The Shape of Things
In the problem The Shape of Things, students use geometric reasoning to solve problems involving two-dimensional objects. The mathematical topics that underlie this problem are the attributes of polygons; symmetry; spatial visualization; mathematical justification, including inductive and deductive reasoning; and formal proof. In each level, students must make sense of the problem and persevere in solving it (MP.1). Each problem is divided into five levels of difficulty, Level A through Level E, to allow access and scaffolding for students into different aspects of the problem and to stretch students to go deeper into mathematical complexity.
In this task, students identify shapes found in a playground scene. The student will draw and name the shapes and describe its parts.
In this level, students are presented with the task of examining, identifying, and comparing the attributes of the different geometric objects and describing their attributes.
This level addresses Common Core standard 2.G.A.1 by having students recognize and draw shapes and describe their attributes.
In this level, students to use line symmetry to reason about a picture half-folded from view. The students are asked to draw the whole picture, even though they see only half of the picture, but know the fold is a line of symmetry.
This level addresses Common Core standard 4.G.A.3 by having students recognize a line of symmetry and identify that, when folded along the line, a figure will have matching parts. It does extend slightly beyond the standard by discussing reflective symmetry, but this can be used as an extension.
In this level, students are given a set of corporate logos. Using their knowledge of rotational symmetry and spatial visualization, the students must determine which logos have rotational symmetry.
Students then must design their own unique logo with rotational symmetry.
In this level, students explore designs for a kite using two sticks as the diagonals of the quadrilateral-shaped kite. They investigate different types of quadrilaterals based on the properties of their diagonals.
In this level, students describe special quadrilaterals, including parallelograms, using the properties of their diagonals (G-CO.C.11). Students are also modeling a kite with quadrilaterals and discussing the properties of those quadrilaterals (G-MG.A.1).
In this level, students are asked to formally prove or disprove a conjecture about the ways a regular hexagon can be divided into two equal-sized pieces.
In this level, students use congruent triangles and properties of shapes inscribed in circles to prove a statement about how a hexagon can be divided equally (G-CO.C.10, G-C.A.2, and G-SRT.B.5).
PROBLEM OF THE MONTH
Download the complete packet of The Shape of Things Levels A-E here.
You can learn more about how to implement these problems in a school-wide Problem of the Month initiative in “Jumpstarting a Schoolwide Culture of Mathematical Thinking: Problems of the Month,” a practitioner’s guide. Download the guide as iBook with embedded videos or Download as PDF without embedded videos.
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