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**A Story of Geometry Grade 8 to Grade 10 Coherence**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X A Story of Geometry Grade 8 to Grade 10 Coherence 2 min. Welcome and introduction.

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Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Objectives Articulate and model the instructional approaches to teaching the content. Examine the coherence of topics and lessons from grade 8 to grade 10. 1 min. Read through the objectives on the slide.

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**Participant Poll Classroom teacher School leader Principal**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Participant Poll Classroom teacher School leader Principal District leader BOCES representative In order for us to better address your individual needs, it is helpful to know a little bit about you collectively. Who of you are classroom teachers? (Call for a show of hands.) School-level leader? Principal? District-level leader? BOCES representative? NOTE TO FACILITATOR: As you poll the participants, take note of the approximate size of each group. This will make it easier for you to re-group the participants for the final portion of this presentation. Regardless of your role, what you all have in common is the need to deeply understand the mathematics of the curriculum and the intentional instructional sequence in which it is brought to life for students. Throughout this session, we ask you to be cognizant of your specific educational role and how you will be able to promote successful implementation in your classroom, school, district, and/or BOCES. Each time we pause to reflect, please do so through the lens of your own professional responsibilities. At the close of this session, you will have the opportunity to share your thoughts, ideas, and concerns with others in a similar role.

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**Agenda Congruence and Rigid Motions Grade 8: Basic Rigid Motions**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Agenda Congruence and Rigid Motions Grade 8: Basic Rigid Motions Translation, Reflection, Rotation Grade 10: Basic Rigid Motions Congruence 1 minute: Read through agenda.

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**Transformations in Geometry under the CCSS**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Transformations in Geometry under the CCSS Transformations, specifically rigid motions, serve as the foundation of the concept of congruence Why is congruence defined in terms of rigid motions? To avoid having to directly measure objects: Are the opposite sides of a rectangle really equal in length? Are two angles positioned differently in space really of equal measure? To develop an intuitive sense of congruence, leading to a definition that can be used with all figures in the plane-not just triangles and polygons. 4 minutes: Say, “The first thing we want to do in grade 8 is explain the need to learn about the basic rigid motions.” Read the first 3 bullets. Say, “How often do students, or people in general, measure with great accuracy? If a measurement is off by a millimeter do you convince yourself that the two lengths are equal? Of course, there must be room for human error, but with this topic we are trying to convince students beyond a shadow of a doubt that two objects are in fact equal in length and/or size.” Read the bullet about angle measures, “Again, how accurate can we be with plastic protractors?” “The real goal of learning about basic rigid motions is to develop a real sense of what it means for two figures to be congruent. And not just triangles! Curved figures as well. Ultimately, we want to remove the textbook school mathematics definition for congruence of ‘same size, same shape.” Same Size & Same Shape

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Grade 8

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Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Translation Translation is defined as a motion that “slides” figures along a vector. A vector is a segment in the plane with a designated starting point and endpoint. 2 minutes: Say, “Translation is defined as the movement of all objects in the plane along a vector.” Read the definition of a vector. Click to show the first vector, “Notice that the arrow of the vector visually displays which way to move where the notation gives that information by writing the segment as AB, the starting point first, followed by the endpoint.” Click to show the second vector. “Notice that in this example the movement is from B to A, therefore the notation displays that same information by putting the B first followed by the A.”

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**Activity Draw the following on a piece of paper:**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Activity Draw the following on a piece of paper: A line, A ray, A segment, A point, An angle, A curved figure, A simple drawing of your choice. 5 minutes: Read the instructions on the slide. Once most participants have finished their drawings tell them to now draw a vector AB, length and direction is their choice. Once the vector is drawn say, “Now trace everything you have on the paper onto your transparency.” Once they have traced the figures, demonstrate on the document camera how to translate along the vector to show that all of the figures in the plane have moved. “Say, these figures did not move randomly, they moved in the exact direction of the vector and the exact length of the segment represented by the vector.” Ask participants, “Did the line change in shape or size? The ray? The segment? The point? The angle? The curved figure? Your simple drawing? In each case, no! That’s why translation is considered to be a rigid motion-a movement without real change, just location.” “Save your drawing, we will use it later.”

