Algebra 1 Summer Institute 2014

The Old Faithful Geyser Prognostication

Summary Goals Participant Handouts

Participants encounter  Interpret the shape, 1. The Old Faithful Data samples of measurement center, and spread of Prognostication data that occurred in visual representations 2. The Old Faithful Data sequence over time. within context. Prognostication Participants construct their  Make conjectures and Extensions own visual displays of the predictions based on the 3. Excel file: Old Faithful data and then make some data Data estimates and projections  Justify their reasoning based on the data.. The and decisions importance of attending to variability in the data and the need to track variation in samples of the data over time are major themes in the tasks in this activity.

Materials Technology Source Estimated Time

Poster paper LCD Projector Focus in High 90 minutes Markers Facilitator Laptop School Mathematics Rulers GeoGebra Reasoning and Excel Sense Making: Statistics and Probability

Mathematics Standards

Common Core State Standards for Mathematics MAFS.6.SP.1: Develop understanding of statistical variability 1.2: Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.

MAFS.6.SP.2: Summarize and describe distributions 2.4: Display numerical data in plots on a number line, including dot plots, histograms, and box plots. 2.5: Summarize numerical data sets in relation to their context, such as by: a. Reporting the number of observations c. Giving quantitative measures of center (median and/or average and variability (interquartile range and/or absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.

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MAFS.912.S-ID.1: Summarize, represent, and interpret data on a single count or measurement variable 1.1: Represent data with plots on the real number line (dot plots, histograms, and box plots). 1.3: Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme points (outliers).

Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics

Instructional Plan

The activity is designed for small-group exploration with many opportunities to share reasoning – in private think time, small group discussion, and whole-class discussions. Prompts of questions from the facilitators might help focus the discussion and push the participants to offer justifications; this activity is purposely not heavily scripted. The main idea is to elicit participants’ thinking and reasoning without too much steering by the facilitators.

This investigation focuses on the distribution of wait times between successive eruptions of the Old Faithful Geyser from 1985. Information on the Old Faithful Geyser can be found at: http://geyserstudy.org/geyser.aspx?pGeyserNo=OLDFAITHFUL

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Old Faithful Data – Minutes Between Blasts (Table can be found in the Excel file named Old Faithful Data)

Participants Instructions: (Slides 2 to 7)

1. Divide participants in groups of 4. Each group will pick two rows of the wait times found in the Old Faithful Data – Minute Between Blasts table. (A group member might be asked to pick two numbers between 1 and 16 to pick the two rows, so that every group does not pick the same two rows. Using a random number generator – like excel- to pick the rows would be best)

2. Decide who in each group will create what type of graphical or tabular representation (bar graph, dot plot, stem and leaf, connected line plot). Technology can be used if preferred.

3. Working individually, look over the data. Is there anything that you notice, or anything that you wonder about in your two samples of data? Jot down your “notices” and “wonders”. Create one type of graphical representation for each the two days of data to help you visualize any patterns in the wait times. Jot down any additional notices or wonders that occur to you.

4. Working in your group of 4, share and compare your graphical representations. On the basis of the data, make a group decision about how long you would expect to wait between blasts of Old Faithful is you showed up at the Yellowstone Park and Old Faithfull has just finished erupting. Be prepared to present your graphs to the other groups in class and to defend your group’s data-based prediction for the expected wait time.

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5. Share your group results with another group of 4. As a new group of 8, try to make a group decision about how long to wait between blasts. Be prepared to share results and points of discussion with the rest of the class.

Facts about Old Faithfull:

Predicting Old Faithfull: It is not possible to predict more than one eruption in advance. Old Faithful is currently bimodal. It has two eruption durations, either a long (over 4 minutes) or more rarely a short (about 2-1/2 minutes). Short eruptions lead to an interval of just over an hour and long eruptions lead to an interval of about 1-1/2 hours. In the past the interval was predicted using more precise measurements of the duration of the current eruption. The duration was timed from the first heavy surge, which lifts water skyward at the start of the eruption until the last small splash above the cone at the very end. The longer the eruption lasted, the longer the interval until the next eruption. The following regression table was used:

Duration Interval 1.5min 51 min 2.0min 58 min 2.5min 65 min 3.0min 71 min 3.5min 76 min 4.0min 82 min 4.5min 89 min 5.0min 95 min

The above table is shown for historic purposes. Predictions are currently based only on short or long durations. This simpler prediction model still results in a 90 percent accuracy of the predictions.

