Alignments

Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate). [See: Coordinates and Game Design.]

Write and interpret numerical expressions. [See: Order of Operations.]

Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. [See: Order of Operations.]

Reason about and solve one-variable equations and inequalities. [See: Simple Inequalities; Compound Inequalities: Solutions & Non-Solutions.]

Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set. [See: Function Composition; Defining Values; Functions Make Life Easier!; Functions: Contracts, Examples & Definitions; Defining Linear Functions; Solving Word Problems with the Design Recipe; Functions for Character Animation; Surface Area of a Rectangular Prism; Problem Decomposition; Sam the Butterfly - Applying Inequalities; Piecewise Functions and Conditionals; Player Animation; The Distance Formula; Collision Detection - Distance and Inequality; Defining Table Functions; Grouped Samples; Linear Regression.]

Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams. [See: Simple Inequalities; Sam the Butterfly - Applying Inequalities.]

Solve real-world and mathematical problems involving area, surface area, and volume. [See: Surface Area of a Rectangular Prism.]

Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems. [See: Surface Area of a Rectangular Prism.]

Understand ratio concepts and use ratio reasoning to solve problems. [See: Making Flags; Bar and Pie Charts.]

Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. [See: Making Flags.]

Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. [See: Making Flags.]

Develop understanding of statistical variability. [See: Histograms; Visualizing the “Shape” of Data; Measures of Center; Box Plots.]

Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. [See: Introduction to Data Science; The Data Cycle; Choosing Your Dataset.]

Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape. [See: Histograms; Visualizing the “Shape” of Data; Measures of Center; Box Plots; Standard Deviation.]

Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number. [See: Box Plots; Standard Deviation.]

Summarize and describe distributions. [See: Visualizing the “Shape” of Data; Measures of Center; Box Plots.]

Display numerical data in plots on a number line, including dot plots, histograms, and box plots. [See: Histograms; Visualizing the “Shape” of Data; Box Plots; Standard Deviation.]

Summarize numerical data sets in relation to their context. [See: Measures of Center; Box Plots; Standard Deviation.]

Summarize numerical data sets in relation to their context by giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. [See: Measures of Center; Box Plots; Standard Deviation.]

Summarize numerical data sets in relation to their context by relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered. [See: Measures of Center.]

Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. [See: Problem Decomposition.]

Solve real-life and mathematical problems using numerical and algebraic expressions and equations. [See: Solving Word Problems with the Design Recipe; Problem Decomposition; Sam the Butterfly - Applying Inequalities; Piecewise Functions and Conditionals; Player Animation; The Distance Formula.]

Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. [See: Defining Values; Functions Make Life Easier!; Solving Word Problems with the Design Recipe; Simple Inequalities; Compound Inequalities: Solutions & Non-Solutions.]

Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. [See: Solving Word Problems with the Design Recipe.]

Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. [See: Solving Word Problems with the Design Recipe; Surface Area of a Rectangular Prism.]

Analyze proportional relationships and use them to solve real-world and mathematical problems. [See: Making Flags; Making Game Images.]

Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. [See: Making Flags.]

Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. [See: Functions Can Be Linear; Defining Linear Functions.]

Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. [See: Functions Can Be Linear; Defining Linear Functions.]

Use random sampling to draw inferences about a population. [See: Probability, Inference, and Sample Size.]

Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. [See: Probability, Inference, and Sample Size.]

Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. [See: Probability, Inference, and Sample Size.]

Draw informal comparative inferences about two populations. [See: Bar and Pie Charts.]

Understand the connections between proportional relationships, lines, and linear equations. [See: Functions Can Be Linear; Defining Linear Functions.]

Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. [See: Functions Can Be Linear; Defining Linear Functions.]

Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. [See: Contracts; The Vertical Line Test.]

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). [See: Functions Can Be Linear; Defining Linear Functions.]

Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. [See: Functions Can Be Linear; Defining Linear Functions.]

Use functions to model relationships between quantities. [See: Functions Make Life Easier!; Defining Linear Functions; Solving Word Problems with the Design Recipe; Functions for Character Animation; Problem Decomposition.]

Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. [See: Functions Can Be Linear; Defining Linear Functions.]

Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. [See: Functions Can Be Linear; Defining Linear Functions.]

Verify experimentally the properties of rotations, reflections, and translations. [See: Making Game Images.]

Understand and apply the Pythagorean Theorem. [See: The Distance Formula.]

Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. [See: The Distance Formula.]

Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. [See: The Distance Formula.]

Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. [See: Scatter Plots; Grouped Samples; Correlations; Linear Regression.]

Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. [See: Scatter Plots; Correlations; Linear Regression.]

Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. [See: Linear Regression.]

Create equations that describe numbers or relationships. [See: Solving Word Problems with the Design Recipe; Surface Area of a Rectangular Prism; Problem Decomposition; Sam the Butterfly - Applying Inequalities.]

Create equations and inequalities in one variable and use them to solve problems. [See: Sam the Butterfly - Applying Inequalities.]

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. [See: Sam the Butterfly - Applying Inequalities.]

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. [See: Sam the Butterfly - Applying Inequalities.]

Interpret expressions that represent a quantity in terms of its context. [See: Defining Values; Functions Make Life Easier!; Solving Word Problems with the Design Recipe.]

Interpret parts of an expression, such as terms, factors, and coefficients. [See: Piecewise Functions and Conditionals; Player Animation.]

Interpret complicated expressions by viewing one or more of their parts as a single entity. [See: Piecewise Functions and Conditionals; Player Animation; Custom Scatter Plots.]

Use the structure of an expression to identify ways to rewrite it. [See: Order of Operations.]

Write expressions in equivalent forms to solve problems. [See: Order of Operations.]

Build a function that models a relationship between two quantities. [See: Function Notation; Defining Linear Functions; Solving Word Problems with the Design Recipe; Functions for Character Animation; Problem Decomposition; Simple Inequalities; Compound Inequalities: Solutions & Non-Solutions; Defining Table Functions.]

Write a function that describes a relationship between two quantities. [See: Function Notation; Defining Linear Functions; Solving Word Problems with the Design Recipe; Functions for Character Animation; Problem Decomposition; Simple Inequalities; Compound Inequalities: Solutions & Non-Solutions; Defining Table Functions.]

Combine standard function types using arithmetic operations. [See: Function Composition.]

Compose functions. [See: Function Composition; Problem Decomposition; Sam the Butterfly - Applying Inequalities; Collision Detection - Distance and Inequality.]

Build new functions from existing functions. [See: Problem Decomposition; Sam the Butterfly - Applying Inequalities; Collision Detection - Distance and Inequality.]

Understand the concept of a function and use function notation. [See: Functions Make Life Easier!; Function Notation.]

Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). [See: Contracts; The Vertical Line Test.]

Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. [See: Contracts; Making Flags; Function Notation; Defining Linear Functions; Solving Word Problems with the Design Recipe; Functions for Character Animation; Problem Decomposition; Simple Inequalities; Compound Inequalities: Solutions & Non-Solutions; Defining Table Functions.]

Interpret functions that arise in applications in terms of the context. [See: Function Notation; Defining Linear Functions; Solving Word Problems with the Design Recipe; Functions for Character Animation; Problem Decomposition; Simple Inequalities; Compound Inequalities: Solutions & Non-Solutions; Defining Table Functions.]

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. [See: Functions Can Be Linear; Defining Linear Functions.]

Analyze functions using different representations. [See: The Vertical Line Test; Functions: Contracts, Examples & Definitions; Functions Can Be Linear; Defining Linear Functions; Solving Word Problems with the Design Recipe.]

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). [See: Functions Can Be Linear; Defining Linear Functions; Problem Decomposition.]

Reason quantitatively and use units to solve problems. [See: Making Flags.]

Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ('or', 'and', 'not'). [See: Compound Inequalities: Solutions & Non-Solutions.]

Use permutations and combinations to compute probabilities of compound events and solve problems. [See: Permutations; Combinations.]

Understand and evaluate random processes underlying statistical experiments. [See: Probability, Inference, and Sample Size.]

Understand statistics as a process for making inferences about population parameters based on a random sample from that population. [See: Probability, Inference, and Sample Size.]

Make inferences and justify conclusions from sample surveys, experiments, and observational studies. [See: Collecting Data.]

Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. [See: Probability, Inference, and Sample Size; Collecting Data.]

Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling. [See: Collecting Data.]

Evaluate reports based on data. [See: Threats to Validity.]

Summarize, represent, and interpret data on a single count or measurement variable. [See: Bar and Pie Charts; Histograms; Visualizing the “Shape” of Data; Measures of Center; Box Plots.]

Represent data with plots on the real number line (dot plots, histograms, and box plots). [See: Histograms; Visualizing the “Shape” of Data; Box Plots; Standard Deviation.]

Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. [See: Histograms; Measures of Center; Box Plots; Standard Deviation.]

Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). [See: Histograms; Visualizing the “Shape” of Data; Measures of Center; Box Plots; Standard Deviation.]

Summarize, represent, and interpret data on two categorical and quantitative variables. [See: Scatter Plots; Correlations.]

Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. [See: Scatter Plots; Correlations.]

Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. [See: Linear Regression.]

Fit a linear function for a scatter plot that suggests a linear association. [See: Linear Regression.]

Interpret linear models. [See: Linear Regression.]

Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. [See: Linear Regression.]

Compute (using technology) and interpret the correlation coefficient of a linear fit. [See: Scatter Plots; Correlations; Linear Regression.]

Distinguish between correlation and causation. [See: Correlations; Linear Regression.]

Initiate and participate effectively in a range of collaborative discussions (one-on-one, in groups, and teacher-led) with diverse partners on grades 9-10 topics, texts, and issues, building on others' ideas and expressing their own clearly and persuasively. [See: Introduction to Data Science.]