# Planning - Probability of sums of two six-sided dice - outline format

## Intended learnings & learners thinkings

### Overview

Investigate the probability of the sums of two six - sided die.

### Focus questions

How does the sum of two six-sided dice vary when they are rolled?

### Background Information

Probability can be determined in one of two ways: theoretical and experimental.

It is very rare that students are able to understand the probability of the outcomes of sums 2 - 12 without charting all possible values.

### Concepts

1. A six sided die has a one in six probability for each (A die has six sides).
2. A fair die has equal probability for each side (Each number appears only once).
3. (Generalization) The probability of an outcome is the number of specific outcomes out of the total number of all possible outcomes of one event.
4. Theory is an idea used to explain or predict an event.
5. Theoretical probability is determined with reasoning - by generating all the possible outcomes or combinations of outcomes in an event.
6. Experiment is a test or set of trials made to try to understand something.
7. Experimental probability is determined by repeating a certain event a number of times and collecting numerous results to determine the probability.
8. The probability of a certain sum of two die is equal to the total number of different sum combinations for each possible sum out of the total number of all possible sums.
9. The sums of two six-sided dice are: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.
10. There is one way for the dice to have a sum of 2, two for 3...six for 7 five for 8... For a total of 36.

### Misconceptions

I can cause a certain number to be rolled (blowing, throw hard, throw a certain way, wishing for it...). It is magic.

### Assessment

Diagnostic: What number(s) do you think will be the most likely to be rolled? The least likely?

Summative: What is the probability for all possible rolls? How do you know?

Generative:

• Explain how to find the experimental probability and theoretical probability of each of the following.
• Sums of dice with the number of sides different than six.
• Sums of dice with two different number of sides.
• Spinners with unequal partitions and or different colors of sections and the probability of getting pairs of colors.
• Create problems with different amounts of different colors of socks in a drawer and pulling out one sock at a time and what kinds of pairs would be made that way.

Bloom’s Taxonomy If learners have never experienced the concept and derive the concept on their own it would be application or possibly synthesis. If they have conceptualized the concept before it is comprehension.

### Objective or outcome

Students predict the outcome of the sum of two dice rolled 36 times, roll the dice 36 times and record the sums, chart the data, draw a conclusion about probability and communicate the difference between theoretical and experimental probability and how to find each.

### Materials

Die, pencil, paper

## Strategies to achieve educational learnings

Based on learning cycle theory & method

### Beginning

• What do you think would happen if you rolled two six-sided dice at a time, summed the dice, recorded it, and repeated the process 36 times.
• How did you made that prediction?
• Display all answers on a board for all to see.
• What makes you believe any are right?
• Suggest they should roll the die, collect the data, and find out.

### Middle

• How to display data.
• If need a hint, suggest.
• Could chart the number of rolls for each sum 2 - 12 (2 - 12 horizontal axis, # rolls vertical axis). OR could write each addend above the sum. See graph data sheet
• If students do not know how to arrange data have them chart the number of rolls for each sum 2 - 12 (2 - 12 horizontal axis, # rolls vertical axis). Data could also include the addends for each sum.
• Students put their data on the board.
• Ask questions like the following to see how they interpret the data.
• What pattern do you see from the data?
• What sum turned up most?
• What are the odds of each sum turning up?
• Analyze the data by having students explain the pattern. It may be necessary to list every pair of addends for each sum 2 - 12 theoretical probability.
• Have students communicate the pattern and compare the experimental probability with the theoretical probability
• Share the data.
• How could the data results be displayed?
• If there are no suggestions to arrange data have them chart the number of rolls for each roll 1 - 6 (1 - 6 horizontal axis, # rolls verticle axis). Students put their data on the board. Analyze the data. Possible questions:
• What number turned up most?
• What number would you predict would turn up most if you did it again?
• What are the odds of a certain number turning up?
• What did you discover from the data?
• Have students communicate the concept in several ways.