Lesson 1: Fractions, Decimals and Percents


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1 Lesson 1: Fractions, Decimals and Percents Selected Content Standards Benchmarks Addressed: N2H Demonstrating that a number can be expressed in many forms, and selecting an appropriate form for a given situation (e.g., fractions, decimals, percents, and scientific notation) N2M Demonstrating number sense and estimation skills to describe, order, and compare rational numbers (e.g., magnitude, integers, fractions, decimals, and percents) N5M Applying an understanding of rational numbers and arithmetic operations to reallife situations GLEs Addressed: Grade 7: 2. Compare positive fractions, decimals, percents, and integers using symbols (i.e., <,, =,, >) and position on a number line (N2M) 7. Select and discuss appropriate operations and solve single and multistep, reallife problems involving positive fractions, percents, mixed numbers, decimals, and positive and negative integers (N5 M) (N3M) (N4M) Grade 9: 3. Apply scientific notation to perform computations, solve problems, and write representations of numbers (N2H) Lesson Focus This lesson contains difficult concepts for many students to be able to grasp and use effectively. The instructor may choose parts that best fit the needs of the students in his or her class. The lesson includes all of the following: Convert between fractions, decimals, and percents Use different methods appropriately when working with percents in problem solving Use of place value in understanding and using scientific notation GEE 21 Connection The skills that will be addressed in this lesson include the following: Understand the relationship between fraction, decimal, and percent of a whole unit Solve problems involving percent in real world situations Demonstrate an understanding of the use of scientific notation in representing very large and very small numbers 1
2 Translating Content Standards into Instruction A. The first topic for this lesson is the relationships among fractions, decimals, and percents. 1. Begin by building from student knowledge and understanding. Ask ¼ of the students to stand. Then ask what fraction of the group is seated. Next, ask 25% of the students to stand. Ask what percent is seated. Repeat the process with.25. Use fractions that will work well with your group and try other topics such as what part of the class is wearing blue. These activities will help determine the level of understanding of your students. 2. Pose the following problem from Teacher Blackline #1, Part A to your students. There are 100 students in the 10 th grade. 35% are in chorus, 2/5 are in 4H, and.4 ride the bus. How many students are in each group? Distribute centimeter grid paper and ask the students to mark off 10 by 10 grids to represent the three groups. Then ask them to shade the part of the 10 th graders who will be in each group. The students need to understand that fractions, decimals, and percents all represent a part of a whole unit. In this example, the unit is the 10 th grade class. 3. The next task is to help students understand conversions between fractions, decimals, and percents. Using Teacher Blackline #1, Parts B and C, work through the What part is shaded? activities. Help students understand the conversions using the method below = = =. 10 = = = =. 40 = 40% = = =. 75 = 75% Discuss the meaning of percent as parts of a hundred. Encourage nontraditional methods that demonstrate conceptual understanding. For example, ¾ changed to a percent may be done in increments of fourths if the student knows that ¼ is 25% and then 3 times 25% is 75%. 4. At the end of the activity, review traditional conversion methods if you think your students are reaching that level of understanding. These conversions methods are listed on the second page of Teacher Blackline #1, Part D. 5. Have students try Teacher Blackline #1, Part E on their own. 2
