section 1 (3/96)eo.ucar.edu/skymath/section4.pdf · section 4 data representation and change the...

Section 4Data Representation

and Change

The big idea in this section centers on how to represent and analyzechanges in temperature over time.

In the first activity, students make a graph of local high and lowt e m p e r a t u res. They identify other locations in the United Stateswhere the temperatures are most similar to and most different fromtheir own. By hand, students construct line graphs to comparepatterns of highs and lows across three geographic locations.

Next, students analyze temperature data collected by theStowAway™ data logger that was launched at the start of this unit.They use the computer to explore line graphs of the temperaturechanges outside their classroom; calculate the amount and rate ofchange between each high and low temperature; and answerquestions such as “Is the rate of change faster from the coldesttemperature to the hottest, or from the hottest to the coldest?” and“What is the relationship between the slope of the graph and the rateof change?”

F i n a l l y, students write a narrative description of the temperatureover the last several weeks, drawing upon the data they havecollected with the StowAway™, their weather logs, and theanalytical techniques they have learned in this unit. Aftercommunicating with their partner class on e-mail, students comparethe temperature changes outside their own classroom with thoseoutside the classroom of their partner class.

How can the temperature fluctuations in students’ location best berepresented graphically? Here students construct line graphs of thehigh and low temperatures they have been collecting. They theninterpret information presented in line graphs.

• Students construct line graphs toshow changes in temperature overtime.

• Students read and interpret linegraphs.

• Students informally compare ratesof change.

Students will need five days’ worth of temperature data from other cities in theUnited States. You can provide five days of the USA Today weather page or getdata from Blue Skies daily for five days.

materials1 copy of Reproducible Master 14 (Learning About Line Graphs) for eachstudent

1 copy of Reproducible Master 15 (Make Your Own Line Graph and Story)for each student

high/low temperature data from students’ weather logs

graph paper (1/4 inch)

rulers or straight edge

This activity will enable you toassess students’ ability to constructand interpret line graphs. UseReproducible Master 14 as anassessment of individual students'ability to read and reason about datashown in a line graph.

math goals ongoing assessment

advance preparation

Line Graphs11

DRAFT Page 81

DRAFT Line Graphs

Section 4: Data Representation and Change

Teacher AMy students worked in groups of three.Each group constructed a line graph ofminimum and maximum temperatures forthe previous week (Monday throughFriday). When we were looking at all ofthe graphs, one group mentioned that wedidn‘t really know exactly what all of thetemperatures were between the high forMonday and the high for Tuesday.

We talked about how we didn’t knowwhether the temperature fluctuated a lotbetween days or if it changed smoothly. Inother words, a lot of data were missing.Most of the students felt this was OKbecause we just wanted to show a trend inthe data. A few students feltuncomfortable with so much data missing.

I couldn’t have planned this better! Thisdebate was the perfect setup for the nextlesson, using the data collected by theStowAway™ Logger.

what students do1. Using Reproducible Master 14

(Learning About Line Graphs),students match events from a storywith points on a corresponding linegraph. Instruct students to fill in the times alongthe horizontal axis of the graph. Tell themthis is an opportunity for them to becomemore comfortable reading line graphs andmore familiar with the parts of a line graph.

2. Students construct a line graph of thelocal high temperatures over the lastweek. Also have students use the same grid toconstruct a line graph of the local lowtemperatures.

3. Students compare and discuss theconstruction and meaning of theirline graphs.Ask students:

• What scale did you use? • How many degrees are represented by

each grid mark on the graph paper?

• Did the high temperatures follow thesame pattern as the low temperatures?

Teachers'

Activity 11 DRAFT Page 83


Teacher BI gave my students one week to findexamples of line graphs in the media. Atthe end of the week, they gave mini–oralreports about the graphs they brought toclass. They described what eachgraph was about and detailed thetrends being represented in each.The sources that seemed to be mostfruitful were business magazines andthe business section of thenewspaper.

What I would like to do next is havestudents focus on the scales used inline graphs. How would the datalook if we used larger increments onthe scale? What if we used smallerincrements? The point is that by changingthe scale you can change the look of thedata and therefore change the message tothe reader. I think students will have funfinding misleading graphs in the media,especially in advertisements.

