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**August 19, 2014 Geometry Common Core Test Guide Sample Items**

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Old or New?????

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Old or New?

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Old or New? Trees that are cut down and stripped of their branches for timber are approximately cylindrical. A timber company specializes in a certain type of tree that has a typical diameter of 50 cm and a typical height of about 10 meters. The density of the wood is 380 kilograms per cubic meter, and the wood can be sold by mass at a rate of $4.75 per kilogram. Determine and state the minimum number of whole trees that must be sold to raise at least $50,000. The diameter of a sphere is 5 inches. Determine and state the surface area of the sphere, to the nearest hundredth of a square inch.

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**Agenda Congruence Similarity Geometry Regents Test Guide**

Grade 8 CCSS-Geometry Grade 8 Geometry Tasks HS CCSS-Geometry High School Geometry Tasks Grade 8 CCSS-Geometry Grade 8 Geometry Tasks HS CCSS-Geometry High School Geometry Tasks

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**Grade 9 – Module 4 Module Focus Session**

February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 8 minutes MATERIALS NEEDED: None Key Changes in CCSS “The concepts of congruence, similarity, and symmetry can be understood from the perspective of geometric transformation. Fundamental are the rigid motions: translations, rotations, reflections, and combinations of these, all assumed to preserve distance and angles… Similarity transformations (rigid motions followed by dilations) define similarity in the same way that rigid motions define congruence, thereby formalizing the similarity ideas of "same shape" and "scale factor" developed in the middle grades.” 2 mins Modules 1 and 2 are where some of the most significant changes are occurring in geometry vs. traditional curriculum. This is due to the way congruence and similarity are defined under the CCSS, which is with the use of ‘geometric transformation’ as stated in this quote. Common Core State Standards for Mathematics, p74

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**Grade 9 – Module 4 Module Focus Session**

February 2014 Network Team Institute Regents example? 2 mins Another change under the CCSS is how we visually see the transformations unfold. We are used to questions such as the one here- it has been common to see the basic rigid motions (rotations, reflections, translations) and dilations on the coordinate plane. While there is nothing mathematically incorrect about using the coordinate plane, the use of the coordinate plane makes us tend towards very specifics cases of transformations

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**A Checklist of “rules”:**

Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute ? A Checklist of “rules”: 4 min This is the scope of rules you’d have to know to successfully answer questions on the Regents. I have a feeling other teachers in the room can attest to this, but I personally found this topic frustrating because it really boiled down to knowing this checklist, memorizing a bunch of formulas essentially, with no further purpose in math ever after, except the test. So, accordingly, I covered it as test prep, just squeezing it in around June 5th, and hoped it would stick for about 1-2 weeks. And I say this to almost as a note to myself, because when I first heard that the CCSS were on their way, that we were going to be responsible for these standards vs. the NY state standards, I wondered what made them so different. And in some respects, as far as learning objectives go, they are not so different. However it’s a topic like this, and the treatment it is receiving under the CCSS that helped me begin to distinguish: this is an example where as a part of traditional curriculum, it was one of those places that was it’s own isolated, little package of information. It was our job to deliver it neatly and for the students to show that they knew it by regurgitating it on a test. I think the way transformations are studied under the CCSS, there is more meaning, they are studied for a greater purpose that extends beyond just this module, this course, etc. It’s one of the instances I can put my finger on that says, ok we are really considering meaning.

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Congruence

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Building From Grade 8

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**Definition of Translation**

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**Definition of Reflection**

The reflection R across a given line L, where L is called the line of reflection, assigns each point on L to itself, and to any point P not on L, R assigns the point R(P) which is symmetric to it with respect to L, in the sense that L is the perpendicular bisector of segment joining P to R(P)

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**Definition of Rotation**

Rotations require information about the center of rotation and the degree in which to rotate. Positive degrees of rotation move the figure in a counterclockwise direction. Negative degrees of rotation move the figure in a clockwise direction. Basic Properties of Rotations: (R1) A rotation maps a line to a line, a ray to a ray, a segment to a segment, and an angle to an angle. (R2) A rotation preserves lengths of segments. (R3) A rotation preserves degrees of angles. When parallel lines are rotated, their images are also parallel. A line is only parallel to itself when rotated exactly 180˚. The rotation Ro of t degrees (-360 < t < 360) around a given point O, called the center of the rotation, is a transformation of the plane defined as follows. Given a point P, the point Ro(P) is defined as follows. The rotation is counterclockwise or clockwise depending on whether the degree is positive or negative, respectively. If P is distinct from O, then by definition, Ro(P) is the point Q on the circle with center O and radius |OP| so that <QOP = t and so that Q is in the counterclockwise direction of the point P.

