Phantom Tollbooth -- Math Activities

Phantom Tollbooth: Math Activity Ideas

Debra K. Borkovitz

October 2003

Chapter 14: The Dodecahedron Leads the Way

1. Measurement

The chapter starts with a sign giving distances to Digitopolis in miles (5), rods, yards, feet, inches, and half inches.

· How long is a rod? Figure out from the other units and measure it out.

· Make a similar table using metric distances, i.e. kilometers, decimeters, centimeters, millimeters, nanometers, etc. Compare and contrast the numbers on the two tables.

· Measure other objects using more than one unit. Can use non-standard units (e.g. toothpicks), body measures (e.g. handspan), etc.

· Extend to area or volume/capacity (e.g. how many gumballs will fit in a plastic Halloween pumpkin etc.)

2. Dodecahedra and Friends

Milo is greeted by a dodecahedron.

· Build a dodecahedron. We have different kinds of plastic materials for building dodecahedra and other polyhedra (3 dimensional shapes whose sides are triangles, squares, pentagons, and other polygons) in room ACW 205. I also have some leftover toothpicks and marshmallows. You can use cardboard with glue or tape; you can find a template for pentagons at http://schools.reap.org.nz/advisor/3Dmodels/pentagoncircles.pdf.

· Suppose you were a tiny bug living on a dodecahedron. Could you take a walk along the edges that starts at a vertex, visits all vertices, and returns to the starting place? Such a walk is called a Hamiltonian Circuit. Can you find Hamiltonian Circuits on cubes and other polyhedra?

· How many colors do you need in order to color the faces of your dodecahedron so that pairs of faces that meet at edges are colored with different colors?

· A dodecahedron is an example of a Platonic Solid, which is a polyhedron built from identical regular polygons (i.e. shapes with straight sides of the same length and all angles congruent, such as equilateral triangles, squares, regular pentagons, etc.). The cube is another platonic solid. Try to build some other platonic solids out of equilateral triangles or hexagons.

· Try counting the number of vertices, edges, and faces in various solids that you make. Look for a relationship between these numbers.

3. Estimation , Big Numbers, and Fermi Problems

The Mathemagician claims to have 4,827,659 hairs on his head.

· Is it plausible for a person to have this number of hairs on his/her head? Or is the number too low or too high? Figure out a strategy for estimating the number of hairs on your head.

· Here is a website that looks at mathematical patterns in African American Hairstyles: http://www.math.buffalo.edu/mad/special/gilmer-gloria_HAIRSTYLES.html. A braided hairstyle can help with the estimation, plus the site makes connections to tessellations (tiling patterns).

· The physicist Enrico Fermi used to like to give estimation questions that seemed impossible on first glance, but with some thought, could yield a very good estimate. Along with, “How many hairs are on your head,” another very famous Fermi questions is, “How many piano tuners are there in Chicago?” My students have investigated questions such as, “How many college students in Boston are hung-over on a Friday morning?” (their choice, not mine, and there are certainly more appropriate questions for younger children, such as how many kernels of popcorn would fill your classroom?…) Invent and answer a Fermi Question. Compare with a friend. Here’s a site for more info: http://mathforum.org/workshops/sum96/interdisc/sheila1.html.

· How long would it take to count to a million? To a billion? How many minutes have you been alive? What is the defense department budget? The Boston Public School’s budget? Play with big numbers to get a better feel for millions, billions, and trillions.

Chapter 15: This Way to Infinity

1. Modeling Addition and Subtraction of Negative Numbers

In Digitopolis, people start out full and eat to get hungrier. Then, as they digest their food, they become full again.

· Eating in Digitopolis can represent a model of addition and subtraction of negative numbers. Being full represents the positive direction and being hungry represents the negative direction. Eating corresponds to adding negative numbers; digesting corresponds to subtracting negative numbers. Note that subtracting negative numbers moves one in the positive direction, toward being full.

· Compare the above model to other models for adding and subtracting positives and negatives, such as thermometers, credits and debits, number lines, colored chips, etc.

2. Order of Operations

The Mathemagician asks Milo to compute

· Compute the above result on both a scientific calculator and on an inexpensive four-function calculator. Note that on Windows computers, the calculator (which can be accessed by following Programs-Accessories) has two views, standard and scientific, which show the differences between these two types of calculators. Which calculator agrees with Milo’s answer of 0?

