<p>In this lesson, students are asked to solve problems using proportional reasoning. Students will:&nbsp;<br>- understand how rates relate to proportions.<br>- set up proportions and apply them to solve problems.<br>- write proportions as equations in the form <i>y = kx</i>, and use the equations to calculate other converted measurements.&nbsp;<br>- determine the constant of variation between different representations.<br>&nbsp;</p>

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

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

<p>- one Conversion Chart (M-7-3-1_Conversion Chart) per student&nbsp;<br>- one Constant of Proportionality worksheet (M-7-3-1_Constant of Proportionality Practice and KEY) per student&nbsp;<br>- one Proportion Practice activity sheet (M-7-3-1_Proportion Practice and KEY) per student&nbsp;<br>- one Lesson 1 Exit Ticket (M-7-3-1_Lesson 1 Exit Ticket and KEY) per student&nbsp;<br>- one Lesson 1 Small Group Practice sheet (M-7-3-1_Small Group Practice and KEY) as needed&nbsp;<br>- one Expansion Work sheet (M-7-3-1_Expansion Work and KEY) as needed</p>

<p>- The Write-Pair-Share activities can be used to pre-test students' knowledge of ratios, rates, and unit rates.&nbsp;<br>- The Constant of Proportionality worksheet (M-7-3-1_Constant of Proportionality Practice and KEY) can be used to evaluate students' ability to recognize the constant, <i>k</i>, in various proportional representations.&nbsp;<br>- The Proportion Practice activity sheet (M-7-3-1_Proportion Practice and KEY) and the Guided Practice activity can be used to evaluate students' abilities to apply proportional reasoning to problem-solving situations.&nbsp;<br>- Use the Lesson 1 Exit Ticket (M-7-3-1_Lesson 1 Exit Ticket and KEY) to quickly evaluate student mastery.<br>&nbsp;</p>

<p>Scaffolding, Active Engagement, Modeling, Explicit Instruction, and Formative Assessment<br>W: Students will learn to represent proportions in different equation forms and apply proportions to solve problems. Students will also learn how to identify the constant of proportionality, or unit rate, in different representations.<br>H: To engage students, brainstorm the differences between the terms <i>ratio</i>, <i>rate</i>, and <i>unit rate</i> before beginning the lesson. They will be asked to provide some conversion rates. These brainstorming activities serve as a precursor to the proportional thinking that will follow.&nbsp;<br>E: The lesson focuses on applying proportions to solve problems. After reviewing the concepts of ratios, rates, and unit rates, students will be given the opportunity to use proportional reasoning to determine converted measurements. Students will create proportions to model each problem, and then write equations in the form <i>y = kx</i> to represent the same proportion. In doing so, the student must find the constant of proportionality. A further section follows, in which students determine the constant, k, from different representations of proportional relationships. Next, students will use proportional reasoning to solve more problems, such as those involving similar figures and scale factors. All of these activities will allow students to assess and apply what they've learned.&nbsp;<br>R: Students will review and revise their grasp of proportions and constants of proportionality while working through lesson problems and activities. The Constant of Proportionality and Proportion Practice sheets provide additional chances for students to study and practice the lesson concepts.&nbsp;<br>E: Student level of understanding may be evaluated by using the results of the Lesson 1 Exit Ticket.<br>T: Use the Extension section for ideas to customize the lesson to match the needs of your students. The Routine section includes strategies for reviewing lesson concepts with students throughout the school year. Students who require further practice may be divided into small groups to work on the Lesson 1 Small Group Practice sheet or any additional suggestions in the Small Group section. Students who are prepared for a challenge beyond the requirements of the standard may use activities in the Expansion section, including the Expansion Worksheet.&nbsp;<br>O: The lesson scaffolds students' understanding of ratios, rates, and unit rates before teaching them how to apply their knowledge of ratios to solving problems. Students initially learn how to set up a proportion using two ratios that are equal to one another. They then learn to write a linear equation to describe the proportion, allowing them to understand the relationship between the different equation forms for proportions. This lesson is intended to provide an overview of using proportions to solve problems. The next two lessons in the unit will look at whether two values are proportionally related and determine the meaning of the points on the line of the graph of a proportional relationship.<br>&nbsp;</p>

