• Use the Is It Proportional worksheet (M-7-3-2_Is It Proportional and KEY.docx) to evaluate student understanding of how to test relations for proportionality.
• The essay activity may be used to assess students’ conceptual understanding of proportionality and representations of proportions.
• Use the Lesson 2 Exit Ticket (M-7-3-2_Lesson 2 Exit Ticket and KEY.docx) to gauge student proficiency with lesson concepts.

Write-Pair-Share Activity

This activity may be used to pre-assess students’ knowledge related to testing for proportionality.

Give the Is It Proportional sheet (M-7-3-2_Is It Proportional and KEY.docx) to all students. Ask them to take 5 minutes to write down some descriptions how they can determine if different representations show a proportional relationship or not. Students should include examples, if possible. Next, have students share ideas and examples with a partner. After about 5 more minutes, the class may reconvene. One member from each group should share their ideas on determining proportionality. Encourage discussion and debate.

“In this lesson, we are going to determine whether two quantities are proportionally related. We will look at written ratios, verbal descriptions, equations, tables, and graphs. The intent of the lesson is for you to be able to look at any form of a relation and determine if it represents a proportion. For each example given, we will determine if it represents a proportional relationship. We will also justify our thinking.”

Give students time to provide answers, discuss, and ask any questions, prior to confirming each answer. This part of the lesson is intended for whole class discussion and participation.

Written Ratios

“Let’s look at some written ratios and decide whether or not they are proportional.”

• Example 1:       and
  • “These ratios are proportionally related because they are equivalent. Each ratio equals the same amount, or .”
  • “Another way to determine that these ratios form a proportion is by investigating the cross-products. In Lesson 1, we learned that for any proportion . Here, 8(24) = 192 and 12(16) = 192; hence, the cross-products are equal, meaning the ratios form a proportion.”
• Example 2:      9:20 and 18:27
  • “These ratios are not proportionally related because they are not equivalent.  and , thus: .
  • 9(27) = 243 and 20(18) = 360; hence, the cross-products are not equal, meaning the ratios do not form a proportion.”

• Example 3:      4:15 and 12:45
  • “These ratios are proportionally related because they are equivalent. The ratio  reduces to the ratio .
  • “12(15) = 180 and 45(4) = 180; hence the cross-products are equal, meaning the ratios form a proportion.”
• Example 4:       and
  • “These ratios are not proportionally related because they are not equivalent.  and , thus .
  • 6(15) = 90 and 12(9) = 108; hence, the cross-products are not equal, meaning the ratios do not form a proportion.”

Verbal Descriptions

“Now we’ll look at some verbal descriptions and decide whether or not they describe a proportional relationship.”

• Example 5:      12 apples: $4.00

3 apples: $1.00

• “This describes a proportional relationship. The ratio of  is equivalent to the ratio of . Each ratio has a value of 3.”

• Example 6:      3 tanks of gas for every 1200 miles driven

7 tanks of gas for every 2800 miles driven

• “This describes a proportional relationship. The ratio  is equivalent to the ratio . Each ratio has a value of .”

• Example 7:      4 pizzas for 16 people

9 pizzas for 42 people

• “This does not describe a proportional relationship. The ratio  is not equivalent to the ratio . The fraction  reduces to , while the fraction  reduces to .”

• Example 8:      35 proposals to 7 employees

105 proposals to 21 employees

• “This describes a proportional relationship. The ratio  is equivalent to the ratio . Each ratio has a value of 5.”

Equations

“Now, we’re ready to determine whether equations represent proportional relationships.”

• Example 9:     
  • “This equation represents a proportional relationship because it has a constant rate of change and a y-intercept of 0. In other words, no amount is added to or subtracted from the term containing the constant rate of change, 7x. Another way to recognize that this equation represents a proportional relationship is to see that it is in the form of y = kx, where k is the constant of proportionality (in this case, 7).”
• Example 10:   
  • “This equation does not represent a proportional relationship because the y-intercept is not 0. The y-intercept is 4, indicating the graph crosses the y-axis at the point, (0, 4), not (0, 0). This equation is not in the form of y = kx, but rather in the form y = mx + b, meaning a constant term has been added to or subtracted from the term with the x.”
• Example 11:   
  • “This equation does not represent a proportional relationship because the
    y-intercept is not 0. The y-intercept is −2, indicating the graph crosses the
    y-axis at the point, (0, −2), not (0, 0). This equation is not in the form of
    y = kx, but rather in the form y = mx + b, meaning a constant term has been added to or subtracted from the term with the x.”
• Example 12:   
  • “This equation represents a proportional relationship because it has a constant rate of change and a y-intercept of 0. In other words, no amount is added to or subtracted from the term containing the constant rate of change, .”