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**Geometry Course Overview**

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Similarity

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Grade 8 – Module 3 – Lesson 1

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**Grade 9 – Module 4 Module Focus Session**

February 2014 Network Team Institute *Lesson 2, Example 2 5 mins Use the handout and ruler to scale the following figure (Lesson 2, Example 2) according to the Ratio Method. -- Allow participants time to create the scale drawing. Step 1. Draw a ray beginning at O through each vertex of the figure . Step 2. Use your ruler to determine the location of A' on OA ; A' should be 3 times as far from O, as A. Determine the locations of B' and C' in the same way along the respective rays. Step 3. Draw the corresponding line segments, e.g., segment A'B' corresponds to segment AB.

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**Triangle Side Splitter Theorem:**

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Grade 8 – Module 3 Lesson 4: Lined Paper Activity leads to an understanding of similarity. Similarity in terms of Dilation Given a dilation with center O and scale factor r, then for any two points P, Q in the plane (when points O, P, Q are not collinear), the lines PQ and P’Q’ are parallel, where P’ = Dilation(P) and Q’ = Dilation(Q), and furthermore, |P’Q’| = r |PQ|.

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***Lesson 5: Scale Factors**

Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute *Lesson 5: Scale Factors Facts known: Ratio and Parallel Methods produce the same scale drawing. Triangle Side Splitter Theorem. 2 mins Once students are familiar with the Ratio and Parallel Methods of creating scale drawings, they establish that both methods in fact create the same scale drawing of a given figure. Proving this yields the Triangle Side Splitter Theorem, which in turn is used in the following lesson to establish the Dilation Theorem. The proof for the Dilation Theorem explains why it is that the the Ratio and Parallel Methods yield scaled or enlarged/reduced versions of a given figure. These two theorems are used over and over again to establish properties of dilations in Topic B lessons.

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**Grade 8 Lesson 2: Properties of Dilations**

Students informally verify the properties of dilations: Lines map to lines, Rays map to rays, Segments map to segments, and Angles map to angles of the same degree. Lesson 7: Informal Proofs of Properties of Dilations Optional Lesson *Informal proof that angles map to angles Informal proofs that lines map to lines, rays to rays, segments to segments.

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**Topic C: Similarity and Dilations**

Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 8 minutes MATERIALS NEEDED: None Topic C: Similarity and Dilations Standards G-SRT.2, G-SRT.3, G-SRT.4, G-SRT.5: Similarity transformations, the AA criterion, prove theorems about triangles, use congruence and similarity to prove relationships in geometric figures. Key Concepts Similarity Transformation. The composition of a finite number of dilations and/or rigid motions. Two figures are said to be similar if a similarity transformation exists mapping one figure onto another. A congruence is a similarity with scale factor 1. 2 mins With a thorough understanding of dilations, we then define a similarity transformation in Topic C as the composition of a finite number of dilations and/or rigid motions, and we say two figures are similar if there exists a similarity transformation that maps one figure onto another. Just as the use of rigid motions allowed us to compare figures other than rectilinear figures, so too does the idea of a similarity transformation allow us to compare and determine whether any two figures are similar to each other.

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***Lesson 15: Angle-Angle Criterion for Two Triangles to be Similar**

Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute *Lesson 15: Angle-Angle Criterion for Two Triangles to be Similar Theorem. Two triangles with two pairs of equal corresponding angles are similar. Use your handout to outline a proof of the theorem. 5 mins The AA criterion for similar traingles is covered in Lesson 15, and the SAS and SSS criteria for similar triangles is covered in Lesson 17. Students prove that a similarity transformation exists for two triangles with two pairs of angles of equal measure. Take a moment to sketch a rough outline of what this proof might look like. Hint: Use a dilation with center A (r < 1…but you can be specific about this)….you will need triangle congruence indicators as well.

