© 2013 UNIVERSITY OF PITTSBURGH Welcome to TNCore Training! Tennessee Department of Education High...

© 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

Algebra II

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

10© 2013 UNIVERSITY OF PITTSBURGH

Tennessee Focus Clusters Algebra 2

Extend the properties of exponents to rational exponents.

Write expressions in equivalent forms to solve problems.

Understand the relationship between zeros and factors of polynomials.

Build a function that models a relationship between two quantities.


The CCSS for Mathematical ContentCCSS Conceptual Category – Number and Quantity

Common Core State Standards, 2010

The Real Number System (N-RN)

Extend the properties of exponents to rational exponents.N-RN.A.1 Explain how the definition of the meaning of rational

exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.

N-RN.A.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.


The CCSS for Mathematical ContentCCSS Conceptual Category – Algebra


Seeing Structure in Expressions (A-SSE)

Write expressions in equivalent forms to solve problemsA-SSE.B.3 Choose and produce an equivalent form of an

expression to reveal and explain properties of the quantity represented by an expression. ★

A-SSE.B.3c Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.

A-SSE.B.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.★

★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.


The CCSS for Mathematical ContentCCSS Conceptual Category – Algebra


Arithmetic with Polynomials and Rational Expressions (A-APR)

Understand the relationship between zeros and factors of polynomials

A-APR.B.2 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).

A-APR.B.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.


The CCSS for Mathematical ContentCCSS Conceptual Category – Functions


Building Functions (F-BF)Build a function that models a relationship between two quantitiesF-BF.A.1 Write a function that describes a relationship between two

quantities.★

F-BF.A.1a Determine an explicit expression, a recursive process, or steps for calculation from a context.

F-BF.A.1b Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

F-BF.A.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.★



Analyzing a

Constructed Response Assessment


Analyzing Assessment Items(Private Think Time)

Four assessment items have been provided:

Car Depreciation

Writing a Polynomial

Patterns in Patterns

One Rocket, Three Equations

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

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


1. Car Depreciation

After you purchase a new car, it begins to lose value. As the years pass, more and more value is lost. This process is known as depreciation.

a. Carmen buys a new car for $24,500. Carmen’s new car loses 14% of its value each year through depreciation. Write a function that can be used to model the value of the car at the end of each year that Carmen owns the car. Justify your equation mathematically.

b. After Carmen buys the car, he adds an audio and speaker system worth $500. The audio system loses 15% of its value the minute it is installed into the car, and 7% of the remaining value each year through depreciation. Write a function that can be used to model the value of the car with the audio system and speakers at the end of each year that Carmen owns the car.


2. Writing a Polynomial

Recall that polynomial functions with only real number zeros can be written in factored form as follows:

where each zn represents some real root of the function, and each pn

is a whole number exponent greater than or equal to 1.

Consider the graph of the polynomial function below.

npn

pp zxzxzxay ...21

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a. Lisa claims that, since the point (0, 6) is on the graph, (x – 6) is a factor of this polynomial. Explain why you agree or disagree with Lisa’s claim. Identify all the zeroes of the function and use that information in your explanation.

b. Suppose a = . Write a function in factored form to represent this graph. Justify your equation mathematically.

-5 -4 -3 -2 -1 0 1 2 3 4 5

-2-10123456789

1011121314


3. Patterns in PatternsLaura creates a design of circles embedded in each other for a poster. The largest circle has a diameter of 28 inches, and each successive circle has a diameter of the previous circle.

a. Write a function that can be used to determine the diameter of any circle drawn in the poster in this way. Explain the meaning of each term in your expression in the context of the problem.

b. Laura eventually draws 10 circles. Write and use a formula for the sum of a series to find the sum of the circumferences of the 10 circles, accurate to two decimal places. Show your work.

28 inches


4. One Rocket, Three Equations

For his science project, Grady designs and launches a model rocket from the rooftop of an empty building. He places a sensor on the rocket to provide him with data about the height of the rocket (from ground level in feet) over time (in seconds) during the flight.From the data returned by the sensor, Grady was able to write and graph a function modeling the height of the rocket over time.

a. Rewrite the function in vertex form. Then describe what this form of the function reveals in the context of the situation.

b. Rewrite the function by factoring completely. Then describe what this form of the function reveals in the context of the situation.


Discussing Content Standards (Small Group Time)

For each assessment item:

With your small group, find evidence in tasks 2 and 4

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


2. Writing a Polynomial


Arithmetic with Polynomials and Rational Expressions (A-APR)

Understand the relationship between zeros and factors of polynomials

A-APR.B.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

Building Functions (F-BF)

Build a function that models a relationship between two quantities

F-BF.A.1 Write a function that describes a relationship between two quantities.★



4. One Rocket, Three Equations


Seeing Structure in Expressions A-SSE

Write expressions in equivalent forms to solve problems

A.SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by an expression. ★

★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.


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 “Missing Function Task” 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...

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