mathematics as communication: graphing information collected over time

MATHEMATICS AS COMMUNICATION: GRAPHING INFORMATION COLLECTED OVER TIMEAuthor(s): MARIAN MOODYSource: The Mathematics Teacher, Vol. 83, No. 9 (DECEMBER 1990), pp. 730-736Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27966947 .

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ACTIVITIES

MATHEMATICS AS COMMUNICATION: GRAPHING INFORMATION COLLECTED OVER TIME By MARIAN MOODY, Oxford High School, Oxford, AL 36203

Teacher's Guide

Introduction: Such recent publications as NCTM's Curriculum and Evaluation Stan dards for School Mathematics (Standards) (NCTM 1989) and Everybody Counts (Na tional Research Council 1989) are increas

ing our awareness that students must be

guided and inspired to be thinkers and prob lem solvers if they are to keep pace with the

ever-changing questions and demands of to morrow's world. "Classroom activities must

encourage students to express their ap proaches, both orally and in writing. Stu dents must engage in mathematics as a hu man activity; they must learn to work

cooperatively in small teams to solve prob lems as well as argue for their approach amid conflicting ideas and strategies" Na tional Research Council 1989, 61).

According to the NCTM's Standards, an

underlying theme of middle school mathe matics is the study of patterns and func tions. "This study requires students to rec

ognize, describe, and generalize patterns and build mathematical models to predict the behavior of real-world phenomena that exhibit the observed pattern. .. . When stu

dents make graphs, data tables,. .. they dis cover that different representations yield different interpretations of a situation"

(NCTM 1989, 98). This activity was created to encourage

students to observe and examine the world around them. It helps students use mathe matics to model real-world problems, to rea son mathematically, to communicate math

ematically, and to solve problems. In

particular, it helps them read and interpret graphs and organize and describe data.

Grade levels: 6-10

Materials: One set of activity sheets for each student, a set of transparencies for dis

cussion, and graph paper

Objectives: Students will (a) create

graphical representations of information collected over a period of time, (6) recognize realistic estimates of numbers in the real

world, and (c) work collaboratively and im

prove communication skills.

Edited by Mary Kim Prichard, University of North Carolina at Charlotte, Charlotte, NC 28223 Nadine Bezuk, San Diego State University, San Diego, CA 92182

Mally Moody, Oxford High School, Oxford, AL 36203

This section is designed to provide in reproducible formats mathematical activities appropriate for students in

grades 7-12. This material may be reproduced by classroom teachers for use in their own classes. Readers who have developed successful classroom activities are encouraged to submit manuscripts, in a format similar to the "Activities" already published, to the editorial coordinator for review. Of particular interest are activities focusing on the Council*s curriculum standards, its expanded concept of basic skills, problem solving and applications, and the uses of calculators and computers.

730 _Mathematics Teacher



Prerequisites: Students should be famil

iar with the concept of events occurring as

functions of time. They should be able to set

up a rectangular grid and to plot points on

the grid.

Directions: The activity can be used in

regular mathematics classes or cocurricular or extracurricular-activity sessions. The

time required for the activities depends on

the level and ability of the students, the amount of class discussion generated by the

teacher and students, and the amount of

material assigned as homework.

Sheet 1: Display an overhead transpar ency of the graph on sheet 1 and make sure

that students understand how the graph

represents the speed of Mrs. Johnson's car.

Ask questions like, How fast was Mrs.

Johnson going at 7:15 a.m.? What time did

she get off the highway? How many times did Mrs. Johnson come to a complete stop while she was driving? Encourage students to think of other questions that they could answer by looking at the graph. Many of

their answers will not be exact. Such an

swers as "about eighty-eight kilometers per hour" and "between ten and twelve min

utes" should be accepted. If students have colored pencils, have

them draw the graphs for problems 2 and 3 in different colors on the same grid as the one used for Mrs. Johnson's trip. After stu

dents complete their graphs, have them ex

change and compare them. Encourage stu

dents to describe and explain their graphs to

each other. After students complete problem 4, ask them to discuss their graphs, as well as the strategies they used to create the

graphs. In creating the other graphs did

they use the story or the graph in problem 1? Are any of the graphs related?

Sheet 2: Distribute sheet 2 and graph paper to each student. Make sure that stu

dents understand how the graph represents the speed of the bus. Divide students into

pairs to work on problems 1 and 2. They should cooperate in creating their graphs and stories. Both students should be pre

pared to discuss their work.

Sheets 3 and 4: Distribute sheets 3 and 4 to each student. Point out to students how these graphs differ from those used on sheets 1 and 2. For example, in problem 1 on sheet

3, plotting how many times the door was

opened at exactly 7:00 a.m. or at exactly 4:00 p.m. would not be useful. Instead, the number of times the door was opened during an interval of time is plotted. Each space on

the horizontal axis represents one hour. The Xs show how many times the door was

opened during that hour. Ask students to discuss how the horizontal axis used in the

graph on sheet 1 differs from this one.

In problem 1 on sheet 4, a scatterplot is used instead of a histogram. One X is used to

represent how many times the lunchroom door was opened. Ask students to speculate on why we are not setting up a graph just like the one on sheet 3. A helpful approach is to use the X to represent the ordered pair that is plotted. If a point is used, many stu dents will try to connect the dots.

Students may work together or individu

ally on these activities. After completing them, encourage students to share their re

sults with the class.

Answers: Answers will vary. Students should be encouraged to present and defend their own answers. Examples of acceptable answers are included here.

Sheet 1: For the possible answers to prob lems 2 and 3, the dotted lines represent Mrs.

Sung-ki's trip.

