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#### MAT-HS.A-APR

Domains are larger groups of related standards. So the Domain Score is a calculation of all the related standards. So click on the standard name below each Domain to access the learning targets and proficiency scales for each Domain's related standards.

# Arithmetic with Polynomials & Rational Expressions

• Perform arithmetic operations on polynomials
• Understand the relationship between zeros and factors of polynomials
• Use polynomial identities to solve problems
• Rewrite rational functions

## Domain Description

An expression is a record of a computation with numbers, symbols that represent numbers, arithmetic operations, exponentiation, and, at more advanced levels, the operation of evaluating a function. Conventions about the use of parentheses and the order of operations assure that each expression is unambiguous. Creating an expression that describes a computation involving a general quantity requires the ability to express the computation in general terms, abstracting from specific instances.

Reading an expression with comprehension involves analysis of its underlying structure. This may suggest a different but equivalent way of writing the expression that exhibits some different aspect of its meaning. For example, p + 0.05p can be interpreted as the addition of a 5% tax to a price p. Rewriting p + 0.05p as 1.05p shows that adding a tax is the same as multiplying the price by a constant factor.

Algebraic manipulations are governed by the properties of operations and exponents, and the conventions of algebraic notation. At times, an expression is the result of applying operations to simpler expressions. For example, p + 0.05p is the sum of the simpler expressions p and 0.05p. Viewing an expression as the result of operation on simpler expressions can sometimes clarify its underlying structure.

A spreadsheet or a computer algebra system (CAS) can be used to experiment with algebraic expressions, perform complicated algebraic manipulations, and understand how algebraic manipulations behave.

## Standards in this Domain

• MAT-HS.A-APR.01 - 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.
• MAT-HS.A-APR.02 - 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).
• MAT-HS.A-APR.03 - 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.
• MAT-HS.A-APR.04 - Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 - y2)2 + (2xy)2 can be used to generate Pythagorean triples.
• MAT-HS.A-APR.05 - Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal's Triangle.
• MAT-HS.A-APR.06 - 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.
• MAT-HS.A-APR.07 - 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.

#### MAT-HS.A-APR.01

 MAT-HS Targeted Standards(A) Concept: Algebra(SSE) Domain: Arithmetic with Polynomials and Rational ExpressionsCluster: Perform arithmetic operations on polynomials MAT-HS.A-APR.01 Add, subtract, and multiply polynomials. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication.

## Student Learning Targets:

### Knowledge Targets

• I can identify polynomials.

### Reasoning Targets

• I can recognize how closure applies under these operations.
• I can

### Skills (Performance) Targets

• I can add, subtract, and multiply polynomials and recognize how closure applies under these operations.

• I can
• I can

## Proficiency Scale

 Score Description Sample Activity 4.0 (advanced) In addition to Score 3.0, the student demonstrates in-depth inferences and applications regarding more complex material that go beyond end of instruction expectations. Shannon, a cabinetmaker, started out with a block of wood, and then she hollowed out the center of the block.  The dimensions of the block and the cutout is shown below. Write the volume of the original block, and the volume of the hole. (ignore units for this problem) Write the polynomial for the volume of the wood remaining. 3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success. 3.0 (proficient) The student can: add, subtract, and multiply polynomials. recognize that when two polynomials are added, subtracted, or multiplied, a polynomial is the result. The student exhibits no major errors or omissions. Simplify the following: (2x2+3x+5)+(5x3-9x-8) (x4+3x3-x2)-(4x3+8) (x+2)(5x2-7x+10) 2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content). 2.0 (progressing) There are no major errors or omissions regarding the simpler details and processes as the student can: recognize and recall specific terminology, such as: polynomial trinomial binomial term combine like terms. However, the student exhibits major errors or omissions regarding the more complex ideas and processes. Simplify the following: 2x2+7x-9+3x2+5x-10 (2x+3)(4x-8) 1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content). 1.0 (beginning) With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) and some of the more complex ideas and processes (Score 3.0 content). 0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content). 0.0 Even with help, the student demonstrates no understanding or skill.

## Resources

 Web Vocab Polynomial

#### MAT-HS.A-APR.02

 MAT-HS Targeted Standards(A) Concept: Algebra(SSE) Domain: Arithmetic with Polynomials and Rational ExpressionsCluster: Perform arithmetic operations on polynomials MAT-HS.A-APR.02 Apply the Remainder Theorem.

## Student Learning Targets:

### Knowledge Targets

• I can use the fact that a is a zero of a polynomial.

• I can
• I can

### Skills (Performance) Targets

• I can apply the Remainder Theorem to a polynomial.

• I can
• I can

## Alg II Solve Polynomial Equations Proficiency Scale

Score   Description Sample Activity

4.0

In addition to Score 3.0, the student demonstrates in-depth inferences and applications regarding more complex material that go beyond end of instruction expectations.
 Describe and correct the error in using synthetic division to divide x3-5x+3 by x-2.
3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success.

3.0

(proficient)

The student can:
• use and apply the remainder theorem/synthetic division to find the zeros of a polynomial function, real and imaginary.
• use and apply long division/synthetic division to find the zeros of a polynomial function.
The student exhibits no major errors or omissions.

Find the zeros:

f(x)= x3-10x2+19x+30

f(x)=2x5+3x4-30x3-57x2-2x+24

f(x)=x3-4x2+25x-100
2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content).

2.0

(progressing)

There are no major errors or omissions regarding the simpler details and processes as the student can:
• recognize and recalls specific terminology, such as:
• factors
• zeros
• imaginary/complex numbers
• synthetic division or long division
• fundamental theorem of Algebra
• possible zeros (p/q)
• use synthetic division to divide polynomials.
• can perform operations with complex numbers.

However, the student exhibits major errors or omissions regarding the more complex ideas and processes.

Synthetically divide to find the remaining zeros.

(6x3-19x2+16x-4)(x-2)

(2x4+7x3-4x2-27x-18)(x+3)
1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content).

1.0

(beginning)

With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) and some of the more complex ideas and processes (Score 3.0 content).
0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content).
0.0 Even with help, the student demonstrates no understanding or skill.

## Resources

 Web Vocab

#### MAT-HS.A-APR.03

 MAT-HS Targeted Standards(A) Concept: Algebra(SSE) Domain: Arithmetic with Polynomials and Rational ExpressionsCluster: Perform arithmetic operations on polynomials MAT-HS.A-APR.03 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. a. Substandard b. Substandard

## Student Learning Targets:

• I can
• I can

• I can
• I can

### Skills (Performance) Targets

• I can generate graphs of a variety of functions (i.e., linear, quadratic, polynomial, absolute value, and exponential), using technology when appropriate.
• I can solve for (isolate) a specific variable.
• I can perform transformations that produce equivalent equations (e.g., adding the same amount to both sides of the equation, etc.).
• I can check the solution.
• I can set a polynomial equal to zero and solve by factoring, quadratic formula, graphing, linear combinations, and substitution.
• I can solve for (isolate) a specific variable.
• I can write a linear system.
• I can perform transformations that produce equivalent equations (e.g., adding the same amount to both sides of the equation, etc.)
• I can check the solution.
• I can solve a quadratic.
• I can multiply and factor polynomials.
• I can solve a linear inequality.
• I can graph a linear inequality.
• I can solve a system of linear inequalities.
• I can graph a system of linear inequalities.
• I can Check the solution of the system.
• I can solve a linear system (substitution, linear combination, graphically).
• I can write a linear system.
• I can identify the number of solutions of a linear system.
• I can identify the most efficient method to solve a system.
• I can check the solution of the system.
• I can find the zeros of a polynomial function when the polynomial is factored.

### Product Targets

• I can use the zeros (x-intercepts) of a polynomial function to sketch a graph of the function.

## Proficiency Scale

 Score Description Sample Activity 4.0 In addition to Score 3.0, the student demonstrates in-depth inferences and applications regarding more complex material that go beyond end of instruction expectations. - 3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success. 3.0 “The Standard.” The student demonstrates no major errors or omissions regarding any of the information and processes that were end of instruction expectations. - 2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content). 2.0 The student demonstrates no major errors or omissions regarding the simpler details and processes but exhibits major errors or omissions regarding the more complex ideas and processes (Score 3.0 content). - 1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content). 1.0 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) and some of the more complex ideas and processes (Score 3.0 content). - 0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content). 0.0 Even with help, the student demonstrates no understanding or skill. -

## Resources

 Web Vocab

#### MAT-HS.A-APR.04

 MAT-HS Targeted Standards(A) Concept: Algebra(SSE) Domain: Arithmetic with Polynomials and Rational ExpressionsCluster: Use polynomial identities to solve problems MAT-HS.A-APR.04 Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x² + y²)² = (x² – y²)² + (2xy)² can be used to generate Pythagorean triples.

## Student Learning Targets:

• I can
• I can

• I can
• I can

### Skills (Performance) Targets

• I can use polynomial identities such as those for x2 – y2, x3 – y3, x3 + y3, (x+y)3 and (x+y)2 to describe numerical relationships.
• I can prove polynomial identities.

• I can
• I can

## Proficiency Scale

 Score Description Sample Activity 4.0 In addition to Score 3.0, the student demonstrates in-depth inferences and applications regarding more complex material that go beyond end of instruction expectations. - 3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success. 3.0 “The Standard.” The student demonstrates no major errors or omissions regarding any of the information and processes that were end of instruction expectations. - 2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content). 2.0 The student demonstrates no major errors or omissions regarding the simpler details and processes but exhibits major errors or omissions regarding the more complex ideas and processes (Score 3.0 content). - 1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content). 1.0 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) and some of the more complex ideas and processes (Score 3.0 content). - 0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content). 0.0 Even with help, the student demonstrates no understanding or skill. -

## Resources

 Web Vocab

#### MAT-HS.A-APR.05

 MAT-HS Targeted Standards(A) Concept: Algebra(SSE) Domain: Arithmetic with Polynomials and Rational ExpressionsCluster: Use polynomial identities to solve problems. MAT-HS.A-APR.05 Know and apply the Binomial Theorem gives 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.

## Student Learning Targets:

• I can
• I can

• I can
• I can

### Skills (Performance) Targets

• I can use Pascal’s Triangle to determine the coefficients of the binomial expansion (x+y)n.
• I can use the Binomial Theorem to find the nth term in the expansion of a binomial to a positive integer power.

