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Objective To investigate the relationship between sample size and the reliability of derived results. 1 materials Teaching the Lesson Key Activities Students take different size samples from a population and create circle graphs for these samples. They determine the more trustworthy of the samples and make predictions. Key Concepts and Skills • Convert fractions to percents. Math Journal 1, p. 180 Study Link 6 4 Teaching Aid Master (Math Masters, p. 430; optional) Geometry Template Class Data Pad small pieces of colored candy [Number and Numeration Goal 5] • Construct circle graphs of class data. crayons or markers (optional) calculator [Data and Chance Goal 1] • Make predictions based on sampling. [Data and Chance Goal 2] slates Key Vocabulary See Advance Preparation sample • population Ongoing Assessment: Recognizing Student Achievement Use the Math Message. [Number and Numeration Goal 2] 2 Ongoing Learning & Practice Students continue to collect data by playing Finish First. Students practice and maintain skills through Math Boxes and Study Link activities. 3 Students practice identifying the whole for use as the denominator when converting fractions to percents. ENRICHMENT Students predict the outcome of an experiment, test the predictions, and summarize the results. Additional Information Advance Preparation For Part 1, you will need candy of different colors, at least 5 pieces per student. Place the candy in a bowl near the Math Message. Draw a large circle on a sheet of paper and mark the center. 400 Math Journal 1, pp. 170, 171, and 181 Study Link Master (Math Masters, p. 167) number cards 4–8 (4 of each; from the Everything Math Deck, if available) materials Differentiation Options READINESS materials Unit 6 Using Data; Addition and Subtraction of Fractions Teaching Masters (Math Masters, pp. 168 and 169) calculator Per partnership: coin or six-sided die Technology Assessment Management System Math Message See the iTLG. Getting Started Mental Math and Reflexes Math Message Fraction-of problems: The bowl contains pieces of candy of several colors. On a half-sheet of paper, explain how you would find the percent of each color in the bowl. 1 If 5 counters are 10 of the set, what is the whole? 50 counters 1 If 8 counters are 2 of the set, what is the whole? 16 counters 3 A set has 40 counters. How many counters are in 8 of the set? What is the whole? 15 counters; 40 counters Study Link 6 4 Follow-Up 2 If 20 counters are 10 of the set, what is the whole? Partners compare answers and resolve any differences. Ask a volunteer to share a strategy for matching the plots and data sets. 100 counters 4 A set has 25 counters. How many counters are in 5 of the 4 set? 5 is what percent of the set? 20 counters; 80% 2 If 12 counters are 3 of a set, what is the whole? 18 counters Ongoing Assessment: Recognizing Student Achievement Math Message Use the Math Message to assess students’ understanding of the concept of percent. Students are making adequate progress if their written responses name the whole as 100% and refer to pieces of a given color as a fraction or percent of the whole. [Number and Numeration Goal 2] 1 Teaching the Lesson Math Message Follow-Up WHOLE-CLASS DISCUSSION Students share how they would find the percent of each color in the bowl. Most will probably suggest counting the total number of candies in the bowl as well as the number of each color in order to find the fraction and then the percent of each color. If looking at a sample is not mentioned, ask: ● How could we find the percent of each color in the bowl without counting every piece? ● Would it help to look at a sample of candies from the bowl and count the number of each color in the sample? ● How many candies should we include in the sample? Tell students that they are going to do a candy-counting experiment. The results will give them information they can use to predict the percent of each color in the bowl. Explain that the total number of candies in the bowl is called the population. To support English language learners, write sample and population on the board and explain their meanings in this mathematical context. Lesson 6 5 401 Student Page Date 6 5 Taking a Small Sample Time LESSON Sampling Candy Colors 1. You and your partner each take 5 pieces of candy from the bowl. Combine your candies, and record your results in the table under Our Sample of 10 Candies. Sample answer: Candy Color red yellow green orange brown Our Sample of 10 Candies Combined Class Sample Count Percent Count Percent 1 5 2 2 10% 50% 20% 20% 13 28 22 12 25 13% 28% 22% 12% 25% 2. Your class will work together to make a sample of 100 candies. Record the counts and percents of the class sample under Combined Class Sample in the table. PARTNER ACTIVITY of Candy Colors (Math Journal 1, p. 180; Math Masters, p. 430) Have each student take 5 pieces of candy from the bowl without looking (10 candies per partnership). Ask partners to count and record the number and percent of each candy color in their small sample on journal page 180. Tell them to put the candy back in the bowl when they have finished recording their results. 