<p><strong>Write-Pair-Share Activity 1</strong><br><br>As an introduction to thinking about proportional relationships, have students describe and provide instances of <i>ratios</i>, <i>rates</i>, and <i>unit rates</i>. Students should also discuss the similarities and differences between the three terms. Allow students 2–3 minutes to record their ideas. Then ask partners to discuss their ideas. After about 5 minutes, ask one person from each pair to offer definitions and examples of the terms. Encourage discussion and debate.<br><br><strong>“</strong><i><strong>Rates</strong></i><strong> and </strong><i><strong>unit rates</strong></i><strong> are types of ratios. A </strong><i><strong>ratio</strong></i><strong> is simply the comparison of one value to another value. A rate is a comparison of two values expressed in different units. A unit rate is a type of rate with a denominator of one; in other words, a unit rate compares the value of one measurement to one of another type of measurement. The Venn diagram illustrates examples of ratios, rates, and unit rates:”</strong></p><figure class="image"><img style="aspect-ratio:495/277;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_20.png" width="495" height="277"></figure><p><strong>Write-Pair-Share Activity 2</strong><br><br>Ask students to think about customary and metric unit conversions they know, such as the fact that 12 inches equals 1 foot. Allow students 2-3 minutes to create a list of some conversions. Pair each student with a partner and allow them time to share their lists. After about 5 minutes, one member from each group should come to the front of the room and write the conversions in the appropriate categories, such as length/distance, weight, capacity, and time. When students have mentioned all they can think of, suggest anything else they missed. Request that students write the list of conversions in their notes or provide them with a prepared copy (M-7-3-1_Conversion Chart).<br><br>Encourage discussion about conversions and rates. Students should grasp that the list of conversions shows rates, which are calculated by comparing a measurement in one unit to a measurement in another unit. It is critical that students learn that these rates can be used in proportional reasoning.<br><br><strong>Converting Values within the Customary System</strong><br><br>After discussing ratios, rates, and unit rates, students may apply proportional reasoning to convert some values.&nbsp;<br><br><strong>"We are now going to apply proportional reasoning to convert measurements within the Customary System."</strong><br><br>Present the following problems:<br><br>5 yd = _____ ft &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (<i>15 ft</i>)<br><br>48 oz. = _____ lb &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(<i>3 lbs</i>)<br><br>30 fl oz. = _____ c &nbsp; &nbsp; &nbsp; &nbsp;(<i>\(3 {3 \over 4} \)&nbsp;c</i>)<br><br>4.5 lbs = _____oz. &nbsp; &nbsp; &nbsp; &nbsp;(<i>72 oz</i>)<br><br>For each example, present the following ideas:&nbsp;<br><br>1. Setting up the proportion.&nbsp;<br>2. To solve the proportion, use one of the methods listed below:&nbsp;<br>a. Fractional reasoning.&nbsp;<br>b. Inverse operations.&nbsp;<br>c. Cross-products&nbsp;<br>3. Identifying the proportionality constant.&nbsp;<br>4. Writing the proportional relationship as an equation in the form <i>y = kx</i>.&nbsp;<br>5. Using the equation to identify more converted measurements.&nbsp;<br><strong>"Let's look at the first problem:</strong><br><br>5 yd = _____ ft<br><br><strong>"To determine the number of feet in 5 yards, we will first set up a </strong><i><strong>proportion</strong></i><strong>. A proportion is an equation that equates two ratios (or fractions). For example, \(1 \over 2\)&nbsp;= \(2 \over 4\)&nbsp;is a proportion since the ratio on the left and right sides of the equal sign have the same value. The best way to set up a proportion is to consider the ratios that have been provided."</strong><br><br><br>1. Setting up the proportion.</p><p><strong>"Our example, 5 yd = ____ ft, can also be expressed as the ratio \(5(yd) \over x (ft)\) &nbsp;. So we have one ratio, but two are required to complete the proportion:</strong><br><br>\(5(yd) \over x (ft)\) =\(? \over ?