Mathematics Standards: Grades 9-12
Algebra – Reasoning with Equations and Inequalities
Grades 9-12
Grade 9-12
Understand solving equations as a process of reasoning and explain the reasoning
1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a
solution. Construct a viable argument to justify a solution method.
2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
Solve equations and inequalities in one variable
3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
4. Solve quadratic equations in one variable.
a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula
from this form.
b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the
equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.
Solve systems of equations
5. Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same
solutions.
6. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
7. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the
line y = –3x and the circle x2 + y2 = 3.
8. (+) Represent a system of linear equations as a single matrix equation in a vector variable.
9. (+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 °— 3 or greater).
Represent and solve equations and inequalities graphically
10. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
11. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions
approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial,
rational, absolute value, exponential, and logarithmic functions.
12. Graph the solutions to a linear inequality in two variables as a half-plane excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear
inequalities in two variables as the intersection of the corresponding half-planes.
Description
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These Standards define what students should understand and be able to do in their study of mathematics. The Standards set grade-specific standards but do not define
the intervention methods or materials necessary to support students who are well below or well above grade-level expectations.
The complexity options for these standards assure that all students, including those with the significant cognitive disabilities, have access to the Common Core State
Standards through appropriate instructional tasks.
Mathematics Standards:
Algebra – Reasoning with Equations and Inequalities
Extended Standards
Grades 9-12
Essence of the Standards:
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Justify a solution method.
Solve linear equations and inequalities.
Solve a system of linear equations with graphs.
Graph and solve a system with a linear relationship and a quadratic relationship.
Most Complex
Least Complex
Understand solving equations as a process of reasoning and explain the reasoning.
A.REI.912.1a Order a given sequence of
steps to solve an equation.
A.REI.912.1b Identify the operation performed
to get to the next step of a given sequence of
steps used to solve an equation.
A.REI.912.1c Identify how many should be
added or taken away from a set to get a given
total.
Solve equations and inequalities in one variable.
A.REI.912.2a Solve linear equations.
A.REI.912.3a Solve for viable solutions to
real-world, 1-step inequality situations.
A.REI.912.2b Solve 1-step linear equations.
A.REI.912.3b Identify whether a given value is
a viable solution to a 1-step inequality.
A.REI.912.4a Locate the coordinate at which
two lines intersect.
A.REI.912.5a Locate the coordinate of the
point(s) at which a line intersects a quadratic
function (e.g., at which two coordinates does
the line intersect the parabola?).
A.REI.912.4b Locate the point on the graph at
which two lines intersect.
A.REI.912.5b Locate the point(s) on the graph
at which a line intersects a quadratic function
(e.g., identify on the graph where the line
intersects the parabola).
A.REI.912.2c Solve for the missing number
within a given number sentence involving
addition or subtraction of numbers less than
10.
A.REI.912.3c Identify viable answer when
given a real world context involving an
inequality (e.g., Jane has a bag of 20
marbles, she gives at least 8 away, how
many could she have left at the end of the
day?).
Solve systems of equations.
A.REI.912.4c Identify whether two lines
intersect.
A.REI.912.5c Identify whether a line
intersects a quadratic function (e.g., does the
line intersect the parabola at one or two
points? Does the line intersect the
parabola?).
Mathematics Standards: Grades 9-12
Unique Instructional Targets:
Building Blocks to Algebra
• Understand and use +, - and = in problems.
• Solve addition and subtraction problems.
• Model and solve problems involving multiplication
or division.
Understand solving equations as a process of
reasoning and explain the reasoning.
• Order a sequence of steps to solve an equation.
Solve equations and inequalities in one variable.
• Use equations to solve real-world problems when a
part is unknown.
• Use inequalities (e.g., < and >) to solve real world
problems where a part is unknown.
High School Grade Band Lessons and
Activities
Lesson 19: Math Story Problems
Lesson 25: Algebra
Unique Supporting Activities:
Instructional Guides / Math Guidelines
Instructional Tools / Math Pack: Numbers
Instructional Tools / Math Pack: Arrays
Standards Connection
Mathematics Standards: Grades 9-12
Standard: Math Standards – Algebra – Reasoning with Equations and Inequalities
Grades 9-12
Extended Standard
Activities/Tasks: (a.)
Activities/Tasks: (b.)
Activities/Tasks: (c.)
• Students will calculate addition and subtraction
problems in the context of a real-world scenario.
• Students will read, write and solve a problem
sentence.
• Students will solve multi-step problems using a
combination of operations in the context of a realworld scenario.
• Students will model multiplication and division
with objects and numbers showing equal groups
in the context of a real-world scenario.
• Students will model addition and subtraction of
two sets of objects in the context of a real-world
scenario.
• Students will select pictures and numbers to
model a problem sentence.
• Students will solve a two-step problem using
operations and models in the context of a realworld scenario.
• Students will count equal numbers of objects in
selected groups or an array.
• Students will count a set of objects in an addition
or subtraction problem by using a sequencing
talking device or model.
• Students will select a number (errorless choice)
within a math problem.
• Students will select numbers and count within a
two-step problem in the context of a real-world
scenario.
• Students will count a set of objects in a group by
using a sequencing talking device or model.
Accommodations:
Assistive Tech: