Number and algebraic methods. The student applies mathematical processes to simplify and perform operations on functions represented in a variety of ways, including real-world situations. The student is expected to:

collect observations and measurements as evidence;

record and organize data using pictures, numbers, words, symbols, and simple graphs; and

connect tabular representations to symbolic representations when adding, subtracting, and multiplying polynomial functions arising from mathematical and real-world situations such as applications involving surface area and volume;

develop and use models to represent phenomena, objects, and processes or design a prototype for a solution to a problem.

compare and contrast the results when adding two linear functions and multiplying two linear functions that are represented tabularly, graphically, and symbolically;

Scientific and engineering practices. The student analyzes and interprets data to derive meaning, identify features and patterns, and discover relationships or correlations to develop evidence-based arguments or evaluate designs. The student is expected to:

determine the quotient of a polynomial function of degree three and of degree four when divided by a polynomial function of degree one and of degree two when represented tabularly and symbolically; and

identify basic advantages and limitations of models such as their size, properties, and materials;

determine the linear factors of a polynomial function of degree two and of degree three when represented symbolically and tabularly and graphically where appropriate.

analyze data by identifying significant features and patterns;

Number and algebraic methods. The student applies mathematical processes to represent, simplify, and perform operations on matrices and to solve systems of equations using matrices. The student is expected to:

use mathematical concepts to compare two objects with common attributes; and

add and subtract matrices;

evaluate a design or object using criteria to determine if it works as intended.

multiply matrices;

multiply matrices by a scalar;

Scientific and engineering practices. The student develops evidence-based explanations and communicates findings, conclusions, and proposed solutions. The student is expected to:

develop explanations and propose solutions supported by data and models;

represent and solve systems of two linear equations arising from mathematical and real-world situations using matrices; and

communicate explanations and solutions individually and collaboratively in a variety of settings and formats; and

represent and solve systems of three linear equations arising from mathematical and real-world situations using matrices and technology.

listen actively to others' explanations to identify important evidence and engage respectfully in scientific discussion.

Number and algebraic methods. The student applies mathematical processes to estimate and determine solutions to equations resulting from functions and real-world applications with fluency. The student is expected to:

Scientific and engineering practices. The student knows the contributions of scientists and recognizes the importance of scientific research and innovation for society. The student is expected to:

estimate a reasonable input value that results in a given output value for a given function, including quadratic, rational, and exponential functions;

explain how science or an innovation can help others; and

solve equations arising from questions asked about functions that model real-world applications, including linear and quadratic functions, tabularly, graphically, and symbolically; and

identify scientists and engineers such as Katherine Johnson, Sally Ride, and Ernest Just and explore what different scientists and engineers do.

approximate solutions to equations arising from questions asked about exponential, logarithmic, square root, and cubic functions that model real-world applications tabularly and graphically.

Recurring themes and concepts. The student uses recurring themes and concepts to make connections across disciplines. The student is expected to:

Modeling from data. The student applies mathematical processes to analyze and model data based on real-world situations with corresponding functions. The student is expected to:

represent domain and range of a function using interval notation, inequalities, and set (builder) notation;

identify and use patterns to describe phenomena or design solutions;

investigate and predict cause-and-effect relationships in science;

compare and contrast between the mathematical and reasonable domain and range of functions modeling real-world situations, including linear, quadratic, exponential, and rational functions;

describe the properties of objects in terms of relative size (scale) and relative quantity;

determine the accuracy of a prediction from a function that models a set of data compared to the actual data using comparisons between average rates of change and finite differences such as gathering data from an emptying tank and comparing the average rate of change of the volume or the second differences in the volume to key attributes of the given model;

examine the parts of a whole to define or model a system;

determine an appropriate function model, including linear, quadratic, and exponential functions, for a set of data arising from real-world situations using finite differences and average rates of change; and

identify forms of energy and properties of matter;

determine if a given linear function is a reasonable model for a set of data arising from a real-world situation.

describe the relationship between structure and function of objects, organisms, and systems; and

describe how factors or conditions can cause objects, organisms, and systems to either change or stay the same.

Matter and its properties. The student knows that objects have physical properties that determine how they are described and classified. The student is expected to:

classify objects by observable physical properties, including, shape, color, and texture, and attributes such as larger and smaller and heavier and lighter;

explain and predict changes in materials caused by heating and cooling; and

demonstrate and explain that a whole object is a system made of organized parts such as a toy that can be taken apart and put back together.

Force, motion, and energy. The student knows that forces cause changes in motion and position in everyday life. The student is expected to:

explain how pushes and pulls can start, stop, or change the speed or direction of an object's motion; and