nek ccssm – hs session # 2 algebra content domain and applications of practice standards...

NEK CCSSM – HS Session # 2Algebra Content Domain and

Applications of Practice StandardsWednesday, November 28, 2012

What changes in your instructional practice have you made that reflect what you have learned about the

CCSSM?

What questions about the CCSSM do you bring with

you today?

Break Enforcer(s) Needed!

Must be willing to lead group in stretch every 20 to 30 minutes!

• The CCSSM HS Content Domain “Algebra”

• The 8 Standards for Mathematical Practice

• How to find rich tasks that support student understanding of “Algebra” and student practice of the Practice Standards

Goals: You will leave this workshop with a deeper

understanding of…

• With a partner, try to recall the 8 practice standards. • Write down the ones you remember best and share what helps you to remember

them. • Make a note of which ones that you forgot and think about ways that may help you

to remember them.• Share with your partner instructional moves that you have used this school year

that elicit the practices in students.• Report out to large group.

Overarching Habits of Mind of a Mathematician: (1) Make sense of problems and persevere in solving them.

(6) Attend to precision.Ability to Communicate Mathematically

(2)Reason abstractly and quantitatively.(3) Construct viable arguments and critique the reasoning of others.

Create Mathematical Models to Problem Solve(4) Model with mathematics.(5) Use appropriate tools strategically.

Recognize and Use Mathematical Structure to Solve Problems(7) Look for and make use of structure.(8) Look for and express regularity in repeated reasoning.

As we work problems throughout the day, we will mark which Practice Standards we used with an X on the

Practice Standards Poster.

I need a volunteer to make sure that we mark the Practice Standards Poster!!!

Modeling

K 1 2 3 4 5 6 7 8 9 10 11 12

The Common Core State Standards in Mathematics

Geometry

Measurement and Data

The Number System

Number and Operations in Base Ten

Operations and Algebraic Thinking

Geometry

Number and Operations Fractions

Expressions and Equations

Statistics and Probability

Algebra

Number and Quantity

Functions

Statistics and Probability

Ratios and Proportional Relationships

F

CC

© Copyright 2011 Institute for Mathematics and Education

Modeling is the Creative Essence of Mathematics

With a partner, discuss what modeling means to you.

What types of tools one could use to model mathematically?

Report out to the large group.

As we go through the problems we work today, we will put a star on the Practice Standards Poster when we work a problem that involves modeling.

AlgebraThe following quotes and examples are from Algebra: Form and Function (McCallum, Connally, Hughes-Hallett et al)

The fact that algebra can be encapsulated in rules sometimes encourages students to try to learn the subject merely as a set of rules. However, both manipulative skill and understanding are required for fluency. Inadequate practice in manipulation leads to frustration; inadequate attention to understanding leads to misconceptions, which easily become firmly rooted.

Algebra Seeing Structure

in Expressions

• Arithmetic with Polynomials and Rational Expressions

• Creating Equations

• Reasoning with Equations and Inequalities

Algebra Seeing

Structure in Expressions

Read HS Algebra Progressions Documentpp. 1 – 5 (stop at Arithmetic with…)Discuss highlights as a group.

Seeing Structure in Expressions

Cluster: Interpret the structure of expressions.  

A-SSE.1Interpret expressions that represent a quantity in terms of its context. ⋆  

a. Interpret parts of an expression, such as terms, factors, and coefficients.

Algebra I & II

Seeing Structure in Expressions

Cluster: Interpret the structure of expressions.  

Look at handout packet

McCallum, Connally, Hughes Hallett

Problems on Algebraic Structure, 2011# 11

Seeing Structure in Expressions

In each of the following, the two expressions are changed by introducing a parentheses. Explain what this difference makes to the calculations and choose values of the variables to illustrate the difference:

2x2 and (2x)2

2l + w and 2(l + w)

3 – x + y and 3 – (x + y)

Seeing Structure in Expressions

Cluster: Interpret the structure of expressions.  

A-SSE.2Use the structure of an expression to identify ways to rewrite it.

For example, see x4−y4 as (x2)2−(y2)2, thus recognizing it as a difference of squares that can be factored as (x2−y2)(x2+y2).

Algebra I & II

Seeing Structure in Expressions

Describe how each expression breaks down into parts.

Look at the forest before the trees!

.5h(a + b)

3(x – y) + 4 (x + y)

R + S RS

Seeing Structure in Expressions

Suppose p and q represent the price in dollars of two brands of MP3 player, where p > q. Which expression in each pair is larger? Interpret your answer in terms of prices.

p + q and 2p

p + 0.05p and q + 0.05q

500 – p and 500 - q

Seeing Structure in Expressions

Guess possible values of x and y that make each expression have the form(x + y) + xy 2(3 + 4) + 3*4 2

10 + 21 2 (2r + 3s) + 6rs 2

Seeing Structure in Expressions

Cluster: Write expressions in equivalent forms to solve problems.

A-SSE.3

Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. ⋆ a. Factor a quadratic expression to reveal the zeros of the function it defines.   Algebra I

Seeing Structure in Expressions

Seeing Structure in Expressions

Seeing Structure in Expressions

Write expressions in equivalent forms to solve problems.

A-SSE.3Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. ⋆ b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.  

Algebra I

Seeing Structure in Expressions

The water spouts are programmed to follow a specific parabolic path. The fountain has spouts with different paths. One of the paths has the formula

y = -x2 + 10x – 19where y is the vertical distance in feet above the surface of the fountain and x the the horizontal distance in feet along the surface of the fountain. Write an equivalent form for the equation that will easily show the maximum height the water will reach and how wide the entire arc of water is.

Seeing Structure in Expressions

Write expressions in equivalent forms to solve problems.A-SSE.3Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. ⋆ c. Use the properties of exponents to transform expressions: Write 1.15t as (1.151/12)12t≈1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.

Algebra I

Seeing Structure in ExpressionsA-SSE.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. See handout (A-SSE-4) Example from Algebra Form and FunctionAlgebra II