#### A Math Forum Project

 ESCOT Problems of the Week: NCTM Standards-Aligned Index

The technical requirements for ESCOT are here, however, every ESCOT PoW does some self-diagnostics to determine if any software installation is necessary.

Each problem is described below including:

• short description of the problem
• Where's the Math
• teacher support page
• archived problem with highlighted solutions
• student version
• applet (in some cases more than one)
• (in some cases additional versions of the problem)

If you have something to share with us as you use any of the links or suggestions on this page (something you tried and changed or a new idea) we would love to hear from you. Please email us.

Fish Farm, Part I: Students try to place male and female fish in different ponds such as to satisfy certain restrictions on the ratios of male to female fish.

Where's the Math: Depending on the student's approach to this problem, many types of math can be incorporated to solve this problem. The concept of ratio is used throughout, and students can solve the problem by experimenting with the number of fish so that the ratios fit the constraints. Algebra can also be used, by coming up with simultaneous equations to represent the conditions given. These can be simplified to one equation in multiple unknowns which requires positive integral solutions, otherwise known as a Diophantine equation.
[teacher support] [archive] [student version] [applet]
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Fish Farm, Part II: Students collect data on randomly chosen fish from a pond. Using this data, they try to determine the ratio of male fish to female fish in the pond.

Where's the Math: This problem allows students to investigate different data collection methods. Students realize that they get different data depending on whether they have a small sample size, but a large number of trials; a large sample size and a small number of trials, etc. With further experimenting, students can discover which method of collecting data gives the most accurate results. In addition, the bonus question deals with the concept of expected value. Given the number of fish in the pond and the size of the sample, students can realize intuitively that the ratio of males to females in their sample should be close to the actual ratio. If they investigate the probability of getting a male or female fish, they can discover the formula for expected value, and why it works.
[teacher support] [archive] [student version] [applet]

Fractris: Students fill up a row in a Tetris-like game by making combinations of certain fractions that add up to 1.

Where's the Math: This problem deals with basic addition of fractions. However, some of the questions also encourage students to investigate different ways to get the fractions to add up to 1. By multiplying all the fractions by 12, all the fractions are converted to integers, and the problem becomes finding all the ways to get the numbers 1-6 to add up to 12. This involves combinatorics and number theory.
[teacher support] [archive] [student version] [applet]

Galactic Exchange: Students are asked to discover the exchange rates among different types of alien currency, and use this information to find out the amount of money needed to "buy" certain types of food.

Where's the Math: This problem allows students to investigate ratio and proportion, by discovering the exchange rate between different alien currencies (i.e. 7 circles have the same value as 3 triangles, etc.) By manipulating these exchange rates algebraically, students can come up with equations which represent the money needed to buy certain products. The use of symbols for coin types encourages a symbolic or alphabetic representation of each type of currency.
[teacher support] [archive] [student version] [applet] [extra practice]
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Graph Zooming: Students use different buttons to zoom a graph. They then investigate what is visible and what isn't, depending on which scale the zoom displays.

Where's the Math: By asking students about the intersection between two lines, this problem involves solving two linear equations graphically. However, the main purpose of the problem is to investigate what happens when a graph is zoomed with different scales. This allows students to become more familiar with Cartesian coordinates, and the representations of certain lines when graphed in such a coordinate system.
[teacher support] [archive] [student version] [applet]

The Hispaniola Water Shortage: Students are asked to use virtual containers of water, of definite sizes, and combine them in various ways to come up with as many different resulting volumes as possible.

Where's the Math: Gives students an introduction to number theory by emphasizing the significance of starting with containers of different parity, as compared to containers of the same parity. Some implied work with modular arithmetic, i.e. using the 3 oz. and 8 oz. containers, whenever one fills up the 8 oz. with the 3 oz., 1 oz. will be left in the 3 oz. container because 8 is congruent to 2 (mod 3). Questions encourage students to find a pattern involving parity.
[teacher support] [archive] [student version] [applet 1] [applet 2]

In the Dark with an Elephant: Students are asked to investigate how the appearance of a graph changes, based on the scale of the graph and the region being viewed.

Where's the Math: Gives students experience in manipulating graphs by changing domain and range values for the viewing window, which can easily be carried over to more powerful tools such as graphing calculators. Allows students to become familiar with the Cartesian coordinate system. Questions encourage thought about how the shape of different areas of a graph are not necessarily representative of the shape of the entire graph.
[teacher support] [archive] [student version] [applet]
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Marabyn: Students change the distance Marabyn rides the bus and the distance she walks to fit constraints about the distance and time that she walks.