3 B. The next part of the lesson deals with applications of percent such as raises, discounts, or taxes. Use Teacher Blackline #2 for these activities. State the following problem to the students. Jessie makes $100 each day as a trained daycare worker. At the end of the year, Jessie is offered a 7% raise in salary. How much will she make each day after the raise? 1. Ask students to draw a 10 by 10 grid and model the raise by shading the appropriate amount. They should be able to determine that the new pay is $ The next step is to allow students to transfer knowledge of percents as parts per hundred into units that are not in 100 parts. One method of doing this is to divide the whole into 10 parts that are each. The following examples provide a demonstration of this method. Current salary (0) divided into 10 equal parts is. A raise is, and it would follow that a 20% raise would be 2(20)=$40. If they get a 15% raise, that is one and then half of another, or $30 total. 3. In problem 2, the current salary of $25,400 can be divided into 10 equal parts of. A 15% raise is one raise () plus a half of another ($1270) for a total of $3810 which will make the new salary $29,210. The new salary is 115% of her old salary. Show the students that multiplying 1.15 times $25,400 will get the new salary of $29, In problem 3, the current price of $45 divided into 10 parts would be $4.50. Then 30% off would be three discounts of $4.50 each. The sale price would be 45 3(4.50) = $
4 5. In problem 4, the original price is unknown, but $14 represents 80% or 8 parts of. $14 divided into eight parts is. Two more parts of each would give the 100% or the original price of $ In problem 5, the original price of $86 divided by 10 would be $8.60. The 15% off would be computed by adding or $8.60 and half of or $4.30 for a total discount of $ The student would pay $ In problem 6, the price of $73.10 divided into 10 parts is $7.31. The 7.5% tax to be added to this would be 5% + 2.5% which is ½ (7.31) + ¼ (7.31). The tax would come to $5.48 and the total bill would be $ Again, at some point you can transfer the process to multiplying by the decimal equivalent of the percent when you feel the students are ready. 8. Have students complete problem 7. C. In the final part of the lesson, students will represent very large and very small quantities using scientific notation. Mathematicians and scientists use a form of shorthand based on powers of 10 to write unusually large and small numbers. This method is called scientific notation. A number is expressed in scientific notation when it is in the form a x 10 n where 1 a < 10 and n is an integer. Work through Teacher Blackline #3 with students. Point out that it is not always necessary to write all the numbers when doing multiplication and division. Scientific notation allows the student to round after two or three decimal places. Allow students an opportunity to do the practice problems at the bottom of Teacher Blackline #3. GEE 21 Connection On the GEE 21 test, students may be required to place fractions, decimals, and percents in order, estimate percent of a region represented on a diagram, compute with percents in problem situations, and use scientific notation. 4
5 Sources of Evidence of Student Learning A. Students should participate in all activities involved in the Teacher Blacklines. B. Have students complete Student Worksheets provided with the lesson. C. Supplement with problems at the end of the Patterns unit from outside resources. GEE 21 Connection Sample items similar to what students might see on the GEE 21 test: (1) Which means the same as 3580? a x 10³ b x 10³ c x 10³ d. 358 x 10³ Connecticut Dept. of Ed. (2) The population of Germany is about 81,700,000. This number can also be written as: a x 10 5 b x 10 6 c x 10 7 d. 817 x 10 6 Louisiana Department of Education (3) Waiters should receive a tip that equals 15% of a customer s bill. If the meal costs $6.50, about how much should the tip be? a. $0.15 b. $0.25 c. $0.65 d. $1.00 e. $2.50 Illinois Department of Education (4) Josh s baseball card collection went up in value by approximately 50%. It is now worth about $225. Choose the amount that he paid for the cards to begin with. a. $100 b. $110 c. $150 d. $400 e. $450 Illinois Department of Education 5
6 (5) The circle graph represents sales of five different types of automobiles. Which automobile accounted for about 25% of the sales? Sales of Automobiles K M Z X L a. K b. L c. M d. X e. Z Illinois Department of Education Attributes of Student Work at the GotIt level A. Students will demonstrate skill and conceptual understanding of the conversions among fractions, decimals and percent. B. Students can understand and solve application problems involving percents. C. Students will understand calculator readout and be able to use scientific notation in representing very large and very small numbers. 6
