Problem Solving: Lesson Planner Skills Maintenance Variables and Proportions Problem Solving: In this lesson, students learn about different situations where percents are used. They learn to use a percent formula to find discounts, tips, interest, and sales tax. Objectives Students will translate percent problems from word statements to formulas. Students will solve percent problems. Homework Students convert percents to decimal numbers, select the correct answer to percent problems, and solve percent word problems by using a formula. In Distributed Practice, students practice operations with decimal numbers and fractions. Lesson 12 Skills Maintenance Name Skills Maintenance Variables and Proportions Activity 1 Convert the percents to decimal numbers. 1. 75% 0.75 2. 4% 0.04 3. 16% 0.16 4. 55% 0.55 5. 2% 0.02 6. 85% 0.85 Activity 2 For each group of shapes, select the two that are similar. 1. Which shapes are similar? (a) A and B (b) A and C (c) B and C 2. Which shapes are similar? (a) A and B (b) A and C (c) B and C 3. Which shapes are similar? (a) A and B (b) A and C (c) B and C A A A B B Date B C C C Unit 2 Unit 2 Lesson 12 81 Skills Maintenance Variables and Proportions (Interactive Text, page 81) Activity 1 Students convert the percents to decimal numbers. Activity 2 Students identify similar shapes. Remind students that similar shapes have proportional dimensions. Unit 2 Lesson 12 295
Lesson 12 Problem Solving: Problem Solving: How do we write percent problems using variables? (Student Text, pages 202 206) Connect to Prior Knowledge Tell students they have seen many examples of everyday problems involving percents and variables. For example, when a store has a sale, items are advertised as a certain percent off. Ten percent, 15 percent, and 20 percent are all common discounts. If there is enough time, distribute ads from the newspaper and have students search for discounts that are represented as percents. Have students convert the percents to decimal numbers. Explain that we convert to decimal numbers to make the problems easier to solve on a calculator or in our heads. 202 Problem Solving: How do we write percent problems using variables? In everyday life, we often find problems that require us to work with percents. Variables can help us solve percent problems. A common type of problem with percents is the discount problem. Before we even know the problem, we can set up a general statement for discount problems. First, we choose variables. d = the amount of the discount c = the original cost of the item p = percent of the discount A discount amount is computed by multiplying the original cost by the percent of the discount. d = c p Let s look at a problem. Problem: James went shopping for a television. The original cost was $400, but it was on sale for 25% off. What is the amount of the discount on the television? We substitute the variables with the price and percent to determine the discount. d = c p d = 400 25% 202 Unit 2 Lesson 12 Link to Today s Concept Tell students that in today s lesson, we look at writing our own percent problems with variables. Engagement Strategy: Teacher Modeling how we set up a percent problem involving discounts in one of these ways: : Use the mbook Teacher Edition for Student Text, pages 202 203. Overhead Projector: Reproduce the variables and formula on a transparency, and modify as discussed. Board: Copy the variables and formula on the board, and modify as discussed. Explain that we first choose variables. Write the variables d, c, and p and explain what each variable represents. Make sure students understand that we can use any letter variable we want. Remind students that we find a discount by multiplying the original cost by the percentage of the discount. Write the equation d = c p. Show the word problem, and explain that we can use the formula to solve it. Show how we substitute the values of the variables with the information we know: the original cost, $400, and the percent of discount, 25 percent. So d = 400 25%. 296 Unit 2 Lesson 12
Continue going over the formula. Explain that we convert 25 percent to 0.25. So d = 400 0.25. After multiplying, we find that d = 100. Remind students that the variable d represents the amount of the discount. Point out that a discount is not the same thing as the sales price. To find the sales price, we subtract the discount from the original price. This is why it is important to pay careful attention to what each problem is asking for. Then have students look at Example 1. This example demonstrates another discount problem, but this time the item is 10 percent off. Point out that the problem is asking for the discount, not the sales price. Students should notice that the p is replaced with 0.10 since p stands for percent off. We also replace the c with 50 for the cost of the item. The discount is $5. Explain Point out that other types of percent problems are used in everyday life. For example, people often use a formula to compute the tip at a restaurant. It is common to leave a 15 percent tip for a meal. In order to work with the percent, we have to change it into a decimal number. So our problem is: d = 400 0.25 d = 100 The amount of the discount on the television is $100. We can use the general statement d = c p as a formula for discounts. But we do need to be careful to adjust the formula for the problem we are trying to solve. For instance, the discount may be a different percent. In addition, the cost changes from problem to problem. We need to make sure we understand what the problem is asking. Example 1 Compute the discount on a $50 sweater that is 10% off. First, we remember our discount formula. d = c p Next, we substitute the variables with the cost of the item and percent of discount. Remember, we change 10% to 0.10 to make the computation easier. d = c p d = 50 0.10 d = 5 The discount amount is $5. Discount problems are just one type of percent problem. Another common percent problem that is used in everyday life involves figuring out the tip for a meal at a restaurant. A common tip is 15%, or 0.15, of the total bill for the meal. Since the percent is the same for most tips, this is a good use of a formula. We only need to adjust the cost of the meal each time. Unit 2 Lesson 12 203 If students are not sure about the answer, prompt them to look about at other students solutions to help with their thinking. Review the answers after all students have held up their solutions. 