Identify functions using sets of ordered pairs, tables, mappings, and graphs

Standards Alignment

• TEKS.111.28.05G primary
  identify functions using sets of ordered pairs, tables, mappings, and graphs;
• TEKS.111.28.05H primary
  identify examples of proportional and non-proportional functions that arise from mathematical and real-world problems; and
• TEKS.111.28.05I primary
  write an equation in the form y = mx + b to model a linear relationship between two quantities using verbal, numerical, tabular, and graphical representations.

Lesson Overview

This Grade 8 mathematics lesson focuses on identifying functions through various representations including sets of ordered pairs, tables, mappings, and graphs. Students will learn to distinguish functions from non-functions, recognize proportional and non-proportional relationships in real-world and mathematical contexts, and write linear equations in the form y = mx + b to model these relationships. The lesson is grounded in TEKS standards and emphasizes conceptual understanding and application through direct instruction, guided practice, and reflection.

Learning Objectives

• Identify functions using sets of ordered pairs, tables, mappings, and graphs.
• Recognize examples of proportional and non-proportional functions from mathematical and real-world problems.
• Write equations in the form y = mx + b to model linear relationships using verbal, numerical, tabular, and graphical representations.

Success Criteria

• Students can determine whether a set of ordered pairs, table, mapping, or graph represents a function.
• Students can classify functions as proportional or non-proportional based on their characteristics.
• Students can write and interpret linear equations in slope-intercept form to represent relationships between two quantities.

Prerequisite Knowledge

Students should understand the concept of ordered pairs, coordinate planes, and basic graph reading skills. They should be familiar with ratios, proportional relationships, and linear relationships from previous lessons.

Key Vocabulary

• Function
• Domain
• Range
• Ordered Pair
• Mapping
• Table
• Graph
• Proportional Function
• Non-Proportional Function
• Linear Equation
• Slope
• y-intercept
• Slope-Intercept Form

Materials and Resources

• Graph paper
• Coordinate plane posters or digital graphing tools
• Sets of ordered pairs worksheets
• Tables showing input-output values
• Mapping diagrams
• Markers or colored pencils
• Calculators (optional)

Teacher Preparation

• Prepare examples of sets of ordered pairs, tables, mappings, and graphs that represent functions and non-functions.
• Prepare real-world problem scenarios illustrating proportional and non-proportional relationships.
• Prepare guided practice problems for writing equations in y = mx + b form.
• Set up graphing tools or materials for student use.

Detailed Lesson Notes

Definition of a Function

A function is a relation between a set of inputs and a set of possible outputs where each input is related to exactly one output. This means that for every x-value (input), there is only one corresponding y-value (output). Understanding this is key to identifying functions from different representations.

Representations of Functions

Functions can be represented in multiple ways: 1. Ordered Pairs: Each pair (x, y) shows an input and its output. To be a function, no two pairs can have the same x-value with different y-values. 2. Tables: Lists inputs and outputs in rows or columns. Check if each input corresponds to exactly one output. 3. Mappings: Visual diagrams where each input is connected by an arrow to exactly one output. 4. Graphs: Plots of points on a coordinate plane. Use the vertical line test to determine if the graph represents a function — if any vertical line crosses the graph more than once, it is not a function.

Proportional vs. Non-Proportional Functions

Proportional functions are linear functions where the ratio of y to x is constant and the graph passes through the origin (0,0). They can be modeled by equations of the form y = mx, where m is the constant of proportionality (slope). Non-proportional functions are linear but do not pass through the origin; their equations have the form y = mx + b where b ≠ 0. Recognizing these differences in tables, graphs, and equations helps students understand real-world relationships.

Writing Linear Equations in y = mx + b Form

To write an equation modeling a linear relationship: 1. Identify the slope (m), which is the rate of change between y and x values. 2. Identify the y-intercept (b), the value of y when x is zero. 3. Use the form y = mx + b to write the equation. This can be done from verbal descriptions, numerical data in tables, or points on a graph. For example, if a table shows that y increases by 3 when x increases by 1, and y is 2 when x is 0, then m = 3 and b = 2, so the equation is y = 3x + 2.

