Forecast Accuracy

• ## Forecast Accuracy

Grades: 6th to 8th,High School
Periods: 3
Author: Mark Roddy

### Materials

• Daily access to weather forecasts either web-based or through TV, newspaper, etc.
• Forecast Accuracy Spreadsheet (optional)
• Needed on the first day: the previous day's high temperature recorded for your community

### Instructional Plan

People are apt to complain about weather forecasts and their accuracy. This is your chance to do something about them - or at least to understand their accuracy. In this lesson, students will gather data and crunch numbers to find out whether the weather forecasters are doing their jobs. Read the story behind Forecast Accuracy here.

Time:  One hour for problem formulation

One to four weeks for data gathering (~10 min per day)

One hour for the final session and possible extensions

First Day (60 minutes):

15 minutes: Introduce the Investigation

Ask students the following questions:

• What do you think was the high temperature yesterday?
• What will be the high temperature today?
• How about tomorrow?
• The next day?

List some responses for each question. Go back to the first list and write down the actual high temperature recorded for yesterday. Ask students the following:

• How accurate were your estimates?
• How far off were they? Over? Under?

Have them check their process with an elbow partner. Then allow volunteers to explain their reasoning to the whole group. Make sure everyone comes to a conclusion about finding the error, the differences between their estimates and actual high temperature. Does it matter if you were too high or too low? Not really; what matters is how far your estimate was from the actual temperature. Error is the numerical distance of the estimate from the true value. When the estimate matches the actual high temperature, the error is zero. This foreshadows the notion of absolute value, but you do not need to label it as such right now.

10 minutes: Determine a Source for High Temperature Forecasts

Ask students the following:

• Do you ever look at weather forecasts and if so, what is your source?

Depending on their age and level of access to technology, they will have different response or will need direction, but you will need to come to a consensus as to an acceptable forecast source. This is the source you will use throughout the investigation, so it needs to be easily accessible, and it should provide forecasts for at least 5 days out. A week or 10 days is better. (Check the Related Resources section at the end of this lesson plan for some possible web-based sources.) Using your selected source, look at a forecast for your community and see how the forecasted high for the present day compares with the estimations the students gave earlier. Use this as an opportunity to solidify their grasp of the process of calculating errors.

15 minutes: Construct a Plan for Determining the Accuracy of Multiday Forecasts

• Are weather forecasts usually right?

As it turns out, although weather forecasts are often the subject of complaint, they are usually quite good for a day or two or three (but keep this to yourself for now). The central question for this investigation is this:

• How does the accuracy of the forecast for daily high temperature change as the length of the forecast increases-that is, for a 1-day, 2-day . . . 9-day forecast?

Let students consider this question. Guide them to think about how they might use mathematics to quantify the accuracy of a high temperature forecast. Give them time to think on their own, with partners, or in groups as you see fit. After they have had some time, ask for responses and explanations.

20 minutes: Use Mathematics to Address the Central Question

Your objective is to guide your class to come up with a method that will allow them to calculate errors for long forecasts (e.g., a 9-day forecast) and short forecast (1-, 2-day forecasts), to find the mean error for any given length of forecast, and then to compare these means so that the accuracy of short and long forecasts may be compared.

Here is a link to a spreadsheet that employs a simple process, recording forecasts for high temperature each day and then recording actual observed highs and calculating errors: Forecast Accuracy Spreadsheet. The linked spreadsheet allows for 1-day, 2-day, and so on up to 9-day forecasts to be recorded every day for 23 days. This may be too long (or too short) for your needs; you may decide to record more or fewer forecasts each day as your circumstances dictate. Regardless, the forecasts are recorded each day, as is the actual observed high temperature. With more experienced students, you might allow the spreadsheet to perform these calculations as the data are recorded each day.

Now that you have a means of organizing data and making the error computations, you have some decisions to make: Do you want to have the whole class work on the same data set-for your town, for example? Or do you want to have individuals or small groups gather data for towns of their own choosing? The former will simplify your life as students work on a single data set. The latter will probably increase students' engagement and enable the class to compare forecast accuracies for different areas and to speculate about these differences. You might also be able to amalgamate the data and compute mean errors based on a larger sample size.

Regardless of whether everyone is working on the same town's forecasts or different ones, every person or every small group should gather their own data. This may produce a variety of results, but this is an opportunity to discuss reasoning and justifications-a hallmark of the mathematical process.

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Data Gathering Days (number of days varies at the teacher's discretion. 5-15 days?)

