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NEK CCSSM – HS Session # 2
Algebra Content Domain and
Applications of Practice
Standards
Wednesday, 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!
Goals: You will leave this
workshop with a deeper
understanding of…
•
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
• 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!!!
The Common Core State Standards in Mathematics
K
1
2
3
4
5
Measurement and Data
6
7
8
Statistics and
Probability
Ratios and
Proportional
Relationships
CC
9
10
11
12
Statistics and Probability
F
Functions
Number and Operations
Fractions
Number and Operations in Base Ten
The Number
System
Number and Quantity
Algebra
Operations and Algebraic Thinking
Expressions and
Equations
Geometry
Geometry
Modeling
© 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.
Algebra
The 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
Document
pp. 1 – 5
(stop at Arithmetic
with…)
Discuss highlights as a
group.
 Seeing Structure
in Expressions
Cluster: Interpret the
structure of expressions.
A-SSE.1
Interpret 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.2
Use 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.3
Choose 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 – 19
where 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.3
Choose 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 Expressions
A-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 Function
Algebra II