© 2013 university of pittsburgh study group 2 – algebra 2 welcome back! let’s spend some...

© 2013 UNIVERSITY OF PITTSBURGH

Study Group 2 – Algebra 2Welcome Back!

Let’s spend some quality time discussing what we learned from our Bridge to Practice exercises.


Part A From Bridge to Practice #1: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.


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.

3

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


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

21

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


Part B from Bridge to Practice #1: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.


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

implications for instruction?



classroom?



Part C From Bridge to Practice #1: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.


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

for instruction?



classroom?



Supporting Rigorous Mathematics Teaching and Learning

Tennessee Department of Education

High School Mathematics

Algebra 2

Engaging In and Analyzing Teaching and Learning through an Instructional Task

Rationale

By engaging in an instructional task,

teachers will have the opportunity to

consider the potential of the task and

engagement in the task for helping learners

develop the facility for expressing a

relationship between quantities in different

representational forms, and for making

connections between those forms.


Question to Consider…

What is the difference between the following types of tasks?

• instructional task • assessment task


Taken from TNCore’s FAQ Document:


Session Goals

Participants will:

• develop a shared understanding of teaching and

learning through an instructional task; and

• deepen content and pedagogical knowledge of

mathematics as it relates to the Common Core State

Standards (CCSS) for Mathematics.

(This will be completed as the Bridge to Practice)


Overview of Activities

Participants will:

• engage in a lesson; and

• reflect on learning in relationship to the CCSS.

(This will be completed as the Bridge to Practice #2)


Looking Over the Standards

• Briefly look over the focus cluster standards.

• We will return to the standards at the end of the lesson and consider:

What focus cluster standards were addressed in the lesson?

What gets “counted” as learning?


Missing Function Task

If h(x) = f(x) · g(x), what can you determine about g(x) from the given table and graph? Explain your reasoning.

x f(x)-2 0-1 10 21 32 4

The Structures and Routines of a Lesson

The Explore Phase/Private Work Time

Generate Solutions

The Explore Phase/Small Group Problem Solving

1. Generate and Compare Solutions

2. Assess and Advance Student Learning

MONITOR: Teacher selects examples for the Share,

Discuss, and Analyze Phase based on:

• Different solution paths to the

same task

• Different representations

• Errors

• Misconceptions

SHARE: Students explain their methods, repeat others’

ideas, put ideas into their own words, add on to ideas

and ask for clarification.

REPEAT THE CYCLE FOR EACH

SOLUTION PATH

COMPARE: Students discuss similarities and

difference between solution paths.

FOCUS: Discuss the meaning of mathematical ideas in

each representation

REFLECT: By engaging students in a quick write or a

discussion of the process.

Set Up of the Task

Share, Discuss, and Analyze Phase of the Lesson

1. Share and Model

2. Compare Solutions

3. Focus the Discussion on

Key Mathematical Ideas

4. Engage in a Quick Write


Solve the Task(Private Think Time and Small Group Time)

• Work privately on the Missing Function Task. (This should have been completed as the Bridge to Practice prior to this session)

• Work with others at your table. Compare your solution paths. If everyone used the same method to solve the task, see if you can come up with a different way.

• Consider what each person determined about g(x).


Expectations for Group Discussion

• Solution paths will be shared.

• Listen with the goals of:– putting the ideas into your own words;– adding on to the ideas of others;– making connections between solution paths; and– asking questions about the ideas shared.

• The goal is to understand the mathematics and to make connections among the various solution paths.


Missing Function Task

If h(x) = f(x) · g(x), what can you determine about g(x) from the given table and graph? Explain your reasoning.

x f(x)-2 0-1 10 21 32 4


Discuss the Task(Whole Group Discussion)

• What do we know about g(x)? How did you use the

information in the table and graph and the

knowledge that h(x) = f(x) · g(x) to determine the

equation of g(x)?

• How can you use what you know about the graphs

of f(x) and g(x) to help you think about the graph of

h(x)?

• Predict the shape of the graph of a function that is

the product of two linear functions. Explain from the

graphs of the two functions why you have made

your prediction.


Reflecting on Our Learning

• What supported your learning?

• Which of the supports listed will EL students benefit from during instruction?

Linking to Research/LiteratureConnections between Representations

Pictures

WrittenSymbols

ManipulativeModels

Real-worldSituations

Oral Language

Adapted from Lesh, Post, & Behr, 1987

Five Different Representations of a Function Language

TableContext

Graph Equation

Van De Walle, 2004, p. 440

The CCSS for Mathematical ContentCCSS Conceptual Category – Number and Quantity

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.

Common Core State Standards, 2010, p. 60, NGA Center/CCSSO

The CCSS for Mathematical ContentCCSS Conceptual Category – Algebra

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 the 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 P 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 – FunctionsBuilding 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.★

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.★

★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.



Bridge to Practice #2: Time to Reflect on Our Learning

1. Using the Missing Function Task:

a. Choose the Content Standards from pages 11-12 of the handout that this

task addresses and find evidence to support them.

b. Choose the Practice Standards students will have the opportunity to use

while solving this task and find evidence to support them.

2. Using the quotes on the next page, Write a few sentences to

summarize what Tharp and Gallimore are saying about the learning

process.

3. Read the given Essential Understandings. Explain why I need to

know this level of detail about quadratics to determine if a student

understands the structure behind quadratics.


Research Connection: Findings by Tharp and Gallimore

• For teaching to have occurred - Teachers must “be aware of the students’ ever-changing relationships to the subject matter.”

• They [teachers] can assist because, while the learning process is alive and unfolding, they see and feel the student's progression through the zone, as well as the stumbles and errors that call for support.

• For the development of thinking skills—the [students’] ability to form, express, and exchange ideas in speech and writing—the critical form of assisting learners is dialogue -- the questioning and sharing of ideas and knowledge that happen in conversation.

Tharp & Gallimore, 1991


Underlying Mathematical Ideas Related to the Lesson (Essential Understandings)

• The product of two or more linear functions is a polynomial function. The resulting function will have the same x-intercepts as the original functions because the original functions are factors of the polynomial.

• Two or more polynomial functions can be multiplied using the algebraic representations by applying the distributive property and combining like terms.

• Two or more polynomial functions can be added using their graphs or tables of values because given two functions f(x) and g(x) and a specific x-value, x1, the point (x1, f(x1)+g(x1)) will be on the graph of the sum f(x)+g(x). (This is true for subtraction and multiplication as well.)

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