The Probability Toolkit focuses learning on fundamental math concepts related to probability theory. Simulated experiments engage students in fascinating learning activities. The experiments are represented graphically and the results are shown using tables and graphs. The Common Core Standards for Mathematics recognizes the importants of emphasizing learning activities that deal with data analysis of chance events. This emphasis represents a departure from a mathematics curriculum that empahsizes memorization of facts. Today the role of data and chance in school mathematics is changing in terms of its importance and the extent to which these concepts relate to other subjects has been recognized. Modifications to the K12 curriculum incorporate probability theory to a much greater extent.
People have a natural approach to understanding the role of data and chance in terms of everyday life. This informal understanding of probability theory becomes the basis for specific explorations which are designed to enhance this informal mathematical knowledge by helping students develop the ability to analyze chance situations presented through random experiments. The approach taken in the design of this app involves the implementation of instructional strategies that focus on these five themes:

Data Collection 
Students conduct experiments and make observations, then record the results in a tabular form. 

Communication 
Through small group instructional approaches students learn to discuss mathematical concepts related to probability. Graphs, for example, are one way in which students learn how to describe chance events mathematically. 

Problem Solving 
The role that understanding probability plays in enhancing problem solving skills cannot be over emphasized. Several investigations involve problem solvking using an understanding of probability theory to evaluate whether an answer is reasonable. 

Logic and Reasoning 
Students (and adults) often make errors in reasoning when analyzing chance events. For example, many people would say erroneously that a coin toss is more likely to result in 'heads' after a series of events that resulted in 'tails' have occurred. 

Connections 
The concepts involved in understanding probability, chance events and data analysis directly relate to many subject areas other than mathematics. Through the development of an understanding of the principles involved in probability theory students will have a greater ability to think critically about certain concepts in science, technology, engineering, social science, art and literature. 
Simulated Experiments in Probability:

Coin Toss
For each trial in the Coin Toss experiment students select the number of coins to be tossed on the iPad screen (150). The coins land with either the heads or tails side facing up. As the coins are tossed a table is built showing the results of each trial. A bar graph is used to compare the number of heads versus tails. The data can also be shown in a circle graph. Coins from the United States, Canada, Australia or Great Britain can be used in the experiment. The experiment shows a situation with two possible outcomes and each outcome has an equal probability. 

Gumball Machine
In setting up the Gumball Machine experiment students can control how many of each color gumball can be put into the machine. As the experiment runs randomly selected gumballs are removed from the machine. In analyzing this experiment students are asked to find the theoretical probability of getting a certain color gumball. When the experiment is complete students compare the experimental probability with the theoretical probability. Tables, bar graphs and circle graphs are used to communicate the results of the experiments. This experiment shows a situation where the ratio of colored gumballs is used to determine the probability of any specific color being selected.


Dice Roll
This experiment involves simulating the roll of a pair of dice. The roll of each die represents an independent event. The probability of getting a specific number when a pair of dice is rolled is determined by analyzing a table showing the number of times the specific number can occur considering all possible outcomes. Tables, bar graphs and circle graphs are used to communicate the results of the experiments. 

Random Numbers
The Random Numbers experiment lets students control the range of numbers from which random numbers are selected. Number theory and probability theory concepts are involved in this experiment because students are asked to determine the probability of getting an even, odd, prime, composite, square or palindrome based on the range setting. For example, if the range is numbers from 110, then the chances of getting an even number is 0.50 since onehalf of the numbers in the range are even. Tables, bar graphs and circle graphs are used to communicate the results of the experiments. 

Spinners
The Spinners experiment lets the students select one of eight different spinner. Each spinner represents different ratios of green, blue, red, yellow, orange, purple, black or white sections. By studying the spinners the students determine the theoretical probability of each color occurring. While running the experiments students will compare the theoretical probability to the actual outcomes. Tables, bar graphs and circle graphs are used to communicate the results of the experiments. 

Binomial Machine
Probability theory and statistics define the binomial distribution as a discrete probability distribution based on the number of successes in a sequence of n independent yes/no experiments. In this experiment a ball is dropped through a set of pegs that are placed in the shape of Pascal's Triangle. At each peg the ball has a 50/50 chance of falling to the right or left. A binomial distribution results and this important concept is the basis for the popular binomial test of statistical significance.




Probability Toolkit
The Probability Toolkit is a fun way to introduce probability theory and statistics. Using simple gestures students set the parameters for probability experiments, such as coin tosses. For this experiment students select the number of coins tossed for each trial. The results for each trial is reflected in a table. Data shown in the table can be represented as a bar graph or circle graph.
Possible Topics:
 Relationship between Tables and Graphs
 Making Comparison (Greatest, Least, etc.)
 Problem Solving
 Independent vs. Dependent Events
 Using Graphs to Describe Data
 Analyzing, Interpreting and Drawing Conclusions from Graphs and Tables





Tables and Graphs
The results of probability experiments are represented in a tables and graphs. For example, in the Gumball Machine experiment, students set the ratio of each color of gumball. The initial set up is 10 red, 10 yellow, 10 green and 10 blue gumballs. Each ball has an equal probability of being randomly selected. As the experiment runs students will observe that experimental results do not always reflect the theoretical probability, however, increasing the number of gumballs in a trial or increasing the number of trials will bring the experimental results closer to the results predicted using probability theory.
 Data Table shows the results for each trial and sum across trials.
 Experimental percentages can be compared to theoretical predictions.
 Data represented in the table can be shown as a bar graph or circle graph by simply tapping an icon.
 Optional sounds and speech functions enhance learning opportunties.

