<p>Students understand that percentages represent a rate per 100 and use them to solve problems. Students will:&nbsp;<br>- consider percentages as a rate per 100.&nbsp;<br>- solve problems involving finding the percent, a given part.&nbsp;<br>- solve problems requiring finding the whole, given a part and the percent.&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 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.&nbsp;<br>- Unit Rate: A rate simplified so it has a denominator of 1.</p>

<p>- one copy of Percents with Hundreds sheet (M-6-7-3_Percents with Hundreds and KEY) for each student&nbsp;<br>- one copy of Estimating and Finding Percentages (M-6-7-3_Estimating and Finding Percentages and KEY) for each student&nbsp;<br>- one copy of Finding the Whole (M-6-7-3_Finding the Whole and KEY) for each student</p>

<p>- Examine students' responses to the Percents with Hundreds worksheet (M-6-7-3_Percents with Hundreds and KEY) to measure their ability level.&nbsp;<br>- Use the Estimating and Finding Percentages worksheet to assess students' comprehension (M-6-7-3_Estimating and Finding Percentages and KEY).&nbsp;<br>- Determine the student's level of mastery based on their responses to the Finding the Whole worksheet (M-6-7-3_Finding the Whole and KEY).</p>

<p>Active Engagement, Metacognition, Modeling, and Explicit Instruction<br>W: Students will study how percentages can provide insight into the world around us. They will learn how to calculate percentages given the whole, as well as how to compute the whole given a percentage and part.&nbsp;<br>H: The lesson begins with a discussion of Facebook, which most students are familiar with. Then, students break down the word <i>percent</i> to see what it really means; prior to this, most students had likely seen and possibly worked with <i>percent</i> but had not recognized the parts that make up the word.&nbsp;<br>E: Students work via teacher-guided examples to investigate different types of problems (from Activities 1, 2, and 3). After experiencing them, they get to study how the two main problem types are related (and, essentially, opposites of one another).&nbsp;<br>R: Activity 2 helps students to continue to review and improve their estimating skills and practice a new skill (finding percentages of nonround numbers). Students can practice both concepts in the lesson with the Finding the Whole activity, which includes problems of both primary types.&nbsp;<br>E: You will evaluate students' work after the lesson. In Activity 2, students can self-evaluate, allowing them to examine both their estimating and computation skills.<br>T: Use the Extension section to personalize the lesson to the needs of the students. The Routine section includes strategies for reviewing lesson concepts throughout the year. The Small Group section is designed for students who might benefit from more practice or learning opportunities. The Expansion section includes suggestions for students who are ready for a challenge that goes beyond the requirements of the standard.&nbsp;<br>O: The lesson starts with a real-world example to engage students. In addition, Activity 1 is designed for students to finish quickly and without the use of calculators, allowing them to gain confidence in their percentage abilities before moving on to more challenging problems. Each activity begins with teacher-guided instruction and examples.&nbsp;</p>

<p>Tell students to assume they had 80 Facebook friends (or 80 other objects that can be easily classified into two groups, such as male and female).&nbsp;<br><br><strong>"If I say 50% of your 80 Facebook friends are males, how many of your friends are males?"</strong> (<i>40</i>) Ask students how they arrived at their answer. Students may understand that 50% means half; if not, explain that when we say 50%, we mean half.&nbsp;<br><br><strong>"If 100% of your 80 Facebook friends are female, how many of your Facebook friends are females?" </strong>(<i>80</i>) Students should understand that 100% means all, so all 80 are female.<br><br><strong>"What if 35% of your 80 Facebook friends are female?"</strong> Students are likely to struggle with this. <strong>"This is the kind of problem we're going to look at; we're going to talk about percentages and how to solve problems using percentages."</strong>&nbsp;<br><br><strong>Activity 1</strong>&nbsp;<br><br>Write the word <i>percent</i> on the board. Underline the two syllables separately.&nbsp;<br><br><u>per</u> <u>cent</u><br><br><strong>“What does the word </strong><i><strong>per</strong></i><strong> mean?”</strong> Students may struggle, so provide some context, such as miles per hour or a store charging, say, $1 per soda. Guide students to the idea that <i>per</i> actually means for <i>each</i>. If a car achieves 55 miles per gallon, it will travel 55 miles for each gallon of gasoline. A store that charges $1 per soda is charging $1 for each soda.<br><br><strong>"Now let's look at some examples of words with </strong><i><strong>cent</strong></i><strong> as the root. Raise your hand if you can provide an example."</strong> Examples include cents (pennies), century, centipede, centimeter, and centigram. Explain how each of these relates to the numeric value 100. <strong>"The root word </strong><i><strong>cent</strong></i><strong> means 100."