4. Students find two other locations inthe United States that havetemperatures similar to and differentfrom their own.Students construct line graphs of the highand low temperatures for the two locationsthey chose.

Tell students to use the same scale theyused on the local graph.

For example:

5. Students compare patterns in thethree sets of line graphs.Ask students to identify what similaritiesand what differences they notice.

6. Using Reproducible Master 15,students create their own line graphsand corresponding stories.

Stories

DRAFT Line Graphs


background informationConstructing Line GraphsTo construct a line graph, first draw the horizontal and verticalaxes. The horizontal axis typically shows the time periodcovered by the data. In the graph below, the time period is sixmonths—January through June—with one data point for eachmonth.

The scale on the vertical axis indicates degrees Fahrenheit. Therange is 0 to 100 degrees, and the scale is marked in 5-degreeintervals. It is also important to label each axis and to title thegraph.

Scales on Line GraphsIt is useful to look at the scale on a line graph before youinterpret the data. On the three line graphs shown on thesepages, the scale on the vertical axis indicates degreesFahrenheit. The range of the scale is 0 to 100 degrees. The scaleon the horizontal axis indicates time. In these graphs, the timeperiod is six months, January through June.

If you wanted to change thelook of the data, you couldchange the scale on thevertical axis. For example, tomake differences in the dataappear smaller, use a largerrange:



To make differences in the data appear larger, use a smallerrange:

Any of these scales is appropriate, but as a reader one shouldconsider the scale being used and how it may change theappearance of the data.

Interpreting Line GraphsWhen interpreting a line graph, the reader is usually looking fora trend over time. In the Kalamazoo graph, it could be notedthat the trend is for the temperatures to go up from Januarythrough June at a fairly consistent rate. Another trend is for thedifferences between the high temperatures and the lowtemperatures to become greater during spring and intosummer.

Another thing to look for in a line graph is the size of thechanges between data points. When did the greatest changeoccur?

In the Kalamazoo graph, the greatest change in hightemperatures occurred between May and June. The smallestchange occurred between January and February. Similarpatterns can be seen in the low-temperature data.

How do the data recorded by the StowAway™ compare with thedata kept by students? In this activity students analyze temperaturedata collected electronically outside their classroom over 18 days.They examine a line graph of the data and mark the hightemperature, low temperature, and amount of change betweenmaximum highs and minimum lows.

Students use graphing software toconstruct line graphs.

• Students interpret data presented inline graphs.

• Students compare patterns andtrends in line graphs.

The StowAway™ data logger was set to collect temperature data at thebeginning of the unit for an 18-day period. Download the temperature datafrom the StowAway™ and print a line graph of the data. (See BackgroundInformation for directions on how to do this.) Export the data to yourspreadsheet and format and print the data.

Copy the graph overlay reproducible master (Reproducible Master 16) onto anoverhead. Put the overhead over a printout of the StowAway™ graph andmake one copy of the graph and the spreadsheet data for each student.

materialsoverhead from Reproducible Master 16 (LogBook® Graph Overlay)

calculators

students’ weather logs

LogBook® software

StowAway™ data logger

Observations during this activityand students’ work will enable youto assess their ability to organize andanalyze temperature data.


advance preparation

Exploring the StowAway™ Data12

DRAFT Page 87

DRAFT Exploring the StowAway™ Data


Teacher AMy students worked in groups of three inthe computer lab. Each group downloadedthe data from the StowAway™ logger intoLogBook® and then into ClarisWorks.This took almost the whole class period,but I felt my students needed thatexperience.

Students had various difficulties attachingthe StowAway™ to the computer. Somegroups attached the gray cable to theprinter port instead of the phone port.Other students didn’t make sure theconnections were tight by pressing thecable firmly into the computer and theStowAway™.

Students all found the LogBook® softwarepretty easy to use. They were fascinatedwhen they ran the cursor around the graphand saw the values displayed. They werealso amazed at the variability of thetemperature over the 18 days.

