<p>This lesson teaches students about the language and notation of ratios. Students will:&nbsp;<br>- describe ratio relationships between two quantities.<br>- ratios can be expressed in a variety of forms, i.e., 3 to 4, 3:4, \(3 \over 4\).<br>- interpret ratios in spoken and written English, such as “For every 5 votes candidate A received, candidate B received 4 votes.”</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 may patterns be used to describe mathematical relationships?&nbsp;<br>- How can mathematics help to quantify, compare, depict, and model numbers?&nbsp;<br>- What makes a tool and/or strategy suitable for a certain task?&nbsp;</p>

<p>- Ratio: A comparison of two numbers by division.</p>

<p>- one copy of the Using Ratios in Problem Solving worksheet (M-6-7-1_Using Ratios in Problem Solving and KEY) for each pair of students&nbsp;</p><p>- Sets of 20 playing cards or other objects that come in two colors, etc. (marbles, Monopoly houses and hotels, game pieces, etc.) Each set should contain 8 objects of 1 color and 12 objects of the other color.</p><ul><li>Activity 3 can be completed without the manipulatives, but they will help engage students and can be a useful tool in completing the Sets with Equal Ratios sheet.</li></ul><p>- one copy of Sets with Equal Ratios sheet (M-6-7-1_Sets With Equal Ratios and KEY) for each set of 20 objects&nbsp;</p><ul><li>Note: If the objects are something other than playing cards, the Sets with Equal Ratios sheet will need to be adjusted as it refers to red and black cards.&nbsp;</li></ul><p>- one copy of the Lesson 1 Exit Ticket (M-6-7-1_Lesson 1 Exit Ticket and KEY) for each student</p>

<p>- Analyze class progress using the results of the Using Ratios in Problem Solving exercises (M-6-7-1_Using Ratios in Problem Solving and KEY).<br>- Use the Sets With Equal Ratios worksheet (M-6-7-1_Sets With Equal Ratios and KEY) to assess students' comprehension of equivalent ratios.&nbsp;<br>- The Lesson 1 Exit Ticket (M-6-7-1_Lesson 1 Exit Ticket and KEY) can be used to evaluate student mastery.</p>

<p>Scaffolding, Active Engagement, Explicit Instruction, and Formative Assessment<br>W: Students will learn about ratios, how to express them, and apply them in real-life situations. Students will practice making sets with predetermined ratios.&nbsp;<br>H: Students actively participate in the lesson by either raising their hands or expressing mathematical observations about their classmates with raised hands. They learn about ratios by investigating what the values in a ratio represent, as well as the relationship between fractions and ratios.&nbsp;<br>E: Students will begin by learning basic ratios and comparing them to their classmates' real-life properties. Following that, they investigate various ways to express ratios and use their knowledge of reducing and creating equivalent fractions to reduce and create equivalent ratios.<br>R: Students will complete two worksheets in groups. Students will discuss, rethink, and update their ideas as they complete these tasks. They also use manipulatives in Activity 3, which is a hands-on activity.&nbsp;<br>E: The worksheet results and corresponding group discussions will evaluate students' grasp of the subject matter.&nbsp;<br>T: Use the Extension section to customize the lesson to match the needs of your students. The Routine section includes options for revisiting the lesson topic throughout the school year. The Small Group section is designed for students who could benefit from more teaching or practice. The Expansion section is intended to challenge students who are ready to go beyond the requirements of the standard.&nbsp;<br>O: The lesson requires students to make observations and calculate ratios instantly. Students analyze the composition of the class before collaborating with a partner on a relatively simple problem-solving worksheet. After demonstrating some success, students are allowed to work in a larger group on a more difficult, creative task that will allow them to make general observations about the concepts covered in the lesson.</p>