</strong>&nbsp;<br><br>After clarifying what each syllable signifies, put them together. <strong>"</strong><i><strong>Per</strong></i><strong> means 'for each' and </strong><i><strong>cent</strong></i><strong> means 'hundred.' So, the word </strong><i><strong>percent</strong></i><strong> implies for </strong><i><strong>each</strong></i><strong> </strong><i><strong>hundred</strong></i><strong>."</strong><br><br><strong>"Suppose you download 200 songs, 48% of them are rock songs. Remember, 48% means you have 48 rock songs for every 100 songs downloaded. How many groups of 100 songs did you download?"</strong> (<i>2</i>) <strong>"For each of these 2 groups, you downloaded 48 rock songs. So you have 48 rock songs for the first 100, and 48 rock songs for the second 100. So, how many rock songs do you have altogether?"</strong> (<i>96</i>)&nbsp;<br><br><strong>"Notice that percentages are really rates; they're rates per 100 of whatever you're talking about."</strong><br><br><strong>"Suppose you download 300 songs, 12% of them are jazz music. How many jazz songs have you downloaded?"</strong> Guide students through the process of determining how many 100s they have (3) and counting 12 for each hundred (12 + 12 + 12 = 36).&nbsp;<br><br>Work through some more examples with the same context, keeping the number of songs as a multiple of 100 (100, 200, 300, etc.)&nbsp;<br><br><strong>"Suppose you download 150 songs, 8% of them are country music songs. How many country music songs do you have?"</strong><br><br>Tell students that the first 100 songs include 8 country songs. <strong>"How about the other 50 songs? How many of them are country songs?"</strong> If students are struggling, explain them that there are 8 country songs for every 100 songs, and 50 is half of 100. Students should understand that because 50 is half of 100, we have half as many (4) country songs. <strong>"So our total number of songs is 8 + 4 = 12."</strong><br><br>If required, provide more examples before having students work on the Percents with Hundreds worksheet (M-6-7-3_Percents with Hundreds and KEY). Students should complete the worksheet without using a calculator or any decimal or fraction multiplication, but using the method describe above. Students should finish the worksheet individually before comparing answers with a partner.<br><br><strong>Activity 2</strong><br><br><strong>"When we talk about percentages, we're usually talking about finding some quantity out of 100. So, if we find, say, 24% of 98, we are actually multiplying 24% by 98. Remember that in mathematics, the word 'of' often signifies multiplication. So, in addition to the strategies you used on the worksheet, which work great with round numbers like finding percentages of 200 or 650, we can also use multiplication."</strong>&nbsp;<br><br>Write "24% of 98" on the board.<br><br><strong>"To figure this out, let's rewrite 24% as a fraction."</strong> Pause here to write, underneath "24% of 98," the fraction 24/100 followed by a multiplication symbol (×). <strong>"We'll also change the word </strong><i><strong>of</strong></i><strong> to a multiplication symbol, and then just copy our 98, the amount we're finding the percentage of."</strong> Write 98 next to the multiplication symbol, followed by an equal sign.&nbsp;<br><br><strong>"To multiply, we'll just multiply 24 by 98 and divide by 100. So, using this multiplication method, what is 24% of 98?"</strong> (<i>23.52</i>)&nbsp;<br><br><strong>"That's an answer we couldn't get with our previous strategy. We can, however, estimate our result to 24% of 98 by rounding the 98 to 100. If the problem was 24% of 100, keeping in mind that percent means for each hundred, we would know that the answer was 24. So our answer of 23.52 makes sense. It's really close to 24, but a little less, because 98 is really close to 100, but a little less."</strong><br><br><strong>"Another example: what is 48% of 90? Does anyone have a method for estimating this percentage?"</strong> Answers may include rounding 90 to 100 (to give an answer of 48), whereas rounding 48% to 50% (to give an answer of 45). <strong>"Based on these estimates, we know the answer should be approximately 45. To calculate the exact number, write 48% as 48 over 100 and then multiply by 90. Multiply 48 by 90 and divide by 100. What is 48% of 90?"</strong> (<i>43.2</i>)<br><br><strong>"So, even though this is another problem that we can't readily solve with mental math, we can still use rounding and mental math approaches to get a good estimate of the answer. This is useful in case we make a mistake when performing multiplication or division. We'll at least know if our answer seems reasonable."</strong>&nbsp;<br><br>Allow students to work in pairs on the Estimating and Finding Percentages worksheet (M-6-7-3_Estimating and Finding Percentages and KEY). Please provide the following instructions.<br><br><strong>"The worksheet contains 20 problems. Each of you will estimate the answer to 10 problems without using a calculator, and you will also answer the remaining 10 problems exactly. Decide with your partner which problems each of you will estimate the answer to. One of you will estimate the answers to problems 1–10 and solve 11–20 precisely, while the other will do the opposite, solving 1–10 and estimating 11–20. Do not compare solutions until each of you has completed all 20 problems, half by estimating and half by calculating the correct answer. Then, compare your answers to ensure that the estimated and actual answers are reasonably close."