### Clarifications

*Clarification 1:*Instruction includes recording sample spaces for repeated experiments using organized lists, tables or tree diagrams.

*Clarification 2:* Experiments to be repeated are limited to tossing a fair coin, rolling a fair die, picking a card randomly from a deck with replacement, picking marbles randomly from a bag with replacement and spinning a fair spinner.

*Clarification 3:* Repetition of experiments is limited to two times except for tossing a coin.

**Subject Area:**Mathematics (B.E.S.T.)

**Grade:**8

**Strand:**Data Analysis and Probability

**Date Adopted or Revised:**08/20

**Status:**State Board Approved

## Benchmark Instructional Guide

### Connecting Benchmarks/Horizontal Alignments

- NA

### Terms from the K-12 Glossary

- Event
- Sample Space

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 7, students determined the sample space for a single experiment. In grade 8, students find the sample space for a repeated experiment. In high school, students will describe events as subsets of a sample space and consider unions, intersections and complements of events.- 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.

- Tossing a coin
- 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

- List
- 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}, $r$_{1}b, $r$_{2}$r$_{1}, $r$_{2}b, 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 r_{1}, r_{2}or b (as shown in the list). - Table
- Tree Diagram (two options shown)

- List

- For example, the repeated experiment of tossing a coin three times has the sample space that can be written as:

### 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 Task 1*

**(***MTR.7.1*)Brianna flips a round, flat game piece with yellow on one side and white on the other side. Make a tree diagram to show the sample space for flipping the game piece four times.

*Instructional Task 2*

**(***MTR.4.1*)List or describe all of the possible outcomes for each experiment.

### Instructional Items

*Instructional Item 1*

Joanna is spinning a spinner twice with 4 equal sections numbered 1 to 4. What are all the possible outcomes in the sample space?

**The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.*

## Related Courses

## Related Access Points

## Related Resources

## Formative Assessments

## Lesson Plan

## Tutorials

## MFAS Formative Assessments

Students are asked to make an organized list that displays all possible outcomes of a compound event.

Students are asked to make a tree diagram to determine all possible outcomes of a compound event.

## Student Resources

## Tutorials

This video explores how to create sample spaces as tree diagrams, lists and tables.

Type: Tutorial

This video shows how to use a sample space diagram to find probability.

Type: Tutorial

The video will show how to use a table to find the probability of a compound event.

Type: Tutorial

This video shows an example of using a tree diagram to find the probability of a compound event.

Type: Tutorial