# Mathematical Modeling Handbook II: The Assessments

### Author: Various

Mathematical Handbook II: The Assessments - What does it mean to be able to mathematically model successfully? Modeling has several aspects as we know from the structure of the modeling cycle. Moreover, performance-based assessments are difficult to construct and difficult to grade. This second Handbook represents our attempt to provide several alternate assessment modes for each of the original modules in Handbook I.

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**TABLE OF CONTENTS** **COULD KING KONG EXIST?**

When watching the movie King Kong, moviegoers are swept away with the idea of a gigantic gorilla capable of running around, climbing with ease, and, most importantly, saving Ann Darrow from harm. But could this animal really exist? In this two-day lesson, students will investigate surface area, volume, and bone strength to determine if his existence is mathematically possible.**A MODEL SOLAR SYSTEM**

In this two-day lesson, students will create several scale models of the Solar System using everyday items. Open with discussing the size of the universe and aim to steer the conversation towards the size of the astronomical bodies. Pose questions that make students think about how large one astronomical body is compared to another. How can they create a model that considers the scale of the bodies?**FOR THE BIRDS**

In this two-day lesson, students are challenged to consider the different physical factors that affect realworld models. Students are asked to find out how long it will take a birdfeeder with a constant stream of birds feeding at it to empty completely.

To begin, explain that the students will be watching over a neighbor's home. This neighbor is an ornithologist (a scientist that studies birds) with a birdfeeder to be looked after. Humans can't come around too often because it will frighten the birds, but they also can't come around too infrequently because the birds will leave if the feeder frequently is empty. The students need to find out when to come back and fill the feeder to ensure that the neighbor and the birds are all happy.**ON SAFARI**

In this two-day lesson, students determine the best way to schedule their time while out on a safari. With only four hours to be out they must use the probabilities of seeing an animal species to determine how much time they should spend there before moving on. The probabilities change with the amount of time spent at a location.

By determining the expected number of animal species seen, students see that the ideal amount of time spent at each location is neither the maximum value nor the minimum value. They justify their conclusion about scheduling using a graph and the slopes of lines from the origin.**CHOOSING A COLLEGE**

In this two-day lesson, students determine their best-matched college. They use decision-making strategies based on their preferences and ranked choices. This lesson guides students through the process of selecting a list of choices and rating these choices based on their preferences in order to find the college most suited to their preferences and requirements.**A TOUR OF JAFFA**

In this two-day lesson, students will model a graph optimization problem called the "Traveling Salesman Problem" (TSP). The TSP seeks to minimize the cost of the route a salesperson should follow to visit a set of cities and return to home. The goal is to find a minimal-cost Hamilton circuit in a complete graph having an associated cost array, M.

To begin, explain the situation to students. They are about to visit a new place such as a zoo, a city, a shopping center, or an amusement park, and they wish to plan their trip beforehand. What should they consider when planning their trip? How would they plan the most efficient route?**GAUGING RAINFALL**

In this two-day lesson, students will estimate the average rainfall for a 16 km by 18 km territory in Rajasthan, India. Rainfall estimations will be based on rain gauges scattered around the territory. Since these placements are varied, students will need to identify each gauge's "region of influence" to estimate the average rainfall.

To begin, explain the situation that needs to be modeled. Meteorologists need to understand average rainfall totals in a region in order to make short-term forecasts. These are usually for relatively shorter periods of time. Climatologists need to understand average rainfall totals for relatively longer periods of time in order to understand, among other things, climate change.**NARROW CORRIDOR**

In this two-day lesson, students are asked to determine whether large, long, and bulky objects fit around the corner of a narrow corridor.

The objective of this lesson is to apply the concept of turning points (maximum or minimum points) and the Pythagorean Theorem to determine the longest object that can go around the corner of a corridor.**TALE OF THE TAPE**

In this two-day lesson, students will model the path of a baseball in flight and use that model to determine how far the ball will travel (in ground distance). Students then use those ideas to apply them to skeet shooting where they determine not just the flight of the clay disk, but also the flight of the pellet and their intersection point.

Ideally, the lesson will involve solving a system of linear equations to determine the function and solving a quadratic equation to find the roots of this function, although other models are encouraged. For some examples, you may be able to factor to solve the resulting quadratic equation, but if the polynomial is prime, the quadratic formula, completing the square, or the graphing calculator can be used.**UNSTABLE TABLE**

Have you ever tried to eat on an unstable, tippy table? No doubt drinks and soup were spilled easily! Restaurant wait staff often fold paper napkins to wedge under one of the legs to stabilize the table.

In this two-day lesson, students learn to stabilize a table without the use of napkins they can rotate it up to 90°. The result is counterintuitive but can be verified mathematically.**SUNKEN TREASURE**

In this two-day lesson, students help the crew of a shipwreck recovery team minimize the amount of work done to remove treasure chests from a ship lost at sea. The divers must move the chests to a rope that is between their locations coming from the recovery team's boat above. The captain of the boat's crew insists on placing the rope in one spot; he doesn't want to waste time and money moving it each time a chest is collected.**ESTIMATING TEMPERATURES**

In this two-day lesson, students will model temperature data. They will use "known temperature stations" in order to estimate temperatures at any given point accurately. Websites that give the temperature at a specific place typically do not give the actual values; they give an estimate based on meteorological data.

