Title: “What’s Your Area Code?”

Problem: How many possible area codes are available if all three of these properties are met:
1)      The first digit can’t be 0 or 1,
2)      The second digit must be 0 or 1
3)      The third digit can be anything

Materials: Pencil, paper, calculator.

Math Topic/Concept: Principles of Counting, Combinations and Permutations, Multiplying Principle of Counting

Classroom Use: Developmental/Evaluation

Classroom Use Comments: Urge students to think of this a digit at a time.  Breaking it down and then using the product rule makes this problem much more manageable.

Grade: 11-12

Grade Cluster: Late High School

Illinois Goal: 10

Standard: 10.C.5b

Source:  “March Problem Calendar”. Mathematics Teacher. Pg. 200-204. March 2001, Volume 94, Number 3.

Answer: 160 possible area codes

Strategies Used: Draw a picture. Use a calculator

Solution: Eight number choices for the first digit, two number choices for the second digit, and ten number choices for the third digit. Applying the product rule, our solution is 8*2*10= 160 possible area codes.

Extensions or related problems: How many of these area codes begin with the number 8? How many of these area codes begin with an odd digit?

Intended Rubric: 

Write-up Submitted by: Scott Epperly

Title:  Sock Probability

Problem:  A trunk contains a mixture of red socks and blue socks, at most 2000 in all.  When two socks are selected randomly without replacement, the probability is one-half that the colors match.  What is the largest possible number of red socks in the trunk?

Math Topic/Concept:  Probability

Materials:  None

Classroom Use: Developmental

Classroom use comments: 

This problem uses thought-provoking statistics.  It should be used after the topic has been introduced and practiced a bit.

Grade:  Statistics course

Grade Cluster: Early HS

Illinois Goal:  10

Standard:  10.C.5.c

Applied? (1-4):  level 2

Source:  Mathematics Teacher – Volume 93-No. 1- January 2000

Answer:  990

Strategies Listed:  Reasoning-possibly acting out some of it. 

Solution:  Let R and B, respectively, denote the numbers of red and blue socks in the trunk.  Because the probability of obtaining a non-matching pair is ½, we have

(R*B)/(R+B choose 2) = ½.

Therefore (by algebraic simplification),  (R+B)(R+B-1)=4RB, which can be rewritten (more algebra) as (R-B)^² = R + B.  The total number of socks in the trunk then is a perfect square.  Let n = R- B, so that n^{^2}  = R + B.  Then (by adding the equations and dividing by 2):  R = (n^{^2} + n)/2

Because R + B <= 2000, then the absolute value of n <= square root of 2000 <45.  Thus, the largest possible value of R occurs when n=44, which gives R = 990 (and B = 946).

Other solution methods (if any):  none that I know of

Extensions or related problems:  many statistic problems could be related to this one

Intended rubric or assessment method: 

Notes (if any): 

Write-up submitted by:  Lisa Wolf

James R. Olsen, Western Illinois University
E-mail: jr-olsen@wiu.edu
updated: December 16, 2001