Module 17: Numerical & Algebraic Expressions

Develop algebraic thinking by examining how numbers, variables, operations, and mathematical properties can be used to represent calculations, patterns, and general relationships.

Numerical Expressions - Interpret, construct, and evaluate expressions involving numbers and operations while paying attention to their mathematical structure.

Algebraic Expressions - Use variables and operations to represent unknown values, changing quantities, patterns, and relationships.

Variables as Unknowns, Changing Quantities, & Generalized Numbers - Explore the different roles variables can play and move beyond the idea that a letter always represents a single missing number.

Properties of Operations as Foundations for Algebra - Connect the commutative, associative, and distributive properties to equivalent expressions and the reasoning used throughout algebra.

Module 18: Equality & Equations

Develop a relational understanding of equality and use that understanding to interpret, create, analyze, and solve equations.

Algebraic Equality - Explore equality as a relationship between two expressions that have the same value rather than as a signal to perform a calculation.

Algebraic Equations - Interpret equations as statements about equal quantities and use reasoning to determine values that make those statements true.

Meaning of the Equal Sign - Examine the equal sign as a symbol expressing equivalence between two quantities or expressions on either side.

True, False, & Open Number Sentences - Determine whether mathematical statements are true or false and identify values that make open number sentences true.

Relational Thinking - Use relationships among numbers and expressions to reason about equality without necessarily carrying out every calculation.

Balance Models for Equations - Use balance models to visualize equality and explain why the same mathematical operation must preserve both sides of an equation.

Module 19: Patterns & Generalizations

Investigate visual and numerical patterns, identify how they grow or repeat, and develop increasingly general ways to describe their structure.

Geometric Patterns - Explore patterns created with shapes and visual arrangements and describe how their structure changes from one stage to the next.

Number Patterns - Identify relationships within sequences of numbers and use those relationships to predict and extend patterns.

Finding Rules in Repetition - Analyze repeating patterns to identify the underlying rule that determines how the pattern continues.

Recursive vs. Explicit Patterns - Compare rules that describe a term using previous terms with rules that determine any term directly from its position.

Generalizing a Pattern with Variables - Use variables and expressions to describe a pattern in a way that applies to any stage or term.

Module 20: Introduction to Functions

Develop the idea of a function as a relationship between quantities and connect function rules to tables, graphs, and other representations.

Inputs, Outputs, & Rules - Understand functions as rules that connect allowable input values to corresponding output values.

Function Tables - Organize input-output relationships in tables and use patterns in the values to understand or identify a function rule.

Graphing Functions - Represent functions on coordinate graphs to visualize how one quantity changes in relation to another.

Multiple Representations of Functions - Connect verbal descriptions, rules, tables, graphs, and other representations of the same functional relationship.

Module 21: Linear & Nonlinear Relationships

Compare different types of functional relationships by examining patterns of change, graphical behavior, and the distinction between constant and changing rates.

Linear Functions - Explore relationships with a constant rate of change and connect their tables, graphs, equations, and real-world meanings.

Other Functions - Examine examples of nonlinear relationships and compare their behavior with familiar linear patterns.

Constant vs. Changing Rate of Change - Determine whether a relationship changes at a constant rate or whether its rate itself varies.

Comparing Linear & Nonlinear Patterns - Use tables, graphs, equations, and patterns to distinguish linear relationships from nonlinear ones.

Module 22: Mathematical Structure & Problem Solving

Strengthen mathematical problem solving by looking for structure, making conjectures, describing relationships precisely, and clearly explaining the reasoning behind a solution.

Geometric Problems - Apply geometric ideas and relationships to solve problems that require visualization, reasoning, and connecting multiple concepts.

Describing Geometry - Use precise mathematical language, properties, and relationships to communicate geometric reasoning clearly.

Looking for Structure - Identify useful patterns, properties, and relationships within a problem that can make unfamiliar mathematics easier to understand.

Making and Testing Conjectures - Form mathematical predictions from observed patterns and test them using examples, counterexamples, and logical reasoning.

Explaining Mathematical Reasoning - Communicate not only an answer but also the mathematical evidence and reasoning that show why the answer makes sense.

Module 23: Foundations of Probability

Develop a conceptual understanding of probability by examining outcomes, sample spaces, experiments, simulations, and the relationship between theoretical and experimental results.

Basic Probability - Understand probability as a measure of how likely an event is to occur and represent likelihood using numbers from 0 to 1.

Probability Rules - Use fundamental probability rules to determine the likelihood of individual, combined, and complementary events.

Experiments & Simulations - Use repeated trials and simulations to model random processes and investigate long-run patterns.

Experimental vs. Theoretical Probability - Compare probabilities predicted by mathematical models with probabilities observed through actual or simulated experiments.

Sample Spaces & Outcomes - Identify all possible outcomes of a chance process and organize them into a sample space for probability analysis.

Module 24: Collecting & Understanding Data

Examine how statistical investigations begin by asking appropriate questions, identifying populations and samples, choosing sampling methods, and considering sources of bias.

Sampling Methods - Compare different methods for selecting samples and consider how the method used can affect the quality of the resulting data.

Types of Data - Distinguish among different kinds of categorical and numerical data and consider how data type influences analysis.

Statistical Questions - Identify questions that anticipate variability in their answers and can be investigated through the collection and analysis of data.

Population vs. Sample - Distinguish the full group of interest from the smaller group actually observed and examine how samples are used to learn about populations.

Bias in Data Collection - Identify ways that sampling, question wording, measurement, and other collection methods can systematically distort data.

Module 25: Displaying Data

Represent categorical and numerical data using appropriate displays and develop the ability to choose, interpret, and critically evaluate graphical representations.

Displaying Categorical Data - Organize and represent data made up of groups or categories so that frequencies and comparisons can be seen clearly.

Bar Charts & Pie Charts - Use bar charts and pie charts to represent categorical data and compare when each display is most useful.

Displaying Numerical Data - Represent quantitative data in ways that reveal patterns in values, frequencies, and distributions.

Histograms & Dotplots - Use histograms and dotplots to examine the shape, concentration, gaps, and spread of numerical data.

Choosing an Appropriate Data Display - Select a graph or display that matches the type of data and the question being investigated.

Interpreting Misleading Graphs - Identify design choices and distortions that can make a graph give a misleading impression of the underlying data.

Module 26: Describing & Comparing Data

Describe distributions using measures of center and spread, graphical summaries, and overall features such as shape and outliers, then use those features to compare data sets meaningfully.

Measures of Center - Use mean, median, and other measures of center to describe a typical or representative value in a data set.

Weighted Averages - Calculate and interpret averages in which some values contribute more heavily than others.

Quartiles & Boxplots - Use quartiles and boxplots to summarize the center and spread of data and identify how values are distributed across four sections.

Standard Deviation - Measure how far data values typically lie from the mean and use standard deviation to describe overall variability.

Measures of Spread - Compare ways of describing variability, including range, interquartile range, and standard deviation.

Comparing Distributions - Compare two or more data sets by considering differences in their centers, spreads, shapes, and unusual values.

Center, Shape, Spread, & Outliers - Describe a distribution as a whole by examining its typical value, overall form, variability, and any unusually distant observations.