3. Finally, your class will count the total number of candies in the bowl and the number of each color. a. How well did your sample of 10 candies predict the number of each color in Answers vary. the bowl? b. How well did the combined class sample predict the number of each color in Answers vary. the bowl? c. Do you think that a larger sample is more trustworthy than a smaller sample? Yes Sample answer: Smaller samples vary more. A larger sample can be trusted to give a better picture of the real situation than a smaller one. Explain your answer. 180 Math Journal 1, p. 180 With samples of 10, percents are easy to calculate because the fractions easily convert to percents. For example, 4 yellows out of 4 40 10 candies is 10 100 40% of the sample. Point out that sample sizes are often chosen to make calculations easy. Have each partnership make a circle graph to show their sample result for 10 candies. They can color the sections of the graph with the candy colors. Students may use the Percent Circle on their Geometry Templates and circles you provide from copies of Math Masters, page 430. NOTE Counting a whole population is rarely possible, which is why one is concerned with practical sampling and sampling sizes. Even if it is possible to count a whole population, it is often tedious and time-consuming. Yellow Green Orange Red Brown Each circle graph shows the result for one sample of 10 candies. Graphing and Predicting on WHOLE-CLASS ACTIVITY the Basis of a Sample (Math Journal 1, p. 180) Ask 10 partnerships to display their circle graphs, report their results, and record them on the board. Discuss the variations in these results. Individual small samples of 10 may have many of one color, few of another color, and none of some colors. Combine the data from the small partner samples to make a large sample. Tally the results by color on the board. Ask students to record the tallies on journal page 180. By choosing 10 pairs of students, the combined sample total is 100—making percent calculations simple. Make a circle graph of the combined sample. Use the large circle that you drew previously. (See Advance Preparation.) Ask 402 Unit 6 Using Data; Addition and Subtraction of Fractions Student Page volunteers to use a Percent Circle to mark the sections showing percents of colors in the combined sample. Date Time LESSON Math Boxes 6 5 1. Find the median and mean for each set 2. Estimate the sum. 3.1 0.72 34.7 of numbers. a. 17, 13, 27, 33, 25 25 23 median: Yellow Green Orange Red Brown Choose the best answer. mean: 1,370 and 1,380 440 and 460 b. 47, 29, 53, 46, 43, 32 44.5 41.6 苵 median: mean: 35 and 40 13 and 15 247 119–121 Fifth Graders’ Pets 3. Draw a circle graph that is divided into the following sectors: 32%, 4%, 22%, 18%, and 24%. Make up a situation for the graph. Give the graph a title. Label each section. Circle graph of the combined samples of 100 candies (title) Sample answer: Pets owned by fifth graders Discuss which is more trustworthy—a sample of 10 candies or the combined sample of 100. Prompt students with questions like the following: ● ● How do the small-sample results compare with one another? Sample answer: The results for our small samples jump all over. One of them shows 50% yellow, and one shows 10% yellow. Guide students to predict and agree on what percent of each color is in the bowl, and record these predictions on the Class Data Pad. Then have them count the total number of pieces of candy in the bowl and the number of each color. Record the results on the board. Ask students to find the percent of each color: 1. Name each color as a fraction of the total number of candy pieces. 18% none 22% gerbil 4% bird 125 126 5. Write or to make the sentence true. 4. Solve. 5.7 7.81 a. 34.2 6 b. 39.05 / 5 203,467 699,842 521,369 a. 662 b. 7,341 231.2 c. c. 3冄6 苶9 苶3 苶.6 苶 How do the results of the larger combined sample compare with the small ones? Sample answer: The combined sample of 100 is better. Every color is represented. The sample for 100 gives a better picture than the sample for just 10. 24% dog 32% cat Description: d. 42 43 e. 626 7,347 203,764 699,428 9 531,399 181 Math Journal 1, p. 181 Adjusting the Activity Have students calculate the combined class sample using all partnerships in the class. If there are 17 partnerships, the whole is 170 pieces of candy. Compare these results with the results for 100 pieces of candy. AUDITORY 2. Use a calculator to divide the number of each color (the numerator) by the total number of candies in the bowl (the denominator). KINESTHETIC TACTILE VISUAL Study Link Master Name 3. Multiply the result by 100. Date STUDY LINK 65 Expect the percent for each color in the bowl to be reasonably close to the percent on the circle graph for the large sample of 100. 1. Time Constructing a Graph from Landmarks Make up a list of data with the following landmarks: Sample answers: 119 122 mode: 15 minimum: 5 median: 10 maximum: 20 Use at least 10 numbers. 