2.

w 105

7:00 7:10 7:20 7:30 7:40 7:50 8:00 8:10 8:20 8:30 8:40 8:50 9:00 Time

7:00 7:10 7:20 7:30 7:40 7:50 8:00 8:10 8:20 8:30 8:40 8:50 9:00 Pm- Time

December 1990 731



4. 120

w 105 3 A

3:00 3:10 3:20 3:30 3:40 3:50 4:00 4:10 4:20 4:30 4:40 4:50 5:00

Sheet 2: 1. The roller coaster will go slowly up the first long hill and will descend rapidly. From then on its speed will fluc tuate until it stops. One acceptable answer is shown here.

0 1 2 3 4 5 Time in Minutes

Sheet 3: Answers will vary.

Sheet 4: 1. A few workers go in and out of the lunchroom in the morning. A few

teachers visit the lunchroom between 10:00 a.m. and 11:00 a.m. and 1:00 p.m. and 2:00 p.m. for a snack. We have two lunch periods, one from 11:15 a.m. to 11:45 a.m. and an other from 12:15 p.m. to 12:45 p.m. Fewer

people eat during the second session.

2.

Full

0 1 2 3 4 5 Time in Seconds

3 and 4. Answers will vary. REFERENCES

National Council of Teachers of Mathematics, Commis sion on Standards for School Mathematics. Curricu lum and Evaluation Standards for School Mathemat ics. Reston, Va.: The Council, 1989.

National Research Council. Mathematical Sciences Ed ucation Board. Everybody Counts: A Report to the

Nation on the Future of Mathematics Education.

Washington, D.C.: National Academy Press, 1989.

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732 Mathematics Teacher



GRAPHING FUNCTIONS OF TIME SHEET 1

Mrs. Sung-ki left her house in the country at 7:00 a.m. to drive to a business

meeting in Birmingham. She drove through a small town that had one red

light and then got on an interstate highway. When she got off the highway in

Birmingham, she drove three blocks to the location of her meeting. She arrived just in time for the 8:00 a.m. meeting. The graph below shows the

speeds at which her car was traveling during her trip.

7:00 7:10 7:20 7:30 7:40 7:50 8:00 8:10 8:20 8:30 8:40 8:50 9:00 Time

1. Examine the graph and notice how it corresponds to the story.

2. Grandpa Joe and Uncle Julio are making the same trip. They are going into town for a leisurely day of shopping. Graph a possible representation of their trip on the same grid as the one used in problem 1.

3. Hot-Rod Millie, who lives near Mrs. Sung-ki, will leave at 7:00 p.m. to attend a rock concert in Birmingham. How might her trip look? Use the same grid that you used in problems 1 and 2.

4. Mrs. Sung-ki's meeting will end at 3:00 p.m. and she will drive home. How do you think her trip could be represented? Graph it on your own graph paper. Be sure to indicate the time and speed on the axes.

From the Mathematics Teacher, December 1990



MORE GRAPHING FUNCTIONS OF TIME SHEET 2

75

I 60

8. 45 52 ? 30 e I 15 If I I ? I a I I 111.I I

10 20 30 40 Elapsed Time in Minutes

50 60

The graph above shows the speed of a school bus making its afternoon round. Notice that the bus leaves the school and speeds up to 45 kilometers per hour until it reaches the neighborhood where the students are to disembark. In the

neighborhood the bus stops and goes until all the students have left the bus. The bus then picks up speed and returns to its garage for the night.

1. How do you think the trip of a roller coaster at an amusement park could

be represented? Graph it below. Be sure to indicate the elapsed time and

the rate on your axes.

2. Next graphically represent a trip of your own choosing and write down the

story of what took place. You will need graph paper. You may select a

story about a snow skier riding a lift up a mountain and skiing down or a

pizza-shop worker making home deliveries, or you may make up your own.

(I prefer that you make up your own.)




COLLECTING DATA OVER TIME SHEET 3

During one twenty-four-hour period I recorded the number of times that our

refrigerator was opened each hour. The results appear below. You will notice that the refrigerator was not opened during the day when no one was home. It was opened when we got in from school and wanted a snack. It was

particularly busy around the supper hour, and it was even opened once during the night when Jerry woke up thirsty.

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Time

1. What do you think a graph of the activity of your refrigerator door would look like? Use your own graph paper to draw it.

2. Post a tally sheet on your refrigerator at home and ask the members of your family to record when they open the door during a twenty-four-hour period. (You can use a magnetic clip to hold the tally sheet and hang a

pencil on a string beside it for their convenience.)

Morning # ^ e# ^ <P

Afternoon

& & & & & J> & & ^ ,** N#' ^ ^ ^

3. Plot the results on graph paper and compare the resulting graph to the

graph of your prediction. Write a paragraph about your findings.




GRAPHING DATA COLLECTED OVER TIME SHEET 4

Many activities in everyday life are functions of time.

1. This is a graph of the number of times the door to our lunchroom was

opened during the hours of one school day. Write a paragraph explaining the graph.

300

S 200

100 " "

7:00 - 8:00 - 9:00- 10:00- 11:00-Noon- 1:00 - 2:00 - 3:00 8:00 9:00 10:00 11:00 Noon 1:00 2:00 3:00 4:00

Clock Time

2. On the left-hand graph below, represent the amount of air in a tire just before it blows out, when it blows out, and shortly after it blows out.

3. On the right-hand graph above, record the number of times you think the

telephone in your house rings each hour between 2:00 p.m. and 9:00 p.m.

How would your graph differ for the hours between 2:00 a.m. and 9:00

a.m.?

4. Make up your own graph for a function of time and see if your classmates can figure out what it represents.




mathematics as communication: graphing information collected over time

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