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**Properties of Translation**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Properties of Translation We have experimentally verified that a translation: Maps lines to lines, rays to rays, segments to segments, and angles to angles. Preserves lengths of segments. Preserves angles measures of angles. 1 minute: Say, “Based on the activity we just completed, we know that the following statements are true.” Read the first bullet, say “we saw that these things did not change under a translation.” Read the second bullet, say “notice that the length of the segment you drew did not increase nor decrease. Read the third bullet, say “the angle that you drew did not change in size, it just simply moved to a new location in the plane. We now have some basic properties of translation, verified by an experimental activity.”

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**Translation of Lines Some properties of translation are highlighted.**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Translation of Lines Some properties of translation are highlighted. Example: What properties can we discuss about translated lines? There are two possible scenarios: A line and its translated image coincide (when the vector belongs to the line or is parallel to the line): A line and its translated image will be parallel (when the vector is not parallel to the line): 3 minutes: Say, “some properties about translation are looked at more carefully, for example the translation of a line.” Read through the bullets and example diagrams on the slide.

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**A Sequence of Translations**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X A Sequence of Translations Imagine life without an “undo” button on your smart device or computer! We want to make sure that when we move things around in the plane, we can put them back where they belong, or “undo” the motion. For that reason, we show students how a translation along a vector can be undone by translating along a vector This is the beginning of the concept of congruence. It shows that a sequence of two translations can map a figure onto itself. 3 minutes: Read the first two bullets on the slide. “Recall that we spoke earlier about a vectors starting and endpoint. By reversing them, we show an inverse transformation.” Read the last bullet point. “In grade 8 students verify properties of rigid motions experimentally, then must think about congruence as a sequence of rigid motions. This is the first of many experiences where students do this.”

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**Activity Take out your paper and transparency.**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Activity Take out your paper and transparency. This time, reflect each of the images you drew by “flipping” your transparency across the line you drew. 3 minutes: Read the instructions on the slide. Demonstrate on the document camera how to reflect across a line by flipping the transparency to show that all of the figures in the plane have moved. “Say, these figures did not move randomly, they moved to the opposite side of the line of reflection. In some cases, if you drew an object on top of the line of reflection, you will see part of your figure on each side. They are the exact distance from the line of reflection as they were before, just on the other side.” Ask participants, “Did the line change in shape or size? The ray? The segment? The point? The angle? The curved figure? Your simple drawing? In each case, no! That’s why reflection is considered to be a rigid motion-a movement without real change, just location.” “Save your drawing, we will use it later.”

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**Properties of Reflection**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Properties of Reflection We have experimentally verified that a reflection: Maps lines to lines, rays to rays, segments to segments, and angles to angles. Preserves lengths of segments. Preserves angles measures of angles. Additional property that is verified: When you connect a point and it’s reflected image, the segment is perpendicular to the line of reflection. Not only is the line of reflection perpendicular to the segment, but it bisects the segment. 2 minute: Say, “Based on the activity we just completed, we know that the following statements are true.” Read the first bullet, say “we saw that these things did not change under a reflection.” Read the second bullet, say “notice that the length of the segment you drew did not increase nor decrease. Read the third bullet, say “the angle that you drew did not change in size, it just simply moved to a new location in the plane. We now have some basic properties of reflection, verified by an experimental activity.” Read the additional property that is verified. Say, “we want students to have a strategy for checking to see if a figure has been reflected, hence the need to talk about the line of reflection being perpendicular to the segment.”

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**Activity Take out your paper and transparency.**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Activity Take out your paper and transparency. This time, rotate each of the images you drew by placing your finger on top of the point you drew and carefully rotate your transparency in one direction and then the other. 3 minutes: Read the instructions on the slide. Demonstrate on the document camera how to rotate the transparency around a point to show that all of the figures in the plane have moved. “Say, these figures did not move randomly, they moved in a circle around the center of rotation. Ask participants, “Did the line change in shape or size? The ray? The segment? The point? The angle? The curved figure? Your simple drawing? In each case, no! That’s why rotation is considered to be a rigid motion-a movement without real change, just location.”

