Mathematics Cumulative Progress Indicators (CPIs) for the end of the designated grade span

Place a "+" for an expectation that represents a strength & a "-" for a weakness

+ or -

STANDARD 4.4     (DATA ANALYSIS, PROBABILITY, AND DISCRETE MATHEMATICS)     ALL STUDENTS WILL DEVELOP AN UNDERSTANDING OF THE CONCEPTS AND TECHNIQUES OF DATA ANALYSIS, PROBABILITY, AND DISCRETE MATHEMATICS, AND WILL USE THEM TO MODEL SITUATIONS, SOLVE PROBLEMS, AND ANALYZE AND DRAW APPROPRIATE INFERENCES FROM DATA.

B.     Probability

By the end of Grade 2, students will:

 1.         Use chance devices like spinners and dice to explore concepts of probability.

·        Certain, impossible

·        More likely, less likely, equally likely

 2.         Provide probability of specific outcomes.

·        Probability of getting specific outcome when coin is tossed, when die is rolled, when spinner is spun (e.g., if spinner has five equal sectors, then probability of getting a particular sector is one out of five)

·        When picking a marble from a bag with three red marbles and four blue marbles, the probability of getting a red marble is three out of seven

Building upon knowledge and skills gained in preceding grades, by the end of Grade 3, students will:

 1.         Use everyday events and chance devices, such as dice, coins, and unevenly divided spinners, to explore concepts of probability.

·        Likely, unlikely, certain, impossible

·        More likely, less likely, equally likely

 2.         Predict probabilities in a variety of situations (e.g., given the number of items of each color in a bag, what is the probability that an item picked will have a particular color).

·        What students think will happen (intuitive)

·        Collect data and use that data to predict the probability (experimental)

Building upon knowledge and skills gained in preceding grades, by the end of Grade 4, students will:

1.         Use everyday events and chance devices, such as dice, coins, and unevenly divided spinners, to explore concepts of probability.

·        Likely, unlikely, certain, impossible, improbable, fair, unfair

·        More likely, less likely, equally likely

·        Probability of tossing “heads” does not depend on outcomes of previous tosses

 2.         Determine probabilities of simple events based on equally likely outcomes and express them as fractions.

 3.         Predict probabilities in a variety of situations (e.g., given the number of items of each color in a bag, what is the probability that an item picked will have a particular color).

·        What students think will happen (intuitive)

·        Collect data and use that data to predict the probability (experimental)

·        Analyze all possible outcomes to find the probability (theoretical)

Building upon knowledge and skills gained in preceding grades, by the end of Grade 5, students will:

 1.         Determine probabilities of events.

·        Event, probability of an event

·        Probability of certain event is 1 and of impossible event is 0

 2.         Determine probability using intuitive, experimental, and theoretical methods (e.g., using model of picking items of different colors from a bag).

·        Given numbers of various types of items in a bag, what is the probability that an item of one type will be picked

·        Given data obtained experimentally, what is the likely distribution of items in the bag

 3.         Model situations involving probability using simulations (with spinners, dice) and theoretical models.

Building upon knowledge and skills gained in preceding grades, by the end of Grade 6, students will:

 1.         Determine probabilities of events.

·        Event, complementary event, probability of an event

·        Multiplication rule for probabilities

·        Probability of certain event is 1 and of impossible event is 0

·        Probabilities of event and complementary event add up to 1

 2.         Determine probability using intuitive, experimental, and theoretical methods (e.g., using model of picking items of different colors from a bag).

·        Given numbers of various types of items in a bag, what is the probability that an item of one type will be picked

·        Given data obtained experimentally, what is the likely distribution of items in the bag  

 3.         Explore compound events.

 4.         Model situations involving probability using simulations (with spinners, dice) and theoretical models.

 5.         Recognize and understand the connections among the concepts of independent outcomes, picking at random, and fairness.

Building upon knowledge and skills gained in preceding grades, by the end of Grade 7, students will:

 1.       Interpret probabilities as ratios, percents, and decimals.

 2.         Model situations involving probability with simulations (using spinners, dice, calculators and computers) and theoretical models.

·        Frequency, relative frequency

 3.         Estimate probabilities and make predictions based on experimental and theoretical probabilities.

 4.         Play and analyze probability-based games, and discuss the concepts of fairness and expected value.

Building upon knowledge and skills gained in preceding grades, by the end of Grade 8, students will:

 1.       Interpret probabilities as ratios, percents, and decimals.

 2.       Determine probabilities of compound events.

 3.       Explore the probabilities of conditional events (e.g., if there are seven marbles in a bag, three red and four green, what is the probability that two marbles picked from the bag, without replacement, are both red).

 4.         Model situations involving probability with simulations (using spinners, dice, calculators and computers) and theoretical models.

·        Frequency, relative frequency

 5.         Estimate probabilities and make predictions based on experimental and theoretical probabilities.

 6.         Play and analyze probability-based games, and discuss the concepts of fairness and expected value.

Building upon knowledge and skills gained in preceding grades, by the end of Grade 12, students will:

 1.         Calculate the expected value of a probability-based game, given the probabilities and payoffs of the various outcomes, and determine whether the game is fair.

 2.         Use concepts and formulas of area to calculate geometric probabilities.

 3.         Model situations involving probability with simulations (using spinners, dice, calculators and computers) and theoretical models, and solve problems using these models.

 4.         Determine probabilities in complex situations.

·        Conditional events

·        Complementary events

·        Dependent and independent events

 5.         Estimate probabilities and make predictions based on experimental and theoretical probabilities.

 6.         Understand and use the “law of large numbers” (that experimental results tend to approach theoretical probabilities after a large number of trials).