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Descriptors: coin toss, probability, theoretical probability, spinner, Maththematics, mathematical model
Title of resource: Spreadsheet Model of a Coin Toss or Binary Spinner
Author: Bob Sargent
Purpose: Enrichment of probability unit in MathThematics Book 2 demonstrating a mathematical model.
Audience: students grades 7-10
Resource Requirements: I have attached an excel spreadsheet. You will need a compatible application.
Simulating coin tosses is a classic example of an educational computer program modeling an event. The attached spreadsheet graphs simulated trials of some sort of 50/50 binary event, a coin flip or a spinner with two equal sides, using the random function. I am not suggesting that students should not flip real coins or spin real spinners. The lesson is about the mathematical model. The question the students must answer is this; “In what ways does the model behave like what is being modeled, and how is it different?” The lesson assumes the students already know about probability. The model does not look like a coin flip or a spinner, yet it yields a similar table of results. Getting students to articulate this, and elaborate on it, is the hoped for outcome of the lesson. By repeatedly recalculating the spreadsheet (pressing F9 in Excel) it is possible to have many trials in rapid succession. By comparing the graphs of 10 coins to the graphs of 100 coins in this way one can see that 100 coins more closely approximate the expected result of 50/50 more often than do 10 coins. This is a graphic way to get at this idea.
Mathematical modeling is one of the Vermont Standards that can be difficult to convey to students. The spreadsheet is part of an enrichment exercise using a simple model of a simple event to illustrate the idea of mathematical modeling. The spreadsheet could be part of a station among a group of stations in a unit on probability. Coin flips and spinners is part of the curriculum in MathThematics, Book 2. Probability and modeling are Vermont Standards.
This site includes a multitude of ideas, samples, lessons, templates, instructions, and links to other resources, including data banks for gathering data, to help elementary teachers build lesson plan ideas around graphing and using Excel.
Real-time Data Collection Tools to Support Constructivist Projects in Science and Math
Intended Purpose: Use for real-time data collection (such as pH, temperature, dissolved oxygen, distance, heart rate, etc.) and use this data to develop and test hypotheses about a myriad of topics.
- Windows or Mac
- 16 MB free system memory
- 50 MB free hard disk space
- USB port, Serial (COM) port, or SCSI port (for connecting sensors and interfaces)
- Windows: Pentium Processor, Windows 98/ME/2000/XP
- Macintosh: PowerPC Processor, MacOS 9.2 or later
- Cost: From $89 for a temperature probe direct-connected to a computer to hundreds or even thousands of $. Some options include:
- Match Graph Kit (PS-2706), $138--A kit with a distance sensor that students can use to explore motion
- Science Sleuth Kit (PS-2717), $229--A portable unit to measure light, temperature, and sound
- Personal Weather Monitor (PS-2759), $249--A portable unit to measure temperature, barometric pressure, relative humidity, absolute humidity, dew point, and relative altitude
- Hydromania Intermediate Bundle (PS-2758), $759--A great kit for exploring stream ecology that includes temperature, pH, conductivity, dissolved oxygen, and turbidity sensors
Abstract: PASCO makes handheld data recorders, data analysis software, and a wide variety of real-time data collection probes for both Windows and Macintosh computers. They also have an extensive set of manuals and lesson plans.
The sensors and handheld data recorders are relatively rugged and very easy to use. For example, both the handheld data recorders and the direct-to-computer connectors automatically recognize which probe is attached and are immediately ready to begin recording data.
Data collection rates can vary from 1000 samples/second to as low as one sample per day (up to 50,000 data points total), enabling students to measure a variety of natural phenomena.
PASCO probeware can be used in many different projects and activities. Probeware is particularly well-suited to constructivist activities--students collect authentic data and are then challenged to make sense of it, to analyze and interpret that data and construct hypotheses and explanations for what it reveals.
For example, students could conduct a stream ecology research project. They could sample the insects and other aquatic life at different sites on a stream. They could also collect temperature, dissolved oxygen, pH, conductivity, and turbidity measurements at those same sites. Then they could use a data analysis tool (such as Excel or TableTop) to explore relationships between the data they collected in an effort to explain why some creatures are more abundant at some sites, for example.
Justification: PASCO probeware is a fantastic resource for the constructivist classroom (or school). It's not inexpensive, but it's rugged and easy to use. Probeware facilitates and encourages student inquiry. It is a highly motivating tool that enables students to use authentic data to increase their understanding of the world around them--students collect REAL data and then interpret that data themselves. (For example, they don't read about and memorize the impact of low dissolved oxygen on aquatic life, they discover and prove it themselves.) Projects can be adapted to upper elementary through high school students, depending on the students' math and science knowledge and skill levels.