# Powers of 2: Many Grains of Wheat

This material is replicated on a number of sites as part of the SERC Pedagogic Service Project

## Summary

In this Spreadsheets Across the Curriculum activity, students build a spreadsheet to examine the relationship between powers of 2 and powers of 10. The objective is for them to develop skill in estimating the numbers obtained when small amounts are doubled many times and thereby gain an understaning of this example of exponential growth. The module addresses the specific problem contained in a well-known fable: If you start with two grains of wheat how many do you have if you double the number of grains 40 times? Additionally, the students build a spreadsheet to calculate the volume of the quantity of wheat in terms of liters, cubic meters, and truckloads. The result sheds light on the plight of the king in the fable.

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## Learning Goals

Students will:
• Create a spreadsheet in Excel and the necessary equations needed to do the calculations.
• Use laws of exponents to relate powers of two to the powers of ten.
• Evaluate the errors involved in using a thousand as an approximation for 2^10, and a million for 2^20 (for example)
• Estimate the volumes of aggregates of grains
• Develop cell equations to convert the volumes to a variety of different units (including liters, cubic meters and truckloads).
• Do this study of exponential growth in the context of a famous story.
In the process the students will:
• Improve their grasp of the size of quantities produced by exponential growth.
• Understand that repeated doubling is exponential growth.
• Improve their skill at estimation
• Gain knowledge of Excel.

## Context for Use

This module is appropriate for any undergraduate-level mathematics course. Students need basic number sense, which includes estimation skills. Exponential growth is included in any list of quantitative literacy topics, and understanding exponential growth depends on one's ability to grasp the magnitude and growth rate of the quantities involved. Grasping the magnitudes is contingent on one's number sense via estimation. Instructions are complete enough for students to do homework activities.

For use of the module:

• 1) The course most likely to use the entire module is the General Math course taken by non-technical majors needing to meet a 6 hour requirement and part of the Basic Studies requirements.
• 2) Nursing majors or elementary education majors needing to review the metric system and basic arithmetic could benefit from the entire module, given a little help with the log function on the calculator.
• 3) Any course which needs to do a lot of conversion, like a low level general science course or informal geometry.

I have used parts of the module in many different settings:

• 1) Any class where a discussion of types of reasoning comes in (eg. a geometry class where you want to convince students that inductive reasoning,trusting pictures, or relying on intuition can lead to errors). More specifically, the paper folding problem in the End of Module slide has been most effective in convincing students that their intuition is faulty when dealing with exponential growth.
• 2) Any class where students are studying logarithms can take our "renaming" of 2 as 10 raised to the power .301 and extend it to rename many other numbers, and eventually provide a chart to be used when the only operations are multiplication, division, exponentiation, and root extraction. Our "calculator generation" has no appreciation of the important breakthrough that logarithms brought to the world of paper-and-pencil calculation, and the creation of the slide rule. I now teach students to make their own slide rules in our history of mathematics class. And, of course, I use the "breakthrough" association that 2 "equals" 10 raised to the power of .301 to create a renaming chart for all natural numbers from one to one hundred.

## Description and Teaching Materials

The module is a PowerPoint presentation with embedded spreadsheets. If the embedded spreadsheets are not visible, save the PowerPoint file to disk and open it from there.

The above PowerPoint files are the student version of the module. An instructor version is available by request. The instructor version includes the completed spreadsheet. Send your request to Len Vacher (vacher@usf.edu) by filling out and submitting the Instructor Module Request Form.

## Teaching Notes and Tips

In this Spreadsheets Across the Curriculum activity, this module will motivate students to appreciate number sense as it relates to exponential growth.

## Assessment

I evaluate students by
• class presentation,