Summary:

At The Irene Erickson Elementary School in Tucson, Arizona, students explore the concept of "halves" using geoboards to represent areas that can be partitioned into two equal parts. First Ms. Richardson leads the class through two examples of square unit counting to prove "halves." In the first example two equal area rectangles convince students they have found "half" in the first example and one of two equal area triangles is determined to be another representation for "half." Students conclude from these demonstrations that one-half means one of two equal-sized parts that may or may not be congruent but must have equal areas. Working in pairs or threesomes, students find many more instances of congruent and non-congruent "halves" on their geoboards and record these findings on geoboard paper. Teammates justify their findings to each other before presenting "proofs" to the entire class.

Standard Connection:

(Practice Standard)—In this lesson, the third Common Core Practice Standard—Construct viable arguments for conclusions reached and critique the reasoning of others—is played out as students attempt to convince each other that they have found halves represented by either congruent or non-congruent polygons. The four boys try to prove that the congruent trapezoids on their geoboard represent one-half of the geoboard. Ms. Richardson's carefully constructed questions guide them all into reasoning that two 1-unit by 2-unit triangles were equal in area to two unit squares. This discovery requires the boys to solve a problem isolated from the larger "area of the trapezoid" problem. The boys critique each other's thinking and create a viable argument for a polygon that is "half" of the geoboard.

(Content Standard)—The domain that captures the mathematics content in this lesson is—Geometry 3.G. Students demonstrate that they can "partition shapes into parts with equal area" and "express the areas" of each newly created area as "a unit fraction of the whole." In this clip students partition the geoboard into two parts with equal parts, describe the area of each part as one half the area of the entire board, and prove that they have created halves.

Questions to Consider:

Having students present findings to the class is of what value to students and to the teacher? What other measurement, geometry, and number concepts can be addressed using geoboards?

3. Construct viable arguments and critique the reasoning of others

3.G Geometry