Appendix A — Answer Key, Chapter 16: The Data Cycle and Circle Graphs

SOL 6.PS.1 · Covers textbook Chapter 16 and the companion workbook. Item numbers match the textbook; workbook items that repeat textbook problems share the same answers, and workbook-only items appear in the final section. Reasoning answers show an acceptable response, not the only wording.

Lesson 16.1 — The Data Cycle

Guided practice

1. Formulate questions; collect or acquire data; organize and represent data; analyze data and communicate results.
2. Collect or acquire data. Choosing a method is part of deciding how to gather the data you need.
3. Organize and represent data.
4. The cycle does not end there. After representing the data you still have to analyze it and communicate the results, and the results usually raise a new question that sends you back to phase 1.
5. Analyze data and communicate results.

Independent practice

6. a) Collect or acquire data b) Formulate questions c) Analyze data and communicate results d) Organize and represent data
7. Formulate questions; collect or acquire data; organize and represent data; analyze data and communicate results.
8. Collect or acquire data. Deciding whom to ask is part of planning the collection.
9. Any statistical question about the class whose answers vary, such as "Which one after-school activity do students in our class do most often?"
10. Formulate: the manager writes, "Which one snack do students most want the store to stock?" Collect: the manager surveys students, having each choose exactly one snack from a fixed list. Organize: the counts go into a table of percents and then a circle graph. Analyze and communicate: the manager observes which sector is largest, concludes which snack to stock, and reports the decision to the school.
11. The class returns to the first phase, because a new question has been formulated. They must then collect data from fifth graders, organize and represent it, and analyze it. The old data cannot answer the new question.
12. Because answering one question almost always produces a new one, and the new question sends you back to the beginning. A straight line would suggest the work stops once a graph is drawn, which is not how data work goes.

Exit ticket 16.1

1. Formulate questions; collect or acquire data; organize and represent data; analyze data and communicate results.
2. Collect or acquire data.
3. Organize and represent data.
4. Raw data is hard to interpret. Representing it — in a table or a graph — makes the patterns visible, and you need to see the patterns before you can make observations and draw conclusions.

Lesson 16.2 — Formulating Good Questions

Guided practice

1. Yes. Different students name different colors, so the answers vary and data must be collected.
2. No. The door has one color, so there is a single answer and nothing varies.
3. Because a circle graph shows parts of one whole. If a student chooses two options, that student is counted twice and the parts add to more than the whole.
4. Sample answer: "Which one type of music — pop, country, hip-hop, or other — is your favorite?"
5. So that every person surveyed has a category that fits, which keeps everyone counted exactly once and makes the parts add to the whole.

Independent practice

6. a) Not statistical b) Statistical c) Statistical d) Not statistical
7. Heights are numerical measurements with many different values, and they are not categories that combine into a whole. A circle graph would need a sector for nearly every student, which shows nothing useful. A line plot is the better display.
8. Sample answer: "Which one after-school club is your favorite: art, coding, drama, or sports?"
9. Four students have no category that fits, so they cannot answer honestly and the parts will not add to the whole. Fix it by adding an "other" category, or by adding "bird" as a fifth choice: "What is your favorite pet: dog, cat, fish, bird, or other?"
10. Sample answer: "Which one lunch is your favorite: pizza, sandwich, salad, or pasta?" Categories: pizza, sandwich, salad, pasta.
11. Sample answer: "Which one cover color do you prefer: blue, green, red, or purple?" It fits a circle graph because the responses are categories, each student chooses exactly one, and the four choices cover everyone once an "other" option is added if needed.
12. When students may circle more than one item, some are counted twice, so the counts add to more than the number of students. A circle graph's sectors must add to exactly one whole, so the graph would be meaningless. The smallest fix is to change "circle all that apply" to "circle exactly one."

Exit ticket 16.2

1. Yes. Different sixth graders have different numbers of siblings, so the answers vary.
2. Sample answers: the responses must be categories rather than measurements; each person must be counted exactly once; the categories must cover everyone.
3. Sample answer: "Which one kind of music is your favorite: pop, country, hip-hop, or other?"
4. It lets one person be counted in more than one category, so the parts add to more than the whole and the sectors cannot fill exactly one circle.

Lesson 16.3 — Collecting Data and Representative Samples

Guided practice

1. Observation, measurement, survey, and experiment.
2. Measurement.
3. Population: all 240 seventh graders. Sample: the 40 students surveyed.
4. Your friends are chosen for convenience, not at random, and they are likely to be similar to you and to each other. Their answers may not reflect the rest of the grade.
5. Any two: selection is random; the sample is large enough; the sample matches the makeup of the population; the setting does not push the answer; the question is neutral.