7 Lesson 1: Fractions, Decimals and Percents Teacher Blackline #1 A. There are 100 students in the 10 th grade. Of those students, 35% are in chorus, 2/5 are in 4H, and.4 ride the bus. How many students are in each group? Mark off three 10 by 10 grids on the graph paper provided by your teacher to represent the different groups of students in the 10 th grade. Fraction: Decimal: Percent: 35% Fraction: 2 5 Decimal: Percent: Fraction: Decimal:.4 Percent: B. What part of the following grid is shaded: Fraction? Decimal? Percent? C. Using the graph paper provided by your teacher, shade the following parts of 10 by 10 grids, if this will assist you in determining the equivalent fraction, decimal, or percent. Fraction: 1 5 Decimal: Percent: Fraction: Decimal: Percent: 30% Fraction: Decimal:.65 Percent: Each time you shade, you are relating to a part of a whole. 7
8 D. Traditional Conversion Rules Changing fractions to decimals Divide the top number (numerator) by the bottom number (denominator). Use the calculator when it is available. It is not always necessary to write all the digits displayed on your calculator. Usually you can round to two digits past the decimal. This is the hundredth place in the place value chart. For instance, take the fraction 2 7. Divide 2 by 7. You see on your calculator screen many numbers behind the decimal. Most of the time, only the first two are important, so write the decimal as.29 since the 5 after the 8 is at least half of 10. Changing decimals to fractions Identify the place value of the last digit on the right. Move the decimal to the left so that you have a whole number and then place it over the place value that was determined at the beginning. For example, in the decimal.295, the place value of 5 is thousandth. The fraction would be written as 295 over This fraction can then be reduced to lowest terms by dividing 295 and 1000 by the greatest common factor of 5 to get 59 over 200. Changing percents to decimals and fractions Move the decimal two places to the left to change the percent to a decimal. Place the percent over 100 and reduce the fraction to change the percent to a fraction in lowest terms. For example, 28% is.28 or The can be reduced to 7 25 because 28 and 100 are both divisible by 4. Changing decimals to percents Move the decimal two places to the right to change a decimal to a percent. For example, 2.57 is 257%. E. Question for thought Stormy ate 35% of the pizza and Remey ate 2. Who ate the most 5 pizza? About what percent was left? Hint: Use one 10 by 10 centimeter grid and use x s to represent Stormy and o s to represent Remey. Change both to equivalent forms such as percents. 8
9 Lesson 1: Fractions, Decimals and Percents Student Worksheet #1 Write each decimal or fraction as a percent = = = = = = = = Write each percent as a decimal and a fraction. Express the fraction in lowest terms % = = % = = % = = 12. George ate 7 pieces out of a 35 piece box of chocolates. What percentage of the chocolates did George eat? 13. Shade the approximate given percentage, fraction, or decimal of the shapes. a. 3/5 b. 30% c A pizza store noted that 6 25 of the customers liked cheese pizza, 28% of the customers liked pepperoni pizza, 11 of the customers prefer 50 sausage pizza, and.26 liked the vegetarian pizza. Rank the pizza preferences of the customers in order from most preferred to least preferred. Use your centimeter grids as needed. Explain your work. 9
10 Lesson 1: Fractions, Decimals and Percents Teacher Blackline #2 1. Jessie makes $100 each day as a trained day care worker. At the end of the year, Jessie is offered a 7% raise in salary. How much will she make each day after the raise? The new salary is what percent of her old salary? Now, let s look at some other ways to look at applications of percents. 2. Leslie currently makes $25,400 per year at her current job. She is interviewing for a promotion that guarantees a 15% raise over her current salary. How much will Leslie make if she gets the promotion? The new salary is what percent of her old salary? 3. Sally finds a sweater that is priced at $45. The store is having a 30% sale on every item in the store. How much will Sally have to pay for the sweater assuming there are no taxes? The sale price is what percent of the original price? 4. Joseph paid 80% of the original price on a book at the bookstore. His price was $14. What was the original price? 5. You receive a flyer advertising 15% off on your next trip to the Shoe Mart. You wish to purchase a pair of Nike running shoes that have a regular price of $86. What will be your price for the shoes before tax? 6. If you must pay a 7.5% tax on the shoes that you bought in the previous example, what will be your total cost? 7. 2 off means you pay % of the original price. 3 27% off means you pay % of the original price. 15% raise means you have % of your original salary. 8 ½ % tax means your total bill will be % of the price of item(s) purchased. 10
