lesson 6 homework practice ordered pairs and relations

Chapter 1, Lesson 6: Ordered Pairs and Relations

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Relations And Functions

Related Pages More On Relations And Functions Graphs Of Functions Algebra Lessons

In these lessons, we will look at ordered-pair numbers, relations, and functions. We will also discuss the difference between a relation and a function, and how to use the vertical line test.

Ordered-Pair Numbers

An ordered-pair number is a pair of numbers that go together. The numbers are written within a set of parentheses and separated by a comma.

For example, (4, 7) is an ordered-pair number; the order is designated by the first element 4 and the second element 7. The pair (7, 4) is not the same as (4, 7) because of the different ordering. Sets of ordered-pair numbers can represent relations or functions.

A relation is any set of ordered-pair numbers.

The following diagram shows some examples of relations and functions. Scroll down the page for more examples and solutions on how to determine if a relation is a function.

Suppose the weights of four students are shown in the following table.

The pairing of the student number and his corresponding weight is a relation and can be written as a set of ordered-pair numbers. W = {(1, 120), (2, 100), (3, 150), (4, 130)}

The set of all first elements is called the domain of the relation. The domain of W = {1, 2, 3, 4}

The set of second elements is called the range of the relation. The range of W = {120, 100, 150, 130}

All functions are relations. A function is a relation in which no two ordered pairs have the same first element.

A function associates each element in its domain with one and only one element in its range.

Example: Determine whether the following are functions a) A = {(1, 2), (2, 3), (3, 4), (4, 5)} b) B = {(1, 3), (0, 3), (2, 1), (4, 2)} c) C = {(1, 6), (2, 5), (1, 9), (4, 3)}

Solution: a) A = {(1, 2), (2, 3), (3, 4), (4, 5)} is a function because all the first elements are different.

b) B = {(1, 3), (0, 3), (2, 1), (4, 2)} is a function because all the first elements are different. (The second element does not need to be unique)

c) C = {(1, 6), (2, 5), (1, 9), (4, 3)} is not a function because the first element, 1, is repeated.

Vertical Line Test

A function can be identified from a graph. If any vertical line drawn through the graph cuts the graph at more than one point, then the relation is not a function. This is called the vertical line test .

Determining Whether A Relation Is A Function

Understanding relations (defined as a set of inputs and corresponding outputs) is an important step to learning what makes a function. A function is a specific relation, and determining whether a relation is a function is a skill necessary for knowing what we can graph. Determining whether a relation is a function involves making sure that for every input there is only one output.

How To Determine If A Relation Is A Function?

A function is a correspondence between a first set, called the domain, and a second set, called the range, such that each member of the domain corresponds to exactly one member of the range.

The graph of a function f is a drawing hat represents all the input-output pairs, (x, f(x)). In cases where the function is given by an equation, the graph of a function is the graph of the equation y = f(x).

The vertical line test - a graph represents a function if it is impossible to draw a vertical line that intersects the graph more than once.

This Algebra 1 level math video tutorial

• defines a relation as a set of ordered pairs and a function as a relation with one to one correspondence.
• models how to determine if a relation is a function with two different methods.
• shows how to use a mapping and the vertical line test.
• discusses how to work with function notation. It is defined as replacing y in an equation that is a function.
• Using a mapping diagram, determine whether each relation is a function.
• Using a vertical line test, determine whether the relation is a function.
• Make a table for f(t) = 0.5x + 1. Use 1, 2, 3, and 4 as domain values.
• Evaluate the function rule f(g) = -2g + 4 to find the range for the domain (-1, 3, 5).

Determine If A Relation Is A Function

This video explains the concepts behind mapping a relation and the vertical line test.

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Curriculum  /  Math  /  7th Grade  /  Unit 1: Proportional Relationships  /  Lesson 6

Proportional Relationships

Lesson 6 of 18

Criteria for Success

Tips for teachers, anchor problems, problem set, target task, additional practice.

Lesson Notes

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Represent proportional relationships in graphs.

Common Core Standards

Core standards.

The core standards covered in this lesson

Ratios and Proportional Relationships

7.RP.A.2 — Recognize and represent proportional relationships between quantities.

7.RP.A.2.A — 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.

7.RP.A.2.D — Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.

Foundational Standards

The foundational standards covered in this lesson

6.RP.A.3.A — Make tables of equivalent ratios relating quantities with whole number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.