Explain to students that temperatures are not measured everywhere and educated estimates need to be made. Have the students imagine they are meteorologists interested in making a model to estimate temperature at a given time and at a given location.**BENDING STEEL**

Metal railroad tracks expand and contract due to weather. In this two-day lesson, using the assumption that a railroad track is secured at both ends, students will use models to estimate how expansion of the track affects the height of the rail off the ground. Sometimes tracks will expand outward along the ground, but this lesson focuses on the case where they expand upward.

Interestingly, very small increases in length as a result of expansion have a large effect on height. Students will investigate this phenomenon using both triangular and arc models.**A BIT OF INFORMATION**

In this two-day lesson, students will learn to use a logarithmic function to model information functions. A significant portion of the secondary curriculum revolves around the analysis of functional relationships. In the context of computers, the notion of sending and receiving information gives way to an interesting relationship between the required length of code and how much information it carries.

In fact, this represents one of very few real world situations where only a logarithmic function can model the relationship.**STATE APPORTIONMENT**

A new country is being formed in this two-day lesson. Students will determine how to allot the representation for the different states in the country, also known as apportionment.

Begin by asking students how democracy works in the US Ask them how a country that is newly forming and wishes to adopt a similar representation system to the US might pick how many representatives each state gets. What different mathematical ways are there to model this?**RATING SYSTEMS**

In this two-day lesson, students will model rating systems like those used in many sports. They are asked to consider the various factors that the human mind employs to "rate" one team over another; they will then model a way to consider these factors in order to make a systematic, mathematical rating method. Note that even professional rating systems often are disputed for their "accuracy": such is the nature of both mathematical modeling and sports!

Begin with the description of the situation: you are trying to compare teams or players, but not every team/player plays the other, so there is no clear "clean-cut" method. How can you devise a system to do this?**THE WHE TO PLAY**

In this two-day lesson, students develop different strategies to play a game in order to win. In particular, they will develop a mathematical formula to calculate potential profits at strategic points in the game and revise strategies based on their predictions.

Allow students to imagine that they are living in the twin islands of Trinidad and Tobago where a popular game called **Play Whe** is played everyday. They can think of the game as an investment opportunity and their goal is always to realize a profit. How can they devise a strategy so that their expenditure is always less than their potential winnings?**WATER DOWN THE DRAIN**

In this two-day lesson, students will collect data from a water dripping experiment. The data that the students collect will be the basis for estimating how much water is wasted from typical leaky faucets. At the beginning of the lesson, the students are faced with a statistic that states leaky faucets in US homes waste $10,000,000 worth of water each year. At the end of the lesson, students will have the opportunity to determine what specifications (homes, faucets, drips/minute) result in that amount of money.**VIRAL MARKETING**

In this two-day lesson, students will model "viral marketing." Viral marketing refers to a marketing strategy in which people pass on a message (such as an advertisement) to others, much like diseases and viruses are spread.

To begin, explain that you are interested in starting your own business and you are researching different marketing strategies to "get the word out." Viral marketing is one strategy that should be considered. What is viral marketing and what can be said about it *mathematically*?**SUNRISE, SUNSET**

In this two-day lesson, students will examine changes in the average monthly sunlight over the course of a year. They will use actual sunrise and sunset data found on the internet in order to calculate the "length of an average day" for the chosen city. Students will model the data with a sine curve. The model will be interpreted and used to make connections to the real world.**SURVEYING THE ANCIENT WORLD**

In this two-day lesson, students will construct and use a simple version of an ancient tool called an **astrolabe**. This tool measures the angle between the tool and the horizontal plane. It was used frequently by ancient surveyors, engineers, astronomers, and seafarers to compute angles and heights.

To introduce the lesson, explain the use of the astrolabe to students and have them imagine that they are ancient surveyors trying to measure the heights of mountains. The astrolabe only measures angles, though. How could ancient surveyors complete their task?**PACKERS' PUZZLE **

In this two-day lesson, students consider ways to estimate the number of spheres that will fit within a container. They also will try to pack as many as possible into differently shaped containers.

The objective of this lesson is to have students use geometric solids so that they can solve basic packing problems that arise in the real world.**FLIPPING FOR A GRADE**

In this two-day lesson, students play different coin-flipping games and try to understand what the outcomes may be. The objective of this lesson is to understand the meaning of expected value and standard deviation and why they are so important.**PRESCIENT GRADING**

In this two-day lesson, students will learn how to approximate test grades given homework grades. They will construct a scatter plot and use the line of best fit to predict grades, as well as examine the effect the correlation coefficient and the residual have on the predictions.**PICKING A PAINTING**

In this two-day lesson, students are asked to choose the best possible painting from a group provided to them. Certain restrictions prevent students from going back to previously viewed paintings, so choosing the best is not as straightforward as just looking at all of them and deciding.

The objective of this lesson is to use ordering and logical thinking to create probabilistic strategies that have greater chances of success than just random selection. Conditional probability is also explored as a way to evaluate the strategies further.**CHANGING IT UP**

In this two-day lesson, students will examine the United States monetary system and make mathematical judgments about how to stock a cash register till (the drawer containing the money that "pops out" of the register). Different situations are modeled, each time refining the initial model.

Introduce the students to the situation to be modeled: a cash register till needs to be stocked with extra coin rolls. Cashiers want to try to run out of all the types of change at about the same time so they need to cash in for new change as rarely as possible. Under-stocking doesn't work because running out of coins too frequently results in longer waiting times for customers, and supervisors have to supply more change for the cashier. The till cannot be overstocked with coin rolls because it will be too heavy and will be very slow to open.

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