Common Misconceptions

- Confusing functions with relations that assign multiple outputs to one input. Emphasize the 'one output per input' rule. - Thinking all linear functions are proportional. Clarify that proportional functions must pass through the origin. - Misinterpreting the slope and y-intercept in equations. Use concrete examples to show their meaning.

Worked Examples

Classroom Activity

Students work in pairs to analyze tables and mappings, marking those that are functions and justifying their choices.

Guided Practice

Independent Practice

1. Foundational: Given the set of ordered pairs {(2, 4), (3, 6), (4, 8)}, is this a function? Explain why or why not.
2. Developing: Look at the table: x: 1, 2, 3; y: 2, 4, 6. Is this a proportional function? How do you know?
3. Application: Given the mapping diagram where each input 1, 2, 3 maps to outputs 3, 6, 9 respectively, write the equation in y = mx + b form that models this relationship.
4. Analysis: A graph passes the vertical line test and goes through points (0, 1) and (2, 7). Is this function proportional or non-proportional? Write the equation in y = mx + b form.
5. Challenge: Given the table: x: 0, 1, 2, 3; y: 1, 4, 7, 10, identify if the function is proportional or non-proportional and explain how to write the equation y = mx + b for this data.

Independent Practice Teacher Answer Key

1. 1. Yes, it is a function because each input (2, 3, 4) has exactly one output (4, 6, 8).
2. 2. Yes, it is proportional because the ratio y/x is constant (2/1=2, 4/2=2, 6/3=2), and the graph would pass through the origin.
3. 3. The outputs increase by 3 for each increase of 1 in input, so slope m = 3. Since input 0 would correspond to output 0, y-intercept b = 0. Equation: y = 3x.
4. 4. It is non-proportional because it does not pass through the origin (y-intercept is 1). Slope m = (7-1)/(2-0) = 6/2 = 3. Equation: y = 3x + 1.
5. 5. The function is non-proportional because y ≠ 0 when x = 0 (y = 1). The slope m = (4-1)/(1-0) = 3. The y-intercept b = 1. Equation: y = 3x + 1.

Guiding Questions

• What makes a relation a function?
• How can you tell if a graph represents a function?
• What is the difference between proportional and non-proportional functions?
• How do you write an equation for a linear function?

Common Misconceptions

• Believing that if a relation has repeated x-values, it can still be a function if the outputs are the same; however, repeated x-values with different outputs mean it is not a function.
• Assuming all linear functions are proportional, forgetting that proportional functions must pass through the origin.
• Confusing the slope with the y-intercept when writing equations.

Differentiation

Assessment

• Identify if the following set of ordered pairs is a function: {(1,2), (2,3), (1,4)}.
• Use the vertical line test on a given graph to determine if it represents a function.
• Classify a function from a table as proportional or non-proportional.
• Write the equation of the line that passes through (0,3) and (2,7).
• Explain why a relation with two outputs for one input is not a function.
• Given a real-world problem, identify if the relationship is a function and write its linear equation.
• Analyze a mapping diagram and determine if it represents a function.
• Compare two graphs and classify each as proportional or non-proportional functions.

Answer Guide

• Is the set {(1, 3), (2, 6), (3, 9), (4, 12)} a function and proportional?
  Answer: Yes, it is a function because each input has one output. It is proportional because the ratio y/x is constant (3).
• Is the table with x: 0,1,2,3 and y: 2,5,8,11 a function and proportional? Write the equation.
  Answer: It is a function but not proportional because it does not pass through the origin. The equation is y = 3x + 2.

Real-Life Application

Ask students to find examples of functions in their home or community, such as recipes (input: ingredients, output: servings) or speed-distance-time relationships, and describe them using ordered pairs or tables.

Homework or Home Connection

• Ask students to find examples of functions in their home or community, such as recipes (input: ingredients, output: servings) or speed-distance-time relationships, and describe them using ordered pairs or tables.

Lesson Summary

Students learned to identify functions from ordered pairs, tables, mappings, and graphs by ensuring each input has exactly one output. They distinguished proportional functions, which have a constant ratio and pass through the origin, from non-proportional functions. Finally, students practiced writing linear equations in the form y = mx + b to model these relationships, connecting mathematical concepts to real-world contexts.

Teacher Notes

Use the exact standards alignment and retrieved-source provenance stored with this enrichment.

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