• I can
• I can

## Proficiency Scale

 Score Description Sample Activity 4.0 In addition to Score 3.0, the student demonstrates in-depth inferences and applications regarding more complex material that go beyond end of instruction expectations. - 3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success. 3.0 “The Standard.” The student demonstrates no major errors or omissions regarding any of the information and processes that were end of instruction expectations. - 2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content). 2.0 The student demonstrates no major errors or omissions regarding the simpler details and processes but exhibits major errors or omissions regarding the more complex ideas and processes (Score 3.0 content). - 1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content). 1.0 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) and some of the more complex ideas and processes (Score 3.0 content). - 0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content). 0.0 Even with help, the student demonstrates no understanding or skill. -

## Resources

 Web Vocab

#### MAT-HS.A-APR.06

 MAT-HS Targeted Standards(A) Concept: Algebra(SSE) Domain: Arithmetic with Polynomials and Rational ExpressionsCluster: Rewrite rational expressions. MAT-HS.A-APR.06 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.

## Student Learning Targets:

• I can
• I can

• I can
• I can

### Skills (Performance) Targets

• I can rewrite rational expressions, a(x)/b(x), in the form q(x) + r(x)/b(x) using inspection or long division (use a computer algebra system for complicated examples).

• I can
• I can

## Proficiency Scale

 Score Description Sample Activity 4.0 In addition to Score 3.0, the student demonstrates in-depth inferences and applications regarding more complex material that go beyond end of instruction expectations. - 3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success. 3.0 “The Standard.” The student demonstrates no major errors or omissions regarding any of the information and processes that were end of instruction expectations. - 2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content). 2.0 The student demonstrates no major errors or omissions regarding the simpler details and processes but exhibits major errors or omissions regarding the more complex ideas and processes (Score 3.0 content). - 1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content). 1.0 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) and some of the more complex ideas and processes (Score 3.0 content). - 0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content). 0.0 Even with help, the student demonstrates no understanding or skill. -

## Resources

 Web Vocab

#### MAT-HS.A-APR.07

 MAT-HS Targeted Standards(A) Concept: Algebra(SSE) Domain: Arithmetic with Polynomials and Rational ExpressionsCluster: Use polynomial identities to solve problems MAT-HS.A-APR.07 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.

## Student Learning Targets:

### Knowledge Targets

• I can recognize that rational expressions are closed under addition, subtraction, multiplication, and division by a nonzero expression.

• I can
• I can

### Skills (Performance) Targets

• I can add, subtract, multiply, and divide rational expressions.

• I can
• I can

## Proficiency Scale

 Score Description Sample Activity 4.0 In addition to Score 3.0, the student demonstrates in-depth inferences and applications regarding more complex material that go beyond end of instruction expectations. - 3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success. 3.0 “The Standard.” The student demonstrates no major errors or omissions regarding any of the information and processes that were end of instruction expectations. - 2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content). 2.0 The student demonstrates no major errors or omissions regarding the simpler details and processes but exhibits major errors or omissions regarding the more complex ideas and processes (Score 3.0 content). - 1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content). 1.0 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) and some of the more complex ideas and processes (Score 3.0 content). - 0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content). 0.0 Even with help, the student demonstrates no understanding or skill. -

## Resources

 Web Vocab

#### MAT-HS.A-CED

Domains are larger groups of related standards. So the Domain Score is a calculation of all the related standards. So click on the standard name below each Domain to access the learning targets and proficiency scales for each Domain's related standards.

# Creating Equations

• Create equations that describe numbers or relationships

## Domain Description

An equation is a statement of equality between two expressions, often viewed as a question asking for which values of the variables the expressions on either side are in fact equal. These values are the solutions to the equation. An identity, in contrast, is true for all values of the variables; identities are often developed by rewriting an expression in an equivalent form.

The solutions of an equation in one variable form a set of numbers; the solutions of an equation in two variables form a set of ordered pairs of numbers, which can be plotted in the coordinate plane. Two or more equations and/or inequalities form a system. A solution for such a system must satisfy every equation and inequality in the system.

An equation can often be solved by successively deducing from it one or more simpler equations. For example, one can add the same constant to both sides without changing the solutions, but squaring both sides might lead to extraneous solutions. Strategic competence in solving includes looking ahead for productive manipulations and anticipating the nature and number of solutions.

Some equations have no solutions in a given number system, but have a solution in a larger system. For example, the solution of x + 1 = 0 is an integer, not a whole number; the solution of 2x + 1 = 0 is a rational number, not an integer; the solutions of  – 2 = 0 are real numbers, not rational numbers; and the solutions of  + 2 = 0 are complex numbers, not real numbers.

The same solution techniques used to solve equations can be used to rearrange formulas. For example, the formula for the area of a trapezoid, A = ((b1+b2)/2)h, can be solved for h using the same deductive process. Inequalities can be solved by reasoning about the properties of inequality. Many, but not all, of the properties of equality continue to hold for inequalities and can be useful in solving them.

Connections to Functions and Modeling. Expressions can define functions, and equivalent expressions define the same function. Asking when two functions have the same value for the same input leads to an equation; graphing the two functions allows for finding approximate solutions of the equation. Converting a verbal description to an equation, inequality, or system of these is an essential skill in modeling.

## Standards in this Domain

• MAT-HS.A-CED.01 - 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.
• MAT-HS.A-CED.02 - Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
• MAT-HS.A-CED.03 - 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.
• MAT-HS.A-CED.04 - 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.

#### MAT-HS.A-CED.01

 MAT-HS Targeted Standards(A) Concept: Algebra(CED) Domain: Creating ExpressionsCluster: Create equations that describe numbers or relationships MAT-HS.A-CED.01 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.

This standard is divided into parts.  Click on these links to see specific proficiency scales.

MAT-HS.A-CED.01.a

MAT-HS.A-CED.01.b

MAT-HS.A-CED.01.c

MAT-HS.A-CED.01.d

#### MAT-HS.A-CED.01.a

 MAT-HS Targeted Standards(A) Concept: Algebra(CED) Domain: Creating ExpressionsCluster: Create equations that describe numbers or relationships MAT-HS.A-CED.01.a Create and solve linear equations in one variable.

## Student Learning Targets:

• I can

• I can

### Skills (Performance) Targets

• I can create a linear equation or inequality given a situation or set of data.

• I can

## Proficiency Scale

Score   Description Sample Activity
4.0 In addition to achieving level 3.0 content, the student makes in-depth inferences and applications that go beyond what was taught
• Write an equation of a parabola that has two x-intercepts and a minimum vertex.

• Create a rational equation given a situation or set or data
3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success.
3.0

Can create equations and inequalities in one variable and use them to solve problems (linear only)

The student exhibits no major errors or omissions.
• An online music club sells compact discs for $13.95 each plus$1.95 shipping and handling per order.  Maria’s total bill was $85.65. Create an equation to model this situation. Solve your equation to determine how many compact discs were sold. • Suppose you are working for a trucking company. Your job is to load a truck with at least 5000 lb of freight. You have loaded 2395 lb of freight, but you have to unload 50 lb that was loaded by mistake. Write and solve an inequality to find how many more pounds you need to load. • Which kind of function best models the data in the table? Write an equation to model the data.  x y 0 5 1 7 2 9 3 11 2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content). 2.0 There are no major errors or omissions regarding the simpler details and processes as the student: • recognizes and recalls specific terminology, such as: (A.SSE.1) • linear • correctly identifies whether the relationship is linear or exponential. • writes the general equations for linear and exponential functions. • pick out relevant information in a contextual problem. However, the student exhibits major errors or omissions regarding the more complex ideas and processes. - 1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content). 1.0 With help, a partial understanding of some of the simpler details and processes and some of the more complex ideas and processes. - 0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content). ## Resources  Web Vocab linear equations quadratic equations rational equations exponential equations inequalities #### MAT-HS.A-CED.01.b  MAT-HS Targeted Standards(A) Concept: Algebra(CED) Domain: Creating ExpressionsCluster: Create equations that describe numbers or relationships MAT-HS.A-CED.01.b Create and solve exponential equations in one variable ## Student Learning Targets: ### Knowledge Targets • I can ### Reasoning Targets • I can ### Skills (Performance) Targets • I can ### Product Targets • I can ## Proficiency Scale  Score Description Sample Activity 4.0 (advanced) In addition to Score 3.0, the student demonstrates in-depth inferences and applications regarding more complex material that go beyond end of instruction expectations. 3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success. 3.0 (proficient) The student can: The student exhibits no major errors or omissions. 2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content). 2.0 (progressing) There are no major errors or omissions regarding the simpler details and processes as the student: However, the student exhibits major errors or omissions regarding the more complex ideas and processes. 1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content). 1.0 (beginning) With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) and some of the more complex ideas and processes (Score 3.0 content). - 0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content). 0.0 Even with help, the student demonstrates no understanding or skill. - ## Resources  Web Vocab #### MAT-HS.A-CED.01.c  MAT-HS Targeted Standards(A) Concept: Algebra(CED) Domain: Creating ExpressionsCluster: Create equations that describe numbers or relationships MAT-HS.A-CED.01.c Create and solve linear inequalities in one variable. ## Student Learning Targets: ### Knowledge Targets • I can ### Reasoning Targets • I can • I can ### Skills (Performance) Targets • I can create a linear equation or inequality given a situation or set of data. ### Product Targets • I can ## Proficiency Scale Score Description Sample Activity 4.0 In addition to achieving level 3.0 content, the student makes in-depth inferences and applications that go beyond what was taught • Write an equation of a parabola that has two x-intercepts and a minimum vertex. • Create a rational equation given a situation or set or data 3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success. 3.0 Can create equations and inequalities in one variable and use them to solve problems (linear only) The student exhibits no major errors or omissions. • An online music club sells compact discs for$13.95 each plus $1.95 shipping and handling per order. Maria’s total bill was$85.65.  Create an equation to model this situation.  Solve your equation to determine how many compact discs were sold.

• Suppose you are working for a trucking company. Your job is to load a truck with at least 5000 lb of freight. You have loaded 2395 lb of freight, but you have to unload 50 lb that was loaded by mistake. Write and solve an inequality to find how many more pounds you need to load.