Independent practice

6. a) Experiment b) Observation c) Survey d) Measurement
7. Any two: the sample of 8 is far too small for a school of 400; students in the library at one moment are a convenience sample, not a random one; students in the library may differ from the rest of the school in ways that affect the answer.
8. The population is the whole group you want to describe. The sample is the smaller part of that group you actually collect data from, used to draw conclusions about the population.
9. Sample answer: "Would you prefer longer recess or more art class?"
10. Advantage: it saves the time and effort of collecting data, and the data set is probably much larger. Disadvantage: it describes teenagers nationally rather than sixth graders at this school, so it may not represent the population the class actually cares about.
11. Sample answer: survey 60 students — 10 chosen at random from each of the six homerooms. It is representative because every sixth grader has an equal chance of being chosen, all six homerooms are included in proportion, and 60 out of 300 is large enough that a few unusual answers will not distort the result.
12. Sample answer: the two classes surveyed different samples, so ordinary variation between samples produced different percents; or one class used a biased method, such as surveying only students in the cafeteria line. I would trust the class whose sample was larger and chosen at random, because that sample is more likely to reflect the whole grade.

Exit ticket 16.3

1. Population: all 500 students. Sample: the 30 students surveyed.
2. A survey, because favorite genre is an opinion you must ask people about.
3. Five students is far too few to reflect 250. One unusual answer changes the percent by 20 percentage points.
4. A sample is representative when it reflects the larger population: chosen at random, large enough, matching the population's makeup, gathered in a neutral setting, and asked with a neutral question.

Lesson 16.4 — Making a Circle Graph

Guided practice

1. Walk $\tfrac{10}{20} = 50\%$; bus $\tfrac{5}{20} = 25\%$; car $\tfrac{3}{20} = 15\%$; bike $\tfrac{2}{20} = 10\%$. Check: $50 + 25 + 15 + 10 = 100$.
2. Walk $180°$; bus $90°$; car $54°$; bike $36°$. Check: $180 + 90 + 54 + 36 = 360$.
3. $\tfrac{3}{4} = 75\%$ and $0.75 \times 360° = 270°$.
4. $100 - (45 + 20 + 15) = 100 - 80 = 20\%$.
5. $0.25 \times 360° = 90°$.

Independent practice

Checks: $40 + 30 + 20 + 10 = 100$ and $144 + 108 + 72 + 36 = 360$.

7. Pretzels $\tfrac{4}{8} = 50\%$, $180°$; popcorn $\tfrac{2}{8} = 25\%$, $90°$; fruit $\tfrac{1}{8} = 12.5\%$, $45°$; cheese $\tfrac{1}{8} = 12.5\%$, $45°$. Checks: $50 + 25 + 12.5 + 12.5 = 100$ and $180 + 90 + 45 + 45 = 360$.
8. Bus $45\%$; walk $32\%$; car $18\%$; bike $5\%$. With a total of 100, each count is already a number out of 100, so the count and the percent are the same number. Check: $45 + 32 + 18 + 5 = 100$.
9. The circle represents the entire data set, and every value is counted in exactly one sector. Together the sectors must fill the whole circle, which is $100\%$ of it.
10. The percents add to $40 + 30 + 20 + 15 = 105$, which is more than $100\%$, so at least one is wrong. Check by writing each count as a fraction of 20 and converting: with 20 students the sectors would be, for example, $8$, $6$, $4$, and $2$ students, giving $40\%$, $30\%$, $20\%$, and $10\%$. Adding the percents and confirming they total $100$ is the check.

To draw it: draw a circle, mark the center, and draw a radius straight up. Center the protractor on the center point with its zero line on that radius, mark $144°$, and draw the new radius. Move the zero line to the new radius and mark $90°$, then $72°$, then $54°$. Label each sector with its activity and percent, and title the graph with the total of 20 students.

12. With 17 students, a count such as 7 gives $\tfrac{7}{17}$, which is not a clean percent and produces a central angle that is not a whole number of degrees, so the sector cannot be measured accurately by hand. A total of 20 works because $\tfrac{7}{20} = 35\%$ and $0.35 \times 360° = 126°$. Better totals are those with denominators of 12 or less or factors of 100, such as 10, 20, 25, or 50.

