Big Ideas: Students will investigate and analyze algebraic relations through a variety of representations, make connections among those representations, graph linear functions, and write equations of lines to include both slope and y-intercepts. Students will also determine if a relation is a function, identify the domain and range of the function, and distinguish between the independent and dependent variables. Essential Questions students should be able to answer: What does a slope tell us about a given situation, and how can it be applied to problems in context? How can the rate of change and y-intercept of a function be determined and applied to make predictions? How can we apply our understanding of independent and dependent variables to real-life scenarios? How is the algebraic representation of a linear function related to its graph? How can we determine if a relation is a function and identify the domain or range of a given mathematical representation? What are the inferences that can be drawn from sets of data points having a positive relationship, a negative relationship, and no relationship?

Introduction to Relations and Functions (00:00 - 02:15)

  • Discussion on the difference between a general mathematical relation and a function.

  • Use of the 'vending machine' analogy to explain the requirement that every input has exactly one output.

Domain, Range, and Variables (02:15 - 04:30)

  • Definitions of domain as inputs (x-values) and range as outputs (y-values).

  • Explanation of independent and dependent variables using real-world examples like sleep and mood or soccer practice and goals.

  • Identification of which axes usually represent which variable type.

Linear Equations and Slope (04:30 - 06:15)

  • Breakdown of the linear equation $y = mx + b$.

  • Analysis of slope (m) as the rate of change or steepness of a line.

  • Discussion on how positive, negative, and zero slopes represent different real-world trends, such as battery drainage or financial growth.

The Y-Intercept and Predictions (06:15 - 07:45)

  • Explanation of the y-intercept (b) as the 'starting value' or initial state of a function.

  • How combined algebraic representations allow for making predictions by substituting values into the equation.

Data Trends and Scatter Plots (07:45 - 09:30)

  • Discussion on positive, negative, and no relationships in data sets.

  • Exploring connections between tables, graphs, and equations (the 'facts, picture, and rule').

  • Tips for identifying independent and dependent variables in word problems.