ALEX Resources

   View Standards     Standard(s): [MA2019] FM-19 (9-12) 7 :

7. Develop and use the Fundamental Counting Principle for counting independent and dependent events.

a. Use various counting models (including tree diagrams and lists) to identify the distinguishing factors of a context in which the Fundamental Counting Principle can be applied.

Example: Apply the Fundamental Counting Principle in a context that can be represented by a tree diagram in which there are the same number of branches from each node at each level of the tree.

In this activity, students will use a teacher-provided menu and generate a list of a three-course meal with all possible combinations containing one appetizer, one main course, and one dessert.  The students will work individually to start the lesson and then move to partner work. The students will work with a partner to determine the best way to create a complete list and answer follow-up questions.  Finally, the students will be asked to use the Fundamental Counting Principle to determine the number of outcomes instead of making a list. This learning activity is intended to be a Before Activity prior to a lesson on the Fundamental Counting Principle.

   View Standards     Standard(s): [MA2019] FM-19 (9-12) 7 :

7. Develop and use the Fundamental Counting Principle for counting independent and dependent events.

a. Use various counting models (including tree diagrams and lists) to identify the distinguishing factors of a context in which the Fundamental Counting Principle can be applied.

Example: Apply the Fundamental Counting Principle in a context that can be represented by a tree diagram in which there are the same number of branches from each node at each level of the tree.

This lesson includes a teaching video and has a Desmos card match activity where students are practicing their skills using the Fundamental Counting Principle. The video and card match activity will allow students to apply their knowledge of the Fundamental Counting Principle by solving problems and self-checking their results.

   View Standards     Standard(s): [MA2019] FM-19 (9-12) 7 :

7. Develop and use the Fundamental Counting Principle for counting independent and dependent events.

a. Use various counting models (including tree diagrams and lists) to identify the distinguishing factors of a context in which the Fundamental Counting Principle can be applied.

Example: Apply the Fundamental Counting Principle in a context that can be represented by a tree diagram in which there are the same number of branches from each node at each level of the tree.

In the learning activity, students will review the Fundamental Counting Principle. After the students have reviewed the Fundamental Counting Principle, the students will use an exit ticket to assess their understanding of the topic. This activity is to be used as an after-activity.

   View Standards     Standard(s): [MA2019] FM-19 (9-12) 7 :

7. Develop and use the Fundamental Counting Principle for counting independent and dependent events.

a. Use various counting models (including tree diagrams and lists) to identify the distinguishing factors of a context in which the Fundamental Counting Principle can be applied.

Example: Apply the Fundamental Counting Principle in a context that can be represented by a tree diagram in which there are the same number of branches from each node at each level of the tree.

Using an everyday lunch menu, this video demonstrates how the counting principle is used to determine the number of possible lunch combinations on the menu. The counting principle is used to multiply the number of lunch items in each category to determine the number of possible combinations. Regents Review materials are designed to help high school students prepare for New York State's Regents exams.

   View Standards     Standard(s): [MA2019] AL1-19 (9-12) 39 :

39. Compute the conditional probability of event A given event B, using two-way tables or tree diagrams.

[MA2019] FM-19 (9-12) 6 :

6. Use multiple representations and methods for counting objects and developing more efficient counting techniques.
Note: Representations and methods may include tree diagrams, lists, manipulatives, overcounting methods, recursive patterns, and explicit formulas.

[MA2019] FM-19 (9-12) 7 :

7. Develop and use the Fundamental Counting Principle for counting independent and dependent events.

a. Use various counting models (including tree diagrams and lists) to identify the distinguishing factors of a context in which the Fundamental Counting Principle can be applied.

Example: Apply the Fundamental Counting Principle in a context that can be represented by a tree diagram in which there are the same number of branches from each node at each level of the tree.

Tree diagrams display all the possible outcomes of an event. Each branch in a tree diagram represents a possible outcome. Tree diagrams can be used to find the number of possible outcomes and calculate the probability of possible outcomes.

   View Standards     Standard(s): [MA2019] FM-19 (9-12) 7 :

7. Develop and use the Fundamental Counting Principle for counting independent and dependent events.

a. Use various counting models (including tree diagrams and lists) to identify the distinguishing factors of a context in which the Fundamental Counting Principle can be applied.

Example: Apply the Fundamental Counting Principle in a context that can be represented by a tree diagram in which there are the same number of branches from each node at each level of the tree.

[MA2019] FM-19 (9-12) 8 :

8. Using application-based problems, develop formulas for permutations, combinations, and combinations with repetition and compare student-derived formulas to standard representations of the formulas.

Example: If there are r objects chosen from n objects, then the number of permutations can be found by the product [n(n-1) ... (n-r)(n-r+1)] as compared to the standard formula n!/(n-r)!

a. Identify differences between applications of combinations and permutations.

b. Using application-based problems, calculate the number of permutations of a set with n elements. Calculate the number of permutations of r elements taken from a set of n elements.

c. Using application-based problems, calculate the number of subsets of size r that can be chosen from a set of n elements, explaining this number as the number of combinations "n choose r."

d. Using application-based problems, calculate the number of combinations with repetitions of r elements from a set of n elements as "(n + r - 1) choose r."

[MA2019] FM-19 (9-12) 9 :

9. Use various counting techniques to determine probabilities of events.

This informational material will introduce three techniques that can be used to count outcomes: (1) Fundamental Counting Principle, (2) permutation, and (3) combinations. It will describe the appropriate situation to use each counting method and describe how to use each method to calculate probabilities of events. The corresponding videos explain each of the three techniques. Practice questions with a PDF answer key are provided.