Your next great Lessn starts here.

Introduction to Linear Functions – Define a linear function – A function with a constant rate of change, represented by a straight line – Characteristics of linear functions – They have a constant slope and y-intercept – Real-life linear function examples – Distance vs. time in travel, or cost vs. quantity in shopping – Understanding function tables – Tables show input-output pairs; we can find the function’s formula | This slide introduces the concept of linear functions, which are fundamental in algebra. A linear function is defined by its constant rate of change, which can be visually represented by a straight line on a graph. Key characteristics include a constant slope, which indicates the steepness of the line, and a y-intercept, where the line crosses the y-axis. Real-life examples help students relate to the concept, such as how distance traveled increases over time at a steady pace, or how the total cost increases with the number of items purchased at a constant price. Encourage students to look at tables of values as a starting point for writing linear functions, identifying the pattern of change between inputs (x-values) and outputs (y-values). This understanding will be crucial as they learn to write equations that model these relationships.

Understanding Tables of Values – Define a table of values – A grid showing two sets of related numbers – Identify patterns in tables – Look for consistent increases or decreases – Predict future table values – Use patterns to forecast upcoming numbers – Writing linear functions – Translate patterns into linear equations | This slide introduces students to the concept of a table of values, which is a fundamental tool in understanding linear functions. A table of values pairs input with output, often representing a function’s behavior. Students should learn to recognize patterns, such as a constant rate of change, which is indicative of a linear relationship. By identifying these patterns, students can predict future values and ultimately write a linear function that models the data. Emphasize the importance of looking for a constant difference between y-values when x-values have a constant difference, as this is a key indicator of linearity. Encourage students to practice with different tables to solidify their understanding.

From Tables to Graphs: Linear Functions – Plot points from a table – Use the table values as coordinates to plot – Points represent a function – Each point shows an input-output pair – Connect tables, points, graphs – Tables show pairs, points show relation, graphs show the function visually – Graphing a linear function – Use the plotted points to draw the line of the function | This slide is aimed at teaching students how to translate data from a table into a graphical representation. Start by explaining how each row in a table corresponds to a point on the graph, with the x-value (input) and y-value (output) forming coordinates. Emphasize that a function represents a specific relationship between inputs and outputs, and this relationship can be visualized when the points are plotted on a graph. Discuss how the table, points, and graph are interconnected and how they all represent the same linear function. Finally, demonstrate how to draw the line that connects the points on the graph, which represents the linear function. Encourage students to practice by providing multiple tables and guiding them through plotting points and drawing lines.

Slope and Y-Intercept of Linear Functions – Define the slope of a line – Slope measures the steepness of a line, rise over run – Finding slope from a table – Compare vertical change to horizontal change between points – What is the y-intercept? – The y-intercept is where the line crosses the y-axis – Writing linear functions – Use slope and y-intercept to write the function | This slide introduces the concepts of slope and y-intercept, which are crucial in understanding linear functions. Start by defining the slope as the ratio of the vertical change (rise) to the horizontal change (run) between two points on a line. Demonstrate how to calculate the slope from two points in a table of values. Explain the y-intercept as the point where the line crosses the y-axis, which is the starting value of the function when x is zero. Conclude by showing how to combine the slope and y-intercept to write the equation of a linear function in slope-intercept form (y = mx + b). Provide examples and encourage students to practice with different tables to solidify their understanding.

Writing Linear Functions from a Table – Understand the formula y=mx+b – The foundation for linear equations, where m is slope and b is y-intercept – Identify slope (m) from a table – Slope (m) is the rate of change between points in the table – Determine y-intercept (b) from a table – Y-intercept (b) is where the line crosses the y-axis – Write the equation from table data – Use m and b to form y=mx+b representing the function | This slide introduces students to the concept of writing linear functions from tables. Begin by explaining the standard form of a linear equation, y=mx+b, where ‘m’ represents the slope of the line and ‘b’ represents the y-intercept. Demonstrate how to calculate the slope by finding the rate of change between two points in the table. Then, show how to determine the y-intercept, which is the value of y when x is zero. Finally, guide students through the process of combining the slope and y-intercept to write the equation of the linear function that the table represents. Provide examples with different slopes and y-intercepts to ensure students understand how to interpret the data from the table and form the corresponding equation.

Practice: Writing Linear Functions from Tables – Practice with diverse tables – Determine slope and y-intercept – Slope (m) is rise over run between points – Write linear functions – Use y = mx + b, where m is slope, b is y-intercept – Apply to real-world scenarios – E.g., budgeting, distance over time | This slide is aimed at providing students with practice problems to solidify their understanding of writing linear functions from tables. Encourage students to identify the pattern of change in the ‘y’ values as the ‘x’ values increase, which will help them find the slope. Once they have the slope, they can look for the y-intercept, which is the value of ‘y’ when ‘x’ is zero. Provide a variety of tables to ensure students get practice with different types of linear relationships. Emphasize the real-world application of these skills, such as budgeting money over time or calculating distance traveled over a period. For the activity, consider having students work in pairs to solve problems and then explain their reasoning to the class.

Class Activity: Crafting Linear Functions – Create a table with a linear pattern – Calculate the slope (m) – The slope is the rate of change between any two points – Find the y-intercept (b) – The y-intercept is where the line crosses the y-axis – Write your linear function – Use the formula y = mx + b | This activity is designed to reinforce students’ understanding of linear functions by having them create their own tables of values and derive the corresponding linear function. Students should start by creating a table with an x (independent variable) and y (dependent variable) column, ensuring that the y-values change at a constant rate as x increases. They will then use two points from their table to calculate the slope (m), which represents the change in y divided by the change in x. Next, they will determine the y-intercept (b), which is the y-value when x is zero. Finally, students will write the linear function in slope-intercept form (y = mx + b). Encourage students to share their functions with the class to compare different linear relationships. Possible variations for different students could include different starting points, slopes, or even challenging them to create a table with a negative slope.

Conclusion: Linear Functions & Homework – Review key linear function concepts – Address any questions – Homework: Real-life linear functions – Find examples like saving money or distance over time – Create tables and equations – Represent your examples in tables and derive equations | This slide wraps up the lesson on linear functions by reviewing the main concepts taught during the class. It’s crucial to ensure that students have a solid understanding of how to write linear functions from tables. Encourage students to ask questions to clarify any doubts. For homework, students are tasked to observe their surroundings and identify real-life situations that can be modeled by linear functions, such as saving money over time or the relationship between distance and time in travel. They should create tables based on these observations and use the tables to write the corresponding linear equations. This exercise will help them apply mathematical concepts to real-world scenarios and deepen their understanding of linear functions.