</strong>&nbsp;<br><br>Make sure students have enough time, and that each pair compares their estimates to their actual answers.&nbsp;<br><br><strong>Activity 3&nbsp;</strong><br><br><strong>"So far, all of the problems we've solved have been ones in which you've been given the whole. For example, knowing how many Facebook friends someone has in total, we've been requested to calculate a part, say, 25%. But it's possible that you'll be provided additional information. For example, you may be given a percentage, such as 25%, as well as a part; for example, Jake may be told that he has 10 family members who are Facebook friends, accounting for 25% of his overall friend list. The question is: How many Facebook friends does Jake have in total? To write it another way:"</strong>&nbsp;<br><br>Write <i>25% of what number equals 10?</i> on the board. Spend some time ensuring that students grasp the difference between that problem and the similar problem <i>25% of 10 is equal to what?</i>&nbsp;<br><br><strong>"For these types of problems, when we're given a percentage and a part and must discover the whole, we're simply moving backward from what we've been doing. So, previously, we would multiply our two numbers together (remembering that one is a fraction above 100, so we'd divide by 100 at the end). Now, we'll divide our two numbers and multiply by 100 at the end. To solve this problem, we'll just divide 10 by 25—making sure to divide by the percentage—and multiply by 100. The equation 10 ÷ 25 × 100 yields 40. So Jake has 40 Facebook friends."</strong>&nbsp;<br><br>Create two columns on the board. Label one <i>Given the percentage and the whole</i>, and another <i>Given the percentage and the part</i>.&nbsp;<br><br>Label the first row <i>What are we asked to find?</i>&nbsp;<br><br><strong>"So, in a problem where we're given the percentage and the whole, what do we need to find?"</strong> (<i>the part</i>) <strong>"And given the percentage and the part, what do we need to find?"</strong> (<i>the whole</i>)<br><br>Label the second row <i>First Step</i>.<br><br><strong>"When given the percentage and the whole, what do we do first?"</strong> (<i>Multiply the whole by the percentage.</i>) <strong>"And when given the percentage and the part?"</strong> (<i>Divide by the percentage.</i>) Pause here to circle the differences (multiply and divide) and underline the similarities (in both situations, we are doing something <i>to</i> the known quantity <i>by</i> the percentage).&nbsp;<br><br>Label the third row <i>Second Step</i>.&nbsp;<br><br><strong>"After multiplying the percentage by the whole, what do we do?"</strong> (<i>Divide by 100</i>). <strong>"And after dividing the percentage by the part?"</strong> (<i>Multiply by 100</i>).<br><br>Point out the differences and similarities again, as well as point out how the two types of problems essentially represent opposites of one another.<br><br>Work throught the following three examples:<br><br>45% of what is 36? (<i>80</i>)<br>65% of what is 46.8? (<i>72</i>)<br>12% of what is 29.4? (<i>245</i>)<br>Make sure students are dividing the part by the percentage and multiplying by 100.&nbsp;<br><br>Have students work independently on the Finding the Whole sheet (M-6-7-3_Finding the Whole and KEY).&nbsp;<br><br>The lesson is centered on a language-based approach, and all of Activity 1 focuses on applying this language-based approach to solve percentage problems. Students who appreciate math and quick problem-solving will enjoy the rest of the lesson. Students who are conceptual learners will appreciate the connections made between the two major problem categories.<br><br><strong>Extension:</strong><br><br>Use these suggestions to personalize the lesson to your students' needs throughout the unit and year.<br><br><strong>Routine:</strong> Because percentages are so common in the news, students can continually bring in new articles that use percentages to illustrate populations. These articles can also be used to introduce the concept of percent change (increase and decrease), which is frequently discussed in financial articles (the change in the Dow Jones Industrial Average is reported as a percentage in virtually every newspaper and new-related Web site.)<br><br><strong>Small Groups:</strong> Use this website, <a href="http://www.webmath.com/wppercents.html">http://www.webmath.com/wppercents.html</a>, to have students generate challenges for one another as practice. Students should incorporate all of the problem types they've learned (this lesson just addresses the first and fourth types on the website). The Web site enables students to construct an infinite number of problems and be confident that they have the correct answer when they check their group members' answers.&nbsp;<br><br><strong>Expansion:</strong> The lesson can be expanded in numerous ways: Students can calculate the percentage given the part and the whole, which requires an understanding of decimals and converting percentages. Students can convert fractions or decimals to percentages. For fractions, this requires learning equivalent fractions as well as division with decimals for fractions that cannot be changed to have a denominator that is a power of 10.&nbsp;</p>