Students will decide if two quantities are proportionally connected. Students will: 
- determine the proportionality of relationships using a range of representations, such as written ratios, verbal descriptions, equations, tables, and graphs. 

- How are relationships represented mathematically? 
- How are expressions, equations, and inequalities used to quantify, solve, model, and/or analyze mathematical problems? 
- How can mathematics help us communicate more effectively? 
- How can recognizing repetition or regularity assist in solving problems more efficiently?
- How can mathematics help to quantify, compare, depict, and model numbers?
- What does it mean to analyze and estimate numerical quantities?
- What makes a tool and/or strategy suitable for a certain task?
 

- Proportion: An equation of the form \(a \over b\) = \(c \over d\) that states that the two ratios are equivalent. 
- Ratio: A comparison of two numbers by division. 
- Unit Rate: A rate simplified so that it has a denominator of 1.

- one Is It Proportional sheet (M-7-3-2_Is It Proportional and KEY) per student 
- one Lesson 2 Exit Ticket (M-7-3-2_Lesson 2 Exit Ticket and KEY) per student 
- copies of Small Group Practice worksheet (M-7-3-2_Small Group Practice and KEY) as needed 
- copies of the Expansion Work sheet (M-7-3-2_Expansion Work and KEY) as needed

- Use the Is It Proportional worksheet (M-7-3-2_Is It Proportional and KEY) to assess students' comprehension of how to test relations for proportionality.
- The essay assignment can be used to evaluate students' conceptual understanding of proportionality and representations of proportion.
- Use the Lesson 2 Exit Ticket (M-7-3-2_Lesson 2 Exit Ticket and KEY) to assess student understanding of lesson concepts.
 

Scaffolding, Active Engagement, Metacognition, and Formative Assessment
W: Students will learn how to identify proportionality in representations. In other words, students will assess whether two quantities are proportionally related.
H: Students will engage with this lesson by exploring strategies to identify if a representation shows a proportionate relationship. Before receiving official training, students will be given the opportunity to write and discuss what they already know about the topic. 
E: The lesson focuses on determining if quantities are proportionally related using various representations. Students will be exposed to many different examples of relationships, and they will discuss and decide whether the relationship is proportional or not. Examples include written ratios, descriptions, equations, tables, and graphs.
R: The Write-Pair-Share Activity provides an opportunity for discussion at the beginning of the lesson. Students will have numerous opportunities to discuss and make decisions regarding the proportionality of a relationship. During these activities and discussions, students will review, rethink, and revise their understanding of proportionality. The Essay Activity will require students to connect the concepts covered in the lesson. 
E: Evaluate student comprehension using the Lesson 2 Exit Ticket.
T: Use the suggestions in the Extension section to personalize the lesson to the students' requirements. The Routine section offers opportunities to review lesson content throughout the school year. The Small Group section is designed for students who would benefit from more learning opportunities on the course topic. The Expansion section is intended to challenge students who are willing to go above and beyond the requirements of the standard. 
O: The lesson is scaffolded to ensure students review their prior knowledge of proportional relationships. Students build on their prior knowledge by working through explicit examples of proportional and nonproportional relationships, discussing the examples and explaining reasons for the presence or absence of proportionality. 
 

Write-Pair-Share Activity

This activity can be used to assess students' prior knowledge of proportionality testing. 

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

"In this lesson, we will assess whether two quantities are proportionally related. We will analyze written ratios, verbal descriptions, equations, tables, and graphs. The goal of this lesson is for you to be able to look at any form of a relation and evaluate whether it reflects a proportion. For each example given, we will examine whether it represents a proportional relationship. We'll also justify our reasoning." 

Allow students time to answer, discuss, and ask any questions before confirming each answer. This part of the lesson is designed 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: \(8 \over 12\) and \(16 \over 24\)
- "These ratios are proportionally related since they are equivalent. Each ratio represents the same amount, or \(2 \over 3\)." 
- "Another way to determine that these ratios form a proportion is to examine the cross-products. In Lesson 1, we discovered that for any proportion \(a \over b\) = \(c \over d\), ab = cd. Here, 8(24) = 192 and 12(16) = 192, so the cross-products are equal, meaning that the ratios form a proportion."