Tables

“Now, we’re ready to look at some tables of values to determine whether they represent proportional relationships.”

“Look at this table and determine if it represents a proportional relationship. How can you tell?”

• Example 13:

• “This table is easy to interpret. We are given the y-intercept, or point at which the graph crosses the y-axis; so we know just by looking at the first row of the table (0, 0) that the relationship satisfies one requirement of proportionality: the y-intercept is zero. As the x-values increase by 1, the
  y-values increase by a constant rate of 3. This satisfies the other requirement of proportionality: a constant rate of change. Thus, we may declare that this table represents a proportional relationship.”
• “We can check our decision by making sure that the ratios of all x-values to corresponding y-values are equivalent. We may write the following: . This statement is true. Each ratio has a value of . We have now confirmed our decision that this table represents a proportional relationship.”

• Example 14:

• “With this table, we see there is a constant rate of change of 6. However, we must confirm that the y-intercept is 0. We may do this by comparing ratios of x-values to corresponding y-values. In a proportional relationship, the ratios will be equivalent, indicating the y-intercept is indeed 0. Let’s compare  and . Are these ratios equivalent?” (No) “The ratio  equals , not . If we wish to find the y-intercept, we would subtract 6 from 8, showing the x-value of 0 to correspond to the y-value of 2, not 0. This again confirms our decision that this table does not represent a proportional relationship.”

Many students will simply check to see if there is a constant rate of change present in the table and then declare the relationship to be proportional. It is important that they understand the table must represent the ordered pair (0, 0). The y-intercept must be at zero. Otherwise, the table simply represents a linear equation that is not proportional. This is an important distinction to make: all proportions are linear, but not all linear equations are proportional. If students are unsure, they should check the equivalence of ratios of x-values to corresponding y-values.

• Example 15:

• “Notice the x-values in this table are not consecutive. For this one, it will be easier to simply compare ratios of x-values to y-values. Let’s compare  and . Are these ratios equal?” (No) “So, we can declare that this table does not represent a proportional relationship. We do not need to look any further.”

• Example 16:  

• “Since this table does not show consecutive x-values, we may again simply wish to compare ratios of x-values to y-values. Let’s compare  and . Both of these ratios have a value of . So, it looks like the table represents a proportional relationship. But let’s make sure by comparing some more ratios. We may write: . Notice that all of the remaining ratios have a value of , as well. Thus, we can declare that this table represents a proportion.”

Graphs

“Graphs are very easy to check for proportionality. There are only two questions we must ask ourselves. 1) Is the graph a straight line? 2) Does the graph cross the y-axis at the point (0, 0)? In other words, does the linear graph pass through the origin? If it does, the graph represents a proportional relationship. If it does not, the graph does not represent a proportional relationship. It is that simple.”

• Example 17:

• “This graph does not represent a proportional relationship. It does not pass through the origin, or the point, (0, 0). In other words, the y-intercept is not zero. It IS a straight line, but it must also pass through the origin to qualify as a proportional relationship. This one fails the test of proportionality.”

• Example 18:

• “This graph represents a proportional relationship. It is a straight line that passes through the origin, or the point, (0, 0). In other words, the y-intercept is zero.”

• Example 19:   

• “This graph does not represent a proportional relationship. Although it touches the origin, the graph is a curve, not a straight line. Thus, it is not proportional. All proportional relationships are linear.”

• Example 20:

• “This graph represents a proportional relationship. It is a line that passes through the origin, or the point, (0, 0). In other words, the y-intercept is zero.”

Essay Activity

Ask students to write a short essay on the meaning of proportionality and ways to discern proportionality of given representations. Students should describe examples in their essays. Each essay may be uploaded to the class discussion board. Students may then have an opportunity to agree or dissent with classmates’ views.

Have students complete the Lesson 2 Exit Ticket (M-7-3-2_Lesson 2 Exit Ticket and KEY.docx) at the close of the lesson to evaluate student level of understanding.

Extension:

Use the suggestions in this section to tailor the lesson to meet the needs of the students.

• Routine: Have students revisit the idea of proportionality when working with linear equations throughout the school year. Students may compare those that are proportional with those that are not. The connection between linear equations with a y-intercept of 0 and proportional relationships is an important one to promote all year long. Students may also determine whether patterns are proportional. For example, students may determine whether the square numbers represent a proportional relationship.
• Small Groups: Students who need additional practice may be pulled into small groups to work on the Small Group Practice worksheet (M-7-3-2_Small Group Practice and KEY.docx). Students can work on the problems together or work individually and compare answers when done.
• Expansion: Students who are prepared for a challenge beyond the requirements of the standard may be given the Expansion Work sheet (M-7-3-2_Expansion Work and KEY.docx). The worksheet includes more representations for which proportionality must be determined.