After exporting the data (some groups hadtrouble remembering to change the formatto Excel), students imported the data intoour spreadsheet application. They wereconfused by the long number shown in thedate column, but we changed the format ofthe number to “Date & Time” and the tablemade much more sense.

what students do1. Ask students to read and interpret

the StowAway™ line graph. Referring to the line graph, ask studentsquestions like the following:• Was the temperature fairly consistent

during this period?• What patterns do you notice?

• Did any unusual weather days occurduring this period? If so can you spotthem on the graph? How?

Suggest that students refer to their weatherlogs as they look over the graph.

2. Students practice reading the Stow-Away™ data from the LogBook®software.Ask students:• What was the highest temperature for a

given date? The lowest?• At what time did the high temperature

occur?

3. Students mark on the line graph thetimes the high temperatures and lowtemperatures occurred each day. Tell students they can find this informationin the spreadsheet.

4. Students make observations aboutthe graph and the temperature data. Encourage students to make observationssuch as these:• On most days, the temperature changed

about 40 degrees through the day.• The highest temperature occurred on

September 1 at about 2:00 PM., it was 105degrees.

• The lowest temperature occurred onSeptember 10 at about 5:00 AM, it was 45degrees.

Teachers'



Teacher BSeveral students noticed an odd pattern inthe StowAway™ graph. On Tuesdaymornings, the temperature would climbuntil 10:30. Then it would drop 10 or 15degrees, and quickly shoot back up 20minutes later. We speculated that cloudscovered the sun or a cold wind alwaysblew at that time, but nobody thoughtthose were likely explanations. Finally,Jennifer suggested that the cafeteria mightget a delivery then and the truck could beblocking the StowAway™ probe from thesun. The following Tuesdayat 10:30, we checked—therewas the truck!

• During the first week of September thehighest temperatures were all in the 90sor low 100s, but during the second weekthe highs were in the 80s or low 90s.

5. Students mark their graphs with datathey recorded in their weather logs.If students notice discrepancies between theStowAway™ data and the temperaturesthey recorded in their weather logs, askthem to think about why the discrepanciesmight have occurred.

6. Students create a data table to recordthe high temperature, low temper-ature, and range for each day. Also have students record the time the highand low temperatures occurred. Forexample:

7. Students look for patterns in the data. You may find that students makeobservations such as the following:• The high temperatures occur around 3:00

to 3:30 each afternoon.

• The low temperatures occur around 4:00or 5:00 in the morning.

• The range between the high and lowtemperatures was almost 50 degrees onSeptember 4.

Stories

Date 9/1 9/2 9/3 9/4 9/5 9/6 9/7

Low 70° 64° 52° 51° 59° 64° 60°

4:34AM 5:01 AM 5:12 AM 4:38 AM 5:06 AM 5:22 AM 5:10 AM

High 105° 93° 96° 100° 97° 100° 97°

3:33 PM 3:48 PM 3:30 PM 2:39 PM 3:41 PM 3:09 PM 2:35 PM

Range 35 29 44 49 38 36 33

DRAFT Exploring the StowAway™ Data


background informationDownloading Data from theStowAway™To download the data from the StowAway™ data logger, plugthe data logger into the gray cable and then plug the other endof the gray cable into the modem port of your Macintosh or the

DB-9 serial port of your IBM-compatible.Start up the LogBook® software. Themenu will show File Edit Logger

Views Windows. Pull down the Loggermenu and select the Readout option. After10 or 15 seconds, a graph of your data willbe displayed and a dialog box allowing itto be saved will appear over it. Nameyour data and save the file. A graph likethe one shown here will appear.

Students can zoom in on the plot to get a closer look at the databy drawing a selection box around the section they'd like andclicking in the box or selecting Zoom In on the View menu. Tozoom out, select Zoom Out from the View menu.

On the Macintosh, students can use the Show Plot Info

command under the Windows menu in LogBook® to closelyexamine their graphs. This command brings up a Plot InfoWindow which shows start time, stop time, minimum recordedtemperature, and maximum recorded temperature. It alsoshows the plot information (time and temperature) pertainingto the mouse location. Students can place the cursor anywhereon the graph to obtain numerical readings.

To download the StowAway™ data for use in other datamanipulation packages (like Microsoft Excel or ClarisWorks),pull down the File menu and select Export Setup. Click on thedesired export format button and click OK. Then pull down theFile menu again and select Export Plot Data. Type in the nameyou want to give your data set and click OK. The data set isnow saved in a file that can be used in another softwarepackage.