• Which kind of function best models the data in the table? Write an equation to model the data.

 x y 0 5 1 7 2 9 3 11
2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content).
2.0

There are no major errors or omissions regarding the simpler details and processes as the student:

• recognizes and recalls specific terminology, such as: (A.SSE.1)

• linear

• correctly identifies whether the relationship is linear or  exponential.

• writes the general equations for linear and exponential functions.

• pick out relevant information in a contextual problem.

However, the student exhibits major errors or omissions regarding the more complex ideas and processes.
-
1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content).
1.0 With help, a partial understanding of some of the simpler details and processes and some of the more complex ideas and processes. -
0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content).

## Resources

 Web Vocab linear equations quadratic equations rational equations exponential equations inequalities

#### MAT-HS.A-CED.01.d

 MAT-HS Targeted Standards(A) Concept: Algebra(CED) Domain: Creating ExpressionsCluster: Create equations that describe numbers or relationships MAT-HS.A-CED.01.d Create and solve exponential inequalities in one variable

• I can

• I can

• I can

• I can

## Proficiency Scale

 Score Description Sample Activity 4.0 (advanced) In addition to Score 3.0, the student demonstrates in-depth inferences and applications regarding more complex material that go beyond end of instruction expectations. 3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success. 3.0 (proficient) The student can: The student exhibits no major errors or omissions. 2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content). 2.0 (progressing) There are no major errors or omissions regarding the simpler details and processes as the student:   However, the student exhibits major errors or omissions regarding the more complex ideas and processes. 1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content). 1.0 (beginning) With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) and some of the more complex ideas and processes (Score 3.0 content). - 0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content). 0.0 Even with help, the student demonstrates no understanding or skill. -

## Resources

 Web Vocab

#### MAT-HS.A-CED.01.e

 MAT-HS Targeted Standards(A) Concept: Algebra(CED) Domain: Creating ExpressionsCluster: Create equations that describe numbers or relationships MAT-HS.A-CED.01.e Create quadratic equations

• I can

• I can

• I can

• I can

## Proficiency Scale

Score   Description Sample Activity

4.0

In addition to Score 3.0, the student demonstrates in-depth inferences and applications regarding more complex material that go beyond end of instruction expectations.
3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success.

3.0

(proficient)

The student can:
• create quadratic equations in one variable and use them to solve problems.
The student exhibits no major errors or omissions.

A square has an unknown side length of x.  A rectangle has a side length that is four feet longer than the square and a width that is two feet shorter than the square.  The areas of both the square and the rectangle are equal.  Find the side length of the square.

2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content).

2.0

(progressing)

There are no major errors or omissions regarding the simpler details and processes as the student can:

• recognize and recall specific terminology such as:
• equation
• create the equation; however, is unable to solve.

However, the student exhibits major errors or omissions regarding the more complex ideas and processes.

 What is the expression that represents the volume of the following prism?
1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content).

1.0

(beginning)

With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) and some of the more complex ideas and processes (Score 3.0 content).
0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content).
0.0 Even with help, the student demonstrates no understanding or skill.

## Resources

 Web Vocab

#### MAT-HS.A-CED.01.f

 MAT-HS Targeted Standards(A) Concept: Algebra(CED) Domain: Creating ExpressionsCluster: Create equations that describe numbers or relationships MAT-HS.A-CED.01.e Create simple rational equations

• I can

• I can

• I can

• I can

## Proficiency Scale

Score   Description Sample Activity

4.0

In addition to Score 3.0, the student demonstrates in-depth inferences and applications regarding more complex material that go beyond end of instruction expectations.
3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success.

3.0

(proficient)

The student can:
• create simple rational equations in one variable and use them to solve problems.
The student exhibits no major errors or omissions.

 Source  Illustrative Math: Basketball Chase and his brother like to play basketball. About a month ago they decided to keep track of how many games they have each won. As of today, Chase has won 18 out of the 30 games against his brother. How many games would Chase have to win in a row in order to have a 75% winning record?
2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content).

2.0

(progressing)

There are no major errors or omissions regarding the simpler details and processes as the student can:

• recognize and recall specific terminology such as:
• Solution
• Equation
• Rational Equation
• Inverse operations
• solve a linear equation.
• solve a simple rational equation.
However, the student exhibits major errors or omissions regarding the more complex ideas and processes.

Solve:

Solve: 0.25(x+10)=14

1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content).

1.0

(beginning)

With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) and some of the more complex ideas and processes (Score 3.0 content). -
0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content).
0.0 Even with help, the student demonstrates no understanding or skill. -

## Resources

 Web Vocab

#### MAT-HS.A-CED.01.g

 MAT-HS Targeted Standards(A) Concept: Algebra(CED) Domain: Creating ExpressionsCluster: Create equations that describe numbers or relationships MAT-HS.A-CED.01.g Create exponential equations

• I can

• I can

• I can

• I can

## Proficiency Scale

 Score Description Sample Activity 4.0 (advanced) In addition to Score 3.0, the student demonstrates in-depth inferences and applications regarding more complex material that go beyond end of instruction expectations. 3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success. 3.0 (proficient) The student can: create exponential equations in one variable and use them to solve problems. The student exhibits no major errors or omissions. During normal breathing, about 12% of the air in the lungs is replaced after one breath. Write an exponential decay model for the amount of the original air left in the lungs if the initial amount of air in the lungs is 500 mL. How much of the original air is present after 240 breaths? The foundation of your house has about 1,200 termites. The termites grow at a rate of about 2.4% per day. How long until the number of termites doubles? 2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content). 2.0 (progressing) There are no major errors or omissions regarding the simpler details and processes as the student can: recognize and recall specific terminology such as: Exponential Decay Growth Rate Equation Annually Semi-Annually Quarterly Compounded create the equation; however, is unable to solve. solve a quadratic equation. However, the student exhibits major errors or omissions regarding the more complex ideas and processes. Solve: 50 = 2(3)x 1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content). 1.0 (beginning) With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) and some of the more complex ideas and processes (Score 3.0 content). 0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content). 0.0 Even with help, the student demonstrates no understanding or skill.

## Resources

 Web Vocab

#### MAT-HS.A-CED.02

 MAT-HS Targeted Standards(A) Concept: Algebra(CED) Domain: Creating ExpressionsCluster: Create equations that describe numbers or relationships MAT-HS.A-CED.02 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

This standard is divided into parts.  Click on these links to see specific proficiency scales.

MAT-HS.A-CED.02.a

MAT-HS.A-CED.02.b

#### MAT-HS.A-CED.02.a

 MAT-HS Targeted Standards(A) Concept: Algebra(CED) Domain: Creating ExpressionsCluster: Create equations that describe numbers or relationships MAT-HS.A-CED.02.a Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

## Student Learning Targets:

• I can
• I can

• I can
• I can

### Skills (Performance) Targets

• I can create an equation in two or more variables to represent relationships between quantities.
• I can graph equations in two variables on a coordinate plane and label the axes and scales.

• I can

## Proficiency Scale

 Score Description Sample Activity 4.0 In addition to achieving level 3.0 content, the student makes in-depth inferences and applications that go beyond what was taught including construct linear functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (A.F.LE.2) - 3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success. 3.0 The student can: create linear equations in two or more variables to represent relationships between quantities   graph equations on coordinate axes with labels and scales The student exhibits no major errors or omissions. Suppose $15 US is worth$150 Mexican pesos. What is an equation that relates US dollars x to Mexican pesos y? The cost to rent a car is $50 plus$0.25 per mile driven.  Write and graph an equation to represent the situation. 2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content). 2.0 There are no major errors or omissions regarding the simpler details and processes as the student: recognizes and recalls specific terminology, such as:  (A.SSE.1)                linear However, the student exhibits major errors or omissions regarding the more complex ideas and processes. - 1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content). 1.0 With help, a partial understanding of some of the simpler details and processes and some of the more complex ideas and processes. - 0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content).

## Alg II Create and Graph Quadratic Functions Proficiency Scale

 Score Description Sample Activity 4.0 (advanced) In addition to Score 3.0, the student demonstrates in-depth inferences and applications regarding more complex material that go beyond end of instruction expectations. Given two parabolic graphs, create the equations for both functions, graph them, and describe the similarities and differences. Function 1:  the turning point is (2,-3) and another random point on the graph is (4,5)   Function 2:  the turning point is (2,-3) and another random point on the graph is (5,6) 3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success. 3.0 (proficient) The student can: create quadratic equations in two variables given information about the function. The student exhibits no major errors or omissions. Given the x-intercepts of 6 and -2, write a quadratic function that would also pass through the point (1,5). 2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content). 2.0 (progressing) There are no major errors or omissions regarding the simpler details and processes as the student can: recognize and recall specific terminology such as: quadratic identify the graph and/or equation of quadratic functions two variables on a coordinate plane and label the axes and scales. However, the student exhibits major errors or omissions regarding the more complex ideas and processes. Given the graph below, write the quadratic function. 1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content). 1.0 (beginning) With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) and some of the more complex ideas and processes (Score 3.0 content). 0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content). 0.0 Even with help, the student demonstrates no understanding or skill.

