Algebra I: Traditional Pathway Standards

• Algebra I helps students develop understanding of linear, quadratic, and exponential functions.
• Instruction focuses on students deepening their understanding of linear and exponential relationships through comparison and contrasting.
• Applying linear models to data that has a linear trend is a key component.
• Students learn analyzing, solving, and using exponential and quadratic functions is critical.
• The Algebra I course overarching elements are solving equations, function notation, rates of change, growth patterns, representation of functions, and modeling.

Traditional Pathway Standards for Algebra I

• Consists of Modeling, Functions, Number and Quantity, Algebra, and Statistics and Probability.
• Standards must be taught throughout the school year in rich instructional instances instead of isolated topics.
• Students use structure to define and make sense of rational exponents.
• Students explore the algebraic structure of the rational and real number systems.
• Numbers in real-world applications often have units attached and are considered quantities.
• Students explore algebraic expressions and polynomials, as well as, properties when working with expressions.
• Setting expressions with overlapping domains as equal yields an equation with an implied solution set (empty or non-empty).
• It refines techniques for finding an equation's solution set and explaining the algebraic steps to do so.

Linear Equations

• Exploration in middle school initially links proportional equations to real-world contexts, graphs and tables.
• It moves towards understanding general linear equations and their associated graphs.
• It extends knowledge to absolute value equations, linear inequalities, and linear equation systems.
• Students examine the functions by solving f(x) = g(x) for two linear functions f and g.
• Tables, graphs, and equations build an enhanced understanding of functions outside of linear ones.
• Connections are made between function representations, while students learn to build functions in modeling contexts.
• The focus remains on linear, exponential and quadratic equations.
• Formal ways gauge model fit and regression techniques quantify linear relationships between quantities.
• It uses graphical representation and context to analyze linear model appropriateness.
• Students analyze residuals to assess linear model fit.

Advances from earlier grades

• Having broadened arithmetic from whole numbers to fractions (grades four to six).
• Arithmetic broadens again from fractions to rational numbers (grade 7), students encounter irrational numbers like 5√ and π.
• Algebra encompasses the real number system.
• Skills about measurement unit, including multiplying and dividing quantities is extended from student's knowledge.
• Conceptual category N-Q has students applying these skills to solve problems with units adds insight.

Algebraic Thinking

• Beginning in middle school, grade six and seven students begin to use properties to generate equivalent expressions.
• By grade seven students begin rewriting expressions in different forms could be helpful, especially in problem solving.
• Algebra emphasizes continuous properties practice, while developing fluency and "mindful manipulation".

Functions

• Expanded from proportional relationships in grade eight to working with linear functions.
• Mastery of linear and quadratic functions is a must in Algebra.
• Encountering functions apply to all functions and widen quantitative relationships.

Analytical Geometry

• In grade eight, there was connecting knowledge about proportional relationships, lines, and equations.
• Solidifying understanding of the analytic geometry of lines by knowing a variables expression in two variables is a line.
• That any line is the graph of a linear equation in two variables. Students use algebra and functions in statistical contexts and might fit informally a quadratic function to data in a modeling context.
• Drawing on middle school skills to apply basic statistics and probability in modeling.
• Algebra opens problem-solving doors previously inaccessible.

Mathematical Practice Standards

• Standards ensure math is meaningful and relevant.
• Standards that define mathematics should include practice standards instruction.

Key explanations

• Mathematical practice consists of, patience from students, discerning useful and relevant extractions of information, and using functions to solve them.
• Extend slope understanding as the rate of change of a linear function and computing the average rate of change over an interval.
• Reasoning by solving equations, understanding it goes beyond rote memorization, using language such as "if...then" to justify reasoning.
• Students discover mathematics through real-world experimentation by applying them to practical situations.
• Graph understanding is expressed through equations as a tool to interpret results and construct diagrams.
• Students understand the specific definitions of rational and irrational numbers.
• The definition of a function is understood by whether every input value has exactly one output value.
• Formulas like (a ± b)2 = a2 ± 2ab + b2 will be developed applying distributive properties.
• It gives insight students that equations come in the form of “something squared," which are expressed equal or more to zero.
• Recognizing a line is an equal output over equal inputs, where
• The expression gives lines an equal difference in outputs than inputs.
• The equation will give lines a generic point.