· Explore the differences between the two calculators. A simpler example to try than the Mathemagician’s is . Why do you get the answer of 9 on one calculator and 7 on another? Which answer is considered correct in mathematics? (Note that order of operations is a convention; the conventional method is not a logical requirement of mathematics, but a language people decided to agree upon). Make up some other examples.

· Figure out a long sequence like the Mathemagician’s that will give 0 on the calculator that didn’t originally give 0. Can you find a sequence that gives the same answer on both calculators?

· Order of operations is often taught incorrectly in schools it’s often confusing because the acronym PEMDAS (parenthesis, exponents, multiplication, division, addition, subtraction), which is used to help students remember the order, seems to imply that multiplication always comes before division and addition before subtraction, but in fact, multiplication and division are at equal levels in the order hierarchy (as are addition and subtraction) and are computed in order from left to right. For more information see http://mathforum.org/dr.math/faq/faq.order.operations.html.

3. Very Large Numbers

Milo wants to know the greatest number and the Mathemagician shows him that no matter what number he thinks of, there is always a bigger number.

· This activity is only loosely connected, but its fun to try to make the largest number you can with a fixed number of digits and some allowable operations. For example, here are some very large numbers that can be made with only three digits . Which of these numbers is biggest? Can you tell without using a calculator (some might overload your calculator!)? About how big are they? Younger children can compare numbers like

4. Very Small Numbers/ More on Infinity

The Mathemagician explains that no matter how small a number Milo can find, half of the number is always smaller.

· The mathematician only talks about positive numbers. Does his argument work for negative numbers too? For zero?

· How many rational numbers (fractions with whole numbers in the numerator and denominator also decimals that are finite in length or repeat) are there between 0 and 1? Find a fraction between and . Then find another one between your new fraction and . Continue. See if you can find a systematic method to find a rational number between two non-equivalent rational numbers.

· Investigate Zeno’s Paradoxes. Here’s a place to start: http://mathforum.org/isaac/problems/zeno1.html.

· More advanced: investigate the countability of the rationals and reals, see http://mathforum.org/library/drmath/view/52830.html.

· Investigate fractals. Fractals are one of the newest math topics out there, with lots of wonderful graphics and some topics (often involving concepts of infinity) that are accessible to elementary and middle school students. Here’s a site especially aimed at elementary and middle grades teachers. http://math.rice.edu/~lanius/frac/

Chapter 16: A Very Dirty Bird

1. Family Size

Milo meets .58 of a boy from an “average” family with 2.58 children.

· Compare with the current figure for average number of children per family. Here is a good table with U.S. information http://www.census.gov/population/socdemo/hh-fam/tabST-F1-2000.pdf. What is the difference between “average per family” and “average per family with children?” Which states have significantly lower or higher numbers of children per family? What might explain these differences?

· Here is UNICEF data on children in the world, including the fertility rate of every country http://www.unicef.org/sowc03/tables/table5.html. Investigate this data. Some ideas: compare regions of the world or copy the data into a spreadsheet and make a scatterplot of life expectancy versus fertility rate or of the percent urban population versus fertility rate. Is there correlation between your variables? What does your data show?

· What kinds of families might not be correctly counted in the U.S. census? Is the real number of children per family likely to be higher or lower than the official number?

2. Averages

In the book, the word average in “average number of children per family” refers to the mean, i.e. the number found by adding up the number of children and dividing by the number of families.

· Invent some data for hypothetical families (e.g. how many families with one child, two children, etc.) so that the mean number of children per family is 2.58. What’s the smallest number of families you can use to make your mean exact? What is the ratio between families with three and two children in your data set? With two and one child? Now make up a different data set with radically different ratios.

· Compare the mean, median, and mode as measures of “average” numbers of children per family for the following data set: 1 family with 9 children and 4 families with 1 child. When might the mean be a misleading measure of the “average?” Here’s another good example: suppose a company has 1 employee making 10 million dollars per year and 1,000 employees making $20,000 per year. What is the average salary at the company?

· Now, suppose that the 9 children from the same family and the 4 only children are all in the same class. If we ask each child how many children are in his/her family and take the mean, we will get nine answers of 9 and four answers of 1, and it will seem like the mean number of children per family is , which is considerably higher than the mean found in the last part. Even though this example is extreme, such bias is found whenever we compute the mean number of children per family by sampling children, instead of sampling families. Here’s more information about this issue: http://science.ntu.ac.uk/rsscse/ts/bts/madsen/text.html.

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