a story of ratios

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NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios

A Story of RatiosGrade 8 – Module 2


N YS C O M M O N CO R E M AT H E M AT I C S C U R R I C U LU M A Story of Ratios

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 ModuleConcept DevelopmentModule Review



Curriculum Overview of A Story of Ratios



Agenda




L1: Why Move Things Around?• 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.

Lesson 1, Concept Development



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.



Lesson 1, Student Debrief



L2: Definition of Translation and Three Basic Properties• 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



Exercise 2

F

G

D

A’

B’

C’

E’

31˚

5



L3: Translating Lines• What happens when a line is translated?

• The line and its image coincide.

OR• The line and its image are parallel.




Exercises 1-4




• 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• 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.




Example 4



L5: Definition of Rotation and Basic Properties• 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.




Problem Set 1



L6: Rotations of 180 Degrees• 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



Exit Ticket 1

A’=(2, 4)

B’=(3, -1)



L7: Sequencing Translations• 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.




Discussion



L8: Sequencing Reflections and Translations• 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.




Discussion



L9: Sequencing Rotations• 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• 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.




Exercise 4



L11: Definition of Congruence and Some Basic Properties• 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.




Exercise 1



L12: Angles Associated with Parallel Lines• 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





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• 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.




L14: More on the Angles of a Triangle• How can we show that the measure of the exterior

angle of a triangle is equal to the sum of the measures of the two remote interior angles? • Use what we know about the sum of the angles of a

triangle.• Use the fact that a straight angle is 180˚.




Exercise 4



L15: Informal Proof of the Pythagorean Theorem• Square within a square proof.

• Applies knowledge learned in Module 1 with respect to the first Law of Exponents.

• Students use what they know about congruence to verify that four right triangles would have equal areas and apply properties of congruence to make claims about angle sums.

• Students know how to determine the measure of an unknown angle of a straight angle.




Proof of Pythagorean Theorem

Looking at the outside square only, the square with

side lengths (a+b), what is its area?

Are the four triangles congruent? How do you

know?

What is the area of one triangle? What is the total

area of the four triangles?




Is the figure inside really a square? How do we

know?

What is the area of the square in the figure?

How can we use areas to show that the

Pythagorean Theorem is true?



L16: Applications of the Pythagorean Theorem• Traditional application of the Pythagorean theorem to determine the:

• length of the hypotenuse of a right triangle, • distance between two points on the coordinate plane, • finding height a ladder can reach as it leans again a wall,• length of diagonal of a rectangle.




Exercise 3



Agenda




Biggest Takeaway

Turn and Talk:• What questions were answered for you?• What new questions have surfaced?



Key Points• Experimental verifications of the basic rigid motions.

• Precision of language in descriptions of sequences.

• Immediate application of concept of congruence for understanding angle relationships.

• Application of congruence and angle relationships to explain a proof of the Pythagorean theorem.

a story of ratios

Documents

module overview

module focus sessionnovember

common core institutegrade

concept of congruence

story of ratiosgrade

minutes materials

minute materials

mathematical models