MA.8.DP.2.1

Benchmark Instructional Guide

Connecting Benchmarks/Horizontal Alignments

• NA

Terms from the K-12 Glossary

• Event
• Sample Space

Vertical Alignment

Previous Benchmarks

• MA.7.DP.2.1

Next Benchmarks

• MA.912.DP.4.1

Purpose and Instructional Strategies

• For mastery of this benchmark, an experiment is an action that can have more than one outcome. Experiments tend to have randomness, or uncertainty, in their outcomes.
  • For example, an experiment can be the action of tossing a coin more than once. Possible outcomes would be whether the coin lands on heads or lands on tails each time.
• For mastery of this benchmark, repeated experiments are restricted to those listed in Clarification 2.
  • Tossing a coin
    Coins are not limited to those with heads or tails.
  • Rolling a die
    Dice are not limited to 6-sided dice.
  • Picking a card from a deck
    Card decks are not limited to a standard 52-card deck.
  • Picking a marble from a bag
    Picking a marble from a bag is not limited to colors. Picking a tile, slip of paper, or other objects from a bag are acceptable for this benchmark.
  • Spinning a spinner Spinning a spinner is not limited to colors.
• Due to some repeated experiments having a large sample space, instruction may focus on repeating experiments that have at most 6 outcomes for each individual repetition.
  • For example, rolling a 6-sided die twice would have 6 outcomes for each individual repetition giving 36 outcomes for the repeated experiment.
  • For example, drawing a card with replacement twice from a deck containing 2 red cards, 1 green card and 1 blue card would have 4 outcomes for each individual repetition giving 16 outcomes for the repeated experiment.
  • For example, tossing a coin three times would have 2 outcomes for each individual repetition giving 8 outcomes for the repeated experiment.
• For repeating experiments that have more than 6 outcomes for each individual repetition, students should understand that a written description is likely the best way to describe the sample space because complete lists, tables and tree diagrams become challenging.
• Instruction includes the understanding that when an experiment is repeated, the full sample space is kept for each repetition. For the experiments of drawing a card from a deck or a marble from a bag, this idea is referred to as “with replacement.”
  • For example, if you are selecting a card from a deck that card must be returned to the deck before selecting another card.
• Students should experience experiments before discussing the theoretical concept of probability. Within this benchmark, students are creating a sample space for an experiment that is repeated more than once.
  • For example, students could roll a die or spin a spinner twice, or randomly select a card from a deck or a marble from a bag twice, with replacement, or students could toss a coin multiple times
• Students should informally explore the idea of likelihood, fairness, and chance while building the meaning of a probability value. In this benchmark, all experiments are fair, meaning that all of the individual outcomes are equally likely.
  • For example, if the experiment is to draw a marble from a bag twice with replacement, then each marble is equally likely to be chosen on each draw.
• Have students practice making models to represent sample spaces to gain understanding on how probabilities are determined. Use familiar tools, including virtual manipulatives such as a coin, fair die, deck of cards and fair spinner (MTR.2.1).
• For repeated experiments, a sample space will typically be represented by a list of outcomes, a written description, a table or a tree diagram. Providing opportunities for students to match situations and sample spaces will assist with building their ability to visualize the sample space for any given experiment.
  • For example, the repeated experiment of tossing a coin three times has the sample space that can be written as:
    • List
      {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
    • Written Description
      The collection of ordered triples in which each element is either H or T.
    • Table
      This representation is not appropriate when an experiment is repeated more than twice.
    • Tree Diagram
      
      
  • For example, the repeated experiment of drawing a marble twice from a bag containing 2 red marbles (each notated as $r$₁ and $r$₂ in order to distinguish them) and 1 blue marble (notated as b) has the sample space that can be written as:
    • List
      {$r$₁$r$₂, $r$₁b, $r$₂$r$₁, $r$₂b, b$r$₁, b$r$₂, bb, $r$₁$r$₁, $r$₂$r$₂}.
    • Written Description
      The collection of ordered pairs in which each element is either r₁, r₂ or b (as shown in the list).
    • Table
      
      
      
    • Tree Diagram (two options shown)
      
      
      

Common Misconceptions or Errors

• Students may incorrectly organize the data using tables or tree diagrams. To address this misconception, start with a small sample space first to ensure students understand the process.

Strategies to Support Tiered Instruction

• Teacher provides examples of small sample spaces and discuss with students how the options are logical, and where they come from. Once students begin to understand small sample spaces, the teacher continues to increase size of sample spaces and have the same conversations.
• Teacher provides examples of situations and has students decide on the sample space necessary.
• Teacher co-creates models with students to represent sample spaces using a coin, die, spinner or a standard deck of cards.
• Instruction includes the use of real-world objects (coin, die, deck of cards, spinner, marbles in a bag, etc.) to demonstrate the possible outcomes for a single experiment and then the possible outcomes if the experiment is repeated.
  • For example, there are three marbles in a bag that are green, yellow and blue. The sample space for the single experiment of drawing a marble in the bag can be written as {$g$, $y$, $b$}. If this experiment is repeated twice, the sample space could be written as {$g$$g$, $g$$y$, $g$$b$, $y$$g$, $y$$y$, $y$$b$, $b$$g$, $b$$y$, $b$$b$}.
• Instruction includes starting with a small sample space first to ensure students understand the process.

Instructional Tasks

Instructional Items

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