## Alg II Create and Graph Exponential Functions Proficiency Scale

 Score Description Sample Activity 4.0 (advanced) In addition to Score 3.0, the student demonstrates in-depth inferences and applications regarding more complex material that go beyond end of instruction expectations. Given two exponential graphs, create the equations for both functions, graph them, and describe the similarities and differences.Function 1:  contains the point (0, 4) and (1, 12)   Function 2:  contains the point (0, 4) and (1, 2) 3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success. 3.0 (proficient) The student can: create exponential equations in two variables given information about the function. The student exhibits no major errors or omissions. Given the points (0, 3) and (-1, 6), create the exponential equation. 2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content). 2.0 (progressing) There are no major errors or omissions regarding the simpler details and processes as the student can: recognize and recall specific terminology such as: exponential identify the graph and/or equation of exponential functions in two variables on a coordinate plane and label the axes and scales. However, the student exhibits major errors or omissions regarding the more complex ideas and processes. Given the graph below, write the exponential function. 1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content). 1.0 (beginning) With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) and some of the more complex ideas and processes (Score 3.0 content). - 0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content). 0.0 Even with help, the student demonstrates no understanding or skill. -

## Alg II Create and Graph Polynomial Functions Proficiency Scale

 Score Description Sample Activity 4.0 (advanced) In addition to Score 3.0, the student demonstrates in-depth inferences and applications regarding more complex material that go beyond end of instruction expectations. Given two polynomial graphs, create the equations for both functions, graph them, and describe the similarities and differences. Function 1:  the zeros are  1, 2, 3, and another random point on the graph is (0,-6)   Function 2:  the zeros are 1, 2,3, and another random point on the graph is (0,18) 3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success. 3.0 (proficient) The student can: create polynomial equations in two variables on a coordinate plane and label the axes and scales. The student exhibits no major errors or omissions. Given the x-intercepts of -4, -1 and 2, write a polynomial function that would also pass through the point (0,-2). 2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content). 2.0 (progressing) There are no major errors or omissions regarding the simpler details and processes as the student can: recognize and recall specific terminology such as: polynomial identify the graph and/or equation of polynomial functions in two variables on a coordinate plane and label the axes and scales.    However, the student exhibits major errors or omissions regarding the more complex ideas and processes. Given the graph below, write the polynomial function. 1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content). 1.0 (beginning) With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) and some of the more complex ideas and processes (Score 3.0 content). - 0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content). 0.0 Even with help, the student demonstrates no understanding or skill. -

## Resources

 Web Vocab coordinate plane scale

#### MAT-HS.A-CED.03

 MAT-HS Targeted Standards(A) Concept: Algebra(CED) Domain: Creating ExpressionsCluster: Create equations that describe numbers or relationships MAT-HS.A-CED.03* 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.

## Student Learning Targets:

• I can
• I can

• I can
• I can

### Skills (Performance) Targets

• I can write and use a system of equations and/or inequalities to solve a real world problem.
• I can use equations and inequalities to represent problem constraints and objectives (linear programming).

• I can
• I can

## Resources

 Web Vocab linear programming

#### MAT-HS.A-CED.04

 MAT-HS Targeted Standards(A) Concept: Algebra(CED) Domain: Creating ExpressionsCluster: Create equations that describe numbers or relationships MAT-HS.A-CED.04 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.

## Student Learning Targets:

• I can
• I can

• I can
• I can

### Skills (Performance) Targets

• I can solve for (isolate) a specific variable.
• I can perform transformations that produce equivalent equations (e.g., adding the same amount to both sides of the equation, etc.)
• I can solve multi-variable formulas or literal equations for a specific variable.

• I can
• I can

## Proficiency Scale

 Score Description Sample Activity 4.0 (advanced) In addition to Score 3.0, the student demonstrates in-depth inferences and applications regarding more complex material that go beyond end of instruction expectations. Solve for h:   3h+g=5h-hg 3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success. 3.0 (proficient) The student can: solve multi-variable formulas or literal equations for a specific variable. The student exhibits no major errors or omissions. Solve the following equation for b: A = 1/2h (b+c) Solve the following equation for a: Solve the following equation for b: 2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content). 2.0 (progressing) There are no major errors or omissions regarding the simpler details and processes as the student can: identify appropriate inverse operations. solve a simple linear equation for y. However, the student exhibits major errors or omissions regarding the more complex ideas and processes. Solve the following equation for y: 2x - 3y = 8 1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content). 1.0 (beginning) With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) and some of the more complex ideas and processes (Score 3.0 content). 0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content). 0.0 Even with help, the student demonstrates no understanding or skill.

## Resources

 Web Vocab literal equation

#### MAT-HS.A-REI

Domains are larger groups of related standards. So the Domain Score is a calculation of all the related standards. So click on the standard name below each Domain to access the learning targets and proficiency scales for each Domain's related standards.

# Reasoning with Equations and Inequalities

• Understand solving equations as a process of reasoning and explain the reasoning
• Solve equations and inequalities in one variable
• Solve systems of equations
• Represent and solve equations and inequalities graphically

## Domain Description

An equation is a statement of equality between two expressions, often viewed as a question asking for which values of the variables the expressions on either side are in fact equal. These values are the solutions to the equation. An identity, in contrast, is true for all values of the variables; identities are often developed by rewriting an expression in an equivalent form.

The solutions of an equation in one variable form a set of numbers; the solutions of an equation in two variables form a set of ordered pairs of numbers, which can be plotted in the coordinate plane. Two or more equations and/or inequalities form a system. A solution for such a system must satisfy every equation and inequality in the system.

An equation can often be solved by successively deducing from it one or more simpler equations. For example, one can add the same constant to both sides without changing the solutions, but squaring both sides might lead to extraneous solutions. Strategic competence in solving includes looking ahead for productive manipulations and anticipating the nature and number of solutions.

Some equations have no solutions in a given number system, but have a solution in a larger system. For example, the solution of x + 1 = 0 is an integer, not a whole number; the solution of 2x + 1 = 0 is a rational number, not an integer; the solutions of  – 2 = 0 are real numbers, not rational numbers; and the solutions of  + 2 = 0 are complex numbers, not real numbers.

The same solution techniques used to solve equations can be used to rearrange formulas. For example, the formula for the area of a trapezoid, A = ((b1+b2)/2)h, can be solved for h using the same deductive process. Inequalities can be solved by reasoning about the properties of inequality. Many, but not all, of the properties of equality continue to hold for inequalities and can be useful in solving them.

Connections to Functions and Modeling. Expressions can define functions, and equivalent expressions define the same function. Asking when two functions have the same value for the same input leads to an equation; graphing the two functions allows for finding approximate solutions of the equation. Converting a verbal description to an equation, inequality, or system of these is an essential skill in modeling.

## Standards in this Domain

• MAT-HS.A-REI.01 - 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.
• MAT-HS.A-REI.02 - Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
• MAT-HS.A-REI.03 - Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
• MAT-HS.A-REI.04 - Solve quadratic equations in one variable.
• MAT-HS.A-REI.05 - 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.
• MAT-HS.A-REI.06 - Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
• MAT-HS.A-REI.07 - 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  +  = 3.
• MAT-HS.A-REI.08 - Represent a system of linear equations as a single matrix equation in a vector variable.
• MAT-HS.A-REI.09 - 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).
• MAT-HS.A-REI.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).
• MAT-HS.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.
• MAT-HS.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.

#### MAT-HS.A-REI.01

 MAT-HS Targeted Standards(A) Concept: Algebra(REI) Domain: Reasoning with Equations and InequalitiesCluster: Understand solving equations as a process of reasoning and explain the reasoning MAT-HS.A-REI.01 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.

## Student Learning Targets:

### Knowledge Targets

• I can understand the different methods to find a solution.
• I can identify if it is a correct solution to a problem.

### Reasoning Targets

• I can justify the solution.
• I can interpret the process used to find a solution.
• I can construct a convincing argument that justifies each step in the solution process assuming an equation has a solution.

### Skills (Performance) Targets

• I can demonstrate the process used to find the correct solution.

• I can
• I can

## Proficiency Scale

 Score Description Sample Activity 4.0 In addition to Score 3.0, the student demonstrates in-depth inferences and applications regarding more complex material that go beyond end of instruction expectations. - 3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success. 3.0 “The Standard.” The student demonstrates no major errors or omissions regarding any of the information and processes that were end of instruction expectations. - 2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content). 2.0 The student demonstrates no major errors or omissions regarding the simpler details and processes but exhibits major errors or omissions regarding the more complex ideas and processes (Score 3.0 content). - 1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content). 1.0 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) and some of the more complex ideas and processes (Score 3.0 content). - 0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content). 0.0 Even with help, the student demonstrates no understanding or skill. -

## Resources

 Web Vocab

#### MAT-HS.A-REI.02

 MAT-HS Targeted Standards(A) Concept: Algebra(REI) Domain: Reasoning with Equations and InequalitiesCluster: Understand solving equations as a process of reasoning and explain the reasoning MAT-HS.A-REI.02 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

## Student Learning Targets:

• I can
• I can

• I can
• I can

### Skills (Performance) Targets

• I can solve simple rational and radical equations in one variable and provide examples of how extraneous solutions arise.

• I can
• I can

## Alg II Solve Radical Equations Proficiency Scale

 Score Description Sample Activity 4.0 (advanced) In addition to Score 3.0, the student demonstrates in-depth inferences and applications regarding more complex material that go beyond end of instruction expectations. Solve the following equation: 3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success. 3.0 (proficient) The student can: solve a radical equation and identify an extraneous solution. provide examples of how extraneous solutions to a radical equation arise. The student exhibits no major errors or omissions. Solve the following equations: 2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content). 2.0 (progressing) There are no major errors or omissions regarding the simpler details and processes as the student can: recognize and recall specific terminology such as: radical radicand index inverse operations extraneous solution identify an extraneous solution to a radical equation when given possible solutions.      However, the student exhibits major errors or omissions regarding the more complex ideas and processes. Solve the following equations: 1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content). 1.0 (beginning) With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) and some of the more complex ideas and processes (Score 3.0 content). 0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content). 0.0 Even with help, the student demonstrates no understanding or skill.

## Alg II Solve Rational Equations Proficiency Scale

Score   Description Sample Activity

4.0

In addition to Score 3.0, the student demonstrates in-depth inferences and applications regarding more complex material that go beyond end of instruction expectations.

 Describe the error in the following work:

3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success.

3.0

(proficient)

The student can:
• solve a rational equation and recognize an extraneous solution.
The student exhibits no major errors or omissions.

Solve the following rational equation for r  and check for extraneous solutions:

2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content).

2.0

(progressing)

There are no major errors or omissions regarding the simpler details and processes as the student can:
• recognize and recall specific terminology such as:
• rational equation
• denominator
• extraneous solution
• solve simple rational equations.
However, the student exhibits major errors or omissions regarding the more complex ideas and processes.

Solve the following rational equation for n:

1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content).

1.0

(beginning)

With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) and some of the more complex ideas and processes (Score 3.0 content).
0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content).
0.0 Even with help, the student demonstrates no understanding or skill.

## Resources

 Web Vocab

#### MAT-HS.A-REI.03

 MAT-HS Targeted Standards(A) Concept: Algebra(REI) Domain: Reasoning with Equations and InequalitiesCluster: Understand solving equations as a process of reasoning and explain the reasoning MAT-HS.A-REI.03 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. .