<p><strong>Activity 1</strong><br><br>Have 10 students raise their hands. Choose randomly, for example, 2 rows of 5 randomly seated students, and so on. Ask the students, <strong>"What fraction of the students with their hands raised are girls?"</strong><br><br>Remind students that the denominator of a fraction should represent the <i>whole</i>—in this case, the number of students raising their hands (10). The numerator should indicate the <i>part</i> we're interested in, which in this case is the number of girls.&nbsp;<br><br>Record the fraction of the students who are girls.&nbsp;<br><br><strong>"What fraction of the students with their hands raised are boys?"</strong> Write this down as well, and let the students lower their hands.<br><br><strong>"Fractions are one method for comparing amounts; they compare a part to a whole. We can also use another comparison, </strong><i><strong>ratios</strong></i><strong>, to compare amounts as well. Ratios frequently compare one part to another part. For example, we could discuss the ratio of girls to boys among the 10 students who raised their hands."</strong>&nbsp;<br><br>Write the girl-to-boy ratio on the board using a colon, for example, 7:3. Do not tell students what the ratio represents.<br><br><strong>"What do you think this represents, thinking back to how many boys and girls had their hands raised?"</strong> Students should understand that one number represents the number of boys, while the other represents the number of girls. Determine which number represent which quantity and label each number accordingly.&nbsp;<br><br><strong>"This is a ratio of the number of girls to the number of boys. The order in which we interpret a ratio is important. If the ratio compares the number of girls to the number of boys, it indicates that the first number represents girls and the second represents boys. How would I write the ratio of boys to girls?"</strong> (<i>3:7</i>)<br><br><strong>"These two ratios compare parts of a whole. We can also compare other quantities. How would I write the ratio of the number of boys to the total numbers of students who raised their hands?"</strong> (<i>3:10</i>)&nbsp;<br><br>Consider alternative ratios that students can create, such as the number of boys and girls in the entire class, different hair colors, and so on.<br><br><strong>Activity 2</strong><br><br><strong>"In addition to writing ratios like 3:8, using a colon to separate the two numbers, we can also write them as fractions. For example, the 3:8 ratio could be written as \(3 \over 8\). It means exactly the same thing as 3:8, and it's vital to specify that it's a ratio. Usually, ratios compare a part to another part. For example, what part of the class is boys and what part is girls? A fraction usually compares a part to a whole. For example, how many boys are in the class compared to how many students in total are in the class? So, while they appear to be the same, it is critical to recognize the difference between them."&nbsp;</strong><br><br><strong>"However, ratios, like fractions, can be reduced. Suppose there are 4 boys and 8 girls. What is the boy-to-girl ratio?"</strong> (<i>4:8</i>) <strong>"We can decrease that to 1:2, just as we would with the fraction \(4 \over 8\). Now, does that mean there is 1 boy and 2 girls?"</strong> (<i>No</i>) <strong>"So what does it mean?"</strong> Help students comprehend that there is 1 boy for every 2 girls. <strong>"In this sense, a ratio is kind of like a rate; for every boy we have, we know we have 2 girls."</strong>&nbsp;<br><br><strong>"So, considering the ratio of 1 boy for every 2 girls, imagine there are 6 boys in a group with this ratio. How many girls must be in the group?"</strong> (<i>12</i>)&nbsp;<br><br><strong>"Suppose there are 20 boys; how many girls must there be?"</strong> (<i>40</i>)&nbsp;<br><br><strong>"Now we'll use it the opposite way. Assume there are 18 girls in the group. How many boys must there be?"</strong> (<i>9</i>)<br><br>Allow students to work in pairs on the Using Ratios in Problem Solving sheet (M-6-7-1_Using Ratios in Problem Solving and KEY).<br><br><strong>Activity 3</strong><br><br>Divide the class into groups based on the number of sets of playing cards (or similar objects) that have been prepared. Give each group a set of 20 playing cards.&nbsp;<br><br><strong>"Each group has a collection of objects to work with. You will use the given ratio to calculate how many different cards may be in a set made out of the cards you have. For example, if the ratio of red to black cards is 7:12, how many different card groups may be created using that ratio? You can absolutely make a group with 7 red cards and 12 black cards. Can you reduce the ratio?"</strong> (<i>No</i>)<br><br><strong>"Could you use more objects? For example, can you multiply each number in the ratio twice to get 14:24? Can you create a set of 14 red and 24 black objects?"</strong> (<i>No</i>) <strong>"Why not?"</strong> Students should remark that they do not have 14 red objects (or 24 black objects).&nbsp;<br><br><strong>"For another example, consider a ratio of 3 red cards to 5 black cards. Can you create a group of 3 red cards and 5 black cards?"</strong> (<i>Yes</i>) <strong>"Can you reduce the ratio 3:5?"</strong> (<i>No</i>) <strong>"Can you double each number in the ratio and make a group with 6 red cards and 10 black cards?"</strong> (<i>Yes</i>) <strong>"Can you triple the ratio and make a group with 9 red cards and 15 black cards?"</strong> (<i>No</i>) <strong>"So, for the ratio of 3:5, there are two possible sets you can make: one with 8 cards in total and one with 16 cards in total."</strong>&nbsp;<br><br>Each group should receive a copy of the Sets with Equal Ratios sheet (M-6-7-1_Sets With Equal Ratios and KEY).&nbsp;<br><br>The lesson begins by connecting new knowledge to current knowledge. The lesson consists of three parts: a class discussion, a small group activity, and a large-group activity. It also includes verbal portions, mathematical portions, and a concrete, hands-on activity (although the cards are not required).<br><br><strong>Extension:</strong><br><br>Use the strategies listed below to adjust the lesson to your students' needs throughout the year.<br><br><strong>Routine:</strong> Ratios appear regularly in everyday life, therefore real-world examples can be used to keep students "fresh" with ratios. The topic can also be combined with other topics, such as the ratio of males and females in the United States Senate, the ratio of state names beginning with a vowel, the ratio of European countries to South American countries, and so on.&nbsp;<br><br><strong>Small Group:</strong> Use this activity with students who could benefit from more practice. Students can create various groups by combining all of the cards from Activity 3 and then selecting a random choice of 20 cards (rather than the predetermined distribution of 8 red and 12 black cards). Students can experiment with ratios in which there are an equal number of cards of each type, as well as ratios in which there are a minimal number of cards of one type (such as 1 or 0 red cards).&nbsp;<br><br><strong>Expansion:</strong> Use this suggestion for students who are ready for greater challenge. Students can investigate multipart ratios (such as 1:2:5). They can also start incorporating mathematics into ratio problems; for example, if the ratio is 1:2:5 and there are 124 total objects, how many are of the second type? (Set up 1<i>x</i> + 2<i>x</i> + 5<i>x</i> = 124, solve for <i>x</i>, and then multiply by 2.)<br><br>Students can also investigate ratios in which one of the values (usually the first) is always reduced to one (to produce a unit ratio). For example, the ratio of a person's "wingspan" to his or her height could be expressed as 1:1.2.</p>