\)&nbsp;<br><br><strong>"Can anyone think of another ratio that might deal with yards and feet?"</strong><br>(<i>There are 3 feet in 1 yard, or \(3(ft) \over 1 (yd)\).</i>)<br><br><strong>"Many of you know that there are 3 feet to every 1 yard. As a ratio, this is expressed as \(3(ft) \over 1 (yd)\) or \(1(yd) \over 3 (ft)\). Which version do you think we should choose for the second ratio in our example?"</strong> (<i>Use \(1(yd) \over 3 (ft)\) so the units match the first ratio.</i>)&nbsp;<br><br><strong>"When writing proportions, remember that the two ratios are EQUAL, so it is important that the numerator and denominator units of our two ratios match. As a result, we must use \(1(yd) \over 3 (ft)\) to complete the proportion."</strong><br><br>\(5(yd) \over x (ft)\)=\(1(yd) \over 3 (ft)\)<br><br><br>2a. Solving the proportion using fractional reasoning<br><br><strong>"Notice, of course, that we still need to find x. In this situation, finding x is simple as long as we remember that the two ratios in our proportion must have the same value. So, if 1 yard equals 3 feet, then 5 yards equals how many feet?</strong> (<i>15</i>) <strong>"In other words, \(5 \over 15\)&nbsp;= \(1 \over 3\), which is an accurate proportion because \(5 \over 15\) does indeed reduce to \(1 \over 3\), meaning that the two fractions are equal."</strong><br><br>\(5(yd) \over 15 (ft)\)=\(1(yd) \over 3 (ft)\)<br><br><br>2b. Solving the proportion using inverse operations<br><br><strong>"A proportion may be too complex to obtain the correct answer only by reasoning. Fortunately, there are additional methods for determining the unknown value in a proportion. A proportion is essentially an algebraic equation, and as you are aware, algebraic equations can be solved for a variable using inverse operations. We can use this method here. Follow along as I demonstrate the solution steps.</strong></p><figure class="image"><img style="aspect-ratio:1317/492;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_21.png" width="1317" height="492"></figure><p><br><br>2c. Solving the proportion using cross-products<br><br><strong>“There is yet another method for determining the unknown value in a proportion. This is accomplished through the use of what is known as cross-products. For any proportion, if \(a \over b\)=\(c \over d\), then </strong><i><strong>ad = bc</strong></i><strong>, (where </strong><i><strong>ad</strong></i><strong> and </strong><i><strong>bc</strong></i><strong> are referred to as cross-products because we are literally multiplying on the diagonal across the equal sign). With this information, we can easily rewrite any proportion so that its cross-products are equal. Let's use cross-products to solve our example proportion again.”</strong></p><figure class="image"><img style="aspect-ratio:650/469;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_22.png" width="650" height="469"></figure><p><br><br>3. Identifying the constant of proportionality<br><br><strong>"At this point, we have studied how to construct a proportion, as well as three distinct ways for solving the proportion. There is one more important idea about proportions to discuss: the constant of proportionality. The constant of proportionality in a proportion is essentially the value of the two ratios, given that one of the ratios is represented as a unit rate. First of all, let's review that a unit rate is a simplified rate with a denominator of 1. Currently, we have been using the following proportion:</strong><br><br>\(5(yd) \over 15 (ft)\)=\(1(yd) \over 3 (ft)\)<br><br><strong>"In this proportion, can we say that either side of the equation represents a unit rate?"</strong> (<i>No, because none of the denominators equals 1.</i>) <strong>"But there's an easy solution. Observe what happens when we turn both ratios in the proportion upside down:</strong><br><br>\(15(ft) \over 5(yd)\)=\(1(yd) \over 3(ft)\)<br><br><strong>"Notice that the proportion is still true, as \(15 \over 5\)&nbsp; = \(3 \over 1\)&nbsp;. The second ratio, however, can now be referred to as a unit rate because the denominator equals 1. Because my proportion is now such that one of the two ratios has a denominator of 1, I am ready to calculate the constant of proportionality. What is the value of each ratio in the proportion?</strong> (<i>3</i>) <strong>Thus, 3 is the constant of proportionality."