Content Area: Math

 

Index: 4.4C Grade 8 CPI 1

 

Standard: 4.4 - Data Analysis, Probability, and Discrete Mathematics

 

Strand: C - Mathematics--Systematic Listing and Counting

 

Cumulative Progress Indicator: 1 - The student will apply the multiplication principle of counting

·        Permutations:  ordered situations with replacement (e.g., number of possible license plates) vs. ordered situations without replacement (e.g., number of possible slates of 3 class officers from a 23 student class)

·        Factorial notation

·        Concept of combinations (e.g., number of possible delegations of 3 out of 23 students)

 

Grade: 8

 

Sample Activities:

 

·        Students determine the number of possible different sandwiches or hamburgers that can be created at local eateries using a combination of specified ingredients. They find the number of pizzas that can be made with three out of eight available toppings and relate the result to the numbers in Pascal's triangle.

 

·        Students determine the number of dominoes in a set that goes up to 6:6 or 9:9, the number of candles used throughout Hannukah, and the number of gifts given in the song "The Twelve Days of Christmas," and connect the results through discussion of the triangular numbers. (Note that in a 6:6 set of dominoes there is exactly one domino with each combination of dots from 0 to 6.)

 

·        Students design different license plate systems for different population sizes; for example, how large would the population be before you would run out of plates which had only three numbers, or only five numbers, or two letters followed by three numbers?

 

·        Students find the number of different ways of making a row of six red and yellow flowers, organize and tabulate the possibilities according to the number of flowers of the first color, and explain the connection with the numbers in the sixth row of Pascal's triangle. (See also Visual Patterns in Pascal's Triangle.)

 

·        Students pose and act out problems involving the number of different ways a group of people can sit around a table, using as motivation the scene of the Mad Hatter at the tea party. (See Mathematics, a Human Endeavor, p. 394.)

 

·        Students count the total number of different cubes that can be made using either red or green paper for each face. (To solve this problem, they will have to use a "break up the problem into cases" strategy.)

 

·        Students determine the number of handshakes that take place if each person in a room shakes hands with every other person exactly once, and relate this total to the number of line segments joining the vertices in a polygon, to the number of two-flavor ice-cream cones, and to triangular numbers.

 

·        Additional Framework Activities