Exit ticket 16.4

1. $\tfrac{4}{20} = 20\%$ and $0.20 \times 360° = 72°$.
2. $\tfrac{5}{25} = 20\%$.
3. $100 - (35 + 25 + 25) = 15\%$.
4. Write the count over the total as a fraction, then multiply that fraction by $360°$. For example, $\tfrac{7}{20} \times 360° = 126°$.

Lesson 16.5 — Analyzing Circle Graphs

Guided practice

1. Dog, at $35\%$.
2. $0.35 \times 80 = 28$ students.
3. $30\% + 20\% = 50\%$, and $0.50 \times 80 = 40$ students.
4. $35\% - 15\% = 20\%$, and $0.20 \times 80 = 16$ more students.
5. Sample answer. Observation: the dog sector is the largest, at $35\%$. Conclusion: a school pet-care club should plan most of its activities around dogs, because dogs are the most commonly owned pet among these 80 students.

Independent practice

6. $0.50 \times \$40 = \$20$.
7. $0.25 \times \$40 = \$10$.
8. $15\% + 10\% = 25\%$, and $0.25 \times \$40 = \$10$.
9. $50\% - 25\% = 25\%$, and $0.25 \times \$40 = \$10$ more.
10. a) True: snacks is $\$10$ and gifts is $\$6$. b) False: gifts and games together are $25\%$, which is one quarter, not half. c) True: savings is $50\%$, the largest sector.
11. Fiction $0.35 \times 400 = 140$; nonfiction $0.30 \times 400 = 120$; graphic novels $0.25 \times 400 = 100$; poetry $0.10 \times 400 = 40$. Check: $140 + 120 + 100 + 40 = 400$. Sample conclusion: the librarian should order more fiction and nonfiction, since together they account for $65\%$ of the reading, and should consider a small poetry display since poetry is only $40$ students.
12. School A: $0.50 \times 20 = 10$ students. School B: $0.25 \times 200 = 50$ students. School B has five times as many soccer fans, even though its sector is smaller, because sector size shows the share of the whole and not the count. You cannot compare two circle graphs without knowing both totals.

Exit ticket 16.5

1. Math, at $40\%$.
2. $0.25 \times 60 = 15$ minutes.
3. $20\% + 15\% = 35\%$, and $0.35 \times 60 = 21$ minutes.
4. Sample answer. Conclusion: this student should start homework with math, because math takes the largest share of the hour. Supporting observation: the math sector is $40\%$, larger than any other sector.

Lesson 16.6 — Choosing the Best Representation

Guided practice

1. A circle graph, because each lunch option appears as a part of one whole.
2. A line plot (dot plot), because the scores are numerical values and the plot shows every value, including repeats and gaps.
3. $20 \div 5 = 4$ symbols.
4. Only if the total is given. The sectors show shares of the whole, so a $25\%$ sector means 5 students out of 20 but 50 students out of 200.
5. A bar graph, because the bars sit on a numbered scale, so exact counts can be read and compared directly.

Independent practice

6. a) Circle graph, because a percent of the class is a share of one whole. b) Bar graph, because exact counts on a numbered scale can be compared by subtracting. c) Line plot (dot plot), because the number of pets is numerical data and every value should be shown. d) Pictograph, because repeated symbols with a key are easy for younger students to read at a glance.
7. $8 \div 2 = 4$; $5 \div 2 = 2.5$; $4 \div 2 = 2$; $3 \div 2 = 1.5$ symbols.
8. Soccer $0.40 \times 20 = 8$; basketball $0.25 \times 20 = 5$; baseball $0.20 \times 20 = 4$; swimming $0.15 \times 20 = 3$. Check: $8 + 5 + 4 + 3 = 20$.
9. If students may choose more than one sport, the counts add to more than the number of students, so the parts do not form one whole. A circle graph would have to show more than a full circle, which is impossible. A bar graph handles overlapping choices correctly.
10. The most common number of siblings is $1$, with 7 students. Students with three or more siblings: $3 + 1 + 1 = 5$ students.
11. A circle graph best supports the claim, because "half of all sixth graders" is a share of the whole and a $50\%$ sector fills half the circle at a glance. A bar graph best shows exactly how many students each category holds, because its numbered scale gives exact counts rather than shares.
12. The circle graph makes the reader think first about shares — one sector looks like "about half" or "about a quarter" — while the bar graph makes the reader think first about counts and differences in height. The circle graph better answers "what fraction of the group chose this?" and the bar graph better answers "how many more chose this than that?"

Exit ticket 16.6

1. A circle graph.
2. A bar graph.
3. $7 \div 2 = 3.5$ symbols, that is, three full symbols and a half symbol.
4. Number of siblings is numerical data, and a line plot puts every value on a number line so repeats, clusters, and gaps are visible. A circle graph is built for categories that combine into a whole, so it would hide the shape of the numerical data.