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Tem

peratu

re (*

F)

12:0009Aug95 12:00

10Aug95 12:00 11Aug95 12:00 12Aug95 12:00 13Aug95 12:00 14Aug95

S/N: 17581EDC Weather

What‘s Been Happening?13

DRAFT Page 91

How fast did the temperature change each day from coldest tohottest over the last two weeks? From hottest to coldest? In thisactivity students calculate the rate of change from the lowtemperature to the high temperature for each day and compare therates. They also examine the temperature changes using the datarecorded in their weather logs. Finally, they explore what eventscaused the rate of change to increase, decrease, or remain constant.

• Students calculate rates of changebetween high and lowtemperatures.

• Students compare rates of change.

• Students relate changes to actualevents.

materialscalculators

copies of the StowAway™ graph, spreadsheet data, and students’ graphsfrom Activity 12

Your observation of students groupwork will inform you about theirability to calculate rates of change.Students’ written narratives willhelp you evaluate theirunderstanding of the relationshipbetween their own observationaldata and data represented in aStowAway™ line graph.


advance preparation

DRAFT What's Been Happening?


Teacher ABefore we even got to do rates of change,my students made two interestingdiscoveries from the StowAway™temperature data. First, they seemedsurprised that the coldest point of the daywas usually 5:00 AM; they thought itwould be closer to midnight. Second, theynoticed that it took less time to warm up(go from coldest to hottest) than it did tocool down. At this point, we decided to e-mail our partner class to see if this was truethere too.

When our partner class replied to ourmessage, we learned that their datashowed that it took about the same amountof time to cool down as to warm up (about12 hours each). Now, we all began towonder why.

what students do1. Students calculate the rate of change

from coldest temperature to hottesttemperature for a day.Choose one day from the StowAway™ data.Ask students to calculate the rate of changefrom the coldest temperature to the hottesttemperature (see Background Information).

2. Students calculate the average rate ofchange from the hottest temperatureto the coldest temperature for theday.

3. Students compare the rates of change.Ask students: • Does it take longer to warm up (go from

coldest to hottest) or to cool down (gofrom hottest to coldest)?

• Do you think this is true for every day?

4. Students calculate the rates of changefor each day logged by theStowAway™.You may want students to work in groups,with each group responsible for severaldays of data.

5. Students compare the rates of changeto the slopes of the graph for eachday.Ask students:

• What is the relationship between the rateof change and the slope of the graph?

• When the slope is steeper is the rate ofchange faster or slower?

• Can you predict the rate of change bylooking at the slope? If so, how?

Teachers'



Teacher BTwo students were so interested in theStowAway™ that I invited them to worktogether and plan a final project using dataloggers. They decided to find out whetherthe temperatures and the pattern of changewere different in different parts of town.We have two StowAway™s, and so theywere able to take them home. One studentlives in a house shaded by many trees; theother, in a south-facing apartment. Beforethey launched the StowAway™s, I hadthem write their expectations about howthe patterns would compare. It will beinteresting to see what they find out.

6. Each student writes a narrativedescription about the temperatureoutside the classroom for the timeperiod recorded by the StowAway™logger. Direct students to use data collected in theirweather logs, the StowAway™ graph, andany other information they have collected.In their narratives, students may addressquestions like the following:• Did a pattern of daily change occur across

the whole period?• Did you find blocks of days that seemed

to have a common pattern? If so, whatwas the pattern?

• Did some days show a different pattern?If so, which days were they? How werethey different? Can you explain thereasons for the difference? Could youremember what the weather on that daywas like, without looking in the log?

• Would you expect to see the samepatterns at a different time of year?

7. Students share and discuss theirnarratives.

8. Students e-mail their partner classand exchange StowAway™ data files. Encourage students to compare the linegraphs for both locations and to discusssimilarities and differences.

Stories

DRAFT What's Been Happening?


background informationRate of ChangeThe average rate of change is the amount of change that occursin a given time period. For the purposes of this unit, studentswill be calculating changes in temperature over hours, or days.