</strong><br><br><br>4. Writing the proportional relationship as an equation in the form of <i>y = kx</i><br><br><strong>"Once we have the constant of proportionality, we may write our proportion as </strong><i><strong>y = kx</strong></i><strong>, where k is the constant of proportionality and x and y are our independent and dependent variables, as usual. For our example, the equation would be:</strong>&nbsp;<br><br><i>y = </i>3<i>x</i>&nbsp;<br><br><strong>"So, how can understanding the equation </strong><i><strong>y = </strong></i><strong>3</strong><i><strong>x</strong></i><strong> help us? What can we do with this equation?"</strong> Allow time for discussion and debate.&nbsp;<br><br><strong>"We may use the equation to determine additional converted measurements. We can state in words: </strong><i><strong>y</strong></i><strong> feet equals 3 times </strong><i><strong>x</strong></i><strong> yards. Alternatively, if we want to calculate the number of yards in a given number of feet, we can substitute the number of feet for </strong><i><strong>y</strong></i><strong> and solve for </strong><i><strong>x</strong></i><strong>."</strong><br><br><strong>"For example, consider the problem:</strong><br><br>_____ yd = 30 ft<br><br><strong>"Substituting 30 into the equation </strong><i><strong>y = </strong></i><strong>3</strong><i><strong>x</strong></i><strong> gives 30 = 3</strong><i><strong>x</strong></i><strong>. Solving for </strong><i><strong>x</strong></i><strong> gives </strong><i><strong>x</strong></i><strong> = 10. Thus:</strong>&nbsp;<br><br>10 yd = 30 ft&nbsp;<br><br><strong>"Now, let's look at the following problem:</strong>&nbsp;<br><br>48 oz = _____ lbs&nbsp;<br><br><br>1. Setting up the proportion.&nbsp;<br><br><strong>"First, we will create a proportion based on the ratio we were given:</strong><br><br>\(48(oz) \over x (lbs)\)=\(? \over ?\)<br><br><strong>"Now we need to conceive of another ratio that compares ounces and pounds. There are 16 ounces in 1 pound, so we may use this as the second ratio:</strong><br><br>\(48(oz) \over x (lbs)\)=\(16(oz) \over 1(lb)\)</p><p><br>2. Solving the proportion<br><br><strong>"Now that we know our proportion, we can solve for </strong><i><strong>x</strong></i><strong>. If we use fractional reasoning, we may realize that 48 is 3 times more than 16. This means that </strong><i><strong>x</strong></i><strong> must be 3 times more than 1 (to keep the ratios equal). Thus, </strong><i><strong>x</strong></i><strong> = 3."&nbsp;</strong><br><br><strong>"We could also use cross-products to solve the proportion. To do this, we would rewrite the proportion as a statement indicating that the cross-products are equal:</strong><br><br>\(48(oz) \over x (lbs)\)=\(16(oz) \over 1(lb)\)</p><p>\(48 \over x \)=\(16 \over 1\)</p><p>48(1) = <i>x</i>(16)</p><p>48 = 16x<br><br><strong>"Now, we solve the new equation for x.</strong><br><br>\(48 \over 16 \)=\(16x \over 16\)</p><p>3 = <i>x</i><br><br><strong>"No matter the method we use to solve the proportion, it is obvious that </strong><i><strong>x</strong></i><strong> = 3.</strong><br><br><strong>Thus:</strong><br><br>48 oz = 3 lbs<br><br><strong>and&nbsp;</strong><br><br>\(48(oz) \over 3 (lbs)\)=\(16(oz) \over 1(lb)\)<br><br><br>3. Identifying the constant of proportionality<br><br><strong>"Now that we have completed our proportion, we may calculate the constant of proportionality, </strong><i><strong>k</strong></i><strong>, and then write an equation in the form </strong><i><strong>y = kx</strong></i><strong>. Remember that to find the constant of proportionality, determine the value of each ratio in the proportion with a denominator of 1. Here, the second ratio is already expressed as a unit rate, and each ratio has a value of 16."</strong><br><br><br>4. Writing the proportional relationship as an equation in the form of <i>y = kx</i><br><br><strong>"Thus:</strong>&nbsp;<br><br><i>y = 16x</i><br><br><strong>"We can express in words:</strong>&nbsp;<br><br><i>y ounces equal 16 times x pounds</i><br><br><strong>"Now, you will try some examples with a partner."</strong> Instruct pairs of students to follow the same steps as listed above (#1-4) for each of the following cases. The answers are shown in the table below. As students work, circulate around the classroom to assess comprehension and answer any questions.</p><figure class="image"><img style="aspect-ratio:1318/335;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_23.png" width="1318" height="335"></figure><p>Note: in each case, point out that the unit rate is the constant of proportionality. This concept will be explored in Lesson 3.&nbsp;<br><br>Provide 5-10 more examples for practice. Monitor students' progress to ensure comprehension.