203 Check for Understanding Engagement Strategy: Look About Tell students that an $80 shirt is on sale for 20 percent off. Tell them that they find the amount of the discount with the help of the whole class ($16). Tell them to use the discount formula from Example 1. Students should write their solutions in large writing on a piece of paper or a dry erase board. When students finish their work, they should hold up their answer for everyone to see. Unit 2 Lesson 12 297
Lesson 12 How do we write percent problems using variables? (continued) Have students look at Example 2 on page 204 of the Student Text. This example shows a formula for finding a 15 percent tip. Explain that we can use whatever variables we like to write the formula. This example uses t for tip, c for the cost of the meal, and p for the percentage of the meal we want to tip. Point out that there are many strategies people use to figure the tip, but all of them are basically the same formula: t = c p. Show students how we substitute the values for the variables and compute the amount of the tip, $7.50. Example 2 shows how to figure a 15% tip on a meal that costs $49.99. Example 2 Compute a 15% tip for a $49.99 meal. First, we choose variables. t = amount of tip c = cost of the meal p = percent of the meal that we want to tip We multiply the cost of the meal by the percent that we want to tip to find the amount of our tip. t = c p Then, we substitute the values we know for the variables. Remember, we convert the 15% to 0.15 to make the computation easier. t = c p t = 49.99 0.15 When calculating a tip, we do not need to find an exact amount. The computation is easier if we use rounding. We round $49.99 to $50. t = 50 0.15 t = $7.50 The tip should be about $7.50. Another type of percent problem is the interest problem. This is very much like the discount problem. We have different amounts of money that we are computing interest for, as well as different percents of interest. Check for Understanding Engagement Strategy: Pair/Share Have students partner with another student. Distribute a menu from a restaurant and ask pairs to imagine they are out for a meal. Ask them to select the items they would eat. Tell them to total the cost of the meal, then compute the tip. Remind students that a tip is generally 15 percent of the total bill. Have pairs volunteer to explain their work. 204 204 Unit 2 Lesson 12 In return, the bank pays us a percentage, or interest, for letting it use our money. Distribute a bank brochure that explains the interest earned on various accounts or interest charges on various loans. We will focus on simple interest, but it is okay to discuss other types of interest. Reinforce Understanding If students need further practice in calculating tips, have them find the amount of a 10 percent and 20 percent tip for the total cost of the meals they selected. Explain Point out that another percentage problem used in everyday life has to do with interest. When we borrow money from the bank, it charges us a certain percentage, or interest, for using it. Likewise, when we put money in a savings account, it is like lending the money to the bank. 298 Unit 2 Lesson 12
Direct students attention to Example 3 on page 205 of the Student Text. This example shows how to compute simple interest at a rate of 5 percent on a balance of $1,000. Show how we choose the variables. Then point out the formula. The interest rate is 5 percent, so we convert it to 0.05. Explain that if the interest rate is two percent, for instance, we would substitute 0.02 for 0.05 in the formula. Point out how we substitute the values for the variables and solve to find the amount of interest earned, $50. Begin to explore the final type of percent problem in this lesson: sales tax. Explain that most states charge a tax on things we buy. This is one way the state earns money for expenses and services it provides. If there s time, look up the current tax rates for various states in the U.S. and distribute that information to students. Have students look over the various rates and rewrite the percents as decimal numbers. Example 3 shows a simple interest of 5% earned on $1,000 in the bank. Example 3 Compute the interest earned on $1,000 when the interest rate is 5%. First, we choose variables. i = amount of interest r = interest rate a = account balance Then we set up the general statement or formula. i = r a We convert 5% to 0.05 and substitute our values for the variables. i = r a i =0.05 $1,000 i =$50 The interest earned on $1,000 at a rate of 5% is $50. The final type of percent problem we will look at in this lesson is the computation of sales tax. We know that there is tax on items we buy at the store. Sales tax varies from state to state, usually somewhere between 5% and 8%. Example 4 shows the amount of tax charged on a $150 purchase if we are shopping in a state with a 6% sales tax. 205 Unit 2 Lesson 12 205 Unit 2 Lesson 12 299