Chapter 16 Review

Part A — Formulating questions (6.PS.1a)

1. Statistical: b and c. Not statistical: a and d.
2. Sample answer: "Which one sport is your favorite: soccer, basketball, baseball, or swimming?"
3. Because the sectors of a circle graph must add to one whole. If someone chooses two options, that person is counted twice and the parts add to more than the whole.
4. Sample answer: "Which one field trip do you most want: the museum, the zoo, the aquarium, or the science center?" Choices: museum, zoo, aquarium, science center.

Part B — Collecting data and representative samples (6.PS.1b, 6.PS.1c)

5. a) Measurement b) Survey c) Observation d) Experiment
6. Population: all 500 students. Sample: the 50 students surveyed.
7. Any two: students at basketball tryouts are far more likely than average to name basketball; the sample is chosen for convenience rather than at random; students who play no sport are left out entirely.
8. Any three: selection is random so everyone has an equal chance; the sample is large enough; the sample matches the makeup of the population, such as covering every homeroom; the setting does not push a particular answer; the question is worded neutrally.

Part C — Making circle graphs (6.PS.1d)

9. Pizza $\tfrac{10}{25} = 40\%$; sandwich $\tfrac{5}{25} = 20\%$; salad $20\%$; pasta $20\%$. Check: $40 + 20 + 20 + 20 = 100$.
10. Dog $\tfrac{4}{10} \times 360° = 144°$; cat $108°$; fish $72°$; bird $36°$. Check: $144 + 108 + 72 + 36 = 360$.
11. Missing percent $100 - (40 + 25 + 20) = 15\%$; angle $0.15 \times 360° = 54°$.
12. With 20 students, each count over 20 converts to a clean percent and a whole number of degrees, because 20 is a factor of 100. With 17 students, a count such as 7 gives $\tfrac{7}{17}$, which is neither a clean percent nor a whole number of degrees, so the sectors cannot be measured accurately.

Part D — Analyzing circle graphs (6.PS.1e)

13. $0.40 \times 200 = 80$ students.
14. $20\% + 15\% = 35\%$, and $0.35 \times 200 = 70$ students.
15. $40\% - 25\% = 15\%$, and $0.15 \times 200 = 30$ more students.
16. Sample answer. Observation: the summer sector is the largest, at $40\%$, which is 80 students. Conclusion: the school should schedule its outdoor festival in a summer theme, because more students prefer summer than any other season. The observation about the $40\%$ sector supports the conclusion.

Part E — Comparing representations (6.PS.1f)

17. a) Circle graph, because the question asks for each trip's share of the whole class. b) Bar graph, because the numbered scale gives exact counts that can be subtracted. c) Line plot (dot plot), because reading minutes are numerical values and every value should appear.
18. $20 \div 5 = 4$; $15 \div 5 = 3$; $10 \div 5 = 2$ symbols.
19. A bar graph shows exact counts on a numbered scale, which a circle graph cannot show unless the total is stated. A circle graph shows each category as a share of one whole, which a bar graph does not display directly.

Part F — Mixed application and reasoning

Checks: $40 + 30 + 20 + 10 = 100$ and $144 + 108 + 72 + 36 = 360$. Sample conclusion: orange juice should be stocked most often, since it is the choice of $40\%$ of the class, twice the share of grape.

21. First, the sample of 6 is far too small to describe a whole grade. Second, friends are a convenience sample rather than a random one, so they are likely to share tastes. Also, 3 out of 6 is $50\%$ of the friends, not of the grade. Fix it by surveying a larger sample chosen at random from a list of all students in the grade, spread across homerooms.
22. Sample answer. Formulate: write "Which one new club should the school add: coding, cooking, robotics, or photography?" Collect: survey a random sample of students across all homerooms, with each student choosing exactly one option. Organize: tally the counts, convert to percents, and build a circle graph. Analyze and communicate: observe which sector is largest, conclude which club to add, and present the graph to the principal.
23. Sector size shows the share of the whole, not the count. A $50\%$ sector of 20 students represents $0.50 \times 20 = 10$ students, while a $50\%$ sector of 200 students represents $0.50 \times 200 = 100$ students. Without the totals, the two identical-looking sectors cannot be compared.

Workbook-only items

Page 2, four phases. 1. Formulate questions. 2. Collect or acquire data. 3. Organize and represent data. 4. Analyze data and communicate results.