<br><br><br><strong>Identifying the Constant of Proportionality in Tables, Graphs, Equations, and Verbal Descriptions</strong><br><br>Now that students have had an opportunity to see the practical use of the constant of proportionality and have a conceptual knowledge of the term, provide several representations of proportional relationships and ask students to identify the constant.&nbsp;<br><br><strong>"Consider this proportional relationship: A driver drives at a speed of 65 miles per hour. The described rate (65) serves as the proportionality constant. Let's look at this relationship in a table:</strong></p><figure class="image"><img style="aspect-ratio:302/235;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_24.png" width="302" height="235"></figure><p><strong>"The proportionality constant is the ratio of the change in </strong><i><strong>y</strong></i><strong>-values to the change in corresponding </strong><i><strong>x</strong></i><strong>-values. Because this table displays </strong><i><strong>x</strong></i><strong>-values that increase by 1, the difference in consecutive </strong><i><strong>y</strong></i><strong>-values illustrates the rate of change, </strong><i><strong>k</strong></i><strong>. The constant of proportionality, or unit rate, is also represented by the </strong><i><strong>y</strong></i><strong>-value given for the </strong><i><strong>x</strong></i><strong>-value of 1."&nbsp;</strong><br><br><strong>"Because we have identified that our constant of proportionality is 65, this proportional relationship can be represented by the equation </strong><i><strong>y</strong></i><strong> = 65</strong><i><strong>x</strong></i><strong>."</strong><br><br><br><strong>"Now, have a look at a graph showing this relationship:</strong></p><figure class="image"><img style="aspect-ratio:276/250;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_25.png" width="276" height="250"></figure><p><strong>"Notice that between any two consecutive points on the graph, there is a a vertical difference of 65 and a horizontal difference of 1. As a result, the ratio of the change in y-values to x-values is \(65 \over 1\)&nbsp;, or simply, 65. Again, because the point (1, 65) is given, we can conclude from this one point that the constant of proportionality is 65."</strong>&nbsp;<br><br>Give them a couple additional examples of tables and graphs and ask them to determine the constant of proportionality from each representation. All representations are not necessarily connected. In other words, display a table indicating one proportion, a graph representing another, an equation expressing yet another proportion, and so on. The first example used representations of the same proportion, allowing students to quickly compare the presence of the constant in a description, table, equation, and graph. Include tables that don't display consecutive x-values. Give students the Constant of Proportionality worksheet (M-7-3-1_Constant of Proportionality Practice and KEY).&nbsp;<br><br><br><strong>Solving with Proportions&nbsp;</strong><br><br><strong>"We used proportional reasoning to find the converted measurements. Now, let's look at some other applications of proportional reasoning."</strong>&nbsp;<br><br>Consider the following examples. Ask students to write a proportion for each and then solve.<br><br><strong>If John orders 4 pizzas for $28, how many can he get for $70?</strong> (<i>\(4 \over 27\)&nbsp;= \(x \over 70\)&nbsp;; 10 pizzas</i>)<br><br><strong>The cafeteria served 580 lunches. 45 of these were sandwiches. If 696 lunches will be served tomorrow, how many would be expected to be sandwiches?</strong> (<i>\(580 \over 45\)&nbsp;= \(696 \over x\)&nbsp;; 54 sandwiches</i>)<br><br><strong>Lisa ran 2.2 miles in 14 minutes. At this rate, how long would it take her to run 3.75 miles?</strong> (<i>\(2.2 \over 14\)&nbsp;= \(3.75 \over x\)&nbsp;; about 23.9 minutes, round to the nearest tenth</i>)<br><br>If further teaching and practice are required, review additional examples. When you're confident that students understand how to correctly set up and solve proportions of this type, present similar figures.<br><br><br><strong>Similar Figures and Proportional Reasoning</strong><br><br><strong>"Another useful application of proportional reasoning is to solve for missing values in mathematically similar figures. Similar figures have corresponding congruent angles and proportional sides. As a result, each pair of corresponding sides has the same ratio. You can use the ratios to create a proportion to solve for missing side lengths."&nbsp;</strong><br><br><strong>“The figures are similar. Find the missing side length using the side ratio to create a proportion.”