## Student Learning Targets:

• I can
• I can

• I can
• I can

### Skills (Performance) Targets

• I can solve for (isolate) a specific variable.
• I can perform transformations that produce equivalent equations (e.g., adding the same amount to both sides of the equation, etc.).
• I can check the solution.
• I can set a polynomial equal to zero and solve by factoring, quadratic formula, graphing, linear combinations, and substitution.
• I can solve for (isolate) a specific variable.
• I can write a linear system.
• I can perform transformations that produce equivalent equations (e.g., adding the same amount to both sides of the equation, etc.)
• I can check the solution.
• I can solve a quadratic.
• I can multiply and factor polynomials.
• I can solve a linear inequality.
• I can graph a linear inequality.
• I can solve a system of linear inequalities.
• I can graph a system of linear inequalities.
• I can check the solution of the system.
• I can solve a linear system (substitution, linear combination, graphically).
• I can write a linear system.
• I can identify the number of solutions of a linear system.
• I can identify the most efficient method to solve a system.
• I can check the solution of the system.
• I can solve linear equations in one variable, including equations with coefficients represented by letters**.
• I can solve linear inequalities in one variable, including inequalities with coefficients represented by letters**.

• I can
• I can

## Proficiency Scale

 Score Description Sample Activity 4.0 In addition to achieving level 3.0 content, the student makes in-depth inferences and applications that go beyond what was taught |3c+4|=6       |x+3|+2<5 3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success. 3.0 The student can: o solves linear equations with one variable:     -Additive Inverse     -Multiplicative Inverse     -Distributive Property     -Combine like terms     -Proportions **including special cases of no solution, infinitely many solutions o solve and graph the solution set of linear inequalities with one variable:     -Apply the properties of linear equation     -Multiplicative Inverse of negative coefficients The student exhibits no major errors or omissions. o  3(h-4)=-1224-6h+2h o  52c-3=37c+4 o  -67y-642 o  Solve for x: ax+bc=d o  -4 13 2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content). 2.0 There are no major errors or omissions regarding the simpler details and processes as the student: o  demonstrates basics processes (HS.A-REI.01)     -one-step equations/inequalities     -two-step equations/inequalities However, the student exhibits major errors or omissions regarding the more complex ideas and processes. o  3x=14 o  -3x+5-2x=10 o  x2=83 o  4x-17 o  Solve for x: x+y=c 1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content). 1.0 With help, a partial understanding of some of the simpler details and processes and some of the more complex ideas and processes. - 0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content).

## Resources

 Web Vocab

#### MAT-HS.A-REI.03a

 MAT-HS Targeted Standards(A) Concept: Algebra(REI) Domain: Reasoning with Equations and InequalitiesCluster: Understand solving equations as a process of reasoning and explain the reasoning MAT-HS.A-REI.03a Solve linear equations in one variable, including equations with coefficients represented by letters. .

## Student Learning Targets:

• I can
• I can

• I can
• I can

### Skills (Performance) Targets

• I can solve for (isolate) a specific variable.
• I can perform transformations that produce equivalent equations (e.g., adding the same amount to both sides of the equation, etc.).
• I can check the solution.
• I can set a polynomial equal to zero and solve by factoring, quadratic formula, graphing, linear combinations, and substitution.
• I can solve for (isolate) a specific variable.
• I can write a linear system.
• I can perform transformations that produce equivalent equations (e.g., adding the same amount to both sides of the equation, etc.)
• I can check the solution.
• I can solve a quadratic.
• I can multiply and factor polynomials.
• I can solve a linear inequality.
• I can graph a linear inequality.
• I can solve a system of linear inequalities.
• I can graph a system of linear inequalities.
• I can check the solution of the system.
• I can solve a linear system (substitution, linear combination, graphically).
• I can write a linear system.
• I can identify the number of solutions of a linear system.
• I can identify the most efficient method to solve a system.
• I can check the solution of the system.
• I can solve linear equations in one variable, including equations with coefficients represented by letters**.
• I can solve linear inequalities in one variable, including inequalities with coefficients represented by letters**.

• I can
• I can

## Proficiency Scale

 Score Description Sample Activity 4.0 In addition to achieving level 3.0 content, the student makes in-depth inferences and applications that go beyond what was taught |3c+4|=6       |x+3|+2<5 3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success. 3.0 The student can: o solves linear equations with one variable:     -Additive Inverse     -Multiplicative Inverse     -Distributive Property     -Combine like terms     -Proportions **including special cases of no solution, infinitely many solutions o solve and graph the solution set of linear inequalities with one variable:     -Apply the properties of linear equation     -Multiplicative Inverse of negative coefficients The student exhibits no major errors or omissions. o  3(h-4)=-1224-6h+2h o  52c-3=37c+4 o  -67y-642 o  Solve for x: ax+bc=d o  -4 13 2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content). 2.0 There are no major errors or omissions regarding the simpler details and processes as the student: o  demonstrates basics processes (HS.A-REI.01)     -one-step equations/inequalities     -two-step equations/inequalities However, the student exhibits major errors or omissions regarding the more complex ideas and processes. o  3x=14 o  -3x+5-2x=10 o  x2=83 o  4x-17 o  Solve for x: x+y=c 1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content). 1.0 With help, a partial understanding of some of the simpler details and processes and some of the more complex ideas and processes. - 0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content).

## Resources

 Web Vocab

#### MAT-HS.A-REI.03b

 MAT-HS Targeted Standards(A) Concept: Algebra(REI) Domain: Reasoning with Equations and InequalitiesCluster: Understand solving equations as a process of reasoning and explain the reasoning MAT-HS.A-REI.03b Solve linear inequalities in one variable, including equations with coefficients represented by letters. .

## Student Learning Targets:

• I can
• I can

• I can
• I can

### Skills (Performance) Targets

• I can solve for (isolate) a specific variable.
• I can perform transformations that produce equivalent equations (e.g., adding the same amount to both sides of the equation, etc.).
• I can check the solution.
• I can set a polynomial equal to zero and solve by factoring, quadratic formula, graphing, linear combinations, and substitution.
• I can solve for (isolate) a specific variable.
• I can write a linear system.
• I can perform transformations that produce equivalent equations (e.g., adding the same amount to both sides of the equation, etc.)
• I can check the solution.
• I can solve a quadratic.
• I can multiply and factor polynomials.
• I can solve a linear inequality.
• I can graph a linear inequality.
• I can solve a system of linear inequalities.
• I can graph a system of linear inequalities.
• I can check the solution of the system.
• I can solve a linear system (substitution, linear combination, graphically).
• I can write a linear system.
• I can identify the number of solutions of a linear system.
• I can identify the most efficient method to solve a system.
• I can check the solution of the system.
• I can solve linear equations in one variable, including equations with coefficients represented by letters**.
• I can solve linear inequalities in one variable, including inequalities with coefficients represented by letters**.

• I can
• I can

## Proficiency Scale

 Score Description Sample Activity 4.0 In addition to achieving level 3.0 content, the student makes in-depth inferences and applications that go beyond what was taught |3c+4|=6       |x+3|+2<5 3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success. 3.0 The student can: o solves linear equations with one variable:     -Additive Inverse     -Multiplicative Inverse     -Distributive Property     -Combine like terms     -Proportions **including special cases of no solution, infinitely many solutions o solve and graph the solution set of linear inequalities with one variable:     -Apply the properties of linear equation     -Multiplicative Inverse of negative coefficients The student exhibits no major errors or omissions. o  3(h-4)=-1224-6h+2h o  52c-3=37c+4 o  -67y-642 o  Solve for x: ax+bc=d o  -4 13 2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content). 2.0 There are no major errors or omissions regarding the simpler details and processes as the student: o  demonstrates basics processes (HS.A-REI.01)     -one-step equations/inequalities     -two-step equations/inequalities However, the student exhibits major errors or omissions regarding the more complex ideas and processes. o  3x=14 o  -3x+5-2x=10 o  x2=83 o  4x-17 o  Solve for x: x+y=c 1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content). 1.0 With help, a partial understanding of some of the simpler details and processes and some of the more complex ideas and processes. - 0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content).

## Resources

 Web Vocab

#### MAT-HS.A-REI.04

 MAT-HS Targeted Standards(A) Concept: Algebra(REI) Domain: Reasoning with Equations and InequalitiesCluster: Understand solving equations as a process of reasoning and explain the reasoning MAT-HS.A-REI.04  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.

## Student Learning Targets:

• I can
• I can

### Reasoning Targets

• I can explain why taking the square root of both sides of an equation can yield two solutions.

### Skills (Performance) Targets

• I can solve for (isolate) a specific variable.
• I can perform transformations that produce equivalent equations (e.g., adding the same amount to both sides of the equation, etc.).
• I can check the solution.
• I can solve for (isolate) a specific variable.
• I can solve a quadratic.
• I can multiply and factor polynomials.
• I can solve quadratic equations in one variable.
• I can transform a quadratic equation to an equation in the form (x-p)2=q by completing the square.
• I can derive the quadratic formula by completing the square on a quadratic equation.
• I can solve quadratic equations in one variable by simple inspection, taking the square root, factoring, and completing the square.
• I can use the quadratic formula to solve any quadratic equation, recognizing the formula produces all complex solutions and write the solutions in the form a ± bi , where a and b are real numbers.

• I can
• I can

## Algebra II Solving Quadratic Equations Proficiency Scale

 Score Description Sample Activity 4.0 In addition to Score 3.0, the student demonstrates in-depth inferences and applications regarding more complex material that go beyond end of instruction expectations. Suppose the quadratic functions f(x)=ax2+bx+c with f(-2)=0 and -b/(2a) =1. Solve f(x)=0. 3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success. 3.0 The student can: solve a quadratic equation using any method. Solve the following quadratic equation using any method:     x2+4x+15=0 2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content). 2.0 There are no major errors or omissions regarding the simpler details and processes as the student can: solve simple quadratic equations. recognize and recalls specific terminology, such as: quadratic equation factor complete the square quadratic formula square root imaginary/complex numbers Solve the following quadratic equations: x2+100=0 Perform the given operation: (3+i)(-6-i) 1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content). 1.0 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) and some of the more complex ideas and processes (Score 3.0 content). - 0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content). 0.0 Even with help, the student demonstrates no understanding or skill. -

## Resources

 Web Vocab completing the square quadratic formula

#### MAT-HS.A-REI.04.b

 MAT-HS Targeted Standards(A) Concept: Algebra(REI) Domain: Reasoning with Equations and InequalitiesCluster: Understand solving equations as a process of reasoning and explain the reasoning MAT-HS.A-REI.04.b 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.

