Random Numbers

Prior Knowledge.

• the notion of probability being between 0 and 1
• the concept of a random event
• the notion of a conjecture

Background:

The concept of randomness is central to developing an understanding of probability. Random numbers are also very useful tools for a statistician or mathematician.

Further information to back up what they will meet:

• FIO Books – Check Number Books for doubling activities.
• National Archive of Virtual Manipulatives (NVLM via Google)

Comments on these exercises

This set of exercises investigates the notion of randomness, getting random numbers in a variety of ways and using these in some straightforward applications. These exercises are beneficial to all students with the more able students expected to extend and explore further.

All of these exercises should be linked to simulation of natural events.

Exercise 1: Are You Random?
Asks students to explore the concept of random numbers by completing a table of random numbers. The term random has a number of meanings, including a slang usage. Existing understanding needs to be drawn out and the mathematical meaning of the term established before this exercise is attempted. Other teaching lessons preceding this exercise could include:

• establishing other probability language
• throwing a dice gives a random number
• random means “I cannot reliably predict the answer” or similar notion

This exercise is practical and aims to form introductory ideas of “what is random” in relation to “me”; a die and random number tables. No instruction in the use of tables is given as current student notions are to be recorded and noted by the teacher. This is thus a formative assessment of student knowledge. No suggestions for testing for randomness are made as the exercise is designed to make students think and come up with their own ideas. Some strategies are developed in later exercises.

Once this exercise is completed, a valuable follow-up is for students to discuss how they tested to see if the numbers were random. Students initially discussing their ideas/strategies in pairs before feeding back to the group is useful here.

Exercise 2: Using Random Number Tables
Asks students to explore random number tables. This exercise focuses on the random number table and requires students to think how they could use this table to gather different sets of (one digit) random numbers each time it is used. It also introduces students to a way to generate random numbers from the table, with a sample method being outlined. Students are then asked to create their own method. Note that the explanation for the sample method is ambiguous. Some students may start counting across three numbers and down seven, then take the next ten numbers. Others may work out that each set of ten numbers has been called a ‘block’ in the exercise… Students may also repeat the use of the ‘three – seven’ pattern, which will also generate the small set of random numbers required, though eventually this would lead students back to the numbers they originally collected.
Any method that chooses a random place to begin is satisfactory.

Exercise 3: Random? How do you Know?
Asks students to explore a simple test of randomness.  Students may need to be introduced to the meaning of the # sign before starting this exercise. A simple method for testing for randomness is suggested. The main idea is to presume that every number has an equally likely chance of being selected so that in the “long run” a similar number of each is found in the set of numbers. Look up the “Law of Large Numbers” on the internet for more information.

The last part of the exercise gets students to use their count to say whether or not the set of numbers is random according to this method. Some students may answer that the set is not, as not every digit appears exactly the same number of times. This may need some discussion.

Exercise 4: Some Random Events
Asks students to revise with a classmate what they have learnt in Exercises 1-3.
This exercise is formative in that it requests current student understandings and experiences.

Exercise 5: Random or Not?
Asks students to decide if an experiment would generate random results or not.  This exercise should be done by small groups of students working together, with a lot of discussion and justification of answers. In marking, students could be challenged to provide an answer, and justify why that answer is correct. The exercise could be used to assess the key competency of communicating using a rubric.

Exercise 6: How Many Do I Need?
Asks students to make a conjecture about the sample size needed to generate random numbers.  This exercise should establish introductory and fundamental ideas that a few selections is not sufficient and that thousands is excessive. A practical interpretation by the student is expected and should be developed experimentally. Note that students should be encouraged to make and test conjectures about mathematics at every opportunity.

This is a good place to introduce the notion of variation. If a sample of students’ results from question 1 were listed on the board, it would be unusual to find two the same. There would also be a lot of variety in the totals of each number in every sample. With larger numbers of trials, less variability will be progressively seen.

Exercise 7: Making a Random Spinner
Asks students to make spinners. Criteria are given for making the spinners, for example using the numbers 1, 2 and 3 design a spinner so 2 comes up twice as often as 1.

Exercise 8: Going on a Random Walk
Asks students to generate a random path on a grid map. Teaching lessons preceding this exercise could include:

• discussing walking and turning in a random way vestablishing what forward turn left and forward turn right means

This is a fun task and can be used on the playing field as well. Using a LOGO programming language this activity can be used to create fascinating classes of patterns. Use Google to find the free software LOGO or similar downloads. Encourage students to invent and be creative.

Exercise 9: Random Walk with No Returns
Asks students to extend the random path in Exercise 8 by not allowing the path to come back on itself.
In three dimensions a more sophisticated version of this simple idea is used to explore how protein molecules, for example, assemble themselves. Two atoms can not be in the same place hence the restriction on “no returns”. The resulting patterns are totally different from Exercise 7. Exploring these with LOGO is a fascinating investigation.

Exercise 10: Random Numbers in the Calculator
Asks students to use a calculator to generate random numbers. Teaching lessons preceding this exercise could include:

• using the calculator to find random numbers

Note that different calculators use different letters to select RAND. Even the same brand notation can vary between models. It is best to check the instruction manual and master your own calculator before working with the class.

Exercise 11: Random Numbers on the Computer
Asks students to generate random numbers using an Excel spreadsheet.
Teaching lessons preceding this exercise could include:

• using the calculator to find random numbers

Note that different spreadsheets use different expressions to generate random numbers and this should be checked and mastered well before use. It is also important to know how to copy a function to an array of cells.

In Excel the function =2*RAND() gives two times a random number to n decimal places. The decrease decimal place button can then be used to reduce this to a whole number. However, this will produce three outcomes (0, 1 and 2) rather than the expected two.

To get around this the standard strategy is to use =2*RAND() + 0.5. This has the impact of changing the rounding from 0 ≤ x < 1 and 1 ≤ x< 2 to 0.5 ≤ x< 1.5 and 1.5 ≤ x< 2.5