For example, if the high temperature is 65 degrees Fahrenheit at3:00 PM and the low is 22 degrees Fahrenheit at 5:00 AM, therange is 43 degrees. The temperature changed 43 degrees from5:00 AM to 3:00 PM—a 10 hour period. So, the rate of change isas follows:

43°/10 hours = + 4.3°/hour

The temperature increased at an average rate of 4.3 degrees per hour.

When rate of change is used to describe a decrease intemperature, the rate of change should be reported as anegative value.

For example, if the temperature changed -43 degrees from 3:00PM. to 5:00 AM—a 14 hour period—then the average rate ofchange is:

43°/14 hours = -3.07°/hour

The temperature decreased at an average rate of 3.07 degrees per hour.

Slope and Rate of ChangeThe slope of a line is a measure of its steepness. For example, inthe graph shown below, Lines a, b, c, and d have differentslopes.

The slope of a line tells us whether the rate of change wasrelatively fast or slow. The steeper the slope, the faster the rateof change. Line a in the graph above shows the fastest rate ofchange; Line b shows a slower rate of change. The direction of

a

b

c

d

12:00 1:00

Time of Day

Temperature (F)

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68

66

64

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the slope tells us if the change was a positive or negative one.Lines a and b show positive changes, from cooler to warmer;Line d shows a negative change. Line c , which is parallel to thex-axis, shows that there was no change at all.

E-mail Your Partner Class DRAFT Page 97

E-mail YourPartner Class

1. Students send their description of the temperatureoutside their classroom for the past two weeks totheir partner class.If possible, have students enclose the data file that wasexported from the StowAway™.

2. When they receive the data from their partnerclass, students discuss the differences betweentheir partner class’s temperatures and their own.If the partner class has included a data file, print out thegraph and have students compare it with their own.

3. Students formulate and send questions about theirpartner class’s data.

4. Students reply to their partner class’s questions.

Were the Predictions Correct?14

DRAFT Page 99

Were the predictions made by forecasters at USA Today foryesterday's high temperature correct? In this activity students gatherdata about yesterday's high temperatures around the nation andmake a temperature map to graphically display their findings. Theycompare their map of the actual high temperatures with that of thepredicted ones and try to explain any differences they find.

• Students read and interprettemperature maps.

• Students locate cities on a nationalmap.

• Students collect and represent datato show how the high temperaturesfrom the previous day differed fromthe predicted highs.

• Students represent national hightemperatures as bands of color on amap.

Obtain at least one copy of the weather page of USA Today from the past twodays.

materialsReproducible Masters 17–22 (regions of the United States) and 23–28 (citycodes)

This activity is an opportunity foryou to evaluate students'collaborative problem solving asthey construct temperature mapsand compare them with a publishedtemperature map for the same timeperiod.


advance preparation

DRAFT Were the Predictions Correct?


Teacher AA couple of the groups in my class hadheated debates about how many cities theyshould sample in order to make their maps.The group in charge of the Northeastregion decided to plot data from all of thecities in the region because the number ofcities there is relatively few. But the groupwith the Southwest region had exactly theopposite problem, with 36 possible cities inCalifornia alone. They finally decided theyshould divide the states in their region intozones and make sure they sampled datafrom each of the zones. For example, theydivided California into 10 zones andsampled data from one city in each of thosezones. They felt that getting data fromeach of the cities they chose would allowthem to paint an accurate picture of thetemperatures in their region.

what students do1. Students read information from a

color temperature map (such as themaps in USA Today). Ask questions like the following:• What information is this map giving us?

• What does the legend indicate?• What do the numbers represent?

• How many temperature bands arerepresented on the map?

• What might explain some of the patterns?• How would an actual high-temperature

map compare with this map of predictedtemperatures?

2. Students discuss how a temperaturemap might be constructed.Encourage students to think about whatquantity of data points might be required tomake an accurate representation and inwhat way the temperature regions might bedelineated.

3. Groups of students gather yester-day’s high-temperature data for sixregions of the United States: North-east, Midwest, Northwest, Southwest,Southern Plains, and Southeast.Give each group one copy of thereproducible master of their region(Reproducible Masters 17-22) and the citycodes for that region (Reproducible Masters23-28). Instruct them to decide how manycities (from those available on-line) theywill include in their set of data in order tomake an accurate representation of theirregion’s temperature data. Then, askstudents tolocate and mark the cities theychose on their regional maps.

