In this unit students experience "randomness" through a related set of activities that link large sequences of tossing coins, random walks, Pascal's Triangle, and probabilities. All of these are a little too complicated for probability trees.
Many people in the general population have poorly developed intuitions of randomness. For instance, some people believe that there must be a cause behind certain events that, from a mathematical point of view, are deemed to be independent, random events. To develop a mathematical intuition in this area, students usually need to experience the process of randomness first hand. Although this initially requires patience, it does lead to important learning.
The aim of this unit is to provide experiences that will help students develop a better understanding of random processes. We first meet random processes by looking at tossing coins. Somewhat surprisingly, in a random series of tosses, large strings of Heads or Tails can occur. This is perhaps counter-intuitive to what might be expected. This is followed by experiencing random walks on both square and triangular grids. Work on grids of the latter type leads to the introduction of Pascal's Triangle. The aim of introducing this is to give students access to certain probability problems.
The learning opportunities in this unit can be differentiated by providing or removing support to students, and by varying the task requirements. Ways to support students include:
This unit is focussed on exploring randomness in mathematical contexts, and as such, is not set in a real world context. You may wish to explore real world applications of randomness in teaching sessions following the unit, for example, in video game programming, lottery winnings.
Te reo Māori kupu such as matapōkere (random) and tūponotanga (probability, chance) could be introduced in this unit and used throughout other mathematical learning.
In this session we look at a large number of tosses of a coin. The main conclusion is that there are more strings of Heads or Tails than students might expect.
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Here we look at random walks on square and triangular grids.
A random walk is a path traced out by a walker who next step is taken randomly. We look at random walks on the rectangular (Cartesian) grid and on a triangular grid. In the former case we use the tosses of two coins to determine the randomness. In the latter case we use spinners.
Generally each direction of a random walk is equally likely but for demonstration purposes on the rectangular grid we have chosen to (i) alternate a vertical and a horizontal movement and (ii) sometimes allow different probabilities in each direction.
In this session students will be generating random walks using two coins and printed grids. To do this it might be useful to have students work in groups of threes. One can toss the 10c coin, one the 20c coin, and the third can record the results and so produce the random walk.
Throughout this Teaching Sequence we have used coins and spinners. If these are not readily available for you, then dice or some other object(s) will do, provided the probabilities are retained.
In this session we look at the number of ways of getting to any particular point in a triangular grid. This produces Pascal’s Triangle. We discuss some interesting facts about that triangle.
A. Pascal’s Triangle is the array of numbers shown below.
The numbers down the side are all ones and the other numbers are found by adding together the two numbers above it to the left and right. The entries in any row are actually the coefficients of the expansion of (x + 1)n, where n is one less than the number of the row.
It is worth noting that the sum of the numbers on each row of the triangle is a power of 2. The power is one less than the number of the row from the top. There is also a pattern called the Hockey Stick pattern. Add all the numbers in the first few terms of a diagonal starting from a 1. The answer is on the row below the last number. The added numbers and the sum form a hockey stick shape.
This pattern of numbers is called Pascal’s Triangle.
Here we use Pascal’s Triangle to solve probability problems. The probability questions are on Copymaster 5 and the answers are at the end of the teaching sequence
A. This session contains more material than can be covered comfortably in one lesson.
Solutions to the questions on Copymaster 5 (all given to 2dp).
Probability of 0 seedlings = 1 x (0.7)6 = 0.12
Probability of 1 seedlings = 6 x (0.7)5 x (0.3) = 0.30
Probability of 2 seedlings = 15 x (0.7)4 x (0.3)2 = 0.32
Probability of 3 seedlings = 20 x (0.7)3 x (0.3)3 = 0.19
Probability of 4 seedlings = 15 x (0.7)2 x (0.3)4 = 0.06
Probability of 5 seedlings = 6 x (0.7)x (0.3)5 = 0.01
Probability of 6 seedlings = 1 x (0.3)6 = 0.00
Probability of two 50s = 10 x (0.4)2 x (0.6)3 = 0.35
Probability of 3 defective = 120 x (0.1)3 x (0.9)7 = 0.06
Probability of 5 sixes = 126 x (1/6)5 x (5/6)4 = 0.02
Proportion of families = 70 x (0.5)4 x (0.5)4 = 0.27
Probability of 7 heads = (0.5)7 = 0.01
Find the probability for each number of defective switches and find the sum of (each probability x corresponding number of switches) So, probability of 1 defective switch x 1 + probability of 2 defective switches x 2 + probability of 3 defective switches x 3 + …
Dear families and whānau,
Recently we have been exploring "randomness" through sequences of tossing coins, random walks, Pascal's Triangle, and probabilities. Ask your child to share their learning with you.
Printed from https://nzmaths.co.nz/resource/investigation-random-processes at 1:32pm on the 26th April 2024