© 2013 UNIVERSITY OF PITTSBURGH Welcome to TNCore Training! Tennessee Department of Education High School Mathematics Geometry Introduction of 2013 CCSS.

© 2013 UNIVERSITY OF PITTSBURGH

Welcome to TNCore Training!

Tennessee Department of Education

High School Mathematics

Geometry

Introduction of 2013 CCSS Training


What this is / What it is not

What it is What it is not

Peer led learning Information updates from TDOE or expert-delivered training

Content focused – we will dive deep into understanding the expectations

Generic discussion of teaching strategies

Focused on building our capacity (knowledge and skill) as educators

Mandating implementation of a recipe for instant success

Designed to meet participants at a range of experience with Common Core

Redundant of other TNCore trainings - or – dependent on you having done anything thus far

Focused on student achievement Focused on compliance

Focused on your learning Focused on preparing you to train others


Core Beliefs

Earning a living wage has never demanded more skills.

This generation must learn more than their parents to do

as well.

All children are capable of learning and thinking at a high level. Children in Tennessee are as talented as any in the country and often capable of

more than we expect.

Our current education results pose a real threat to state and national competitiveness and security. Improving the skills of our children is vital for the

future of Tennessee and America.

Tennessee is on a mission to become the fastest improving state in the nation. Doing so

will require hard work and significant learning for all. We must learn to teach in ways we

were not taught ourselves.

There is no recipe that will deliver a successful transition. Preparing for Common Core

will demand effective leadership focused on student

growth.

PARCC is coming in two years. We need to use the

transition wisely to make sure our students and our state are

ready.


Norms

• Keep students at the center of focus and decision-making• Be present and engaged – limit distractions, if urgent matters

come up, step outside• Monitor air time and share your voice - you’ll know which

applies to you!• Challenge with respect – disagreement can be healthy, respect

all intentions• Be solutions oriented – for the good of the group, look for the

possible• Risk productive struggle - this is safe space to get out of your

comfort zone• Balance urgency and patience - we need to see dramatic

change and change will happen over time

• Any other norms desired to facilitate your learning?


Emily Barton Video


Supporting Rigorous Mathematics Teaching and Learning

Tennessee Department of Education

High School Mathematics

Geometry

Deepening Our Understanding of CCSS Via

A Constructed Response Assessment


Session Goals

Participants will:

• deepen understanding of the Common Core State Standards (CCSS) for Mathematical Practice and Mathematical Content;

• understand how Constructed Response Assessments (CRAs) assess the CCSS for both Mathematical Content and Practice; and

• understand the ways in which CRAs assess students’ conceptual understanding.


Overview of Activities

Participants will:

• analyze Constructed Response Assessments (CRAs) in order to determine the way the assessments are assessing the CCSSM;

• analyze and discuss the CCSS for Mathematical Content and Mathematical Practice;

• discuss the CCSS related to the tasks and the implications for instruction and learning.


The Common Core State Standards

The standards consist of:

The CCSS for Mathematical Content

The CCSS for Mathematical Practice


Tennessee Focus Clusters Geometry

Understand congruence in terms of rigid motions. Prove geometric theorems. Define trigonometric ratios and solve problems

involving right triangles. Use coordinates to prove simple geometric

theorems algebraically.


The CCSS for Mathematical ContentCCSS Conceptual Category – Geometry

Common Core State Standards, 2010

Congruence (G-CO)

Understand congruence in terms of rigid motions

G-CO.B.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

G-CO.B.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

G-CO.B.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.




Congruence (G-CO)

Prove geometric theoremsG-CO.C.9 Prove theorems about lines and angles. Theorems include:

vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

G-CO.C.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

G-CO.C.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.




Similarity, Right Triangles, and Trigonometry (G-SRT)

Define trigonometric ratios and solve problems involving right triangles

G-SRT.C.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

G-SRT.C.7 Explain and use the relationship between the sine and cosine of complementary angles.

G-SRT.C.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.★

★Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific modeling standards appear throughout the high school standards indicated with a star (★). Where an entire domain is marked with a star, each standard in that domain is a modeling standard.




Expressing Geometric Properties with Equations (G-GPE)

Use coordinates to prove simple geometric theorems algebraically

G-GPE.B.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).