Lesson 12 How do we write percent problems using variables? (continued) Have students look at Example 4 on page 206 of the Student Text. It demonstrates the amount of tax paid on a $150 purchase if the tax rate is 6 percent. Point out the variables in the problem. Explain what you would do if the tax rate was a different percent. Show students how we set up the formula t = p c and substitute the values for the variables. Multiply to solve the problem. The tax is $9. Example 4 Compute the sales tax on $150 when the sales tax is 6%. First, we choose variables. t = tax p = percent charged for tax c = cost of the purchase Then we set up the general statement or formula. t = p c Now we substitute the values in the formula. t = p c t = 0.06 150 t = $9 The tax is $9 on the $150 purchase. Writing a formula using variables to represent the unknowns in the problem helps us organize our thinking. 206 206 Unit 2 Lesson 12 Problem-Solving Activity Turn to Interactive Text, page 82. Reinforce Understanding Use the mbook Study Guide to review lesson concepts. 300 Unit 2 Lesson 12
Problem-Solving Activity Name Date Problem-Solving Activity (Interactive Text, pages 82 83) Have students turn to Interactive Text, pages 82 83, and complete the activities. First students solve percent problems. They must convert each percent to a decimal number, and then substitute values for variables. The formulas are written for the students. Students will have an opportunity to solve at least one interest, tax, tip, and discount problem. Then students solve word problems involving percents. They write the formulas themselves, select the variables, and then solve the formulas by substituting in values from the problem. Monitor students work as they complete the activities. Problem-Solving Activity Solve the percent problems by substituting the value for the variables. You may use a calculator. 1. What is the discount on a $500 TV that is 15% off? Answer $75 D = discount, p = percent off, c = cost D = p c $75 = 0.15 $500 2. What is the tax on a $300 item at a tax rate of 7%? Answer $21 T = tax amount, r = tax rate, c = cost T = r c $21 = 0.07 $300 3. What is a 15% tip on a $100 meal at a restaurant? Answer $15 T = tip, r = percent, c = cost of meal T = r c $15 = 0.15 $100 4. How much interest will you earn on $1,500 at 2%? Answer $30 I = interest, r = rate, a = account balance T = r c $30 = 0.02 $1,500 5. What is the discount on a $250 item at 20% off? Answer $50 D = discount, p = percent off, c = cost D = p c $50 = 0.20 $250 6. How much tax will you pay for a $75 item at a 6% rate. Answer $4.50 T = tax, r = tax rate, c = cost T = r c $4.50 = 0.06 $75 Less 82 Unit 2 Lesson 12 Watch for: Do students know which numbers to substitute for which variables? Can students convert each percent to a decimal number? Lesson 12 Problem-Solving Activity Name Problem-Solving Activity Date Can students perform the computation accurately? You may allow calculators for the second activity. Do students have a clear understanding of what the answer represents? (For example, in a discount, it s the amount of the discount. You would subtract it from the original cost to get the sales price.) Can students write the correct formula for each type of percent problem? Solve the percent problems. Remember to convert the percent to a decimal number before you multiply. You may use a calculator. 1. Everything in a department store is on sale for 10% off. Write an equation to describe the discount on a $500 digital camera. $500 0.1 = $50 2. Roseanne has a coupon for 5% off her monthly phone bill. Write an equation to describe the discount on her $175 bill. $175 0.05 = $8.75 3. The teacher promised the class bonus points for the semester. She would give each student a bonus of 10% of the total homework points for the semester. Write an equation that describes how many bonus points Bonnie will get if she earned 150 homework points during the semester. 150 0.10 = 15 4. The New Jax Band is donating 1% of all ticket sales from a concert tour to charity. Write an equation that describes how much they will donate to the charity if they sell $1,500 in tickets. $1,500 0.01 = $15 Unit 2 Reinforce Understanding Remind students that they can review lesson concepts by accessing the online mbook Study Guide. Reinforce Understanding Use the mbook Study Guide to review lesson concepts. Unit 2 Lesson 12 83 Unit 2 Lesson 12 301
Lesson 12 Homework Homework Go over the instructions on page 207 of the Student Text for each part of the homework. Activity 1 Students convert percents to decimal numbers. Activity 2 Students select the correct answer to percent problems. These are multiple choice questions. Activity 3 Students solve percent word problems by writing formulas and solving them. Activity 4 Distributed Practice Students practice operations with decimal numbers and fractions to continue to improve their skills. Activity 1 Convert the following percents to decimal numbers. 1. 35% 0.35 2. 15% 0.15 3. 5% 0.05 4. 20% 0.2 5. 10% 0.1 Activity 2 Select the correct answer for each of the percent problems. 1. What is the discount on a $200 item that is 15% off? b (a) $3 (b) $30 (c) $170 3. What tip would you pay on a $60 meal at a 15% tip rate? a (a) $9 (b) $90 (c) $69 2. What is the tax on a $500 item at a 6% tax rate? b (a) $3 (b) $30 (c) $470 4. How much interest will you earn at 2% on $1,000? a (a) $20 (b) $200 (c) $800 Activity 3 Solve the word problems involving percents. 1. What is the discount on a $100 item that is 10% off? $10 2. What is the discount on a $200 item that is 10% off? $20 3. How much would a 15% tip be on a $150 meal? $22.50 4. How much interest will you earn on $500 at 1%? $5 5. How much interest will you earn on $1,000 at 2%? $20 6. What is the tax on a $375 item at a 5% tax rate? $18.75 Activity 4 Distributed Practice Solve. 1. 1 3 4 1 8 6 2. 1.29 + 15.09 16.38 3. 332.11 219.34 112.77 4. 7.8 3.9 3.9 5. 7. 2 3 1 2 11 3 5 2 1 3 21 6 6. 2.1 7 14.7 8. 0.44 0.4 1.1 207 Unit 2 Lesson 12 207 302 Unit 2 Lesson 12