Standards for Mathematical Practice

• Modeling is its own conceptual category.
• Although the modeling is not a specific standard it covers all mathematics courses.
• Standards are marked with a star symbol to show real-world modeling situations applied.
• Algebra lays the foundation for higher mathematics curriculum by covering it first.

Algebra I Structure

• Conceptual category, domains, clusters, and then standards.
• Standards considered new for teachers will be thoroughly discussed.

Conceptual Category: Modeling

• High mathematics is marked with a star indicating they are modeling standards
• Modeling is beyond application moving into real-world mathematical problems from students questions
• Students new issues rise as to which values are known, unknown, what tables or data can be made and that a functional relationship is present.
• Determining solutions can be revised and may include calculators, software, spreadsheets with prior derived models.
• Equations are used when answering specific input knowledge.
• Consideration problems are genuine since students need to care under investigation.
• Mathematics is the tool for answering question under consideration.
• Examine problems and formulate equations, tables, graphs etc to compute, interpret, validate and report the results.
• Modeling gives a concrete base which abstracts mathematics and serves to motivate students.

Functions

• Describes situation where one quantity determines another.
• Return on investment is a function of the amount of time money is invested.
• Functions build mathematical models.
• School mathematics have numerical inputs and outputs defined in an algebraic expression.
• Time to drive 100 miles in a car is a function of car's speed in miles per hour.
• T(v) is used to express algebraic relationships whose name is T.
• Inputs to a function is the domain.
• Domain becomes the function when it makes sense in context.
• Defining characteristic is that the input value determines the output or that the output depends on the input.
• The trace of a seismograph represents a function (can be described various ways).
• I'll give you a state, you give me the capital city"
• Each person has an identification number.
• Recursive rule is expressed as f(n + 1) = f(n) + b, f(0) = a.

Interpreting Functions

• Emphasizes the concept of function, and uses function notation with general principles.
• Focusing on linear, arithmetic, geometric, and exponential sequences.
• Function comes from one set (domain) to another (range), giving the domain exactly one element of range.
• The output of f is shown by f (x), corresponding to input X.
• Its key graph is y = f(x).
• Function notation evaluates an outputs domain for a function.
• Can interpret if it has statements in term of a context
• Sequences are defined whose domains has a subset of integers.
• Fibonacci is an example with formulas provided.
• Interpreting functions has applications for key graph features and charts.
• Graphs and charts should describe relationships with intercepts, increasing/decreasing, positive/negative ranges, with end behaviors.
• Relate the domain to graph, use applicable relation.
• If function h provides person-hours for engine building it gives domain as a positive integer.
• Estimate rates symbollicily and calculate over a specified interval.
• Standards define and use math with functions.
• It's with an understanding of functions, in context and in a language.
• Functions use languages with a domain and correlating range.
• When looking at a problem the core question is "Does each element of the domain correspond to exactly one element of the range?"
• Sequences become a new Algebra topic that act like functions whose domain can be an integer subsets.
• Explored number patterns in grades four and five that lead to ratios in grades six and seven.
• This will formalize previously learned understandings
• Terms will be computed by adding 3 to previous term.
• For sequences, a graph and recursive rule be made
• Of course this rule can have a description which is adapted from the UA Document.
• Algebra should provide understanding of working sequences and expressions.
• Functions are analyzed in using varying representation and graphing.
• Includes graphing by functions expressed at a minimum as described.
• Is meant to reveal what is different when the functions are expressed in another way.
• Compare properties and use different representations or descriptions to classify them.
• Algebra focuses on quadratic/exponential relations with their related standards.
• The class uses F and L with functions by hand identifying graph features.

