Summary:

"Halfway," "half-dollar," and "half-hour" are terms we use without giving much thought to the precise meaning of the word "half." Constance Richardson's fourth and fifth grade students at the Irene Erickson Elementary School in Tucson, Arizona, will move from imprecise notions of "half" to more precise mathematical thinking about this word. In answering the question "What does 'half' mean to you?" students use terms like "equal" and "same size." Using geoboards to show areas, Ms. Richardson asks students to find many ways as to represent "half." Common halves (i.e. congruent rectangles and congruent triangles on the 25-peg geoboard) are demonstrated during whole class discussions. Then, students work in pairs and threesomes to find less obvious "halves." Precise area counts of 8-square units are given as "proofs" that halves have been found on the geoboards.

Standard Connection:

(Practice Standard)—In this lesson, Common Core Practice Standard #6—attend to precision—is demonstrated. Students learn that precision is important in calculations and measurements, and also in the language you use to communicate with others. Using light-hearted banter, Ms. Richardson asks students to explain precisely where the rubber band should be placed to represent one half on the geoboard. She also works with the class to develop an informal consensus definition for an area of one half, which means dividing the geoboard into two parts that are equal in area. This informal defining and "proving" are foundational to the formal mathematical arguments and proofs made in later grades. Indeed, Ms. Richardson's insistence that students use precise, unambiguous language to "prove" to peers that halves have been found for congruent and non-congruent constructions is consistent with the way mathematics is warranted by mathematicians.

(Content Standard)—The domain that captures the mathematics content in this lesson is—Geometry 3G. 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 and prove the area of each part is precisely one half the area of the entire board. In addition, Measurement and Data 3.MD content is foundational in this lesson as students use the knowledge that "a unit square" has "one square unit" of area, thus making geoboard halves equal to "eight square units."

Questions to Consider:

What role should "conversation" (i.e. dialogue) play in mathematical proofs? How could this lesson be extended to help students better understand unit fractions? When else might you use geoboards as tools to clarify concepts?

6. Attend to precision

3.G Geometry

3.MD Measurement and Data