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*Lesson 15, Discussion:

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**Grade 9 – Module 4 Module Focus Session**

February 2014 Network Team Institute 4 mins In a nutshell: Dilate about A with r < 1…in fact specifically so that r = DE/AB , so that B goes to B’ and C goes to C. Since we have dilated B and C by the same scale factor, B’C’ is a proportional side splitter; by the Triangle Side Splitter Theorem, we know that B'C'||BC. Since B'C'||BC, then m∠AB'C'=m∠ABC because corresponding angles of parallel lines are equal in measure. Then △AB'C'≅△DEF by ASA. Thus, a similarity transformation exists that takes △ABC to triangle △DEF; triangle △ABC is similar to △DEF. Since the triangles are similar, we can confirm that the Angle Angle criterion between two triangles guarantees that the triangles are similar. Once the proof is established, students can use what they know about the length relationships between similar triangles to solve for unknown sides, much like problems found in current text in similarity units.

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***Lesson 17: SAS and SSS Criteria for Two Triangles to be Similar**

Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute *Lesson 17: SAS and SSS Criteria for Two Triangles to be Similar 4 mins The SAS and SSS criteria for two triangles to be similar is covered in Lesson 17. Take a moment to sketch a rough outline of what this proof of why the SAS criterion is enough to determine that two triangles are similar. Hint: Use a dilation about A. -- Allow participants to consider for a few moments. The proof of this theorem is simply to take any dilation with scale factor r = A'B’/AB = A'C/'AC. This dilation maps △ABC to a triangle that is congruent to △A'B'C' by the Side-Angle-Side Congruence Criterion.

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**Grade 9 – Module 4 Module Focus Session**

February 2014 Network Team Institute *Lesson 17: Discussion 3 mins Take a moment to sketch a rough outline of what this proof of why the SSS criterion is enough to determine that two triangles are similar. Hint: Use a dilation about A. -- Allow participants to consider for a few moments. The proof of this theorem is simply to take any dilation with scale factor r = A'B’/AB = B'C’/BC = A'C’/AC. This dilation maps △ABC to a triangle that is congruent to △A'B'C' by the Side-Side-Side Congruence Criterion.

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**Grade 9 – Module 4 Module Focus Session**

February 2014 Network Team Institute *Lesson 17: Discussion 3 mins Take a moment to sketch a rough outline of what this proof of why the SSS criterion is enough to determine that two triangles are similar. Hint: Use a dilation about A. -- Allow participants to consider for a few moments. The proof of this theorem is simply to take any dilation with scale factor r = A'B’/AB = B'C’/BC = A'C’/AC. This dilation maps △ABC to a triangle that is congruent to △A'B'C' by the Side-Side-Side Congruence Criterion.

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**Grade 9 – Module 4 Module Focus Session**

February 2014 Network Team Institute 4 mins The SAS and SSS criteria for two triangles to be similar is covered in Lesson 17. Take a moment to sketch a rough outline of what this proof of why the SAS criterion is enough to determine that two triangles are similar. Hint: Use a dilation about A. -- Allow participants to consider for a few moments. The proof of this theorem is simply to take any dilation with scale factor r = A'B’/AB = A'C/'AC. This dilation maps △ABC to a triangle that is congruent to △A'B'C' by the Side-Angle-Side Congruence Criterion.

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**Grade 9 – Module 4 Module Focus Session**

February 2014 Network Team Institute 4 mins The SAS and SSS criteria for two triangles to be similar is covered in Lesson 17. Take a moment to sketch a rough outline of what this proof of why the SAS criterion is enough to determine that two triangles are similar. Hint: Use a dilation about A. -- Allow participants to consider for a few moments. The proof of this theorem is simply to take any dilation with scale factor r = A'B’/AB = A'C/'AC. This dilation maps △ABC to a triangle that is congruent to △A'B'C' by the Side-Angle-Side Congruence Criterion.

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1.1 WELCOME TO COMMON CORE HIGH SCHOOL MATHEMATICS LEADERSHIP SUMMER INSTITUTE 2015 SESSION 3 17 JUNE 2015 SIMILARITY IN MIDDLE SCHOOL.

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