Reading Charts, Graphs & Tables Meeting ... - MathCounts

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Copies of the Reading Charts, Graphs & Tables problem set (Problems and answers ... According to the mileage chart below, what was the total length of the  ...
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Reading Charts, Graphs & Tables Meeting (Interpreting Data Representations)

16% 8% 52% 8% 16%

Topic

In this set of problems, students are asked to interpret a variety of methods of displaying data.

Materials Needed

♦ Copies of the Reading Charts, Graphs & Tables problem set (Problems and answers can be viewed below. Complete solutions and a more student-friendly version of the problems—with pictures and larger font—are available for download from www.mathcounts.org on the MCP Members Only page of the Club Program section.) ♦ Calculators OPTIONAL ♦ Paper ♦ Markers ♦ Rulers ♦ Protractors ♦ Compasses

Meeting Plan

Money Spent

The problems for this meeting do not require advanced math. There are a variety of data representations (bar graph, table, pie chart, stem-and-leaf plot, box-and-whisker plot, etc.) for students to interpret. The difficulty level of the problems increases as students work through the problems. 1. The amount of money spent on each of the four days of Kia’s vacation is shown in the bar graph shown. What was the average amount of money spent per day of the vacation? 2004-2005 School Handbook Workout 2-5

$300 $200

$260

$293

$183

$124

$100 Fri

Sat

Sun

Vacation Days

Mon

2. On their summer vacation, the Charen family flew from New York City to Raleigh and then to Atlanta on their way to New Orleans. Returning from New Orleans, they flew to St. Louis and then to Detroit on their way to New York City. According to the mileage chart below, what was the total length of the family’s round-trip? 2004-2005 School Handbook Warm-Up 1-2 Distance Between Cities (Miles)

Atlanta

Detroit

New Orleans

New York City

Raleigh

St. Louis

Atlanta

-

732

Detroit

732

-

466

866

400

547

1077

606

707

542

New Orleans

466

1077

-

New York City

866

606

1300

1300

869

674

-

496

938

Raleigh

400

707

869

St. Louis

547

542

674

496

-

828

938

828

-

Who is the Best Oscars Host?

3. The circle graph shows the results of a 2002 survey of 4050 people. How many people chose Billy Crystal as the best Oscars host? Express your answer to the nearest hundred. 2004-2005 School Handbook Workout 1-4

Whoopi Goldberg

Other 16% 8%

Bob Hope 8%

Billy Crystal 52%

16% Johnny Carson

2009–2010 MATHCOUNTS Club Resource Guide

37

Driving Time to School (Minutes)

4. Twenty-five teachers were asked the length, in minutes, of their daily drive to school. Their responses are recorded in this stem-and-leaf plot. What percent of the teachers take 25 or more minutes to get to school? 2004-2005 School Handbook Workout 3-2

0

5 6 6 7 9

1

0 1 2 8 9 9

2

0 2 3 4 5 7

3

1 2 2 2 2 8

4

1 3

5. James wants to take summer classes but wants to go to class only between 9 a.m. and noon each day. The table below lists the available courses and class times. What is the maximum number of different courses James can take? 2004-2005 School Handbook Workout 7-2 Course

Days

Times (a.m.)

Course

Days

Times (a.m.)

Archery

MWF

9 - 9:50

Karate

MWF

9 - 10:20

TTh

9 - 10:20

Karate

TTh

9 - 10:50

MWF

11 - 11:50

Drama

MWF

9 - 9:50

MTWThF

11 - 11:50

Physics

TTh

8:30 - 9:50

Ceramics

MWF

10 - 10:50

Math Magic

TTh

10:30 - 11:50

Ceramics

MWF

11 - 11:50

Cartooning Diving

6. This graph shows the raise schedule for Mr. Bright’s employees. If the linear pattern continues, what is the amount of the raise for the sixth year of service? 2005-2006 School Handbook WarmUp 3-3 modified

Raise

Archery

$4000 $3500 $3000 $2500 $2000 $1500 $1000 $500 $0

1 2 3 4 5 6

Yrs. of Service

7. The box-and-whisker plot below shows the speed of the 25 fastest roller coasters in the world, in mph. What is the largest possible number of roller coasters that could have a speed greater than 80 mph? 2005-2006 School Handbook Warm-Up 10-1 74 75.5

8. The chart here shows the prices for different seat-types at a stadium for professional soccer. If every seat in the stadium is sold for a game, what is the average price per seat? 2005-2006 School Handbook Warm-Up 14-6

120

94

80

Section

Price

No. of Seats Avail.