## Student Learning Targets:

• I can
• I can

### Reasoning Targets

• I can explain why taking the square root of both sides of an equation can yield two solutions.

### Skills (Performance) Targets

• I can solve quadratic equations by using square roots.
• I can solve quadratic equations by factoring.
• I can solve quadratic equations by completing the square.
• I can find the number of solutions of a quadratic equation.

• I can
• I can

## Proficiency Scale

 Score Description Sample Activity 4.0 In addition to achieving level 3.0 content, the student makes in-depth inferences and applications that go beyond what was taught Use the method of completing the square to derive the quadratic formula. Solve a quadratic equation that has a complex solution and write in the form a + bi 3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success. 3.0 The student can: Solve quadratic equations by: using square roots factoring using the quadratic formula (include equations with zero, one, or two real-solutions) The student exhibits no major errors or omissions. Solve the quadratic equation by finding square roots.            3x2 - 12 = 0 Solve the quadratic equation by factoring.         x2 - 14x + 45 = 0 Use the Quadratic formula to solve the equation.           5x2 - 13x  =  -9 2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content). 2.0 There are no major errors or omissions regarding the simpler details and processes as the student: However, the student exhibits major errors or omissions regarding the more complex ideas and processes. x2 = 49 (4t + 1)(t - 2) = 0 Identify a, b, and c as it’s used in the quadratic formula x2 + 6x = - 5 1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content). 1.0 With help, a partial understanding of some of the simpler details and processes and some of the more complex ideas and processes. - 0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content).

## Resources

 Web Vocab completing the square quadratic formula

#### MAT-HS.A-REI.05

 MAT-HS Targeted Standards(A) Concept: Algebra(REI) Domain: Reasoning with Equations and InequalitiesCluster: Solve systems of equations MAT-HS.A-REI.05 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.

## Student Learning Targets:

• I can
• I can

• I can
• I can

### Skills (Performance) Targets

• I can solve for (isolate) a specific variable.
• I can perform transformations that produce equivalent equations (e.g., adding the same amount to both sides of the equation, etc.).
• I can check the solution.
• I can set a polynomial equal to zero and solve by factoring, quadratic formula, graphing, linear combinations, and substitution.
• I can solve for (isolate) a specific variable.
• I can write a linear system.
• I can perform transformations that produce equivalent equations (e.g., adding the same amount to both sides of the equation, etc.)
• I can check the solution.
• I can solve a quadratic.
• I can multiply and factor polynomials.
• I can solve a linear inequality.
• I can graph a linear inequality.
• I can solve a system of linear inequalities.
• I can graph a system of linear inequalities.
• I can check the solution of the system.
• I can solve a linear system (substitution, linear combination, graphically).
• I can write a linear system.
• I can identify the number of solutions of a linear system.
• I can identify the most efficient method to solve a system.
• I can check the solution of the system.
• I can produce, with justification, from a system of two equations an equivalent simpler system.

• I can
• I can

## Proficiency Scale

 Score Description Sample Activity 4.0 In addition to Score 3.0, the student demonstrates in-depth inferences and applications regarding more complex material that go beyond end of instruction expectations. - 3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success. 3.0 “The Standard.” The student demonstrates no major errors or omissions regarding any of the information and processes that were end of instruction expectations. - 2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content). 2.0 The student demonstrates no major errors or omissions regarding the simpler details and processes but exhibits major errors or omissions regarding the more complex ideas and processes (Score 3.0 content). - 1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content). 1.0 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) and some of the more complex ideas and processes (Score 3.0 content). - 0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content). 0.0 Even with help, the student demonstrates no understanding or skill. -

## Resources

 Web Vocab

#### MAT-HS.A-REI.06

 MAT-HS Targeted Standards(A) Concept: Algebra(REI) Domain: Reasoning with Equations and InequalitiesCluster: Solve systems of equations MAT-HS.A-REI.06 Solve systems of linear equations exactly and approximately, focusing on pairs of linear equations in two variables.

## Student Learning Targets:

• I can
• I can

• I can
• I can

### Skills (Performance) Targets

• I can solve systems of equations using graphing (exactly and approximating).
• I can solve systems of equations using substitution.
• I can solve systems of equations using linear combination (or elimination).
• I can explain when there is one solution, no solution, or infinitely many solutions that may arise from a system.

• I can
• I can

## Proficiency Scale

 Score Description Sample Activity 4.0 (advanced) In addition to achieving level 3.0 content, the student makes in-depth inferences and applications that go beyond what was taught Solve a  3x3 system of equations 2x-y+5z=16x-3y+2z=5x+2y+z=7 3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success. 3.0 (proficient) The student can: solve systems of equations using graphing (exactly and approximating) solve systems of equations using substitution solve systems of equations using linear combination (or elimination) explain when there is one solution, no solution, or infinitely many solutions that may arise from a system. The student exhibits no major errors or omissions. Solve the following system of equations using your preferred method. x – 2y = –9 x + 3y = 16 Explain when one solution, no solutions, and infinitely many solutions will arise. 2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content). 2.0 (progressing) There are no major errors or omissions regarding the simpler details and processes as the student can: recognize and recall basic terminology. such as: linear combination (elimination) substitution solution infinitely many solutions no solution system of equations ordered pair determine if an ordered pair is a solution to a system. approximate a solution to a system of linear equations that are graphed. However, the student exhibits major errors or omissions regarding the more complex ideas and processes. 1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content). 1.0 (beginning) With help, a partial understanding of some of the simpler details and processes and some of the more complex ideas and processes. - 0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content).

## Resources

 Web Vocab linear combination elimination method substitution method

#### MAT-HS.A-REI.07

 MAT-HS Targeted Standards(A) Concept: Algebra(REI) Domain: Reasoning with Equations and InequalitiesCluster: Solve systems of equations MAT-HS.A-REI.07 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.

## Student Learning Targets:

• I can
• I can

• I can
• I can

### Skills (Performance) Targets

• I can solve for (isolate) a specific variable.
• I can perform transformations that produce equivalent equations (e.g., adding the same amount to both sides of the equation, etc.).
• I can check the solution.
• I can set a polynomial equal to zero and solve by factoring, quadratic formula, graphing, linear combinations, and substitution.
• I can solve for (isolate) a specific variable.
• I can write a linear system.
• I can perform transformations that produce equivalent equations (e.g., adding the same amount to both sides of the equation, etc.)
• I can check the solution.
• I can solve a quadratic.
• I can multiply and factor polynomials.
• I can solve a linear inequality.
• I can graph a linear inequality.
• I can solve a system of linear inequalities.
• I can graph a system of linear inequalities.
• I can check the solution of the system.
• I can solve a linear system (substitution, linear combination, graphically).
• I can write a linear system.
• I can identify the number of solutions of a linear system.
• I can identify the most efficient method to solve a system.
• I can check the solution of the system.
• I can solve a system containing a linear equation and a quadratic equation in two variables algebraically and graphically.

• I can
• I can

## Proficiency Scale

 Score Description Sample Activity 4.0 In addition to Score 3.0, the student demonstrates in-depth inferences and applications regarding more complex material that go beyond end of instruction expectations. - 3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success. 3.0 “The Standard.” The student demonstrates no major errors or omissions regarding any of the information and processes that were end of instruction expectations. - 2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content). 2.0 The student demonstrates no major errors or omissions regarding the simpler details and processes but exhibits major errors or omissions regarding the more complex ideas and processes (Score 3.0 content). - 1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content). 1.0 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) and some of the more complex ideas and processes (Score 3.0 content). - 0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content). 0.0 Even with help, the student demonstrates no understanding or skill. -

## Resources

 Web Vocab

#### MAT-HS.A-REI.08

 MAT-HS Targeted Standards(A) Concept: Algebra(REI) Domain: Reasoning with Equations and InequalitiesCluster: Solve systems of equations MAT-HS.A-REI.08 Represent a system of linear equations as a single matrix equation in a vector variable.

• I can
• I can

• I can
• I can

• I can
• I can

• I can
• I can

## Proficiency Scale

 Score Description Sample Activity 4.0 In addition to Score 3.0, the student demonstrates in-depth inferences and applications regarding more complex material that go beyond end of instruction expectations. - 3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success. 3.0 “The Standard.” The student demonstrates no major errors or omissions regarding any of the information and processes that were end of instruction expectations. - 2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content). 2.0 The student demonstrates no major errors or omissions regarding the simpler details and processes but exhibits major errors or omissions regarding the more complex ideas and processes (Score 3.0 content). - 1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content). 1.0 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) and some of the more complex ideas and processes (Score 3.0 content). - 0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content). 0.0 Even with help, the student demonstrates no understanding or skill. -

## Resources

 Web Vocab

#### MAT-HS.A-REI.09

 MAT-HS Targeted Standards(A) Concept: Algebra(REI) Domain: Reasoning with Equations and InequalitiesCluster: Solve systems of equations MAT-HS.A-REI.09 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).

• I can
• I can

• I can
• I can

• I can
• I can

• I can
• I can

## Proficiency Scale

 Score Description Sample Activity 4.0 In addition to Score 3.0, the student demonstrates in-depth inferences and applications regarding more complex material that go beyond end of instruction expectations. - 3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success. 3.0 “The Standard.” The student demonstrates no major errors or omissions regarding any of the information and processes that were end of instruction expectations. - 2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content). 2.0 The student demonstrates no major errors or omissions regarding the simpler details and processes but exhibits major errors or omissions regarding the more complex ideas and processes (Score 3.0 content). - 1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content). 1.0 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) and some of the more complex ideas and processes (Score 3.0 content). - 0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content). 0.0 Even with help, the student demonstrates no understanding or skill. -

## Resources

 Web Vocab

#### MAT-HS.A-REI.10

 MAT-HS Targeted Standards(A) Concept: Algebra(REI) Domain: Reasoning with Equations and InequalitiesCluster: Represent and solve equations and inequalities graphically MAT-HS.A-REI.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).

## Student Learning Targets:

• I can
• I can

• I can
• I can

### Skills (Performance) Targets

• I can find any solution to an equation in two variables from the graph of that equation.
• I can graph the solution(s) of any linear or exponential function.
• I can verify an ordered pair is a solution to the equation.

