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

## Intended learnings & learners thinkings

See for more information on what to include in planning

### 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

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

### Instructional Procedure

### Beginning

Ask.

- 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

Ask

- 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.
- Ask.
- 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.

### End

Ask.

- What sum would you predict would turn up most if you did it again?
- What would happen for dice with different amounts of sides?
- What would happen with spinners that have different sized areas of colors on different spinners?
- What if they had a sock drawer with three white and three black socks?

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