G-GPE.B.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

G-GPE.B.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

G-GPE.B.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.★

★Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific modeling standards appear throughout the high school standards indicated with a star (★). Where an entire domain is marked with a star, each standard in that domain is a modeling standard.


Analyzing a

Constructed Response Assessment


Analyzing Assessment Items(Private Think Time)

Four assessment items have been provided: Park City Task Getting in Shape Task Lucio’s Ride Task Congruent Triangles Task

For each assessment item:• solve the assessment item; and

• make connections between the standard(s) and the assessment item.


1. Park City TaskPark City is laid out on a grid like the one below, where each line represents a street in the city, and each unit on the grid represents one mile. Four other streets in the city are represented by , and .a. Dionne claims that the figure formed

by , , , and is a parallelogram. Do you agree or disagree with Dionne? Use mathematical reasoning to explain why or why not.

b. Triangle AFE encloses a park located in the city. Describe, in words, two methods that use information in the diagram to determine the area of the park.

c. Find the exact area of the park.


2. Getting in Shape Task

. Points A (12, 10), J (16, 18), and Q (28, 12) are plotted on the coordinate plane below.

a. What are the coordinates of a point M such that the quadrilateral with vertices M, A, J, and Q is a parallelogram, but not a rectangle?

b. Prove that the quadrilateral with vertices M, A, J and Q is a parallelogram.

c. Prove that the quadrilateral with vertices M, A, J and Q is not a rectangle.

d. Determine the perimeter of your parallelogram.


3. Lucio’s RideWhen placed on a grid where each unit represents one mile, State Highway 111 runs along the line x + 3, and State Highway 213 runs along the linex - .

The following locations are represented by points on the grid:• Lucio’s house is located at (3, –1).• His school is located at (–1, –4).• A grocery store is located at (–4, 0).• His friend’s house is located at (0, 3).

a. Is the quadrilateral formed by connecting the four locations a square? Explain why or why not. Use slopes as part of the explanation.

b. Lucio is planning to ride his bike ride tomorrow. In the morning, he plans to ride his bike from his house to school. After school, he will ride to the grocery store and then to his friend’s house. Next, he will ride his bike home. The four locations are connected by roads. How far is Lucio planning to ride his bike tomorrow if he plans to take the shortest route? Support your response by showing the calculations used to determine your answer.


4. Congruent Triangles

a. Locate and label point M on such that it is of the distance from point S to point U. Locate and label point T on such that it is of the distance from point S to point N. Locate and label point Q on such that it is of the distance from point N to point U.

b. Prove triangles TNQ and QMT are congruent.


Discussing Content Standards (Small Group Time)

For each assessment item:

With your small group, find evidence in tasks 3 and 4

for the content standard(s) that will be assessed.


3. Lucio’s Ride





G-GPE.B.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

G-GPE.B.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.★

★ Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific modeling standards appear throughout the high school standards indicated with a star (★). Where an entire domain is marked with a star, each standard in that domain is a modeling standard


4. Congruent Triangles





G-GPE.B.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.


David Williams Video


Determining the Standards for

Mathematical Practice Associated with

the Constructed Response Assessment


Getting Familiar with the CCSS for Mathematical Practice(Private Think Time)

• Count off by 8. Each person reads one of the CCSS

for Mathematical Practice.

• Read your assigned Mathematical Practice. Be

prepared to share the “gist” of the Mathematical

Practice.

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The CCSS for Mathematical Practice

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

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Common Core State Standards for Mathematics, 2010, NGA Center/CCSSO


Discussing Practice Standards(Small Group Time)

Each person has a moment to share important

information about his/her assigned Mathematical

Practice.

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Bridge to Practice:Practice Standards

Choose the Practice Standards students will have the opportunity to use

while solving these tasks we have focused on and find evidence to support

them.

Using the Assessment to Think About Instruction

In order for students to perform well on the CRA, what are the implications

for instruction?

• What kinds of instructional tasks will need to be used in the

classroom? • What will teaching and learning look like and sound like in the

classroom?

Complete the Instructional TaskWork all of the instructional task “Building a New Playground” and be prepared to talk about the task and the CCSSM Content and Practice Standards associated with it.

© 2013 UNIVERSITY OF PITTSBURGH Welcome to TNCore Training! Tennessee Department of Education High School Mathematics Geometry Introduction of 2013 CCSS.

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