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Big Idea Algebra provides language through
which we communicate the patterns in mathematics. |
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4.3 A.
Patterns |
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Descriptive Statement: Algebra provides the language
through which we communicate the patterns in mathematics. From the
earliest age, students should be encouraged to investigate the
patterns that they find in numbers, shapes, and expressions, and by
doing so, to make mathematical discoveries. They should have
opportunities to analyze, extend, and create a variety of patterns
and to use pattern-based thinking to understand and represent
mathematical and other real-world phenomena. |
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Essential Questions |
Enduring Understandings |
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- How can change be best represented
mathematically? (4.5C1; 4.5F1; 4.5F2; 4.5F3; 4.5F4)
- How can patterns, relations, and functions be used as tools to
best describe and help explain real-life situations? (4.5C1) |
- The symbolic language of algebra is used to
communicate and generalize the patterns in mathematics.
- Algebraic representation can be used to generalize patterns and
relationships. |
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Areas of Focus/Cumulative Progress Indicators |
Comments and Examples |
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By the end of Grade 12: |
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1.
Use models and algebraic formulas to represent and analyze
sequences and series.
·
Explicit formulas for nth terms
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Sums of finite arithmetic series
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Sums of finite and infinite geometric series
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2.
Develop an
informal notion of limit. |
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3.
Use inductive reasoning to form generalizations.
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4.3 B. Functions and Relationships |
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Descriptive Statement: The function concept is one of the
most fundamental unifying ideas of modern mathematics. Student begin
their study of functions in the primary grades, as they observe and
study patterns. As students grow and their ability to abstract
matures, students form rules, display information in a table or
chart, and write equations which express the relationships they have
observed. In high school, they use the more formal language of
algebra to describe these relationships. |
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Essential Questions |
Enduring Understandings |
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- How are patterns of change related to the
behavior of functions? (4.5F1; 4.5F2; 4.5F3; 4.5F4) |
- Patterns and relationships can be represented
graphically, numerically, symbolically, or verbally. (4.5E1) |
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Areas of Focus/Cumulative Progress Indicators |
Comments and Examples |
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By the end of Grade 12: |
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1.
Understand relations and functions and select, convert flexibly
among, and use various representations for them, including equations or
inequalities, tables, and graphs. |
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2.
Analyze and
explain the general properties and behavior of functions of one variable, using
appropriate graphing technologies.
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Slope of a line or
curve
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Domain and range
·
Intercepts
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Continuity
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Maximum/minimum
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Estimating
roots of equations
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Intersecting
points as solutions of systems of equations
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Rates of change
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3.
Understand and
perform transformations on commonly-used functions.
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Translations,
reflections, dilations
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Effects on linear
and quadratic graphs of parameter changes in equations
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Using graphing
calculators or computers for more complex functions
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4.
Understand and
compare the properties of classes of functions, including exponential,
polynomial, rational, and trigonometric functions.
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Linear vs.
non-linear
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Symmetry
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Increasing/decreasing on an interval
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4.3 C. Modeling |
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Descriptive Statement: The function concept is one of the
most fundamental unifying ideas of modern mathematics. Student begin
their study of functions in the primary grades, as they observe and
study patterns. As students grow and their ability to abstract
matures, students form rules, display information in a table or
chart, and write equations which express the relationships they have
observed. In high school, they use the more formal language of
algebra to describe these relationships. |
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Essential Questions |
Enduring Understandings |
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- How are mathematical models used to describe
physical relationships? (4.5E2)
- How are physical models used to clarify mathematical
relationships? (4.5E3) |
- Mathematical models can be used to describe and
quantify physical relationships. (4.5E2)
- Physical models can be used to clarify mathematical relationships.
(4.5E3) |
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Areas of Focus/Cumulative Progress Indicators |
Comments and Examples |
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By the end of Grade 12: |
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1.
Use functions to
model real-world phenomena and solve problems that involve varying quantities.
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Linear, quadratic,
exponential, periodic (sine and cosine), and step functions (e.g., price of
mailing a first-class letter over the past 200 years)
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Direct and inverse
variation
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Absolute value
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Expressions,
equations and inequalities
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Same function can
model variety of phenomena
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Growth/decay and
change in the natural world
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Applications in
mathematics, biology, and economics (including compound interest)
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2.
Analyze and
describe how a change in an independent variable leads to change in a dependent
one. |
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3.
Convert recursive
formulas to linear or exponential functions (e.g., Tower of Hanoi and doubling). |
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4.3 D. Procedures |
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Descriptive Statement: Techniques for manipulating
algebraic expressions - procedures - remain important, especially
for students who may continue their study of mathematics in a
calculus program. Utilization of algebraic procedures includes
understanding and applying properties of numbers and operations,
using symbols and variables appropriately, working with expressions,
equations, and inequalities, and solving equations and inequalities. |
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Essential Questions |
Enduring Understandings |
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- What makes an algebraic algorithm both
effective and efficient? (4.5D1) |
- Algebraic and numeric procedures are
interconnected and build on one another to produce a coherent whole.
- Reasoning and/or proof can be used to verify or refute conjectures
or theorems in algebra. (4.5D1; 4.5D3; 4.5D4; 4.5D5) |
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Areas of Focus/Cumulative Progress Indicators |
Comments and Examples |
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By the end of Grade 12: |
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1.
Evaluate and
simplify expressions.
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Add and subtract
polynomials
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Multiply a
polynomial by a monomial or binomial
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Divide a
polynomial by a monomial |
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2.
Select and use
appropriate methods to solve equations and inequalities.
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Linear equations –
algebraically
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Quadratic
equations – factoring (when the coefficient of x2 is 1) and using the
quadratic formula
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All types of
equations using graphing, computer, and graphing calculator techniques |
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3.
Judge the
meaning, utility, and reasonableness of the results of symbol manipulations,
including those carried out by technology. |
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