The essential concepts students need to demonstrate or understand to achieve the lesson objective

• Graph a proportional relationship in a coordinate plane using a table of values.
• Identify key features of the graph of a proportional relationship: a straight line that passes through the origin.
• Understand the graph of a proportional relationship to represent the set of equivalent ratios of the two quantities, represented as coordinate points.
• Interpret the coordinate point  $$(x,y)$$ as it relates to the context of the situation.

Suggestions for teachers to help them teach this lesson

• Lessons 6 and 7 focus on representing proportional relationships in the coordinate plane. Students may need to review concepts and skills around the coordinate plane in order to fully access this lesson.

Lesson Materials

• Graph Paper (2-3 sheets per student)
• Ruler (1 per student)

Unlock features to optimize your prep time, plan engaging lessons, and monitor student progress.

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

25-30 minutes

An amusement park sells tickets that can be used to go on rides and play games. The cost of the tickets is proportional to the number of tickets purchased. The table below shows how many tickets can be purchased for different amounts of money.

Graph the relationship in a coordinate plane. What features do you notice about the graph of a proportional relationship?

Guiding Questions

Student response.

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When Lili’s grandmother uses her rice cooker to make sushi rice, she uses a specific ratio of water to uncooked rice. The graph below shows this relationship.

a.   Record the coordinate points from the graph. What do these ordered pairs represent?

b.   What is the ratio of water to uncooked rice that Lili’s grandmother uses?

c.   If Lili added 6 cups of water to 5 cups of uncooked rice, did she use her grandmother’s ratio? Explain how you know by looking at the graph.

d.   What does the point (0, 0) represent in this situation? Why does it make sense that it is included on the graph?

A set of suggested resources or problem types that teachers can turn into a problem set

15-20 minutes

Give your students more opportunities to practice the skills in this lesson with a downloadable problem set aligned to the daily objective.

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

5-10 minutes

The graph below shows the relationship between the cost of gasoline and the number of gallons of gasoline purchased at a gas station.

a.   Is the cost of gasoline proportional to the amount of gasoline purchased? Explain how you know using the graph.

b.   How many gallons of gasoline can you purchase with $15?

c.   How much can you expect to pay for 3 gallons of gasoline?

d.   What are the coordinates of point P, shown in the graph? Describe what point P means in context of the situation.

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

• Include review problems of prior skills such as graphing coordinate points in the coordinate plane, determining and setting up appropriate scale intervals, naming coordinate points in the coordinate plane, distinguishing between points on the $$x$$ -axis and those on the $$y$$ -axis, etc.
• Include problems where students are given a table of values and are asked to represent the relationship in a graph.
• Include problems where students are given a graph of a proportional relationship and must write a table of values to accompany it. 
• Include problems where students interpret coordinate points  $$(x,y)$$ from both graphs and tables
• Illustrative Mathematics Art Class, Variation 1

Topic A: Representing Proportional Relationships in Tables, Equations, and Graphs

Solve ratio and rate problems using double number lines, tables, and unit rate.

7.RP.A.1 7.RP.A.2

Represent proportional relationships in tables, and define the constant of proportionality.

7.RP.A.2 7.RP.A.2.B

Determine the constant of proportionality in tables, and use it to find missing values.

7.RP.A.2.A 7.RP.A.2.B

Write equations for proportional relationships presented in tables.

7.RP.A.2.B 7.RP.A.2.C

Write equations for proportional relationships from word problems.

7.RP.A.2 7.RP.A.2.C

7.RP.A.2 7.RP.A.2.A 7.RP.A.2.D

Interpret proportional relationships represented in graphs.

7.RP.A.2 7.RP.A.2.D

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Topic B: Non-Proportional Relationships

Compare proportional and non-proportional relationships.

Determine if relationships are proportional or non-proportional.

Topic C: Connecting Everything Together

Make connections between the four representations of proportional relationships (Part 1).

7.RP.A.2 7.RP.A.2.A 7.RP.A.2.B 7.RP.A.2.C 7.RP.A.2.D

Make connections between the four representations of proportional relationships (Part 2).

Use different strategies to represent and recognize proportional relationships.

Topic D: Solving Ratio & Rate Problems with Fractions

Find the unit rate of ratios involving fractions.

Find the unit rate and use it to solve problems.

7.RP.A.1 7.RP.A.3

Solve ratio and rate problems by setting up a proportion.

Solve ratio and rate problems by setting up a proportion, including part-part-whole problems.

Solve multi-step ratio and rate problems using proportional reasoning, including fractional price increase and decrease, commissions, and fees.

Use proportional reasoning to solve real-world, multi-step problems.

7.RP.A.1 7.RP.A.2 7.RP.A.3

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