Where's the Math: This problem deals primarily with the distance formula, d = rt. For one question, students also formulate their own equation to represent additional constraints. Using the slope of lines on the graph, students can also calculate Marabyn's walking and riding speeds. With this data, an inequality can be formulated which represents the minimum and maximum walking distances.
[teacher support] [archive] [student version] [applet]

Marathon Graphing: Students are asked to use data and best-fit lines to predict the winning women's marathon times from various years.

Where's the Math: This problem was designed to get students to investigate how a best-fit line can be used to approximate actual data. Also, it was designed to show the limitations of such a best-fit line. To create a line to fit the data, students could move the line using the applet to get a visual representation of how the line fit the data, and then create a linear equation which represents the line. Also, students solve linear equations graphically, by observing where two lines representing two different sets of data intersect.
[teacher support] [archive] [student version] [applet 1] [applet 2]

Mosaic: Students use blocks of variable length (fractional, not decimal) to fill up a map of the USA like a mosaic. They then use the size of the blocks to estimate the area of the map.

Where's the Math: Like Fractris, this problem deals with basic addition of fractions. In addition, question #2 brings up basic ideas of calculus, i.e. using smaller and smaller rectangles to approximate an irregular shape. This question challenges students to come up with that idea independently.
[teacher support] [archive] [student version] [applet]
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Polyrhythm: Students investigate different rhythms, both aurally and visually, using the applet. They then quantify these rhythms based on the accented beats.

Where's the Math: The first few questions and the investigation deal mainly with ratios and their properties.To find the phrase length of a complicated rhythm, however, the concept of the LCM is needed, since the total phrase length must be evenly divisible by each of its constituent ratios.
[teacher support] [archive] [student version] [applet]

Pythagoras' Mystery Tablet: Students use an applet to compute the area of a square based on its side length. They then use this information to determine whether it is possible to get exact side lengths for certain areas.

Where's the Math: The goal of this problem was to get students to investigate the concept of irrational numbers through a familiar concept like area. Students realized that certain areas, like 4, had an exact side length, while others, like 2, did not. This led to questions about the categorization of such numbers, based on how "easy" it was to get an exact area. Without actually introducing the mathematical nomenclature, students discovered the existence of certain irrational numbers.
[teacher support] [archive] [student version] [applet]

Rumors: Students use a simulation to discover how the number of students who are told about a rumor varies with time

Where's the Math: In conjunction with the simulation, a graph is generated by the applet that illustrates the relationship between number of people and time. Students are asked to investigate how the shape of the graph changes when the parameters of the rumor-spreading are changed, i.e. how many people spread the rumor, how the rumor is spread, etc. The bonus question introduces the concept of exponential functions to students, as compared with linear functions for the introduction.
[teacher support] [archive] [student version] [applet]
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Scale 'n Pop: Students resize a balloon by using fractional scaling factors, to try to get the balloon to fit between two walls, yet still be big enough to be popped by a pair of nails.

Where's the Math: This problem allows students to investigate the properties of fractions. The applet allows a visual representations of basic properties of fractions, and makes it easier to see things like the fact that a smaller numerator makes the fraction smaller, but a smaller denominator makes the fraction bigger. The problem also illustrates how fractions can be used to scale a set amount, and the data table displays that scaling. This also illustrates conversion between fractions and decimals.
[teacher support] [archive] [student version] [applet]

Search and Rescue, Part I: Students fly a helicopter to different locations by specifying headings and distances.

Where's the Math: The concept of vectors is introduced in an interactive way. By specifying a distance and a direction (instead of an angle from the origin, an angle relative to true north), a vector corresponding to the helicopter's movement is specified. By becoming familiar with the behavior of vectors in the simulation, students can investigate adding vectors by realizing that, in one movement, they can end up in the same spot as with two other, different movements. The relationship between the distances and angles of added vectors, as well as other vector properties, can also be experimented with in this simulation.
[teacher support] [archive] [student version] [applet 1] [applet 2]

Search and Rescue, Part II: Students are asked to place a helicopter rescue base between two campgrounds. The location of the base should correspond to certain constraints, such as the population of each campground, and the fact that a minimal distance is favorable. The location is determined by dragging a dot across the map, and the heading and distance information is automatically computed and displayed.

Where's the Math: Students investigate the concept of locus. By dragging the dot across the map, students can discover that, even if the distance between the base and each of the two campgrounds is required to be the same, there are an infinite number of locations for the base. However, all these locations lie on a certain line: the perpendicular bisector of the line connecting the two campgrounds. This result can be obtained directly from certain theorems in geometry dealing with equidistance and perpendicular bisectors; conversely, students can discover these theorems on their own by experimenting with the applet. The problem also deals with the concepts of minimum distance and proportion, which result from students trying to make the base twice as close to one campground, while still making the resulting distances as small as possible.
[teacher support] [archive] [student version] [applet]

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Descriptive text of problems provided by Yoni Kahn

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