West, center

$10

1000

West, end

$8

1500

East, center

$8

800

East, end

$6

1700

9. The Green team and Red team each has 10 members. Each member had two turns to earn points. A team member could earn zero through 40 points on each turn. The double stem-and-leaf plot shows the scores for the team members, where 6 | 3 | 5 represents 36 points for a Green team member and 35 points for a Red team member. How many more points than the Red team did the Green team earn? 2006-2007 School Handbook Warm-Up 5-10

38

Red

Green 6 5 4 7 6 4 2 1 5 5 5 4 4 2 9 8 8 6 0 0

0 1 2 3 4

5 6 6 7 2 2 3 4 0 4 4 4 5 5 5 6 8 8 9 0

2009–2010 MATHCOUNTS Club Resource Guide

11. The graph shows the daily amount of food needed to feed a puppy based on the puppy’s maximum adult weight. For which maximum adult weight does a puppy have the greatest percent increase of food intake from “Up to 2 Months” to “4-6 Months?” 2005-2006 School Handbook Workout 3-8

Eric’s Driving Pattern

65 55 45 35 0

7:00

6:00

Number of Cups of Food

Eric’s Speed (mph)

10. For what percent of the period from 6:00 a.m. through 9:30 a.m. was Eric driving at least 50 miles per hour? Express your answer to the nearest whole number. 2006-2007 School Handbook Warm-Up 2-10

9:00

8:00

Time (a.m.)

6

Up to 2 Months

5 4

2-4 Months

3 2

4-6 Months

1 0

30

50

80

Maximum Adult Weight (lbs)

12. This chart was created from the results of a radio station survey. What percent of the males surveyed listen to the station? 2004-2005 School Handbook Warm-Up 4-8 Listen to Station

Yes

No

Total

Male

39%

13%

52%

Female

29%

19%

48%

Total

68%

32%

100%

Answers: $215; 3184 miles; 2100 people; 40%; 5 courses; $8500; 12 roller coasters; $7.72; 18 points; 57%; 50 pounds; 75%

Possible Next Steps

Once students have worked through the problems, you can have them convert some of the data representations into different formats. For instance, the pie chart for #3 can be turned into a histogram. Similarly, the stem-and-leaf plot for #4 and the data for #12 can be converted into a pie charts.

2009–2010 MATHCOUNTS Club Resource Guide

39

Reading Charts & Graphs Meeting Problem Set Money Spent

1. __________ The amount of money spent on each of the four days of Kia’s vacation is shown inthe bar graph below. What was the average amount of money spent per day of the vacation? 0405 WO2-5: $215

$300

$293

$260 $183

$200

$124

$100 Fri

Sat

Mon

Sun

Vacation Days 2. __________ On their summer vacation, the Charen family flew from New York City to Raleigh and then to Atlanta on their way to New Orleans Returning from New Orleas, they flew to St. Louis and then to Detroit on their way to New York City. According to the mileage chart below, what was the total length of the family’s round-trip? 0405 WU1-2; 3184 miles Distance Between Cities (Miles) Atlanta

Detroit

New Orleans

New York City

Raleigh

St. Louis

Atlanta

-

732

466

866

400

547

Detroit

732

-

1077

606

707

542

New Orleans

466

1077

-

1300

869

674

New York City

866

606

1300

-

496

938

Raleigh

400

707

869

496

-

828

St. Louis

547

542

674

938

828

-

3. __________ The circle graph shows the results of a 2002 survey of 4050 people. How many people chose Billy Crystal as the best Oscars host? Express your answer to the nearest hundred. 0405WO1-4; 2100 people

Who is the Best Oscars Host?