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**Properties of Rotation**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Properties of Rotation We have experimentally verified that a rotation: Maps lines to lines, rays to rays, segments to segments, and angles to angles. Preserves lengths of segments. Preserves angles measures of angles. 2 minute: Say, “Based on the activity we just completed, we know that the following statements are true.” Read the first bullet, say “we saw that these things did not change under a rotation.” Read the second bullet, say “notice that the length of the segment you drew did not increase nor decrease. Read the third bullet, say “the angle that you drew did not change in size, it just simply moved to a new location in the plane. We now have some basic properties of rotation, verified by an experimental activity.”

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Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Congruence With each new rigid motion that is learned, students immediately begin sequencing the motion with a known motion. For example: The first sequence is two translations. Once reflection is learned, students sequence two reflections. Then, students sequence a translation and a reflection. Once rotation is learned, students sequence two rotations. Then, students sequence a translation and a rotation, or a rotation and a reflection, etc. Congruence is defined in terms of a sequence of rigid motions, performed using a transparency, that shows the mapping of one figure onto another. 3 minutes: Read through the bullets on the slide.

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Grade 10

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Header July 2013 Network Team Institute Rigid Motions in Grade 10 Students enter Grade 10 with an intuitive sense of congruence and have experimentally verified properties of rigid motions They know that “same size, same shape” is not a precise way of describing congruence Defining congruence with the use of rigid motions captures all types of figures In Grade 10, students formalize the concepts from Grade 8 through language The visual/experiential understanding of how each rigid motion actually “works” is put into explicit parameters Students think about the plane and the rigid motions in the plane more abstractly Constructions are used in the application of rigid motions 3 mins Say, “We now examine how rigid motions are studied in Grade 10.” Read through the bullet points. - Emphasize that ‘all’ types of figures means not just triangles or simple figures constructed out of triangles.

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Header July 2013 Network Team Institute Grade 10: Reflection In Grade 8, students have used transparencies to experimentally verify the properties of a reflection AND that the line of reflection is the perpendicular bisector of any segment that joins a pair of corresponding points between the figure and its image In Grade 10, students clearly define reflection and how to: Determine the line of reflection by construction Reflect a figure across a line by construction. 2 mins Say, “In Grade 8, students experimentally verified that lines map to lines, segments to segments, etc. and that reflections are distance preserving and angle preserving. In Grade 10, they will determine the line of reflection using what they know about the construction of a perpendicular bisector, and they will also reflect a figure across a line of reflection by construction.”

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**Grade 10: Determining the Line of Reflection**

Header July 2013 Network Team Institute Grade 10: Determining the Line of Reflection Use the construction of a perpendicular bisector to determine the line of reflection for the following figures: 5 mins If you are comfortable constructing a perpendicular bisector, take a few moments to consider how to determine the line of reflection between the triangles. If you are not familiar with the construction, we will provide you with a set of instructions to guide you through the construction.” 1. Draw segment CG (or AE or BF) 2. Draw circle CC: center C, radius CG, and circle CG: center G, radius GC. 3. Label the points of intersections as Y and Z. 4. Draw line YZ.

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**Grade 10: Determining the Line of Reflection**

Header July 2013 Network Team Institute Grade 10: Determining the Line of Reflection 5 mins Say, “Students now know the “why” behind the intuitive understanding and experimentation of a reflection. We can show, by construction, that the perpendicular bisector of any segment that joins a pair of corresponding points between the two triangles is in fact the line of reflection.” - Read the definition of reflection (perhaps remind them that there is a “breakdown” of the definition in the lesson). - Emphasize that we discuss a reflection as a “transformation of the plane”- that unlike in Grade 8, when focus on what is happening to the figure, we must broaden our perspective and consider how the whole plane is affected by the transformation. - Point out the use of function notation and the use of lambda as the symbol that indicates reflection. The use of this language and notation contributes to the “precision” of learning reflections in Grade 10. Say, “Each point on one side of the line of reflection is mapped an equal distance away across the line. This mapping explains why reflections are distance preserving and angle preserving. Students now have the language to support their intuitive understanding.”

