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Algebra
• Seeing Structure in
Expressions
 Arithmetic with
Polynomials
and Rational
Expressions
• Creating Equations
• Reasoning with
Equations and
Inequalities
Algebra
 Arithmetic with
Polynomials
and Rational
Expressions
Read HS Algebra
Progressions
Document
pp. 5 – 7
Discuss highlights as a
group.
 Arithmetic with
Polynomials and
Rational Expressions
Perform arithmetic operations
on polynomials.
A-APR.1
Understand that polynomials
form a system analogous to the
integers, namely, they are
closed under the operations of
addition, subtraction, and
multiplication; add, subtract,
and multiply polynomials.
Algebra I & II
 Arithmetic with
Polynomials and
Rational Expressions
Perform arithmetic
operations on polynomials.
What type of skills do
students need in order to
add, subtract, and
multiply polynomials?
 Arithmetic with
Polynomials and
Rational Expressions
Understand the relationship
between zeros and factors of
polynomials.
A-APR.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).
Algebra II
 Arithmetic with
Polynomials and
Rational Expressions
Applying the
Remainder Theorem
Is x – 4 a factor of the
polynomial
p(x) = 5x3 – 13x2 – 30x + 8?
Rather than divide p(x) by
x – 4, apply the Remainder
Theorem as a shortcut:
If p(4) = 0, then we know that
x – 4 is a factor.
p(4) = 5(4)3 – 13(4)2 – 30(4) + 8
p(4) = 5(64)-13(16)-120 + 8
p(4) = 0
So x – 4 is a factor of p(x)
 Arithmetic with
Polynomials and
Rational Expressions
Understand the relationship
between zeros and factors
of polynomials.
The zero-factor principle
states that
If A*B = 0, then either A = 0,
or B = 0 (or both)
 Arithmetic with
Polynomials and
Rational Expressions
Understand the relationship
between zeros and factors of
polynomials.
A-APR.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.
Algebra II
 Arithmetic with
Polynomials and
Rational Expressions
Understand the relationship
between zeros and factors of
polynomials.
Example:
Find a polynomial that has
degree 4 and has zeros at
t = -1, t = 0, t = 1, and t = 2
 Arithmetic with
Polynomials and
Rational Expressions
Use polynomial identities
to solve problems.
A-APR.4
Prove polynomial identities
and use them to describe
numerical relationships.
Algebra II
 Arithmetic with
Polynomials and
Rational Expressions
Using polynomial identities:
x2 – y2 = (x + y)(x – y)
so
16 – 25 = (4 + 5)(4 - 5)
(7 – 3)(7 + 3) = 49 – 9
Show the area model for
(x + y)2
Where does the 2xy appear?
(105)2 = (100 + 5)2
= 1002 + 2(5)(100) + 52
= 10000 + 1000 + 25
= 11025
 Arithmetic with
Polynomials and
Rational Expressions
Use polynomial identities to
solve problems.
A-APR.5
(+) Know and apply the
Binomial Theorem for the
expansion of (x+y)n in powers
of x and y for a positive
integer n, where x and y are
any numbers, with
coefficients determined for
example by Pascal's Triangle.*
Algebra II (+) Honors or
Year 4
 Arithmetic with
Polynomials and
Rational Expressions
Rewrite rational expressions.
A-APR.6
Rewrite simple rational
expressions in different forms;
write a(x)b(x) in the form
q(x)+r(x)b(x), where
a(x), b(x), q(x), and r(x)
are polynomials with the
degree of r(x) less than the
degree of b(x),
using inspection, long division,
or, for the more complicated
examples, a computer algebra
system.
Algebra II
 Arithmetic with
Polynomials and
Rational Expressions
Use long division to
determine the quotient of
(x3 -1)/(x2+1)
Express the fraction in
quotient form.
How does this quotient
relate to the graph of
f(x) = (x3 -1)/(x2+1) ?
 Arithmetic with
Polynomials and
Rational Expressions
Rewrite rational expressions.
A-APR.7
(+) Understand that rational
expressions form a system
analogous to the rational
numbers, closed under
addition, subtraction,
multiplication, and division
by a nonzero rational
expression; add, subtract,
multiply, and divide rational
expressions.
Algebra II (+) Honors or
Year 4
Algebra
• Seeing Structure in
Expressions
• Arithmetic with
Polynomials and
Rational Expressions
 Creating
Equations
• Reasoning with
Equations and
Inequalities
Algebra
 Creating
Equations
Read HS Algebra
Progressions
Document
pp. 8 – 10 (stop at
Reasoning with…)
Discuss highlights as a
group.
 Creating Equations
Create equations that
describe numbers or
relationships.
A-CED.1
Create equations and
inequalities in one variable
and use them to solve
problems. Include equations
arising from linear and
quadratic functions, and
simple rational and
exponential functions. ⋆
Algebra I & II
 Creating Equations
Suppose a friend tells you she
paid a total of $16,368 for a
car, and you'd like to know the
car's list price (the price before
taxes) so that you can compare
prices at various dealers. Find
the list price of the car if your
friend bought the car in:
Arizona, where the sales tax
is 6.6%.
New York, where the sales tax
is 8.25%.
A state where the sales tax is r.
 Creating Equations
Michelle will get a final course
grade of B+ if the average on
four exams is greater than or
equal to 85 but less than 90.
Her first three exam grades
were 98, 74, and 89. What
fourth exam grade will result in
a B+ for the course?
 Creating Equations
Create equations that
describe numbers or
relationships.
A-CED.2
Create equations in two or
more variables to
represent relationships
between quantities; graph
equations on coordinate
axes with labels and scales.
⋆
Algebra I & II
 Creating Equations
Observations show that the
heart mass H of a mammal is
0.6% of the body mass M, and
that the blood mass B is 5% of
the body mass.
(a) Write a formula for M in
terms of H
(b) Write a formula for M in
terms of B
(c) Write a formula for B in
terms of H. Is this
consistent with the
statement that the mass
of blood in a mammal is
about 8 times the mass
of the heart?
 Creating Equations
A borehole is a hole dug deep
in the earth for oil or mineral
exploration. Often temperature
gets warmer at greater depths.
Suppose that the temperature
in a borehole at the surface is
4 oC and rises by 0.02oC with
each additional meter of depth.
Express the temperature T in oC
in terms of depth d in meters.
Graph the equation on a set of
axes.
What does the T-intercept
represent in terms of the
problem? If the temperature is
24oC, how deep is the hole?
 Creating Equations
Create equations that describe
numbers or relationships.
A-CED.3
Represent constraints by
equations or inequalities, and
by systems of equations and/or
inequalities, and interpret
solutions as viable or nonviable
options in a modeling context.
For example, represent
inequalities describing
nutritional and cost constraints
on combinations of different
foods. ⋆
Algebra I & II
 Creating Equations
A newly designed motel
has S small rooms
measuring 250 ft2 and L
large rooms measuring
400 ft2 of available space.
The designers have 10,000
ft2 of available space.
Write an equation relating
S and L.
 Creating Equations
Create equations that
describe numbers or
relationships.
A-CED.4
Rearrange formulas to
highlight a quantity of
interest, using the same
reasoning as in solving
equations. For example,
rearrange Ohm's law V=IR
to highlight resistance R. ⋆
Algebra I & II