing students opportunities to compare and develop strategies for accurately determining the shortest distance between any two points on the coordinate plane without standard measuring practices.

• Promote procedural fluency by encouraging students to draw the triangle, label the sides and hypotenuse with known lengths, and substitute the values into the formula to find the missing length. 

Key Understandings

Misconceptions 

• The Pythagorean Theorem can be used to find the distance between any two points creating a slanted segment on a coordinate plane, regardless of the direction of the slant.

• A right triangle created by connecting the vertical line drawn through the highest endpoint and the horizontal line drawn through the lowest endpoint of a slanted segment can be used to prove that the Pythagorean Theorem determines the distance of any slanted line on the coordinate plane. 

• When finding the length of a vertical or horizontal segment drawn on grid paper, some students may miscount the units. For example, students will count the unit they start on as 1 instead of 0. Make sure students are counting the spaces between the two points.

• Students may forget to take the square root of the final value when using the Pythagorean Theorem to solve a mathematical situation. 

• Students may substitute the values into the wrong location within the formula.

• Students may use negative values and incorrectly simplify the Pythagorean Theorem when solving problems with coordinate pairs including negative integers. 

  Knowledge Connections

Prior Knowledge

Leads to 

• Plot integer and rational valued ordered pairs as coordinates in all four quadrants (6.A.1.1).

• Justify the Pythagorean Theorem using measurements, diagrams, or dynamic software to solve problems in two dimensions involving right triangles (PA.GM.1.1).

• Compare and order real numbers; locate real numbers on a number line. Identify the square roots of perfect squares to 400 or, if it is not a perfect square root, locate it as an irrational number between two consecutive positive integers (PA.N.1.4). 

• 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 (G.C.1.2).

• Apply the distance formula, the Pythagorean Theorem, and the Pythagorean Theorem converse to solve problems (G.RT.1.1).

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

•  Use coordinate geometry and algebraic reasoning to represent and analyze line segments and polygons, including determining lengths of line segments (G.2D.1.6).