We just began a new module this week.  This module introduces points, lines, line segments, rays, and angles, as well as the relationships between them.  Students construct, recognize, and define these geometric objects before using their new knowledge and understanding to classify figures and solve problems.  With angle measure playing a key role in the work throughout the module, students learn how to create and measure angles, as well as how to create and solve equations to find unknown angle measures.  In these problems, where the unknown angle is represented by a letter, students explore both measuring the unknown angle with a protractor and reasoning through the solving of an equation.  This connection between the measurement tool and the numerical work lays an important foundation for success with middle-school geometry and algebra.  Through decomposition and composition activities, as well as an exploration of symmetry, students recognize specific attributes present in two-dimensional figures.  They further develop their understanding of these attributes as they classify two-dimensional figures.

Topic A begins with students drawing points, lines, line segments, and rays, as well as identifying these in various contexts and within familiar figures.  Students recognize that two rays sharing a common endpoint form an angle (4.MD.5).  They create right angles through a paper-folding activity, identify right angles in their environment, and see that one angle can be greater (obtuse) or less (acute) than a right angle.  Next, students use their understanding of angles to explore relationships between pairs of lines as they define, draw, and recognize intersecting, perpendicular, and parallel lines (4.G.1).

In Topic B, students explore the definition of degree measure, beginning with a circular protractor.  By dividing the circumference of a circle into 360 equal parts, they recognize one part as representing 1 degree (4.MD.5).  Through exploration, students realize that, although the size of a circle may change, an angle spans an arc, representing a constant fraction of the circumference.  By carefully distinguishing the attribute of degree measure from that of length measure, the common misconception that degrees are a measure of length is avoided.  Armed with their understanding of the degree as a unit of measure, students use various types of protractors to measure angles to the nearest degree and to sketch angles of a given measure (4.MD.6).  The idea that an angle measures the amount of turning in a particular direction is explored as students recognize familiar angles in varied contexts (4.G.1, 4.MD.5).

Topic C begins by decomposing 360  using pattern blocks, allowing students to see that a group of angles meeting at a point with no spaces or overlaps add up to 360 .  With this new understanding, students now discover that the combined measure of two adjacent angles on a line is 180  (supplementary angles), that the combined measure of two adjacent angles meeting to form a right angle is 90  (complementary angles), and that vertically opposite angles have the same measure.  These properties are then used to solve unknown angle problems (4.MD.7).

An introduction to symmetry opens Topic D as students recognize lines of symmetry for two-dimensional figures, identify line-symmetric figures, and draw lines of symmetry (4.G.3).  Given one half of a line-symmetric figure and the line of symmetry, students draw the other half of the figure.  This leads to their work with triangles.  Students are introduced to the precise definition of a triangle and then classify triangles based on angle measure and side length (4.G.2).  For isosceles triangles, a line of symmetry is identified, and a folding activity demonstrates that base angles are equal.  Folding an equilateral triangle highlights multiple lines of symmetry and establishes that all interior angles are equal.  Students construct triangles given a set of classifying criteria (e.g., create a triangle that is both right and isosceles).  Finally, students explore the definitions of familiar quadrilaterals and classify them based on their attributes, including angle measure and parallel and perpendicular lines (4.G.2).  This work builds on Grade 3 reasoning about the attributes of shapes and lays a foundation for hierarchical classification of two-dimensional figures in Grade 5.  The topic concludes as students compare and analyze two-dimensional figures according to their properties and use grid paper to construct two-dimensional figures given a set of criteria.

 

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