Algebra Seeing Structure in Expressions Arithmetic with Polynomials and Rational Expressions Creating Equations Reasoning with Equations and Inequalities

Algebra• Seeing Structure in

Expressions

• Arithmetic with Polynomials and Rational Expressions

• Creating Equations

Reasoning with Equations and Inequalities


Understand solving equations as a process of reasoning and explain the reasoning.

A-REI.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.

Algebra I


Understand solving equations as a process of reasoning and explain the reasoning.

A-REI.2

Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

Algebra II



Solve equations and inequalities in one variable

A-REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

Algebra I


Solve equations and inequalities in one variable

A-REI.4Solve 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.

Algebra I


On your own:

Derive the quadratic formula by completing the square of the equation

ax2 + bx + c = 0

using either the area model or the algebraic model


Solve equations and inequalities in one variableA-REI.4

Solve quadratic equations in one variable.

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.

Algebra I


Solve the equationy = 3x2 + 2x + 5 using any method.

Express the answer, if it is complex, in the form x = a + bi and x = a - bi


Solve systems of equations.

A-REI.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.

Algebra I


Show graphically that the solution for the system of equationsy = x + 3y = 4x

has the same solution set if you replace y = x + 3 by the sum of y = x + 3 and a multiple of y = 4x.

What general rule for solving systems of equations can you derive from this fact?



A-REI.6

Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Algebra I



A-REI.7Solve 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.

Algebra I



A-REI.8

(+) Represent a system of linear equations as a single matrix equation in a vector variable.

Honors or Year 4



A-REI.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).

Honors or Year 4


Represent and solve equations and inequalities graphically.

A-REI.10Understand 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).

Algebra I



A-REI.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. ⋆

Algebra I & II


The population of a country is initially 2 million people and is increasing at 4% per year. The country's annual food supply is initially adequate for 4 million people and is increasing at a constant rate adequate for an additional 0.5 million people per year. Based on these assumptions, in approximately what year will this country first experience shortages of food?If the country doubled its initial food supply and maintained a constant rate of increase in the supply adequate for an additional 0.5 million people per year, would shortages still occur? In approximately which year?If the country doubled the rate at which its food supply increases, in addition to doubling its initial food supply, would shortages still occur?



A-REI.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.

See Illustrative Math Fishing Expedition

Algebra I

http://illustrativemathematics.org/illustrations/644

Look back at Practice Standards Poster

Which problems had the most X’s?

Computer Algebra ActivityWork with a partner and on your own laptop, go to

http://illustrativemathematics.org/• Go to the HS Algebra Domain and read through the

standards • Check out the illustrations. Which SMPs would they

reinforce?• Which illustrations would you use in your classroom?• Check out the modeling standards (those with a star)• Why do you think these standards were chosen as

modeling standards?

http://illustrativemathematics.org/

http://illustrativemathematics.org/

Documents

Algebra Seeing Structure in Expressions Arithmetic with Polynomials and Rational Expressions Creating Equations Reasoning with Equations and Inequalities