Lessons

What About Medians?

Discipline

Mathematics

Grades

9, 10, 11, 12

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Common Core State Standards for Mathematics

Standards for Mathematical Practice

CCSS.Math.Practice.MP8

Look for and express regularity in repeated reasoning.

High School - Geometry

Congruence

Prove geometric theorems

HSG-CO.C.10

Prove theorems about triangles.

*Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.*Make geometric constructions

HSG-CO.D.12

Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).

*Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.*Ohio Mathematics Academic Content Standards (2001)

Geometry and Spatial Sense Standard

Benchmarks (8–10)

E.

Draw and construct representations of two- and three-dimensional geometric objects using a variety of tools, such as straightedge, compass and technology.

H.

Establish the validity of conjectures about geometric objects, their properties and relationships by counter-example, inductive and deductive reasoning, and critiquing arguments made by others.

Grade Level Indicators (Grade 8)

1.

Make and test conjectures about characteristics and properties (e.g., sides, angles, symmetry) of two-dimensional figures and three-dimensional objects.

Grade Level Indicators (Grade 10)

3.

Make, test and establish the validity of conjectures about geometric properties and relationships using counterexample, inductive and deductive reasoning, and paragraph or two-column proof, including:
a. prove the Pythagorean Theorem;
b. prove theorems involving triangle similarity and congruence;
c. prove theorems involving properties of lines, angles, triangles and quadrilaterals; and
d. test a conjecture using basic constructions made with a compass and straightedge or technology.

Principles and Standards for School Mathematics

Geometry Standard

Use visualization, spatial reasoning, and geometric modeling to solve problems

Expectations (6–8)

draw geometric objects with specified properties, such as side lengths or angle measures;

Expectations (9–12)

draw and construct representations of two- and three-dimensional geometric objects using a variety of tools;

Students draw a triangle and the three medians of the triangle using Geometer's Sketchpad®. The students identify and measure the line segments between the vertices of the triangle and the centroid and between the centroid and the midpoints of the opposite sides using dynamic geometry software. They draw conclusions about the relationships among the lengths of those line segments, as well as relationships among the areas of the triangles formed. Activity sheets and suggestions for assessment are included. (author/sw/jk)