Page 2, phase table. Recording pages read: collect or acquire data. Writing the club question: formulate questions. Presenting a circle graph: analyze data and communicate results. Turning a tally chart into percents: organize and represent data.

Page 3, cycle in action. Sample answers. Formulate: write "Which one lunch is the favorite of sixth graders at our school?" Collect: survey every sixth grader, each choosing exactly one lunch, and record tallies. Organize: build a table of counts and percents and draw a circle graph. Analyze and communicate: observe the largest sector, conclude which lunch to serve more often, and present the graph to the cafeteria manager.

Page 3, explain. Answering one question raises a new one, which sends you back to the first phase, so the process loops rather than ending.

Page 3, challenge. The class returns to phase 1, formulating the new question, and then must collect new data from seventh graders before representing and analyzing it.

Page 5, circle S or N. a) N b) S c) S d) N e) N f) S

Page 5, fits a circle graph when. 1. categories 2. exactly one time 3. everyone (every response)

Page 5, rewrites. Sample answers. "Which one sport is your favorite: soccer, basketball, baseball, or swimming?" and "Which one kind of music is your favorite: pop, country, hip-hop, or other?"

Page 7, method table. Classroom temperature each hour: measurement. Favorite music genre: survey. Students wearing sneakers Monday: observation. Plants under a lamp or by a window: experiment.

Page 7, fill in. Population; sample.

Page 7, school of 500. Population: all 500 students. Sample: the 50 students surveyed.

Page 8, representative or biased. Basketball tryouts: B — the setting favors basketball and the sample is not random. 60 random names from all sixth graders: R — every student has an equal chance and the sample is large. Five closest friends: B — a convenience sample that is far too small. Ten students from each of six homerooms: R — the sample is spread across the whole grade in equal shares.

Page 8, fix the question. Sample answer: "Would you prefer longer recess or more art class?"

Page 8, three factors. Any three: random selection; large enough sample; sample matches the population's makeup; neutral setting; neutral wording.

Page 10, mid-chapter check. 1. Formulate questions; collect or acquire data; organize and represent data; analyze data and communicate results. 2. S. 3. Observation. 4. Population: all 240 seventh graders; sample: the 40 surveyed. 5. Chess club members are chosen for convenience and share an interest, so their answers will not reflect the whole grade. 6. Sample answer: "Which one after-school club is your favorite: art, coding, drama, or sports?"

Page 11, get-to-school table.

Checks: percents add to $100$; angles add to $360$.

Page 12, elective table. Art 20, $40\%$, $144°$; music 15, $30\%$, $108°$; drama 10, $20\%$, $72°$; dance 5, $10\%$, $36°$; totals $100\%$ and $360°$. The drawn graph should show these four sectors, each labeled with its elective and percent, under a title naming the total of 50 students.

Page 12, missing percent. $100 - (45 + 20 + 15) = 20\%$.

Page 12, find the error. The percents add to $105\%$, but the sectors of a circle graph must add to exactly $100\%$.

Page 14, transportation table. Bus $90$; car $50$; walk $40$; bike $20$; total $200$.

Page 14, answers. Most common: bus. Least common: bike. Walk or bike: $30\%$ of 200, or 60 students. Bus exceeds car by $20\%$ of 200, or 40 students.

Page 15, observation or conclusion. Bus sector larger than car sector: O. Most students live far away: C. Ten percent ride a bike: O. School should add a bus route: C.

Page 15, allowance. Savings $20; snacks $10; gifts and games together $10; savings exceeds snacks by $10.

Page 15, think. School B. School A has $0.50 \times 20 = 10$ soccer fans and School B has $0.25 \times 200 = 50$, so the larger sector represents far fewer students.

Page 17, sport table.

Checks: $40 + 25 + 20 + 15 = 100$ and $144 + 90 + 72 + 54 = 360$.

Page 17, match the question. Percent who walk: circle graph. How many more play piano than guitar: bar graph. Pets per student showing every value: line plot (dot plot). Poster for younger students: pictograph.

Page 18, justify it. Sample answers. 1. "A circle graph is best, because the question asks for each trip's share of the class, and this graph shows each category as part of one whole." 2. "A bar graph is best, because the question asks how many more, and this graph shows exact counts on a numbered scale." 3. "A line plot is best, because the question asks for every student's value, and this graph plots each value on a number line."

Page 18, explain. A bar graph shows exact counts on a numbered scale; a circle graph shows each category's share of the whole.

Page 20–21, review. Same answers as the textbook Chapter 16 Review above, items 1–23.