Common Misconceptions:

Old Faithful plays on the hour every hour The legend dates back to the early history of Yellowstone Park. Old Faithful has never played every hour on the hour. It is just as likely to play on the hour as it to play 17 minutes after the hour or 23 minutes after the hour or ......

Old Faithful plays hourly Old Faithful's intervals vary from 35 minutes to 2 hours. You might see one interval of 60 minutes but it is very unlikely that you will see two in a row. This misconception seems to date back to the 1870 Washburn expedition where one of the members reported that Old Faithful played "nearly hourly". No geyser, including Old Faithful, plays at set times and intervals. There is always some variation.

Old Faithful is slowing down and becoming less regular Part of this statement is true; part is not. Because of changes in circulation that resulted

4 Algebra 1 Summer Institute 2014 from the 1959 Hebgen Lake and 1983 Borah Peak earthquakes, as well as other local and smaller earthquakes, the average interval between eruptions has been lengthening during the last several decades. Old Faithful has slowed from an average of about 65 minutes from 1870 through 1947 to an average of 90-92 minutes from 2001 through 2010. The range of intervals has also increased slightly, from about 60 minutes through the 1960s to about 80 minutes in more recent decades. However, regularity, as measured by the difference between actual intervals and average intervals or standard deviation, decreased during the most recent decade.

Old Faithful is getting shorter Measurements show that Old Faithful is still as tall as it has ever been, 90-180 feet. There are a number of reasons people may feel that Old Faithful is getting shorter. They may have seen an exceptionally tall eruption last time and this time saw a more ordinary or even a short eruption. The first time they saw an eruption they may have been more excited and their excitement caused them to over estimate its height. Or they may have first seen it a number of years ago when it was possible to get closer to the geyser. To accommodate the summer crowds, the boardwalk has since been moved further back from Old Faithful. Many people don't realize just how far from the geyser the boardwalk is. With nothing to judge the distance by, most people severely underestimate the height of Old Faithful. Its not until they get back farther from the geyser and see the buildings around the geyser that they realize just how big the geyser is.

Old Faithful erupted late or early It is only possible to make predictions of the time Old Faithful may erupt. The geyser will erupt when it is ready. It’s the prediction that is early or late not the geyser.

Points of Discussion 1. If the average and median of the sample are different, ask what might cause them to be different. 2. It could happen that interval width choices yield seemingly different visual displays of the distribution, since the shape of the frequency distribution can be quite sensitive to the interval-width choice. 3. The usefulness of taking account of variation is a very important point to highlight. Center values alone (average, median, mode) can mask what is really going on in the data set. Because the Old Faithful data distribution is not mound shaped – e.g., it is not unimodal- a central value can hide the real story of variability in these data. 4. The data should appear to form two clusters. Ask participants to make comparisons across the set of graphs they have constructed. 5. Ask participants to compare successive blasts. They might discover an up and down pattern. Once they notice this pattern, they can raise other questions and conjectures, such as: a. Why are these alternating times happening? b. Do all geysers behave this way, or is it just Old Faithful? c. What is Old Faithful doing this year? Is the up-down behavior for the wait times similar to the data set?

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Optional Extension Tasks (2 different ones) (Slides 8, 9)

1. Participants could make a scatter plot of Old Faithful wait time plotted against the immediately previous wait time. Draw reference lines (horizontal and vertical) that are the median wait time for each set of blasts.

In inspecting this scatter plot, participants may notice that the second and fourth “quadrants” created by the median cross hairs are very full of data, whereas the first and third “quadrant” are relatively devoid of data. Can they find the short wait times followed by the long wait times, or the long wait times followed by the short wait times in the graph? Where do these show up?

2. Participants could create a dot plot of the entire two weeks of data on wait times. Ask them to consider the following questions: a. Compare the data in your own two chosen sample days with the data from the entire data set. Do your two days appear to be representative of the entire data set? Explain b. Would you change your conclusion about your expected wait time on the basis of the entire two weeks’ wait times? Why or why not? c. Are there other data about Old Faithful that you think might provide important information for making a prediction for wait time?

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