Teachers'



Teacher BOnce they had completed coloring the classmap, my students compared it with theUSA Today forecast map. Theyimmediately saw many differencesbetween their map and the forecast map.Many students saw slight differences thatthey attributed to errors in theirdemarcation strategies. But the actualtemperatures in the Midwest wereconsiderably lower than the predicatedones. Students were all very excited thatthe predictions were not completelycorrect.

4. Students learn how to log in to UM-Weather Using Telnet to view temper-ature data for the preceding day.See Background Information forinstructions.

5. Using colors that match the USAToday legend, students color-codetheir regional maps.Remind students of the earlier discussionabout how to make a temperature map.Encourage them to work together to decidehow to draw the boundaries for eachtemperature region.

6. Students cut and paste their regionalmaps in order to make a class map ofthe entire country.Ask students: What method did you use todelineate regions? Did the number of citiessampled seem to affect the construction ofthe temperature bands?

7. Students discuss the differences andsimilarities between their map andthe map of yesterday’s USA Todaypredictions.Discuss the following questions:• In what ways does your map resemble

the predictions? • In what ways does it differ?

• Why do you think the differencesoccurred?

Help students consider how their methodsmay have differed from the method used bythe map makers. Or, perhaps thepredictions were incorrect.

Stories



background informationWeather MapsA weather map is an analyzed chart that depicts weatherconditions over an area at a particular time. In this unit,students construct temperature maps—a weather map withonly temperature data.

Below is one example of a temperature map from the Blue Skiessoftware.

In the above representation, temperatures are marked bynumerals in many locations, but the overall patterns of thetemperatures in the United States are shown by colored bands,each representing a different temperature range.

In the United States, on the day the above map was generated,the temperatures ranged from the lows of 35 degrees to 40degrees in the southeastern corner of Oregon to the highs of 85degrees to 90 degrees in the southeastern part of the countryand in the southeastern California/southwestern Arizonaregion. Temperatures are generally cooler from southeast tonorthwest, although some fluctuations in this trend occur in themiddle of the country.



Launching UM-Weather Using TelnetFrom the Window to Blue Skies menu, launch UM-WeatherUsing Telnet by clicking on the icon (at right). The Telnetprogram will load and access the UM Weather site.

Press <RETURN> and select 1 (U.S. forecasts and climate data)from the WEATHER UNDERGROUND MAIN MENU.

Select 2 (Print climactic data for selected city).



Using the city codes on Reproducible Masters 23–28, choose acity from which you’d like climactic data. For example, thescreen below represents data from Casper, Wyoming, onJanuary 10, 1996. The HIGH YESTERDAY is the importantpiece of data for students to recognize. Some students mightwant to keep track of other pieces of information.

Section Assessment Page 105


Students use the Section 4 Quiz found at the end of thecollection of Reproducible Masters for this assessment.

Assessment GoalsThe items in this assessment test students’ ability to constructand interpret data displays. To perform well on thisassessment, students should be able to

• construct bar graphs of data with positive and negativevalues,

• construct line graphs, and

• interpret data presented in bar and line graphs.

Answer KeyMany good answers are possible for these open-endedquestions. A scoring rubric is in development. Shown beloware examples of the kinds of graphs students should make,along with answers to questions related to the data displays.

1a. A line graph is useful for showingchange in temperature over time,as in the following example.

1b. The temperature was dropping,then it shot up and stayed high forat least 8 minutes. It dropped backdown to where it was before for at least 8 minutes; then itshot up again for at least another 16 minutes. Then itdropped even lower than ever.

1c. The dog might have been in the doghouse between 3:38and 4:02 and between 4:10 and 4:42. Or maybe the sun wasvery strong at those times, but something blocked the sunfrom the doghouse the othertimes.

2a. A bar graph is useful forcomparing temperatures atdifferent locations, as in thefollowing example.

2b. Meltopolis

2c. Phoester.

Section 4 Assessment

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Lowest January Temperature in Five Places

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Salsalito Cugaville Helixton Meltopolis Phoester

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