Student Actions

Teacher Actions

• Develop a deep and flexible conceptual understanding by knowing how and when to apply appropriate counting procedures to determine sample space for a variety of mathematical situations.

• Develop the ability to make conjectures, model, and generalize when examining patterns among possible outcomes of an event and drawing conclusions about necessary counting procedures for determining the resulting probabilities of an event.

• Use and connect mathematical representations by having the students use multiple representations to determine the size of the sample space.

• Present engaging tasks that promote reasoning and problem solving where students explore a variety of mathematical situations and develop strategies for determining appropriate procedures, involving permutations and combinations, to calculate probabilities.

• Encourage productive struggle as the students explore and discuss how to determine the sample size and whether to use a multiplication or addition principle. 

Key Understandings

Misconceptions 

• Use an addition principle when two or more events have no common outcomes, the total number of outcomes can be calculated by adding up the possible outcomes for each event.

• Apply sample space, in a probability model for a random process, as a list of the individual outcomes 

• Order of outcomes does not matter for situations involving a combination but order of outcomes does matter for situations involving permutations. For example, it does not matter the order in which pizza toppings are placed on a pizza but it does matter the order in which a sequence of numbers are entered into a lock.

• When outcomes in a combination can repeat (or combination with replacement), the formula is

(r + n -1)! / r!(n -1)!

• When outcomes in a combination can not repeat (or combination without replacement), the formula is

n! / r! (n -r)! 

• When outcomes in a permutation can repeat (or combination with replacement), the formula is n^r.

• When outcomes in a permutation can not repeat (or combination without replacement), the formula is 

n! / (n -r)! 

• Probability is determined by the ratio of: possible combinations or permutations / total combinations or permutations.

• Students may confuse when to use the addition or multiplication principle to calculate the possible outcomes.

• Students may confuse sample size and sample space.

• Students may misidentify independent and dependent events when analyzing situations with and without the replacement of values.

• Students may confuse properties of permutations and combinations and incorrectly determine the outcomes of a mathematical situation.

• Students may incorrectly set up the ratio for determining probability from mathematical situations involving permutations and combinations.

• Students may not recognize when it is necessary to use factorials for determining outcomes for mathematical situations involving permutations and combinations.

  Knowledge Connections

Prior Knowledge

Leads to 

• Determine the theoretical probability of an event using the ratio between the size of the event and the size of the sample space; represent probabilities as percents, fractions and decimals between 0 and 1 (7.D.2.1) 

• Calculate probability as a fraction of sample space. Express probabilities as percents, decimals and fractions (7.D.2.2)

• Calculate experimental probabilities as a percentage and make predictions using the experimental probability (PA.D.2.1)

• Determine the randomness of outcomes to support conclusions from the sample (PA.D.2.2)

• Define, compare, and contrast the probabilities of dependent and independent events. (PA.D.2.3)

• Determine the probability of the union of events, the intersection of events, and the complement of an event (A1.D.2.2)

• Use simulations and experiments to calculate experimental probabilities (A1.D.2.3)

• Apply probability to make informed decisions (A1.D.2.4)

• Describe events as subsets of a sample space. (S.P.1.1)

• Describe the relationship between theoretical and empirical probabilities using the Law of Large Numbers. (S.P.1.2)

• Use counting techniques (e.g., permutations and combinations) to solve mathematical and real-world problems, including determining probabilities of compound events (S.P.1.3) 

• Use characterizations to determine if two events are independent (S.P.2.1)

• Analyze decisions and strategies using probability concepts (S.P.3.1)