Example 2: 9:20 and 18:27
- “These ratios are not proportionally related since they are not equivalent. \(9 \over 20\) and \(18 \over 27\) = 0.666…, thus \({9 \over 20}\neq {18 \over 27}\)” 
- "9(27) = 243 and 20(18) = 360; thus, the cross-products are not equal, indicating that the ratios do not form a proportion." 

Example 3: 4:15 and 12:45
- "These ratios are proportionally related since they are equivalent. The ratio \(12 \over 45\) reduces to the ratio \(4 \over 15\)."
- "12(15) = 180 and 45(4) = 180; thus, the cross-products are equivalent, implying that the ratios form a proportion."

Example 4: 6/12 and 9/15
- "These ratios are not proportionally related because they are not equivalent. \(6 \over 12\) = \(1 \over 2\) and \(9 \over 15\) = \(3 \over 5\), thus \({6 \over 12}\neq {9 \over 15}\)”
- "6(15) = 90 and 12(9) = 108; thus, the cross-products are not equal, indicating that 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 represents a proportional relationship. The ratio of \(12 \over 4\) is the same as the ratio of \(3 \over 1\). 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 represents a proportional relationship. The ratio \(3 \over 1200\) is equivalent to the ratio \(7 \over 2800\). Each ratio has a value of \(1 \over 400\)."

Example 7: 
4 pizzas for 16 people
9 pizzas for 42 people

- "This does not represent a proportional relationship. The ratio \(4 \over 16\) is not equivalent to the ratio \(9 \over 42\). The fraction \(4 \over 16\) reduces to \(1 \over 4\), whereas the fraction \(9 \over 42\) reduces to \(3 \over 14\)."

Example 8: 
35 proposals to 7 employees
105 proposals to 21 employees

- "This represents a proportional relationship. The ratio \(35 \over 7\) is equivalent to the ratio \(105 \over 21\). Each ratio has a value of 5."


Equations 

"Now, we're ready to determine whether equations represent proportional relationships." 

Example 9: y = 7x
- "This equation represents a proportional relationship with a constant rate of change and a y-intercept of 0. In other words, no amount is added or removed from the term containing the constant rate of change, 7x. Another way to recognize that this equation represents a proportional relationship is that it takes the form y = kx, where k is the proportionality constant (in this case, 7)."

Example 10: y = 3x + 4
- "This equation does not represent a proportional relationship because the y-intercept is not 0. The y-intercept is 4, indicating that the graph crosses the y-axis at (0, 4), not (0, 0). This equation is not in the form of y = kx, but rather in the form y = mx + b, indicating that a constant term has been added to or subtracted from the term with the x."

Example 11: y = \(1 \over 2\)x – 2
- "This equation does not represent a proportional relationship because the y-intercept is not 0. The y-intercept is -2, indicating that the graph crosses the y-axis at (0, -2) not (0, 0). This equation is not in the form of y = kx, but rater in the form y = mx + b, meaning a constant term has been added to or subtracted from the term with the x."

Example 12: y = \(3 \over 4\)x
- "This equation illustrates a proportional relationship because the rate of change is constant and the y-intercept is 0. In other words, no amount is added or subtracted from the term containing the constant rate of change, \(3 \over 4\)x."


Tables 

"Now, we're ready to look at some tables of values to determine whether they represent proportional relationships."


"Look at this table and see if it indicates a proportional relationship. How can you tell?"

Example 13:

- "This table is easy to interpret. We are provided the y-intercept, or the point at which the graph crosses the y-axis; hence, simply looking at the first row of the table (0, 0), we can see that the relationship satisfies one of the proportionality requirements: the y-intercept is zero. As x-values increase by 1, y-values increase at a constant rate of 3. This satisfies the other requirement for proportionality: a consistent rate of change. Thus, we might conclude that this table represents a proportional relationship."
- "We can check our decision by ensuring that the ratios of all x-values to their corresponding y-values are equivalent. We could write the following: \(1 \over 3\) = \(2 \over 6\) = \(3 \over 9\) = \(4 \over 12\) = \(5 \over 15\). This statement is true. Each ratio has a value of \(1 \over 3\). We have now confirmed that this table represents a proportional relationship."

