Mathematical Content for Activities for Sixth Grade Urban Students
Department of Mathematics
York College (CUNY)
Jamaica, NY 11451
Email: firstname.lastname@example.org (for additions, suggestions, and corrections)
The ideas below provide suggestions for how to motivate students in urban schools to learn and master some important ideas in the NY State 6th grade curriculum. The contexts mentioned below concern issues involving the location of a medical clinic and the way a spell checker works. Connections are also made to mathematical biology and homeland security.
1. Plot the coordinates of locations along a line and in two dimensions.
2. Learn to find the maximum, minimum, sum, and average for a finite set of numbers.
3. Work with the arithmetic of signed numbers.
4. Develop problem solving skills.
5. Develop modeling skills.
6. Gain understanding concerning different ways of computing distance and different ways of finding an optimum (best) answer.
7. Work with square roots and absolute value of numbers.
Goals tied to specific 6th grade standards
6.PS.1; 6.PS.2; 6.PS.3; 6.PS.7; 6.PS.9; 6.PS.13; 6.PS.15; 6.PS.17; 6.PS.22;
6.CN.2; 6.CN.5; 6.CN.7;
6.N.6; 6.N.13; 6.N14; 6N.15;
Problems such as those in the New York State Testing Program, Mathematics Book 1; Grade 6 listed below are addressed:
2-page 5; 7-page 7; 12-page 9; 13-page 10; 25-page 15;
One-dimensional facility location problems
a. If all the homes of a town are located along a straight line highway, where should a new medical clinic (MC=medical clinic or medical center) be built?
b. If all of the homes of a town are located at integer coordinate points of a grid, where should a medical clinic (MC=medical clinic or medical center) be located?
Figure 1 below shows a simplified example. A town with exactly 5 homes (or apartment complexes) which are located at the mile posts indicated. (The zero mile post is located where there is a road which cuts the highway shown, though there is no home located at this site.
1. If the medical clinic were placed at -4, how far would a resident at D have to travel to get there?
2. If the medical clinic were placed at -4, how far would the resident at E have to travel to get there?
Ideas to convey:
Distance is always either positive or zero. Two locations (objects) are at distance 0 when they are identical.
3. a. What is the farthest that anyone living at A. B, C, D, or E would have to go to reach the clinic if it were located at -4?
b. What is the farthest that anyone living at A. B, C, D, or E would have to go to reach the clinic if it were located at 5?
4. a. What is the shortest distance anyone living at A, B, C, D, or E would have to go to reach the clinic if it were located at -4?
b. What is the shortest distance anyone living at A, B, C, D, or E would have to go to reach the clinic if it were located at 5?
5. What is the average (mean) distance between the houses and a medical center located at 2?
6. a. If instead of miles, the numbers represent intervals of 3 blocks (that is, from 0 to -1 is three blocks) where there are 20 blocks to a mile, how far apart in miles are:
i. E and D?
ii. C and B?
b. What is the distance between C and A in feet?
c. What is the distance between A and D in feet?
d. How long is a typical block along Guy Brewer Boulevard? How would you measure how long a block is? What might be a good approach to estimating the length of an "urban block"?
7. a. Complete the entries in the following table:
|Distance MC to A||Distance MC to B||Distance MC to C||Distance MC to D||Distance MC to E||Total Distance||Mean Distance|
b. Graph the total distance against the location MC of the medical center.
(One can plot the location of the medical center along the horizontal axis of a coordinate system and the total distance to a medical center at the given location along a vertical axis.)
c. Where should the MC be located to minimize the maximum distance between any house and the MC?
d. Where should the MC be located to minimize the total distance between the houses and the MC?
e. Where should the MC be located to minimize the mean (average) distance between the houses and the MC?
Questions to think about:
1. Is it realistic to assume that all the homes or apartment complexes have the same number of people living in them?
2. What is the difference between considering any point along the line as a site for the MC versus having the MC located at a point with integer coordinates?
3. We have neglected the width of streets that might cross the road along which the houses are located. How can this reality be handled?
4. Based on what was learned above, how does one locate the optimal position for a medical center in a more general setting of houses which lie along a line?
5. How does one treat the problem of what to do if there are many houses at the same location?
In what follows one can motivate this work by assuming there is a hospital located at (0, 0). Imagine that an accident occurs at one of the points labeled with a letter. How far would a helicopter have to fly to get to the hospital from an accident site? How far would an ambulance have to travel to get from an accident site to the hospital?
Furthermore, we can repeat the type of situations mentioned in one dimension. If there were housing blocks or homes at the 5 sites shown, what would be the best place to put a medical center or supermarket.
Two-dimensional facility location problems
Note: In an urban setting getting between two points rarely can be accomplished by moving along a straight line. This fact shows that it is often necessary to use a distance based on moving parallel to the urban "grid" (taxicab distance) rather than crow flies (helicopter or Euclidean distance). The taxicab distance from (2,3) to (-3, 5) is given by |2-(-3)| + |3-5| = 7. The Euclidean distance between the same two points would be Ã29.
1. a. What are the coordinates of the points A, B, C, D, and E (see Figure 2)?
b. What are the coordinates of the points A, B, C, D, and E if the point labeled (0, 0) is moved to the point 4 units above E?
(Note: This exercise shows that the coordinates that points are given depends on the where the coordinate axes are placed. The same points will be named differently in different coordinate systems.)
c. Find the Euclidean or crow flies distance between:
i. A and B; C and D; D and E; and A and E. (How many distances do these 5 points determine?)
ii. If you compute the distance from A to B and add it to the distance from B to C how do this this number compare with the distance from A to C?
d. Find the taxicab distance between:
i. A and B; C and D; D and E; and A and E. (How many distances do these 5 points determine?)
ii. If you compute the crow flies distance from A to B and add it to the crow flies distance from B to C, how does this number compare with the crow flies distance from A to C? (Does your answer change if you use taxicab distance?)
e. How do the crow flies distance and taxicab distance between A and B compare? Repeat this question for: B and C and C and D. Can you make a guess about what is always true about the relationship between the Euclidean distance between two points and the taxicab distance between the same two points?