5, 7, 7, 8, 8, 9, 10, 13, 14, 15, 15, 15, 20 2. Draw and label a bar graph to represent your data. Minutes Needed to Get Ready for Bed Playing Finish First Number of Students 2 Ongoing Learning & Practice PARTNER ACTIVITY (Math Journal 1, pp. 170 and 171) Partners continue to collect the needed data for the specified number of games of Finish First. They record their results on journal page 171 and on the classroom tally sheet. For detailed instructions, see Lesson 6-2. Reminder: The game data will be used in the next lesson. 3. 8 7 6 5 4 3 2 1 0 (title) 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Minutes Describe a situation in which these data might actually occur. The number of minutes it takes to get ready for bed Practice 4. 305 29 8,845 5. 524 81 443 6. 671 132 88,572 7. 7,356 4 1,839 Math Masters, p. 167 Lesson 6 5 403 Teaching Master Name Date LESSON Time Math Boxes 6 5 Identify the Whole 65 (Math Journal 1, p. 181) In the following number stories, find the whole using parts-and-total diagram. Write the fraction for the given part, and rename the fraction as a percent. Example: Two girls each have 5 hats. Three of their hats are purple. What percent of the hats are purple? 3 Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 6-7. The skill in Problem 5 previews Unit 7 content. 3 Solution: 2 º 5 10 hats; 3 out of 10 1 0 ; Rename 10 as a fraction with 100 as 30 30 10 º 3 the denominator, 10 º 10 100 ; 100 0.30, or 30%. Reminder: To use a calculator to convert a fraction to a percent, divide the numerator by the denominator. Use your fix key to round to the nearest hundredth, or multiply the decimal by 100 to display the percent. Lamont, Jose, and Kenji are recycling soda cans. Lamont collects 13 cans. Jose collects 20 cans, and Kenji collects 17 cans. What percent of the cans does Jose collect? 1. Unit: cans Fraction: Whole: 20 50 Study Link 6 5 13 20 17 50 Percent: 40% Unit: pieces Fraction: 17 50 Whole: Percent: INDEPENDENT ACTIVITY (Math Masters, p. 167) Jacqui and Edna decide to share their hot lunches. They put together their fried potatoes and their onion rings. There are 33 pieces of fried potatoes and 17 onion rings. What percent of the lunches are the onion rings? 2. INDEPENDENT ACTIVITY Home Connection Students make up a list of data to fit a given set of landmarks. Students then construct a bar graph. 50 34% The boy’s club is having a popcorn sale. Each of the 10 members of the club is given 5 boxes of popcorn, but Edward sells only 3. What percent of the 5 boxes remain for Edward to sell? 3. Unit: boxes Fraction: Whole: 2 5 Percent: 5 40% 3 Differentiation Options Math Masters, p. 168 READINESS Identifying the Whole SMALL-GROUP ACTIVITY 15–30 Min (Math Masters, p. 168) To explore converting fractions to percents, have students find the parts and totals contained in number stories, write the fractions, and convert them to percents. Students can use calculators but should not use the percent key at this time. ENRICHMENT Teaching Master Name LESSON 65 1. Date Investigating Sample Size Time Choose a specific outcome or event for one of the following actions. 1 2 Example: The coin will land heads up. Rolling a die Example: The die will land with a 4 on the top. 1 6 Predict the results of 10 trials and 100 trials. Report your predictions as the fraction of the total you think will result in a favorable outcome, or favorable event. For example, the coin will land heads up about 12 of the time, or the die will land with a 4 on the top about 16 of the time. Event 10 trials 100 trials 1,000 trials Sample answers: Prediction Result Prediction Result Prediction Coin will land heads up 6 10 5 10 60 100 51 100 500 1,000 3. Perform 10 trials. Record the results first with tally marks on a separate piece of paper and then in the table as a fraction. 4. Repeat for 100 trials. Record the results first with tally marks on a separate piece of paper and then in the table as a fraction. 5. How do your predictions compare with the actual results? Sample answer: They are very close. 6. To apply students’ understanding of the relationship between sample size and the reliability of predictions, have students conduct the experiment on Math Masters, page 169. Students choose an event with random outcomes— flipping a coin or rolling a six-sided die. They predict and compare the results of 10 and 100 trials. Then they predict the results for 1,000 trials, explain their prediction, and design an approach to collect the actual results. Discuss students’ predictions and their prediction methods. Consider implementing one of their designs for collecting the actual results for 1,000 trials. Have partners report their findings to the class. Predict the results for 1,000 trials, and explain your prediction. Sample answer: Because the results have been very close 1 1 to 2, I predict the coin will land heads up about 2 of the time. 7. 15–30 Min (Math Masters, p. 169) Investigating Sample Size Flipping a coin 2. PARTNER ACTIVITY On the back of this page, name two ways you and your partner could get data on the actual results for 1,000 trials. Math Masters, p. 169 404 Unit 6 Using Data; Addition and Subtraction of Fractions