**Modeling Activities**

**Unless otherwise indicated, work on a solution to the problems posed in the activity.**

**Additional questions to be addressed in all activities:**

Is the attention to the real world realistic?

Is the modeling question clearly stated?

Is the modeling question phrased in a way that there value to answer in the minds of students?

How does the work on the activity fit with the modeling cycle? Be specific.

What content standards are evident in the student activity? Give evidence.

What mathematical practices are evident as students work? Give evidence.

Could you use or modify this problem for the grade level at which you teach?

Estimating Crowds…………………………………………………………….page 2

Traffic Jam…………………………………………………………………….page 3

Manhattanville………………………………………………………………...page 4

Cash for Gas…………………………………………………………………..page 6

Ceramic Tile…………………………………………………………………...page 7

Flu Epidemic………………………………………………………………….page 8

For the Birds…………………………………………………………………..page 9

Railing………………………………………………………………………....page 13

**Estimating the size of crowds**

**An open field measuring 50 feet by 100 feet reserved for spectators was described as “full of standing adults listening to a government speaker.” Estimate the size of the audience. Explain your reasoning.**

Traffic Jam

An accident at an intersection caused a traffic jam12 miles long on a straight stretch of a two lane freeway. Traffic officers trying to relieve the congestion of cars have interest in the number of cars in the backup of traffic. How many cars do you think were in the traffic jam? Explain your thinking and show all calculations

Adapted from the Illustrative Mathematics Project:

Manhattanville:

**The New Fire Station**

Welcome to Manhattanville—the simplified version of a small town with streets running parallel and perpendicular to one another. The grid blocks are squares. What you learn in Manhattanville may be useful in other settings that require finding best locations. The streets of Manhattanville are two-way streets represented by lines of the grid. Houses are represented by points located at the intersections of some streets. Fire trucks can only travel along grid lines. (Diagonal movement by vehicles is not allowed.)

1. What are important criteria to be considered when determining the location for a fire station?

2.What criteria would make a location a “best” location for the fire station?

Adapted from Mathematics Modeling Our World: Course 2, Unit 1, 2010, Comap Inc.

3.Welcome to the village of Beatenpath. All the houses in this village are along a single straight road. How will you measure distance between the houses?

4. Where would you locate the fire station in the village of Beatenpath?

You might start by exploring some possibilities. What if there are only two houses in BeatenPath? Three houses? Four houses? Five houses?Six houses?n houses?

5. Determine a solution for fire station placement in Beatenpath based on your definition of “best location.” Provide a viable argument for your solution?

Adapted from Mathematics: Modeling our World, COMAP Inc., Lexington, MA, 2010-2012, Course 2, Unit 1, pp. 2-99 Used with permission.

Cash for Gas

Read this task online at: You might prefer to first read the student task located at the end of the document and then read the full document from the beginning. Answer questions posed for all activities.

Ceramic Tile

A homeowner is going to tile an entire rectangular floor that measures 14 ft. by 16 ft. with 1 ft. by 1 ft. tile. The homeowner wants to figure out how much tile to get.

- How much tile should the homeowner get?

- Do you think your answer is realistic? Share any concerns or cautions you have regarding your answer to question 2.

Flu epidemic

A population containing 20,000 people is likely to get hit with a seasonal flu. Statistics suggest that about 10% of this population would be naturally immune to the virus. The statistics also suggest that about 7% of the susceptible population could be stricken with the flu per day. The effects of the flu last five days. A person who gets the flu once will not get it again.

- How long would it take for the flu to completely run its course if no immunization intervention program is offered?

- The Health Service would like to institute an immunization program in an attempt to knock out the epidemic quickly, within a month if possible. However, they would like to inoculate as few people as possible each day. Records indicate that about 0.5% of people that have been inoculated still do get sick with the flu. How many people would need to be inoculated each day so that there would be no new cases after for 30 days? As a simplifying assumption, assume that every susceptible person could safely get a flu shot.

FOR THE BIRDS

Student Name:______Date:______

Your neighbor, an ornithologist, has to leave for the weekend to do a research study. She has asked you to make sure her birdfeeder always has food in it so that the birds keep coming back throughout the day. Refilling too seldom will cause the birds to look elsewhere for food; refilling too much will scare off the birds.

© Ken Hutchinson | Dreamstime.co

Leading Question

How often should you feed the birds so they keep coming back?

Adapted from: *Teachers College Columbia Univ.Mathematical Modeling Handbook*, COMAP Inc., 2011 pp. 11-18.

Railing

A carpenter is repairing an existing deck on the back of a house. The top of the hand railing on one side of the stairs from the deck into the yard should be replaced. This top piece is in the shape of an arc. The carpenter does not want to remove any of the existing railing until she has the replacement part.

The building supply store sells arc shaped railings that would match the existing railings, but the carpenter needs two measurements to get the right sized replacement. The store refers to these measurements as the “span” and “diameter” of the existing railing. The span is the straight line length that the railing must cover. The diameter refers to the diameter of the circle that would contain the arc shape as part of its circumference. It is easy for the carpenter to measure the span using his measuring tape. She does not know how she might determine the diameter of the existing railing. Can you help her?

(Based on a true story.)

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