Whoopi Goldberg

Other 16% 8%

Bob Hope 8%

Billy Crystal 52%

16% Johnny Carson

4. __________ Twenty-five teachers were asked the length, in minutes, of their daily drive to school. Their responses are recorded in this stem-and-leaf plot. What percent of the teachers take 25 or more minutes to get to school? 0405WO3-2; 40%

Driving Time to School (Minutes) 0

5 6 6 7 9

1

0 1 2 8 9 9

2

0 2 3 4 5 7

3

1 2 2 2 2 8

4

1 3

Copyright MATHCOUNTS, Inc. 2009. 2009-2010 MATHCOUNTS Club Resource Guide Problem Set

5. __________ James wants to take summer classes but wants to go to class only between 9 a.m. and noon each day. The table below lists the available courses and class times. What is the maximum number of different courses James can take? 0405WO7-2; 5 courses Course

Days

Times (a.m.)

Course

Days

Times (a.m.)

Archery Archery

MWF TTh

9 - 9:50 9 - 10:20

Karate Karate

MWF TTh

9 - 10:20 9 - 10:50

Cartooning

MWF

11 - 11:50

Drama

MWF

9 - 9:50

MTWThF

11 - 11:50

Physics

TTh

8:30 - 9:50

MWF MWF

10 - 10:50 11 - 11:50

Math Magic

TTh

10:30 - 11:50

Diving Ceramics Ceramics

6. __________ This graph shows the raise schedule for Mr. Bright’s employees. If the linear pattern continues, what is the amount of the raise for six years of service? 0506 WU3-3; $8500 with change

$4000 $3500 $3000 $2500 $2000 $1500 $1000 $500 $0 1 2 3 4 5 6

7. __________ The box-and-whisker plot below shows the speed of the 25 fastest roller coasters in the world. What is the largest possible numver of roller coasters that could have a speed greater than 80 mph? 0506 WU10-1; 12 roller coasters

74 75.5

120 80

94

8. __________ The chart here shows the prices for different seat-types at a stadium for professional soccer. If every seat in the stadium is sold for a game, what is the average prese per seat? 0506WU14-6; $7.72 Section West, center West, end East, center East, end

Price $10 $8 $8 $6

No. of Seats Avail. 1000 1500 800 1700

Copyright MATHCOUNTS, Inc. 2009. 2009-2010 MATHCOUNTS Club Resource Guide Problem Set

9. __________ The Green team and Red team each has 10 members. Each member had two turns to earn points. A team member could earn zero through 40 points on each turn. The double stem-and-leaf graph shows the scores for the team members, where 6 | 3 | 5 represents 36 points for a Green team member and 35 points for a Red team member. How many more points than the Red team did the Green team earn? 0607 WU5-10; 18 points

Eric’s Speed (mph)

10. __________ For what percent of the period from 6:00 a.m. through 9:30 a.m. was Eric driving at least 50 miles per hour? Express your answer to the nearest whole number. 0607 WU2-10; 57%

Green 6 5 4 7 6 4 2 1 5 5 5 4 4 2 9 8 8 6 0 0

0 1 2 3 4

Red 5 6 6 7 2 2 3 4 0 4 4 4 5 5 5 6 8 8 9 0

Eric’s Driving Pattern

65 55 45 35 0

6:00

7:00

8:00

Time (a.m.)

9:00

Number of Cups of Food

11. __________ The graph shows the daily amount of food needed to feed a puppy based on the puppy’s maximum adult weight. For which maximum adult weight does a puppy have the greatest percent increase of food intake from “Up to 2 Months” to “4-6 Months?” 6

Up to 2 Months

5 4

2-4 Months

3 2

4-6 Months

1 0

30

50

80

Maximum Adult Weight (lbs)

12. __________ The chart below was created from the results of a radio station survey. What percent of the males surveyed listen to the station? 0405WU4-8; 75% Listen to Station Male Female Total

Yes 39% 29% 68%

No 13% 19% 32%

Total 52% 48% 100%

Copyright MATHCOUNTS, Inc. 2009. 2009-2010 MATHCOUNTS Club Resource Guide Problem Set