• I can
• I can

## Proficiency Scale

 Score Description Sample Activity 4.0 In addition to achieving level 3.0 content, the student makes in-depth inferences and applications that go beyond what was taught Given a quadratic function, identify two solutions. Given more than one function on a graph, determine the solution of the system. 3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success. 3.0 The student can: Find any solution to an equation in two variables from the graph of that equation. Graph the solution(s) of any linear or exponential function. Verify an ordered pair is a solution to the equation. The student exhibits no major errors or omissions. Given the following function:  y=3x+2,  create a graph on a coordinate plane to show all solutions of the function. Determine if the given point (4, ½ ) is a solution to the       function y=212x Given the following function, identify three solutions. 2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content). 2.0 There are no major errors or omissions regarding the simpler details and processes as the student can: Recognize and recall basic vocabulary such as: solution ordered pair coordinate plane Graph a function, given a table of solutions or technology. Evaluate using order of operations. Plot ordered pairs on a coordinate plane. However, the student exhibits major errors or omissions regarding the more complex ideas and processes. - 1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content). 1.0 With help, a partial understanding of some of the simpler details and processes and some of the more complex ideas and processes. - 0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content).

## Resources

 Web Vocab

#### MAT-HS.A-REI.11

 MAT-HS Targeted Standards(A) Concept: Algebra(REI) Domain: Reasoning with Equations and InequalitiesCluster: Represent and solve equations and inequalities graphically MAT-HS.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. Note: In Algebra I, only linear and exponential will be the focus.

## Student Learning Targets:

• I can
• I can

### Reasoning Targets

• I can explain why the intersection of y = f(x) and y = g(x) is the solution of f(x) = g(x) for any combination of linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

### Skills (Performance) Targets

• I can use technology to graph the equations and find their points of intersection.
• I can use tables of values or successive approximations to find solutions.

• I can
• I can

## Proficiency Scale

 Score Description Sample Activity 4.0 In addition to achieving level 3.0 content, the student makes in-depth inferences and applications that go beyond what was taught On the graphs of f(x) and g(x), interpret where f(x)>g(x)or where f(x)

## Resources

 Web Vocab intersection

#### MAT-HS.A-REI.12

 MAT-HS Targeted Standards(A) Concept: Algebra(REI) Domain: Reasoning with Equations and InequalitiesCluster: Represent and solve equations and inequalities graphically MAT-HS.A-REI.12 Graph the solutions to a linear inequality in two variables as a half-plane. Graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

## Student Learning Targets:

• I can
• I can

• I can
• I can

### Skills (Performance) Targets

• I can graph the solutions to a linear inequality in two variables, including dashed or solid boundary lines when appropriate.
• I can graph the solution set to a system of linear inequalities in two variables as their corresponding intersection.

• I can
• I can

## Proficiency Scale

 Score Description Sample Activity 4.0(advanced) In addition to achieving level 3.0 content, the student makes in-depth inferences and applications that go beyond what was taught 3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success. 3.0(proficient) The student can: graph the solutions to a linear inequality in two variables, including dashed or solid boundary lines when appropriate. graph the solution set to a system of linear inequalities in two variables as their corresponding intersection. The student exhibits no major errors or omissions. 2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content). 2.0(progressing) There are no major errors or omissions regarding the simpler details and processes as the student can: recognize and recall basic vocabulary, such as: linear inequality. linear equation. understand when to use a solid line vs a dashed line. understand that there is a region of solutions and how to identify it. graph a linear equation as the boundary line. solve for Y in a linear equation or inequality. However, the student exhibits major errors or omissions regarding the more complex ideas and processes. 1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content). 1.0(beginning) With help, a partial understanding of some of the simpler details and processes and some of the more complex ideas and processes. - 0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content).

## Resources

 Web Vocab half-plane

#### MAT-HS.A-SSE

Domains are larger groups of related standards. So the Domain Score is a calculation of all the related standards. So click on the standard name below each Domain to access the learning targets and proficiency scales for each Domain's related standards.

# Seeing Structure in Expressions

• Interpret the structure of expressions
• Write expressions in equivalent forms to solve problems

## Domain Description

An expression is a record of a computation with numbers, symbols that represent numbers, arithmetic operations, exponentiation, and, at more advanced levels, the operation of evaluating a function. Conventions about the use of parentheses and the order of operations assure that each expression is unambiguous. Creating an expression that describes a computation involving a general quantity requires the ability to express the computation in general terms, abstracting from specific instances.

Reading an expression with comprehension involves analysis of its underlying structure. This may suggest a different but equivalent way of writing the expression that exhibits some different aspect of its meaning. For example, p + 0.05p can be interpreted as the addition of a 5% tax to a price p. Rewriting p + 0.05p as 1.05p shows that adding a tax is the same as multiplying the price by a constant factor.

Algebraic manipulations are governed by the properties of operations and exponents, and the conventions of algebraic notation. At times, an expression is the result of applying operations to simpler expressions. For example, p + 0.05p is the sum of the simpler expressions p and 0.05p. Viewing an expression as the result of operation on simpler expressions can sometimes clarify its underlying structure.

A spreadsheet or a computer algebra system (CAS) can be used to experiment with algebraic expressions, perform complicated algebraic manipulations, and understand how algebraic manipulations behave.

## Standards in this Domain

• MAT-HS.A-SSE.01 - Interpret expressions that represent a quantity in terms of its context.
• MAT-HS.A-SSE.02 - Use the structure of an expression to identify ways to rewrite it. For example, see x4 - y4 as (x2)2 - (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 - y2)(x2 + y2).
• MAT-HS.A-SSE.03 - Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
• MAT-HS.A-SSE.04 - Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.

#### MAT-HS.A-SSE.01

 MAT-HS Targeted Standards(A) Concept: Algebra(SSE) Domain: Seeing Structure in ExpressionsCluster: Interpret the structure of expressions MAT-HS.A-SSE.01 Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression,such as terms, factors, and coefficients. b. Interpret complicated expressions by viewing one or more of their parts as a single entity.For example, interpret P(1+r)n as the product of P and a factor not depending on P.

## Student Learning Targets:

### Knowledge Targets

• I can identify the different parts of an expression and explain their meaning within the context of a problem.

• I can
• I can

### Skills (Performance) Targets

• I can use algebraic expressions, equations or inequalities involving one or two variables to represent geometric relationships.
• I can interpret expressions that represent a quantity in terms of its context.
• I can interpret expressions and make sense of the multiple factors and terms by explaining the meaning of the individual parts.

• I can
• I can

## Proficiency Scale:

 Score Description Sample Activity 4.0 In addition to Score 3.0, the student demonstrates in-depth inferences and applications regarding more complex material that go beyond end of instruction expectations. - 3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success. 3.0 “The Standard.” The student demonstrates no major errors or omissions regarding any of the information and processes that were end of instruction expectations. - 2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content). 2.0 The student demonstrates no major errors or omissions regarding the simpler details and processes but exhibits major errors or omissions regarding the more complex ideas and processes (Score 3.0 content). - 1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content). 1.0 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) and some of the more complex ideas and processes (Score 3.0 content). - 0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content). 0.0 Even with help, the student demonstrates no understanding or skill. -

## Resources:

### Web:

What is a variable?

### Vocabulary:

 expression term factor coefficient

#### MAT-HS.A-SSE.02

 MAT-HS Targeted Standards(A) Concept: Algebra(SSE) Domain: Seeing Structure in ExpressionsCluster: Interpret the structure of expressions MAT-HS.A-SSE.02 Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²).

## Student Learning Targets:

### Knowledge Targets

• I can recall the order of operations.
• I can understand and identify a power and its parts (base and exponent).
• I know the properties of exponents (i.e. product of powers, power of a power, power of a product).
• I can recognize a zero or negative exponent.
• I can recall the order of operations.
• I can understand and identify a power and its parts (base and exponent).
• I know the properties of exponents (i.e. product of powers, power of a power, power of a product).
• I can recognize a zero or negative exponent.

### Reasoning Targets

• I can analyze an algebraic expression.
• I can decide what to do next based on my knowledge of the order of operations.
• I can analyze an algebraic expression.
• I can decide what to do next based on my knowledge of the order of operations.

### Skills (Performance) Targets

• I can factor algebraic expressions.
• I can combine like terms to simplify an algebraic expression.
• I can perform basic operations.
• I can simplify algebraic expressions.
• I can apply order of operations.
• I can simplify an algebraic expression.
• I can rewrite algebraic expressions in equivalent forms such as factoring or combining like terms.
• I can use factoring techniques such as common factors, grouping, the difference of two squares, the sum or difference of two cubes, or a combination of methods to factor an expression completely.
• I can simplify expressions by combining like terms, using the distributive property and using other operations with polynomials.

• I can

## Proficiency Scale

 Score Description Sample Activity 4.0 In addition to achieving level 3.0 content, the student makes in-depth inferences and applications that go beyond what was taught. Factor the following algebraic expressions. x3+2x2-4x-8 6x3-14x2-12x 3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success. 3.0 The student: uses the structure of an expression to identify ways to rewrite it, using the following techniques to factor: factor by grouping. difference of perfect squares. greatest common factor. factor a trinomial. The student exhibits no major errors or omissions. Factor the following algebraic expressions. 4x2-8x+3 2x2-15x+18 3x2-5x-2 x3+8x2+6x+48 9x2-49 2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content). 2.0 There are no major errors or omissions regarding the simpler details and processes as the student: combines like terms to simplify an expression. (A.SSE.1) & (A.APR.1) uses the following techniques to factor: greatest common factor. factor a trinomial with a leading coefficient of 1. However, the student exhibits major errors or omissions regarding the more complex ideas and processes. Simplify the following algebraic expressions. 3x2+2x-6x2+4y+6+2x Factor the following algebraic expressions. 6x2+3x x2+4x+3 x2-3x-4 1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content). 1.0 With help, a partial understanding of some of the simpler details and processes and some of the more complex ideas and processes. - 0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content).