Once they are proficient, students will need about 5-10 minutes each day to record the forecasts and to enter the verification temperatures. These are the actual high temperatures that were observed each day at their selected site.

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Final Session (50 minutes)

15 minutes: Compute Errors across All Forecasts

After the data-gathering period is over, students will have raw data (forecasts and verification temperatures) and will be able to compute the errors (the differences between the two). If they have not already done so, give students time to calculate all errors for the forecasts they have recorded.

15 minutes : Mean Errors for Short and Long Forecasts

Remind students of the investigation's central question and ask them how the errors look to them. Give individuals and groups time to consider these questions:

• Are the errors larger as the length of the forecast increases?
• What is the magnitude of a typical error for a 1-day forecast? A 7-day forecast?
• How can we use mathematics to provide a defensible answer?

Calculating a mean of the errors for the 1-day forecasts, the 2-day forecasts, and so on and comparing these means is, of course, a useful approach and should be part of the discussion.
This is, however, not the only option. The emphasis should be on sense making and mathematical reasoning and justification.

Spend some time considering your data and come to an understanding of how the mean errors change as the length of the forecast increases.

10 minutes: Conclusion

Having established some measure of the way the errors tend to increase as the length of the forecast increases, take some time to consider the question of how much error is acceptable. Surely one or two degrees is OK, and 10 degrees is not; but where do you and your students
want to draw the line? This is an opportunity to use mathematics in a practical sense.

10 minutes: End with Some Extension Questions

Are forecasts better in some places than in others? If you have had groups investigating the accuracy of forecasts in more than one location, this is an opportunity to consider whether all forecasts are created equal. Do people on the East Coast get more accurate forecasts than those on the West Coast? Much of the U.S. is in the midst of the midlatitude westerlies so most of our weather tends to move from west to east. Thus, people on the West Coast have the relatively data-poor Pacific ocean as the source of the weather systems that are coming for them. Does that result in less accurate forecasts? This is something that can be investigated, and mathematics is the right tool for the job. Students might come up with their own questions. For example, are certain communities served with more accurate forecasts than others?

### Assessments and Extensions

Assessments

1. Table 1 contains the 1-day, 2-day, and 3-day forecasts (FCs) that were made for the high temperatures in Springfield on March 11, 12, and 13. Table 2 has the mean errors for these 1-day, 2-day, and 3-day forecasts. Table 1 is missing the actual observed daily high temperatures and subsequent errors in the forecasts. Use what you know about finding the errors and their means to supply possible observed daily high temps and FC errors that fit the mean errors given in table 2.
2. You are trying to write a report that explains the errors observed in these forecasts and observed high temperatures. As part of this report, you need an x-y plot that shows how much the mean error increases as the length of the daily forecasts increases; that is, the mean errors get larger as you go from the 1-day forecasts to the 2- and 3-day forecasts. Sketch an x-y plot that will show this relationship. Be sure to label the x and y axes and provide a reasonable scale for each axis. Plot the points given by the data in table 2 above and draw a simple best-fit line that suits the data.

Extensions

Check out the final section of the Instructional Plan for some extension ideas.

This investigation focuses on forecasts for high temperatures. Forecasts for other variables could be examined-for example, the low temperature, whether there will be rain or snow, or more simply, whether it will be sunny or cloudy-although these are more difficult to quantify.

### Questions and Reflections

See questions embedded within the Instructional Plan.

### Objectives and Standards

Learning Objectives

1. (Concept) Students will understand how weather forecast data may be used to analyze the accuracy of the forecasts.
2. (Skill) Students will be able to gather, organize, and store simple numerical data.
3. (Skill) Students will be able to use subtraction to calculate errors.
4. (Skill) Students will be able to plot and interpret error data.
5. (Disposition) Students will continue to develop a positive disposition regarding the use of mathematics and science to make sense of the world.

Common Core State Standards for Mathematics

• 6.NS.7.C Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation.
• 6.SP.5.C Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern.

• 7.NS.1.C Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.

• 8.EE.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph.
• 8.SP.2 Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.

High School

• S-ID.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
• S-IC.4 Use data from a sample survey to estimate a population mean or proportion; . . .

Common Core State Standards for Mathematical Practice

• SMP 1 Make sense of problems and persevere in solving them.
• SMP 4 Model with mathematics.

### Related Resources

 Yahoo WeatherWebsite TimeAndDate.comWebsite Forecast Accuracy StoryStory