</strong></p><figure class="image"><img style="aspect-ratio:398/224;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_26.png" width="398" height="224"></figure><p>Allow students time to write the proportion and determine the solution.&nbsp;<br><br><strong>"The following proportion can be used to solve for the missing height:</strong><br><br>\(x \over 28\)&nbsp;= \(36 \over 24\)&nbsp;<br><br><strong>"Using cross-products, this proportion simplifies to 24</strong><i><strong>x</strong></i><strong> = 1008, with </strong><i><strong>x</strong></i><strong> = 42. Thus, the larger triangle has a height of 42 inches."&nbsp;</strong><br><br><strong>"The proportional relationship is expressed by the equation </strong><i><strong>y</strong></i><strong> = 1.5</strong><i><strong>x</strong></i><strong>, where 1.5 is the proportionality constant and </strong><i><strong>x</strong></i><strong> is the smaller triangle's dimension. Thus, the base of the smaller triangle multiplied by 1.5 gives the base of the larger triangle. The height of the smaller triangle multiplied by 1.5 gives the height of the larger triangle."</strong><br><br><br><strong>Guided Practice</strong><br><br>Write 2-3 more examples of similar figures on the board. In each case, ask students to find the missing side length(s) using proportions created from the side length ratios. Keep an eye on students while they are working. To assess comprehension, ask each student or small group questions regarding the process being used. Assist as needed.&nbsp;<br><br>Students should work with a partner to complete the Proportion Practice activity sheet (M-7-3-1_Proportion Practice and KEY).<br><br><br><strong>Proportional Reasoning and Scales</strong><br><br><strong>"Scale drawings and maps also represent proportional relationships. Suppose a map has a key in which 1 inch = 60 miles."</strong>&nbsp;<br><br>Have students respond to the following questions.<br><br><strong>What do you think a 2-inch segment on the map would represent in real life?</strong> (<i>120 miles</i>)&nbsp;<br><strong>... a segment of \(3 {1 \over 2} \) inches?</strong> (<i>210 miles</i>)<br><strong>... a segment of 7.5 inches?</strong> (<i>450 miles</i>)<br><strong>What is the distance on a map between two cities that are 90 miles apart in real life?</strong> (<i>\(1 {1 \over 2} \) inches</i>)&nbsp;<br><strong>How long would the segment have to be to represent the width of a state that is 300 miles across?</strong> (<i>5 inches</i>)&nbsp;<br><strong>If a mountain region is 45 miles long in real life, how long would it be on a map with this scale?</strong> (<i>\(3 \over 4\) inch</i>)<br>Have students complete the Lesson 1 Exit Ticket (M-7-3-1_Lesson 1 Exit Ticket and KEY) at the end of the lesson to assess their understanding.<br><br><strong>Extension:</strong><br><br>The Routine section provides suggestions for how to review lesson concepts throughout the school year. The Small Group section includes suggestions for providing additional learning opportunities to students who would benefit from them. The Expansion section presents a challenge for students who are willing to go beyond the requirements of the standard.&nbsp;<br><br><strong>Routine:</strong> Throughout the school year, students should identify proportional relationships in the real world. For example, cumulative savings that increase by a constant rate each month represent a proportional relationship. Additionally, when working with patterns, students should recognize sequences that are proportional. The relationship between patterns and proportionality is very important. Such a discussion might be substituted for any part of this lesson or added as an extension. Students will gain additional experience evaluating proportional relationships when they determine proportionality in lesson 2 and analyze the significance of points on a proportional relationship graph in lesson 3.&nbsp;<br><br><strong>Small Groups:</strong> Students that require further practice may be divided into small groups to complete the following activity: Small Group Practice (M-7-3-1_Small Group Practice and KEY). Students can work on the problems together or independently, and then compare answers when done.<br><br><strong>Expansion:</strong> Students who are ready for a greater challenge may be assigned the Expansion Work worksheet (M-7-3-1_Expansion Work and KEY). The worksheet offers additional problems about proportionality and the constant of proportionality, as well as prompts students to create their own representations of proportional relationships.</p>