Example 14:

- "With this table, we can see that the constant rate of change of 6. However, we must confirm that the y-intercept is 0. We can do this by comparing x-value ratios to their corresponding y-values. In a proportional relationship, the ratios will be equivalent, implying that the y-intercept is truly 0. Let's compare \(1 \over 8\) and \(2 \over 14\). Are these ratios equivalent?" (No) "The ratio \(2 \over 14\) equals \(1 \over 7\), not \(1 \over 8\). To find the y-intercept, subtract 6 from 8, which shows that an x-value of 0 corresponds to a y-value of 2, not 0. This again confirms our decision that the table does not represent a proportional relationship." 

Many students will just check the table for a constant rate of change and then declare the relationship to be proportional. It is important that they realize that the table must contain the ordered pair (0, 0). The y-intercept must be at zero. Otherwise, the table is simply a linear equation that is not proportional. This is an important distinction to note: while all proportions are linear, 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 how the x-values in this table are not consecutive. For this one, it will be simpler to compare x-value ratios to y-values. Let's compare \(2 \over -8\) and \(5 \over -14\). Are the ratios equal?" (No) "Therefore, we can conclude that this table does not represent a proportional relationship. We don't need to look much further."

Example 16:

- "Because this table does not display consecutive x-values, we may simply compare ratios of x-values to y-values. Let's compare \(2 \over 10\) and \(5 \over 25\). Both ratios have a value of \(1 \over 5\). So, it looks like the table represents a proportional relationship. But let's be sure by comparing some additional ratios. We may write: \(7 \over 35\) = \(9 \over 45\) = \(11 \over 55\) = \(15 \over 75\). Notice that all of the remaining ratios have a value of as well. Thus, we might say that this table represents a proportion."


Graphs 

"Graphs are very easy to check for proportionality. There are just two questions we must ask ourselves. 1) Does the graph represent a straight line? 2) Does the graph intersect the y-axis at (0,0)? In other words, does the linear graph go through the origin? If it does, the graph shows a proportional relationship. If not, then the graph does not represent a proportional relationship. It is that simple."

Example 17: 

"The graph does not represent a proportional relationship. It does not go 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 be classified as a proportional relationship. This one fails the proportionality test."

Example 18:

"The graph represents a proportional relationship. It's a straight line that goes through the origin, or the point (0, 0). In other words, the y-intercept equals zero."

Example 19:

"The graph does not represent a proportional relationship. Although it reaches the origin, the graph is a curve, not a straight line. Therefore, it is not proportional. All proportional relationships are linear."

Example 20:

"The graph represent a proportional relationship. It's a line that goes through the origin, or the point (0, 0). In other words, the y-intercept equals zero."


Essay Activity 

Ask students to write a brief essay on the definition of proportionality and how to determine proportionality of given representations. Students should provide examples in their essays. Each essay can be published to the class discussion board. Students may then have an opportunity to agree or disagree with their classmates' viewpoints. 


Students should complete the Lesson 2 Exit Ticket (M-7-3-2_Lesson 2 Exit Ticket and KEY) at the end of the lesson to assess their understanding.


Extension:

Use the suggestions in this section to customize the lesson to match the needs of your students.

Routine: Have students revisit the concept of proportionality as they work with linear equations throughout the school year. Students can compare those that are proportional with those that are not. The relationship between linear equations with a y-intercept of 0 and proportional relationships is important to emphasize all year long. Students can also determine if patterns are proportional. For example, students can decide if the square numbers show a proportional relationship. 

Small Groups: Students who require further practice may be divided into small groups to complete the Small Group Practice worksheet (M-7-3-2_Small Group Practice and KEY). Students can work on the problems together or independently, and then compare their answers when done.

Expansion: Students who are prepared for a challenge beyond the requirements of the standard may be assigned the Expansion Work sheet (M-7-3-2_Expansion Work and KEY). The worksheet contains additional representations for which proportionality must be calculated.