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A Story of Ratios. Grade 8 – Module 2. Session Objectives. Examine the development of mathematical understanding across the module using a focus on concept development within the lessons.
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A Story of Ratios Grade 8 – Module 2
Session Objectives • Examine the development of mathematical understanding across the module using a focus on concept development within the lessons. • Identify the big ideas within each topic in order to support instructional choices that achieve the lesson objectives while maintaining rigor within the curriculum.
Agenda Introduction to the Module Concept Development Module Review
Agenda Introduction to the Module Concept Development Module Review
L1: Why Move Things Around? Lesson 1, Concept Development • We want to move things around in the plane to avoid direct measurement. Are a pair of angles equal in measure? Are the opposite sides of a rectangle the same size? • Students describe intuitively what motion would be required to move a figure on the plane so that it rests on top of a copy of the figure located somewhere else on the plane. Students “map” a figure onto another.
Exploratory Challenge 1 Slide the original figure to the image (1) until they coincide. Slide the original figure to (2), then flip it so they coincide. Slide the original figure to (3), then turn it until they coincide.
L2: Definition of Translation and Three Basic Properties Lesson 2, Concept Development • Translation is defined as a transformation along a vector. • Translations have three basic properties: • Translations map lines to lines, rays to rays, segments to segments, and angles to angles. • Translations preserve the lengths of segments. • Translations preserve the measures of angles.
Exercise 2 G F D A’ C’ 31˚ B’ 5 E’
L3: Translating Lines Lesson 3, Concept Development • What happens when a line is translated? • The line and its image coincide. OR • The line and its image are parallel.
Lesson 3, Student Debrief • This summary is not accurate as it includes information about corresponding angles. Which is a concept of Topic C. Please correct and we will do the same.
L4: Definition of Reflection and Basic Properties Lesson 4, Concept Development • A reflection is defined as a transformation that occurs across a line. • Reflections have three basic properties: • Reflections map lines to lines, rays to rays, segments to segments, and angles to angles. • Reflections preserve the lengths of segments. • Reflections preserve the measures of angles.
L5: Definition of Rotation and Basic Properties Lesson 5, Concept Development • A rotation is defined as a transformation that around a center a given degree. • Rotations have three basic properties: • Rotations map lines to lines, rays to rays, segments to segments, and angles to angles. • Rotations preserve the lengths of segments. • Rotations preserve the measures of angles.
L6: Rotations of 180 Degrees Lesson 6, Concept Development • Students know that a point (a, b) on the coordinate plane, after a rotation of 180˚ will be located at (-a, -b). • The point, center, and image of point are collinear.
Exit Ticket 1 A’=(2, 4) B’=(3, -1)
L7: Sequencing Translations Lesson 7, Concept Development • Now that we know we can move things around in the plane and they remain rigid, can we move things back? • Basis for conceptual understanding of congruence. • The image of a figure can be mapped back onto its original position by naming the vector that would move it there. For example, a translation along vector AB can be moved back to its original position if you translate the image along vector BA.
L8: Sequencing Reflections and Translations Lesson 8, Concept Development • A figure has been translated along a vector and reflected across a line. Will its image be in the same place if the figure had been reflected first and translated second? • The order in which we perform our basic rigid motions has an effect on the final location of an image. • Supports the development of conceptual understanding of congruence.
L9: Sequencing Rotations Lesson 9, Concept Development • What happens when a figure is rotated twice around the same center compared to when it is rotated twice around two different centers?
L10: Sequences of Rigid Motions Lesson 10, Concept Development • Given two triangles in the plane, (one an image of the other), can we describe a sequence of rigid motions that would map one triangle onto the other? • Focus on precision of language and descriptions of sequences.
L11: Definition of Congruence and Some Basic Properties Lesson 11, Concept Development • Congruence is defined as a sequence of basic rigid motions that maps one figure onto another. • A congruence enjoys the same properties as the individual basic rigid motions.
L12: Angles Associated with Parallel Lines Lesson 12, Concept Development • When two lines are cut by a transversal, angles are formed. • Corresponding angles • Alternate interior angles • Alternate exterior angles • When those lines are parallel, the angles formed are equal in measure and the proof lies within our understanding of rigid motions. • Corresponding angles: translation • Alternate interior and exterior angles: rotation
Exploratory Challenge 2 The angles are equal in measure. A translation along the transversal will map one angle onto the other. The angles are equal in measure. A translation along the transversal will map one angle onto the other. The angles are equal in measure. A rotation around the midpoint of the segment of the transversal between the parallel lines will map one angle onto the other.
L13: Angle Sum of a Triangle Lesson 13, Concept Development • Students are shown 3 informal proofs as to why the angle sum of a triangle is 180˚. (2 in this lesson and 1 in the next.) • Students work through a proof of the angle sum of a triangle that relies on understanding the basic rigid motions and their properties.