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**Grade 10: Mapping over the Line of Reflection**

Header July 2013 Network Team Institute Grade 10: Mapping over the Line of Reflection 8 mins Say, “How can we use the construction of the perpendicular bisector to map a figure across the line of reflection and determine its image? We use an extension of the basic construction- constructing a perpendicular to the line from a point not on the line- to map key points (vertices of the triangle) across DE. Take a few moments to try and map the reflection of triangle ABC. A’ has already been located for you.” Construct circle CA: center A, with radius such that the circle crosses DE at two points (labeled F and G). Construct circle CF: center F, radius FA and circle CG: center G, radius GA. Label the [unlabeled] point of intersection between circles F and G as point A'. This is the reflection of vertex A across DE. Repeat steps 1 and 2 for vertices B and C to locate B' and C'. Connect A'B'C' to construct the reflected triangle.

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**Grade 10: Mapping over the Line of Reflection**

Header July 2013 Network Team Institute Grade 10: Mapping over the Line of Reflection 5 mins Review the solution (note, only constructions through two vertices have been shown– all three constructions are difficult to make sense of) Ask participants to discuss the 8th grade to 10th grade progression of reflections among themselves and then ask for one or more people to share out their thoughts. Ask participants what kinds of instructional strategies they envision implementing; ask administrators how they think this progression will play out.

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Header July 2013 Network Team Institute Grade 10: Rotation In Grade 8, students experimented with a model of a rotation, spinning figures on transparencies to verify that rotations were indeed distance preserving and angle preserving. In Grade 10, students clearly define rotation and learn to: Determine the center of rotation Determine the angle of rotation 2 mins Say, “In Grade 8, students saw first-hand that rotations were distance-preserving and angle-preserving by manipulating figures on a transparency. Now they learn the precise definition and of rotation and incorporate their knowledge of constructions to apply rotations.

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**Grade 10: Determining the Angle of Rotation**

Header July 2013 Network Team Institute Grade 10: Determining the Angle of Rotation 5 mins Say, “How can we determine the angle of rotation? Take a few minutes to consider a strategy and verify this strategy with your protractor.”

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**Grade 10: Determining the Angle of Rotation**

Header July 2013 Network Team Institute Grade 10: Determining the Angle of Rotation 3 mins Say, “You should have determined that joining any pair of corresponding points with the center of rotation determines the angle of rotation. We should point out that by default, we measure the angle of rotation in the counterclockwise direction, but it is acceptable to measure the angle of rotation in the clockwise direction. We indicate the difference by using a negative measure or writing the positive measure accompanied by “CW”.

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**Grade 10: Determining the Center of Rotation**

Header July 2013 Network Team Institute Grade 10: Determining the Center of Rotation 7 mins Say, “How can we determine the center of rotation? If you would like a hint, let us know. If you would like the instructions on how to determine the center of rotation, please let us know so we can provide you with a copy.” - Make participants aware that in the Rotations lesson, students have instructions provided to them.

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**Grade 10: Determining the Center of Rotation**

Header July 2013 Network Team Institute Grade 10: Determining the Center of Rotation 5 mins - Review solution Say, “We have examined how to find the angle of rotation and the center of rotation. If we think back to Grade 8, students know that these parameters are important, but they “saw” them from the vantage point of pressing a finger down on the center of spinning. Let us now examine the precise definition of a rotation.”

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**Grade 10: Rotation Header July 2013 Network Team Institute 5 mins**

- Read the definition of rotation. - Emphasize that we discuss a rotation as a “transformation of the plane”- that unlike in Grade 8, when focus on what is happening to the figure, we think of the entire plane being subject to the rotation. - Point out the use of function notation and the new notation in general. The use of this language and notation contributes to the “precision” of learning reflections in Grade 10. - Be prepared to discuss the concept of a counterclockwise half-plane.

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Header July 2013 Network Team Institute Grade 10: Translation In Grade 8, students experimented with a model of a translations, sliding figures on transparencies to verify that translations were distance preserving and angle preserving. In Grade 10, students clearly define translation and learn to: Apply a translation by constructing parallel lines 2 mins Say, “In Grade 8, students experimentally verified the properties of a translation by sliding figures around on a transparency. They visualized that lines map to lines, segments to segments, etc. and that reflections are distance preserving and angle preserving. In Grade 10, they apply a translation with the help of the construction of parallel lines.”

