Problem:  The nine trees in my large back yard form a rough circle.  I've decided to use four of the trees as fence posts.  This will create a quadrilateral region where my two dogs, Peanuts and Duke, can play.  How many distinct  quadrilaterals  can be formed by joining any four of the trees?

Math Topic/Concept:  Probability: Counting principle: combinations
    Lesser: geometry/ shape definition

Materials:  calculator, pencil, paper

Classroom Use: (Introductory/Developmental/Evaluation)

Classroom use comments*:   this might be paired with other combinations problems

Grade:  7-8

Grade Cluster: (EarlyElem/LateElem/MS-Jr.High/EarlyHS/LateHS)

Illinois Goal:  10

Standard:  10C3a

Applied? (1-4):  2

Source:  www.forum.swarthmore/libr
  Swarthmore university’s web site: past problems of the week

Answer:  126

Strategies Listed:  draw a picture:

Solution: (9*8*7*6)/(1*2*3*4) = 126  (This is 9C4, or 9 choose 4.)

Extensions or related problems*:  Change the numbers

Intended rubric or assessment method: ISAT “student friendly” rubric

Write-up submitted by:  M. K. Robbins

Problem:  After waiting in line for hours, Mary and her cousin have gotten tickets to the hottest game show on television, “Fortune Cookies.” The cousins love the part of the game where a contestant draws four fortune cookies from a basket.  Concealed in each fortune cookie is one of these numbers: 0, 1, 6, 7, 9, or 11. If the total of the numbers inside the cookies is exactly 20, the contestant wins a big prize!  How many different ways can a contestant win the big prize?

Math Topic/Concept: Problem Solving, addition, counting, permutations.

Classroom Use: (Introductory)

Classroom use comments*:  Discuss about getting a 7,7,6, and 0 the same as getting a 7,7,0, and 6?  Is there another to get 20 with these same numbers?

Grade:  6-8

Grade Cluster: (MS-Jr.High)

Illinois Goal: 10

Standard: 10.C.3b

Applied? (1-4):  1

Source:  The Problem Solver 7 (Creative Publications) ISBN # 0-88488-685-9

Answer:  54 different ways

Strategies Listed:  Organized List or Make a Table

Solution: There are 4 possible combinations of these numbers that equal 20.
 (7+7+6+0, 11+9+0+0, 9+9+1+1, 6+6+7+1, 11+1+7+1) You could then set
 up a table and show all the combinations for each number.
Draw 1 Draw 2 Draw 3 Draw 4, respectively:
7 7 6 0
7 7 0 6
7 6 0 7
7 6 7 0
7 0 7 6
7 0 6 7

Extensions or related problems*:  If they still draw 4 cookies, but have the cookies labeled 0, 1, 5, 6, 7, 9, or 11, how many different ways can a contestant score 20 points?

Related Problem  By hitting a target, Juanita can score 11,16,23,26,29, or 40 points with each arrow.  Find at least three ways that she can score exactly 100 points.  What is the least number of arrows that she can shoot to score exactly 100 points? (4 are the least amount of arrows. (11,23,26,40 or 16,26,29,29) other solutions are 11,11,23,26,29,and11, 11,16,16,23,23.)

Intended rubric or assessment method: Classroom discussion

Write-up submitted by:  Denise Mann and Jenni Robinson

Problem:  What are your initials?  Do you have anything with your monogram on it?  A monogram is a design that is made up of one or more letters, usually the initials of a name.  Monograms often appear on stationary, towels, shirts, or jewelry.  How many different three-letter monograms can you make with the letters of the alphabet?  Use a calculator to compute the total number.  Don’t forget to allow for repeat letters in the combination.

Math Topic/Concept:  Combinations

Materials:  Examples of monograms—pictures, towels.   Calculator

Classroom Use: (Introductory)

Classroom use comments*:  Discuss monograms before.

Grade:  7

Grade Cluster: (MS-Jr.High)

Illinois Goal:  10

Standard:  10C3

Applied? (1-4):  2

Source:  Hot words, Hot topics.  (Creative Publications)  ISBN  0-7622-0629-2

Answer:  17,576 different monograms

Strategies Listed:  Make a table or organized list

Solution:  26 x 26 x 26 = 17576.  Because you have a choice of 26 letters of the alphabet for you first initial and since you can repeat letters you have 26 choices for the second and third initial.

Extensions or related problems*:  What if there were only 2 letters?  What if you could not have any two initials the same?

Intended rubric or assessment method:  Classroom Discussion

Write-up submitted by:  Jenni Robinson and Denise Mann

Problem:  You can buy stickers for digits from 0 through 9 at most hardware stores.
  They can be used for house addresses and room numbers. Suppose you
  buy stickers for all the digits in all the integers from 1 through 100.
A) What digit will be used the most?  B) What digit will be used the least?

Math Topic/Concept: Number Sense, Probability

Materials:  Cards labeled 0 –9 (two each) (optional)

Classroom Use: (Introductory)

Grade: 7

Grade Cluster: (MS-Jr.High)

Illinois Goal: 10

Standard:  10.C.3

Applied? (1-4):  2

Source: Transition Mathematics (Scott Foresman)

Answer:  A) 1   B) 0

Strategies Listed:  Organized List, Make a Table

Solution:  1 will be used one extra time because of the number 100 (the only hundreds
  digit).  0 will be used only in the ones digit (except for 100).

Extensions or related problems*: For lower grades supply a 100 number chart.

Intended rubric or assessment method: Answer with explanation.

Write-up submitted by: Denise Mann and Jenni Robinson

James R. Olsen, Western Illinois University
E-mail: jr-olsen@wiu.edu
updated Aug. 20, 2001