## Alg II Equivalent Expressions Proficiency Scale

 Score Description Sample Activity 4.0 (advanced) In addition to Score 3.0, the student demonstrates in-depth inferences and applications regarding more complex material that go beyond end of instruction expectations. Describe the error in each of the following: 3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success. 3.0 (proficient) The student can: recognize a suitable factoring technique given the structure of the expression. use factoring techniques such as common factors, grouping, the difference of two squares, the sum or difference of two cubes, or a combination of methods to factor an expression completely. rewrite algebraic expressions in equivalent forms such as factoring or combining like terms.   The student exhibits no major errors or omissions. Factor: x3+3x2-4x-12Factor and Simplify: 2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content). 2.0 (progressing) There are no major errors or omissions regarding the simpler details and processes as the student can: simplify expressions by combining like terms, using the distributive property and using other operation with polynomials. understand that like terms can be combined using the distributive property and using other operations with polynomials. understand factoring techniques such as common factors, grouping, the difference of two squares, the sum or difference of two cubes, or a combination of methods to factor an expression completely.   recognize equivalent forms of the same expression.   However, the student exhibits major errors or omissions regarding the more complex ideas and processes. Factor: x2+5x-6  Factor and Simplify: 1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content). 1.0 (beginning) With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) and some of the more complex ideas and processes (Score 3.0 content). 0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content). 0.0 Even with help, the student demonstrates no understanding or skill.

## Resources

 Web Vocab

#### MAT-HS.A-SSE.03

 MAT-HS Targeted Standards(A) Concept: Algebra(SSE) Domain: Seeing Structure in ExpressionsCluster: Write expressions in equivalent forms to solve problems MAT-HS.A-SSE.03 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. a. Factor a quadratic expression to reveal the zeros of the function it defines. See: MAT-HS.A-SSE.03a b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. c. Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.

## Student Learning Targets:

### Knowledge Targets

• I can recall the order of operations.
• I can understand and identify a power and its parts (base and exponent).
• I can recognize the properties of exponents (i.e. product of powers, power of a power, power of a product).
• I can recognize a zero or negative exponent.
• I can identify a radical.
• I can identify if a radical expression is in simplest form.
• I can recognize the properties of radicals, i.e., product, quotient, and sum of radicals.
• I can identify perfect squares.
• I can define a logarithm.
• I can recall properties of logarithms and exponents.
• I can recognize common logarithms and natural logarithms.
• I can identify an asymptote.
• I can recall the change of base formula.

### Reasoning Targets

• I can apply the distributive property.
• I can analyze what type of factoring technique that should be used for a given expression. (i.e. factor trinomials, factor difference of squares, factor sum and difference of cubes, factor by grouping, factor out the greatest common factor)
• I can combine like terms in an expression or equation.
• I can analyze an algebraic expression.
• I can decide what to do next based on my knowledge of the order of operations.
• I can distinguish the difference of when to use each property of radicals.
• I can evaluate the properties of radicals.
• I can convert from a logarithm to exponential expressions.
• I can identify an asymptote.
• I can expand and condense logarithms.
• I can evaluate logarithms and exponential expressions.
• I can distinguish between a natural log and common log.

### Skills (Performance) Targets

• I can simplify a polynomial expression.
• I can multiply polynomials.
• I can justify the steps taken in manipulating an expression.
• I can complete all necessary steps in order to simplify the expression.
• I can distribute and factor an expression.
• I can simplify an algebraic expression.
• I can apply factor using perfect square.
• I can do prime factorization.
• I can rationalize a denominator.
• I can solve for (isolate) a specific variable.
• I can perform transformations that produce equivalent equations (e.g., adding the same amount to both sides of the equation, etc.).
• I can check the solution.
• I can set a polynomial equal to zero and solve by factoring, quadratic formula, graphing, linear combinations, and substitution.
• I can write a linear system.
• I can perform transformations that produce equivalent equations (e.g., adding the same amount to both sides of the equation, etc.)
• I can check the solution.
• I can solve a quadratic.
• I can multiply and factor polynomials.
• I can solve a linear inequality.
• I can graph a linear inequality.
• I can solve a system of linear inequalities.
• I can graph a system of linear inequalities.
• I can check the solution of the system.
• I can solve a linear system (substitution, linear combination, graphically).
• I can write a linear system.
• I can identify the number of solutions of a linear system.
• I can identify the most efficient method to solve a system.
• I can check the solution of the system.
• I can apply basic properties of exponents and logarithms to rewrite algebraic expressions.
• I can graph logarithmic and exponential functions.
• I can solve logarithmic and exponential equations.
• I can apply the change of base formula to evaluate a logarithmic expression.
• I can apply basic properties of exponents and logarithms to rewrite algebraic expressions.
• I can graph logarithmic and exponential functions.
• I can solve logarithmic and exponential equations.
• I can apply the change of base formula to evaluate a logarithmic expression.
• I can graph and determine the equation of an asymptote.
• I can choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
• I can write expressions in equivalent forms by factoring to find the zeros of a quadratic function and explain the meaning of the zeros.
• I can complete the square in a quadratic expression to convey the vertex form and determine the maximum or minimum value of the quadratic function, and to explain the meaning of the vertex.
• I can use properties of exponents (such as power of a power, product of powers, power of a product, power of a quotient) to write an equivalent form of an exponential function to reveal and explain specific information about its approximate rate of growth or decay.

• I can
• I can

## Proficiency Scale

 Score Description Sample Activity 4.0 In addition to Score 3.0, the student demonstrates in-depth inferences and applications regarding more complex material that go beyond end of instruction expectations. - 3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success. 3.0 “The Standard.” The student demonstrates no major errors or omissions regarding any of the information and processes that were end of instruction expectations. - 2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content). 2.0 The student demonstrates no major errors or omissions regarding the simpler details and processes but exhibits major errors or omissions regarding the more complex ideas and processes (Score 3.0 content). - 1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content). 1.0 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) and some of the more complex ideas and processes (Score 3.0 content). - 0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content). 0.0 Even with help, the student demonstrates no understanding or skill. -

## Resources

 Web Vocab

#### MAT-HS.A-SSE.03.a

 MAT-HS Targeted Standards(A) Concept: Algebra(SSE) Domain: Seeing Structure in ExpressionsCluster: Write expressions in equivalent forms to solve problems MAT-HS.A-SSE.03.a Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. a. Factor a quadratic expression to reveal the zeros of the function it defines.

## Student Learning Targets:

• I can

• I can

### Skills (Performance) Targets

• I can solve a quadratic equation by factoring methods and the zero product property.

• I can
• I can

## Proficiency Scale

 Score Description Sample Activity 4.0 In addition to achieving level 3.0 content, the student makes in-depth inferences and applications that go beyond what was taught. Solve the following polynomials by factoring: 3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success. 3.0 The student: can solve a quadratic equation by factoring can use the zero product property The student exhibits no major errors or omissions. Solve the following polynomials by factoring. x2+8x+7=0 2x2-15x+18=0 3x2+2x+1=7x+3 4x2-8x+3=0 2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content). 2.0 There are no major errors or omissions regarding the simpler details and processes as the student: The student: can solve a quadratic equation using the zero product property However, the student exhibits major errors or omissions regarding the more complex ideas and processes. Solve the following polynomials by the zero product property. (x+5)(x-3)=0 7x(x-1)=0 (3x-5)(x-9)=0 1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content). 1.0 With help, a partial understanding of some of the simpler details and processes and some of the more complex ideas and processes. - 0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content).

## Resources

 Web Vocab

#### MAT-HS.A-SSE.04

 MAT-HS Targeted Standards(A) Concept: Algebra(SSE) Domain: Seeing Structure in ExpressionsCluster: Write expressions in equivalent forms to solve problems MAT-HS.A-SSE.04 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.

## Student Learning Targets:

• I can
• I can

• I can
• I can

### Skills (Performance) Targets

• I can develop the formula for the sum of a finite geometric series when the ratio is not 1.
• I can use the formula to solve real world problems.

• I can
• I can

## Proficiency Scale

 Score Description Sample Activity 4.0 In addition to Score 3.0, the student demonstrates in-depth inferences and applications regarding more complex material that go beyond end of instruction expectations. - 3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success. 3.0 “The Standard.” The student demonstrates no major errors or omissions regarding any of the information and processes that were end of instruction expectations. - 2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content). 2.0 The student demonstrates no major errors or omissions regarding the simpler details and processes but exhibits major errors or omissions regarding the more complex ideas and processes (Score 3.0 content). - 1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content). 1.0 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) and some of the more complex ideas and processes (Score 3.0 content). - 0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content). 0.0 Even with help, the student demonstrates no understanding or skill. -

## Resources

 Web Vocab

#### MAT-HS.C-DDD.00.z

 MAT-HS Targeted Standards(C) Concept: Algebra(DDD) Domain: DDDNameCluster: CCC PASTE STANDARD HERE. MAT-HS.C-DDD.00.z SUB-STANDARD HERE.

• I can
• I can

• I can
• I can

• I can
• I can

• I can
• I can

## Proficiency Scale

 Score Description Sample Activity 4.0 In addition to Score 3.0, the student demonstrates in-depth inferences and applications regarding more complex material that go beyond end of instruction expectations. - 3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success. 3.0 “The Standard.” The student demonstrates no major errors or omissions regarding any of the information and processes that were end of instruction expectations. - 2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content). 2.0 The student demonstrates no major errors or omissions regarding the simpler details and processes but exhibits major errors or omissions regarding the more complex ideas and processes (Score 3.0 content). - 1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content). 1.0 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) and some of the more complex ideas and processes (Score 3.0 content). - 0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content). 0.0 Even with help, the student demonstrates no understanding or skill. -

## Resources

 Web Vocab

### Z

#### zBLANK

 MAT-HS Targeted Standards(X) Concept: Concept(XX) Domain: DomainCluster: Cluster. MAT-HS.X-XX.XX Standard Description.

• I can
• I can

• I can
• I can

• I can
• I can

• I can
• I can

## Proficiency Scale

 Score Description Sample Activity 4.0 In addition to Score 3.0, the student demonstrates in-depth inferences and applications regarding more complex material that go beyond end of instruction expectations. - 3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success. 3.0 “The Standard.” The student demonstrates no major errors or omissions regarding any of the information and processes that were end of instruction expectations. - 2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content). 2.0 The student demonstrates no major errors or omissions regarding the simpler details and processes but exhibits major errors or omissions regarding the more complex ideas and processes (Score 3.0 content). - 1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content). 1.0 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) and some of the more complex ideas and processes (Score 3.0 content). - 0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content). 0.0 Even with help, the student demonstrates no understanding or skill. -

## Resources

 Web Vocab

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