Math - 2017-18
A.4abdf and *A.4ace - Linear Equations
A.4 The student will solve multistep linear and quadratic equations in two
variables, including
a) solving literal equations (formulas) for a given variable;
b) justifying steps used in simplifying expressions and solving equations, using field properties and axioms of equality that are valid for the set of real numbers and its subsets;
d) solving multistep linear equations algebraically and graphically;
f) solving real-world problems involving
equations and systems of equations.
Graphing calculators will be used both as a primary tool in solving problems and to verify algebraic solutions.
A.4 The student will solve
a) multistep linear equations in one variable algebraically;
c) literal equations for a specified variable;
e) practical problems involving equations
and systems of equations.
Bloom's Level: Apply, Evaluate
BIG IDEAS
- I can find how far and how
fast a bus travels, the number and combinations of fruits that can be
purchased, and how long it takes to drain a swimming pool.
- I can translate real-life situations into equations to find unknown values.
- I will be able to write symbolic representations of the
way numbers behave and will know that in order to maintain equality, an
operation performed on one side must also be performed on the other side.
UNDERSTANDING THE STANDARD
- A solution to an equation is the value or set of
values that can be substituted to make the equation true.
- The solution of an equation in one variable can
be found by graphing the expression on each side of the equation separately and
finding the x-coordinate of the point
of intersection.
- Real-world problems can be interpreted,
represented, and solved using linear and quadratic equations.
- The process of solving linear and quadratic
equations can be modeled in a variety of ways, using concrete, pictorial, and
symbolic representations.
- Properties of real numbers and properties of
equality can be used to justify equation solutions and expression
simplification.
- The zeros or the x-intercepts of the quadratic function are the real root(s) or
solution(s) of the quadratic equation that is formed by setting the given
quadratic expression equal to zero.
- A system of linear equations with exactly one
solution is characterized by the graphs of two lines whose intersection is a
single point, and the coordinates of this point satisfy both equations.
- A system of two linear equations with no
solution is characterized by the graphs of two lines that are parallel.
- A system of two linear equations having infinite
solutions is characterized by two graphs that coincide (the graphs will appear
to be the graph of one line), and the coordinates of all points on the line satisfy both equations.
- Systems of two linear equations can be used to
model two real-world conditions that must be satisfied simultaneously.
- Equations and systems of equations can be used
as mathematical models for real-world situations.
- Set builder notation may be used to represent
solution sets of equations.
2016 VDOE Curriculum Framework - AI.4 Understanding
· A solution to an equation is the value or set of values that can be substituted to make the equation true.
· Each point on the graph of a linear or quadratic equation in two variables is a solution of the equation.
· Practical problems may be interpreted, represented, and solved using linear and quadratic equations.
· The process of solving linear and quadratic equations can be modeled in a variety of ways, using concrete, pictorial, and symbolic representations.
· Properties of real numbers and properties of equality are applied to solve equations.
· Properties of Real Numbers:
Associative Property of Addition
Associative Property of Multiplication
Commutative Property of Addition
Commutative Property of Multiplication
Identity Property of Addition (Additive Identity)
Identity Property of Multiplication (Multiplicative Identity)
Inverse Property of Addition (Additive Inverse)
Inverse Property of Multiplication (Multiplicative Inverse)
Distributive Property
· Properties of Equality:
Multiplicative Property of Zero
Zero Product Property
Reflexive Property
Symmetric Property
Transitive Property of Equality
Addition Property of Equality
Subtraction Property of Equality
Multiplication Property of Equality
Division Property of Equality
Substitution
· Quadratic equations in one variable may be solved algebraically by factoring and applying properties of equality or by using the quadratic formula over the set of real numbers (Algebra I) or the set of complex numbers (Algebra II).
· Literal equations include formulas.
· A system of linear equations with exactly one solution is characterized by the graphs of two lines whose intersection is a single point, and the coordinates of this point satisfy both equations.
·
Equations
and systems of equations can be used as mathematical models for practical
situations.
· Solutions and intervals may be expressed in different formats, including set notation or using equations and inequalities.
- Examples may include:
Equation/ Inequality |
Set Notation |
x = 3 |
{3} |
x = 3 or x = 5 |
{3, 5} |
y≥ 3 |
{y: y ≥ 3} |
Empty (null) set ∅ |
{ } |
ESSENTIALS
The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to
A.4b_{1} Solve equations using the field properties of the real numbers and properties of equality to justify the solution.
· A.4a_{3 }Apply the properties of real numbers and properties of equality to simplify expressions and solve equations.
A.4d_{1} Solve multi-step linear equations in one variable.
· A.4a_{2 }Solve multistep linear equations in one variable algebraically.
A.4d_{2} Determine if a linear equation in one variable has one, an infinite number, or no solutions.
· A.4a_{1 }Determine whether a linear equation in one variable has one, an infinite number, or no solutions.
A.4a_{2} Confirm algebraic solutions to linear equations using a graphing calculator.
A.4a_{1} Solve a literal equation (formula) for a specific variable.
· A.4c_{1 }Solve a literal equation for a specified variable.
·
A.4e_{3 }Solve
practical problems involving equations and systems of equations.
KEY VOCABULARY
literal equation (formula), expression, equation,
properties of real numbers, properties of equality, quadratic equation,
quadratic function, quadratic expression, roots, zeros, linear equation,
system, substitution, elimination, ordered pair, coordinate, intersection,
intercepts, solution, infinite, parallel, coincide, simultaneous