Forms and Functions

• Students use algebraic forms of functions and see the differences with it's context.
• Graphs can use the equation and factor it crosses.
• The function can show the function crosses to obtain where f(x) = 0.
• Modeling relations use F-BF-1.
• F - BF1 states function relations use expressions in context
• A decaying exponential can be added to functions.
• BF2 is translated with geometric sequences and an explicit recursive code or formula will arise for them.
• Existing models build functions from values.
• Identifying its effects and explaining expressions for them.
• It's used to recognize old and even functions.
• Inverse function have to be found
• Its' given equations must write an inverse or a simple function for it.
• Applying reasoning can model quantative and functional change.
• Functions will give insight at hands, with predictions at hand, and changes that are made.
• Function creations will occur modeling standards.

Exponential Growth

• With a multiplicative factor greater than 1 over time, it's understood that the quantity grows exponentially.
• Initial bacteria doubles after days, with populations expressed.
• Can specify to problems students face after using exponential functions.

Questions To Consider

• What will occur by Jun 30 from its algae growth?
• "When will the lake be covered halfway"
• Will there be write of a function showing percentage of area, as time goes on?

Solutions

• June 30 implies, population double. Half the lake covered June 29.
• Formula P(t) represents percentage of algae in lake P(29) = P is known to know
• Formula for percentage comes to(1.86
• Algebra course presents sequence styles with recursive, can help find patterns in it.
• Simple doubling leads to exponential from amples linear, and exponential examples will be available."
• Using squares can illustrate sequence growth.
• BF2 implies even odd functions to come later, to implement functions/output towards output under transformation.
• Such understanding affects, and is to variables.
• In Algebra, F-BF4 gives finding solutions for inversions of simple case.
• Students use show, Celsius, and Fahrenheit.
• In contrast to y, an inverse expression, and context.

Linear Quadratic Exponential Models

• Construct quadratic models and solve construct problems with that.
• Distinguish ones of linear functions from those of exponential functions.
• That its' exponential functions' factors grow with equal intervals
• Situations realize quantities change as a unit and another rate
• Quantities increase, and show their present rates
• In graphs, create the linear equations
• That equation from, variable from exhibits x - y.
• Variables show the change by rate
• Students prove constant functions rate growth and linearity functions are such patterns.
• For equations exhibit constant ratio functions its x illustrations its y is always this equation"

Number and Quantity

• Learning from examples will help. They now have extending.
• This helps those of standard.

Real Number System

• First students introduced exponents, understanding basic properties by algebra.
• Any justified examples use properties of the set.

Further Properties

• Explained by whole functions
• Students extend such radicals with radicals -
• That's also a standard that product or is rational for algebra.
• Closed is a rational application that's shown over course more. Is shown in algebra

Quantities And Units

• To use a guide on multi-step problems, also is the choice of displays.
• Also, to limit the level of accuracy for data.

Example Task

• Felicia plans her freeway trip and notes car fuel.
• She wonders if gas station before exit exists from her.
• Her car gets the distance, and to reach that, how much.

Solutions

• First Felicia has to work her needed miles.
• Gas needed after 90m
• Felica needs 63.50. To make work, travel.

Algebra

• Students go expressions, model with variable solutions, and their world expressions.
• Using it will result as being a representation of satisfying.
• Function category that is expressed.
• Is the produce of expression when produced.
• An expression for solution
• Functions show same expression.
• Points interject by finding values of equation.
• As such, solve expressions, and algebra will have purposes.

Seeing equations in structure.

• Linear, it gives a quantity in terms over course.

Expressions Explain:

• parts of expression, factor and coefficients
• entity: not is an example of this,

Keypoint Reminders

• It express an ambiguous.
• Expression will also ask for the ability from its terms.
• Reading comprehension involves analysis of structures.
• This analysis transforms in meaningful new expressions that add prices.
• The standard gives a constant factor with more complex expressions

Graphs in use

• Choose a format with which you can properly explain it.
• Factor quadratic express, and to define a function
• If you do you must find properties
• Use it to reveal or define it
• With them an expression that could the annual will reveal itself as they monthly rate.
• Students can explain equations with variables in exponent expression- it’s important they apply distributive cases

To note are:

• It will yield factor cases
• Such things are simplified, so emphasis transforms suitable results for purpose.

Standard Expressions

• First, a rich investigation is about for sum
• tale speaks and Gauss.

The key is:

• Students expression show what they know
• If term will always high students are meant to try them. This is very useful
• To what happens