There are two fundamental "distance functions" which pick up where our discussion of the difference between "crow flies" and "taxicab" distance leaves off.
More general distance problems
Named for Richard Hamming who used the idea which now bears his name in conjunction with the design of error correcting codes, Hamming distance is a remarkably easy idea.
Given two strings of the same length, the Hamming distance between them is the number of columns that the two strings differ in.
These two strings have Hamming distance 3. (They differ in columns 2, 4, and 6).
These two strings have Hamming distance 4. (They differ in positions 2, 3, 4 and 6.)
Given the two points (1, 0, 2, 2, -1, 3) and (1, 0, 5, 5, -1, 6) in 6-dimensional space, one can compute their taxicab distance, Euclidean distance, and Hamming distance. The Hamming distance here is 3. The strings differ in positions 3, 4 and 6.
The major limitation of Hamming distance is that it only applies to strings of the same length.
There are many approaches to finding the distance between two strings of different lengths. The intuitive idea behind what has come to be called edit distance or Levenschtein distance is the minimum number of insertions, deletions, or substitutions necessary to change one string into the other.
To illustrate the ideas involved with insertions, deletions and substitutions, the following examples should help:
a. In going from the string ADX to ADXY an insertion occurred.
b. In going from the string UVZR to VZR (or UVR) a deletion occurred.
c. In going from the string UMSA to MNSA a substitution occurred.
The minimum number of such operations necessary to change one string into another is called the edit distance between the two strings.
Here are some suggestions for getting this circle of ideas across to 6th grade students. Following these suggestions are some examples of the very exciting ways that these distance concepts are being applied.
A word chain is a transformation of one word into another using "transformation rules."
Here is an example:
Change warm into cold
This word transformation required 5 steps.
Can you find a transformation which requires only 4 steps?
Can you describe what the transformation rules based on the examples that have been used?
(Answer: Change one letter in a column)
Using this transformation rule, if two words have k letters each, what is the smallest number of moves that will transform one word into another?
Using this transformation rule, if two words have k letters each, can one always transform one word into the other?
Which do you think is an "easier" task: transforming two 4-letter words into each other or transforming two 6-letter words into each other? Explain your reasoning.
Can you think of other transformation rules?
(Answer: Changes must be made in a column adjacent to the column where the previous change was made. One can think of the first and the last columns as being adjacent or not adjacent to get two different approaches.)
Using the word chain notation one gets to a variety of deep and useful ideas.
a. The insulin molecules in various species are not exactly alike. One would like to be able to say which "species" are relatively alike by seeing how far apart their insulin molecules are. (One can use the same idea for any molecule that is shared by a large variety of species, another example being hemoglobin.)
b. How does a spell checker work?
Word processors come equipped with a dictionary. If one types a string of symbols which is not in the dictionary, then the word processor's spell checker provides suggestions concerning what words you might have meant rather than the word you typed.
The simplest approach to a spell checker is to have the computer provide those words in its dictionary which are Hamming distance 1 from words that are in its dictionary. Thus, if one enters "work," the spell checker would suggest wore, word, work, worm, worn, or warp.
However, in many cases the reason a spelling is incorrect is that a word is typed using a phonetic spelling and this can often be far from the true spelling in terms of the symbols used. For example, if someone enters the string "farmacie," the word the person had in mind was probably pharmacy. Thus, a more sophisticated strategy in designing a word processor than that mentioned above is to convert the word into a "standard" string based on phonetics and then look for words in the dictionary that are close to this phonetic string, assuming that original string's phonetic representation is not in the word processor's dictionary.
One such system is Soundex, which was patented in 1918 by Margaret K. Odell and Robert C. Russell. Here is a somewhat specialized version of that system for representing names, which gives the flavor of the idea. Students really enjoy doing this, which is good practice for getting across the concept of an algorithm. The algorithm for a name leads to a string which has length 4 and begins with a letter, followed by 3 digits.
(Preprocessing): If two or more letters with the same number are next to each other in the original name, or adjacent except for any intervening h and w, then omit all but the first of these letters.
Now starting with this reduced string of letters:
1. Copy the first letter of the string (name).
2. Delete all occurrences of the following letters, unless it is the first letter: a, e, h, i, o, u, w, y.
3. Assign numbers to the remaining letters (after the first) as follows:
b, f, p, v = 1
c, g, j, k, q, s, x, z = 2
d, t = 3
l = 4
m, n = 5
r = 6
(What is meant here is that if your name is Rapfter, the p and f which are next to each other are represented only by the digit 1, rather than 11. Rapfter is coded: R136)
4. The final result is the first four characters obtained from this procedure, using zeros on the right if there are fewer than four characters in the converted string.
Can you figure out how many different Soundex strings are possible?
Genome research and homeland security
The idea of edit distance can be used to find the "distance" between the genomes of two species or the distance between two drugs. If drug A has good pain killing properties but unpleasant side effects, one can try to look for a "nearby" molecule that also provides pain relief but without the bad side effects. If one can find a way of representing a person's face by a string, say, by computing numbers associated with the face (e.g. distance between the eyes, length of the nose, etc.), then one can use edit distance to try to determine if a particular person is on a list of people who should be admitted to a particular building.
This work was supported in part by the Teacher Academy of York College. Specific funding was provided by: FIPSE (46274-07 01) and the Fund for PS (72042-07 01) to the Teacher Academy of CUNY.