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**Grade 10: Applying a Translation**

Header July 2013 Network Team Institute Grade 10: Applying a Translation Given the experience students enter Grade 10 with, they can visualize the image of the figure under a translation, provided the vector. 2 mins - Read the bullet point. Refer to Grade 8 experience.

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**Grade 10: Applying a Translation**

Header July 2013 Network Team Institute Grade 10: Applying a Translation To apply the translation, we must construct the line parallel to each side in the direction and at a distance equal to the length of the vector. 2 mins - Read the bullet point. - Explain that constructing a line parallel to each side of a triangle is a more advanced application of the task we will start with (constructing a line parallel to a given line through a point not on the line).

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**Grade 10: Applying a Translation**

Header July 2013 Network Team Institute Grade 10: Applying a Translation Follow the instructions to construct the line parallel to AB through P. 5 mins Say, “In order to successfully translate a triangle, where we would need to create three parallel lines (in the direction and distance away from each side equal to the length of the vector) from the three vertices of the triangle. We prepare ourselves for that task by first learning to translate a single point. Follow the steps above to create a line parallel to AB through the given point P.”

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**Grade 10: Applying a Translation**

Header July 2013 Network Team Institute Grade 10: Applying a Translation Line PQ is parallel to line AB. 2 mins Review solution

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**Grade 10: Translating a Segment**

Header July 2013 Network Team Institute Grade 10: Translating a Segment The translation of a segment might look like this: 2 mins Say, “Obviously, as the figure becomes more complex, the translation becomes increasingly challenging.” - Emphasize that the steps for each additional point in a more complex figure is the same as a single point.

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**Grade 10: Translating a Triangle**

Header July 2013 Network Team Institute Grade 10: Translating a Triangle The translation of a triangle might look like this: 2 mins Say, “Obviously, as the figure becomes more complex, the translation becomes increasingly challenging.”

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**Grade 10: Translation Header July 2013 Network Team Institute 2 mins**

- Read through the definition for translation and use the application from the last several slides to help illustrate the definition.

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Header July 2013 Network Team Institute Congruence Once students are comfortable with rigid motions, they study the link between the concept of rigid motions and congruence We want students to be able to use the language around congruence in a clear way Congruent. Two figures in the plane are congruent if there exists a finite composition of basic rigid motions that maps one figure onto the other figure. 2 mins - Read through the bullet points.

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**Congruence Why can’t a triangle be congruent to a quadrilateral?**

Header July 2013 Network Team Institute Congruence Sample Question: Why can’t a triangle be congruent to a quadrilateral? 2 mins - Read question, allow a few moments for participants to formulate an answer.

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**Congruence Why can’t a triangle be congruent to a quadrilateral?**

Header July 2013 Network Team Institute Congruence Sample Question: Why can’t a triangle be congruent to a quadrilateral? Sample Answer: 2 mins - Take answers from audience A triangle cannot be congruent to a quadrilateral because there is no rigid motion that takes a figure with three vertices to a figure with four vertices.

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Header July 2013 Network Team Institute Coherence It is important that students leave Grade 8 with a solid, intuitive understanding of the rigid motions The physical manipulation of and visual understanding of rigid motions in Grade 8 needs be put into careful language in Grade 10 Properties of rigid motions that make obvious sense need are married with construction, and eventually used in reasoning The “careful use of language” is mentioned frequently in Grade 10. Ultimately, we want students to understand that Geometry exists as a axiomatic system- that the establishment of a new fact comes strictly from basic assumptions or existing facts These assumptions and existing facts appear throughout Module 1, and certainly in the topic of Rigid Motions. 2 mins Read through bullet points Ask audience if they would like to share any observations about coherence

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Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Biggest Takeaway A solid understanding of how rigid motions behave in Grade 8 will lay the groundwork for Grade 10. 2 mins Read through bullet points

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Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Key Points The hands-on, experiential understanding and experimental verification of properties in Grade 8 are formalized through language and construction in Grade 10. 2 mins Read through bullet points

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