Reading Charts, Graphs & Tables Meeting Solutions (2009-2010 MCP Club Resource Guide) Problem 1. Kia spent a total of $183 + $260 + $293 + $124 = $860 in four days of vacation. That’s $860 ÷ 4 = $215 per day, on average. Problem 2. The first leg of the Charen family’s trip is from New York City to Raleigh, a distance of 496 miles, according to the mileage chart. The next leg is from Raleigh to Atlanta, a distance of 400 miles, then Atlanta to New Orleans, a distance of 466 miles. The return trip goes from New Orleans to St. Louis, 674 miles, then St. Louis to Detroit, 542 miles, and finally Detroit to New York City, 606 miles. The total distance of the round trip is 496 + 400 + 466 + 674 + 542 + 606 = 3184 miles. Problem 3. The circle graph shows that 52% of the people chose Billy Crystal as the best Oscars host. To calculate 52% of the 4050 people who were surveyed, we multiply as follows: 0.52 × 4050 ≈ 2100 people, to the nearest hundred. Problem 4. In the stem-and-leaf plot given, the tens digit of each number of minutes is written to the left of the vertical line and the ones digits for all numbers in that range are written to the right of the line. Fifteen of the teachers take less than 25 minutes to get to school, and 10 teachers take 25 minutes or more. Ten out of 25 is equal to 40/100 = 40%. Problem 5. James can take three 50-minute classes that meet on Mondays, Wednesdays and Fridays. He also can take two 80-minute classes that meet Tuesdays and Thursdays. That’s a maximum of 5 courses. The classes could be Drama, Ceramics and Cartooning on MWF and Archery and Math Magic on TTH. Problem 6. Since the pattern is linear, we can draw a line through the dots to continue the pattern. We also can continue the linear pattern of dots by going “up three and right one” from (3, 4000) until we get to (6, ?). The amount of the raise corresponding to six years of service is $8500. Problem 7. In a box-and-whisker plot, the dividing line inside the box represents the median of the data. Since the speeds of an odd number of roller coasters are given, we know for sure that at least one roller coaster has a speed of 80 mph. With 25 roller coasters, there could be at most 12 with speeds greater than this median of 80 mph. Problem 8. There are a total of 1000 + 1500 + 800 + 1700 = 5000 seats at the stadium. The total ticket sales if every ticket is sold is ($10 × 1000) + ($8 × 1500) + ($8 × 800) + ($6 × 1700) = $10,000 + $12,000 + $6400 + $10,200 = $38,600. The average price per seat is $38,600 ÷ 5000 = $7.72. Problem 9. Since we are interested only in how many more points the Green team had over the Red team, we can simply cross off equal numbers on both sides of the double stem-and-leaf graph. We can then look at the remaining numbers and find the differences for each row. In the single digits, Red had 9 more points. In the tens, Green had 19 more points. In the twenties, Green had 3 more points. Red had an extra 35 and Green had an extra 40. By our tally of differences, Red had 9 + 35 = 44 points, and Green had 19 + 3 + 40 = 62 points. In all, Green had 62 – 44 = 18 more points than Red.

Copyright MATHCOUNTS, Inc. 2009. MATHCOUNTS Club Resource Guide Solution Set

Problem 10. Eric was driving at least 50 miles per hour (meaning 50 mph or more) from 6:30 to 8:30. That’s 2 hours out of the 3.5 hours he was driving, or 4 out of 7 half hours. To convert 2/3.5 or 4/7 to a percent, we divide the numerator by the denominator and then multiply by 100. Expressing our result to the nearest whole number, we get 57%. Problem 11. Puppies with a maximum adult weight of 30 pounds start with 1.5 cups of food when they are in the “Up to 2 Months” age group. They get 2.75 cups of food when they are in the “4-6 Months” age group. That is a factor of 2.75/1.5 = 1.83, which represents an increase of 83%. Similarly, the puppies with a maximum adult weight of 50 pounds go from 2.25 cups of food to 4.5 cups of food. That is a factor of 4.5/2.25 = 2, which is an increase of 100%. The puppies with a maximum adult weight of 80 pounds go from 3.5 cups to 5.5 cups. That is a factor of 5.5/3.5 = 1.57, which is an increase of 57%. The puppies with a maximum adult weight of 50 pounds have the greatest percent increase of food intake from “Up to 2 Months” to “4-6 Months.” Problem 12. It is easy to read the answer to a different question directly from the chart. While it is true that 39% of the people surveyed were males who said they listen to the station, the question asks what percent of the males surveyed listen to the station. We need to compare this 39% to the 52% of the people surveyed who were male. What percent is 39 out of 52? It is 39 × 52 = 0.75 or 75%.

Copyright MATHCOUNTS, Inc. 2009. MATHCOUNTS Club Resource Guide Solution Set