11 Lesson 1: Fractions, Decimals and Percents Student Worksheet #2 Solve the following problems involving percent. In each problem, think about whether your answer should be more or less than the amount given in the problem. 1. Dorrie was able to save 20% of her $35,000 that she earned last year on her job. How much money did Dorrie save? 2. It is expected that the waitress should receive 15% of the price of a meal as a tip, if service is satisfactory. Lennie took his family out to dinner and the price of the food totaled $62. How much should Lennie leave the waitress? 3. The Acme Bonding Company had such a good year that it was noted that the company took in 12% more this past year than in years past. The board of directors voted to give all employees a 6% raise as a result of the company s good fortune. Cissy currently makes $27,000 as a yearly salary. What will be Cissy s new salary after the raise? 4. You bought a backpack at 25% off the regular price of $64. The sales tax is 8 ½% in the parish where you made your purchase. What was the total cost of the backpack? 5. If 2.4% of America s enlisted personnel are college graduates, what percent are not college graduates? 6. Suppose 5% of 800 people polled regarding an election expressed no opinion. How many people had no opinion? 7. Frank purchased a new tennis racket for $ He had to pay sales tax of. Find the amount of tax and then the total price he had to pay. 8. The original selling price of a cook top was $ This price was increased by 20%. The increased price was then discounted by. What was the final selling price? Explain how you obtained your answer. 11
12 Lesson 1: Fractions, Decimals and Percents Teacher Blackline #3 Scientific Notation Example 1: Write in scientific notation: 3,600,000, x ,507, x 10 6 Example 2: Write in scientific notation: x x 107 You may use your calculator to write a number in scientific notation. Enter the number 20,570,000,000. On the screen you will see E10. The is the a part, and the E10 means an exponent of 10. You can then rewrite the number in scientific notation which would be x How do you change numbers in scientific notation to standard numbers? Example 3: 8.07 x 10 5 rewritten would be 807,000 Move the decimal point 5 spaces to the right for a positive 5 exponent. Example 4: 4.321E 4 rewritten would be x 104 Move the decimal point 4 places to the left for a negative exponent of 4. Student Practice 1. Write the following numbers using scientific notation. a. 123,400 b. 1,350,000,000 c d. 2,365,389 X e ,000, Write the actual amount that represents the numbers given in scientific notation. a X 10 6 b X 105 c X
13 Lesson 1: Fractions, Decimals and Percents Answer Keys Teacher Blackline #1 A. 35 students are in chorus, 40 students are in 4H, 40 students ride the bus = = = 40% 40 2 = = 40% B. C = =. 25 = 25% = =. 20 = 20% = = = = 65% E. Remey ate the most pizza or 40%. 25% remains. Student Worksheet # % 2. 45% % % % % 7..3% % = % = = a. The student should make the equivalent of 5 divisions and shade 3 of the 5. b. The student should make 10 divisions and shade 3 o9 the 10. c. The student may chose to divide into 8 parts and shade 6 or they may choose to make 4 parts by drawing perpendiculars and shade Student work should reflect changing all values to either percent, decimal, or fraction with a common denominator. #1 Pepperoni (28%); #2 Vegetarian (26%); #3 Cheese (24%); #4 Sausage (22%) 13
14 Teacher Blackline #2 1. $ % 2. $29, % 3. $ % 4. $ $ $ % 73% 115% Student Worksheet #2 1. $7, $ $28, $ % $5.00; $ $ Multiply the original price, $550, by 20%, which equals $110. Add $550 and $110 together to get $660. This new price is then discounted by, so multiply $660 by and get $66. Subtract $66 from $660, which leaves $594, the final selling price. Teacher Blackline #3  Student Practice 1. a x 10 b x 10 c x d x 10 e x a. 2,305,000 b c GEE 21 Practice 1. b 2. c 3. d 4. c 5. b 14
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