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Stage:
Subject
Secondary
mathematics
Year
Term
9
Probability simulations
Module contents
Module focus
Curriculum focus
Comparing experimental and theoretical probabilities using biased
data.
Learning objectives
By the end of the lesson pupils will:

be able to construct frequency graphs and polygons from a
generated set of data;
 be able to compare the effect biased data has on experimental and
theoretical probabilities.
Learning outcomes
Most pupils will:

generate a set of data from a rolling dice simulation and create a
frequency graph and polygon using a graphical calculator;
 analyse the frequency graph, identify which data is biased, why,
and describe how this experimental evidence differs from their
original theoretical probability.
Pupils making slower progress will:

with support, generate a set of data from a rolling dice simulation
and create a frequency graph using a graphical calculator;
 analyse the frequency graph and identify which data is biased and
why.
Pupils making faster progress will:

generate a number of sets of data from a rolling dice simulation and
create corresponding frequency graphs and polygons using a
graphical calculator;
 investigate a number of frequency graphs to identify biased data
and compare and contrast the frequencies graphically by
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superimposing the line y = 1/6 over STAT PLOTs of the relative
frequency plotted against the dice score.
References
Strategy Framework references
Handling data
Processing and representing data, using ICT as appropriate

Find summary values that represent the raw data, and select the
statistics most appropriate to the problem; find the median and
quartiles for large data sets; estimate the mean, median and
interquartile range of a large set of grouped data.
 Select, construct and modify, on paper and using ICT, suitable
graphical representation to progress an enquiry, including;
- frequency polygons;
- line graphs for time series;
- scatter graphs to develop further understanding of correlation;
- lines of best fit by eye, understanding what they represent;
identify key features present in the data.
Probability

Estimate probabilities from experimental data; understand relative
frequency as an estimate of probability and use this to compare
outcomes of experiments.
 Compare experimental and theoretical probabilities in a range of
contexts; appreciate the difference between mathematical
explanation and experimental evidence.
The Framework for teaching mathematics can be found at:
www.standards.dfes.gov.uk/keystage3/respub/mathsframework/forewo
rd.
National Curriculum references
Ma4 Handling data
Collecting data
3 Pupils should be taught to:
b gather data from secondary sources, including printed tables and
lists from ICT-based sources.
Interpreting and discussing results
5 Pupils should be taught to:
a relate summarised data to the initial questions.
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The National Curriculum programme of study can be found at:
www.nc.uk.net/nc/contents/Ma-3-2-POS.html.
Use of ICT
Teacher use of ICT
This module will give you the opportunity to:

use probability simulation software as a whole-class activity to
generate data;
 input data into a graphical calculator to draw statistical graphs;
 use a probability simulation application on a graphical calculator.
Requirements
Hardware


Computer
Data projector and large screen display and/or interactive
whiteboard (IWB)
 Graphical calculator, overhead projector and LCD display panel
Software

Mathematics software, such as MicroSMILE Probability or TI
Interactive
Other




Individual whiteboards and pens
Graphical calculators (at least one between two pupils)
Printed copies of help sheets (optional)
Printed copied of the pupil worksheet Loaded dice
Trial or viewer versions of the software you need may be available
from the Software downloads page.
Lesson preparation
A key element of this module is a lesson for you to adopt and adapt to
meet the needs of your class. It is designed to help you evaluate the
impact of using ICT for learning and teaching. Display and discuss the
objectives and key vocabulary for this lesson with the pupils.
Load the simulation software onto the computer and familiarise
yourself with the software and lesson resources before the lesson. This
lesson and its resources are based on using a Texas graphical
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calculator and associated application ProbSim. You will need to load
the ProbSim probability simulation application on to the teacher and
pupils' graphical calculators. If you are using a different model of
graphical calculator, investigate if the calculator provides probability
simulations or use the probability simulation software to provide data
for your pupils to use during the lesson. Alternatively, if you have
access to an ICT suite pupils could access the simulation software and
generate data themselves.
If you are using the pupil helpsheets during the lesson ensure you
have printed enough copies (at least one copy per pair of pupils).
For ICT support, visit the ICT skills guidance tab.
Vocabulary
biased, discrete, experimental probability, limit, relative
frequency, theoretical probability
Health & Safety
All standard safety procedures with computers need to be in
place. Information can be found at http://schools.becta.org.uk.
ICT skills guidance
The guidance in this section supports the ICT skills described in the
Module contents.
Pupils' prior knowledge and skills
Pupils should already:

know how to compare two distributions using the range and one or
more of the mode, median and mean;
 be able to estimate probabilities from experimental data;
 understand that if an experiment is repeated there may be, and
usually will be, different outcomes;
 understand that increasing the number of times an experiment is
repeated generally leads to better estimates of probability.
Starter
Display the probability simulation programme Loaded dice on a large
screen display or IWB. The pupils will be looking at two columns with a
dice at the top of each column. Explain to pupils that one of the two
dice is biased, which means that when it is rolled, it will not behave in
the expected way. Establish that the probability of throwing each
number on a conventional six-sided dice is one-sixth. The object of this
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simulation is to roll each dice in turn and decide which of the two dice
is loaded and why.
Roll each dice ten times and ask pupils to write on their individual
whiteboards the dice that they think is loaded. Roll each dice a further
10 times and ask pupils to vote again. Ask pupils if any of them have
changed their minds and if so, why?
After a 100 rolls of each dice select the dice that most pupils think is
the biased dice.
Discuss the results window, which displays the loading of the dice in
terms of the probability of obtaining each result. Ask pupils if they can
change each of the relative frequencies from fractions to decimals to
make comparisons between the results. They can do this with the help
of their calculators.
The results from this simulation will be used in the main part of the
lesson. Record the data on the board or pupils can record the
summary data directly into the list facility of their graphical calculators.
Model how to do this for your pupils or, alternatively, hand out the
resource Helpsheet inputting data for guidance.
Main
Using the teacher graphical calculator and whole-class display facility,
model to the pupils how to produce the bar graphs on the graphical
calculator for each set of dice data collected during the starter activity.
Ask pupils to recreate the same steps on their own graphical
calculators as you do this. Ensure all pupils have produced the correct
graphs then discuss the graphs with your pupils and ask them if they
give an indication which of the dice is biased.
Ask questions similar to the example that follows.

Which features of the graphs are the same and which are different?
Now model how to begin the application ProbSim. Ensure pupils are
confident with how to use ProbSim before handing out the pupil
worksheet Loaded dice.
Ask pupils to work in pairs on the task.
Plenary
Ask pupils to compare their results with another pair of pupils and
discuss any observations they made. Ask questions that will probe
pupils' understanding of the outcome of weighting a six-sided dice.
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Ask the following questions.


How could you weight the dice so that it rolls even numbers?
Is it possible to weight the dice so that the median score is the
same as the minimum score?
Encourage pupils to share their outcomes from the main activity to
support their responses.
Assessment
In assessing for learning you should consider the following points.
1. Ensure objectives are expressed in language that pupils
understand.
2. Give pupils clear success criteria related to these objectives.
3. Give pupils opportunities to discuss their successes and challenges
focusing on the objectives.
4. Provide oral and written feedback to pupils.
5. Encourage pupils to explain their thinking and reasoning in a secure
environment.
6. Provide time for pupils to reflect upon what they have learned and
understood and identify any difficulties.
You can find information on assessment for learning at:
www.standards.dfes.gov.uk/keystage3/respub/afl_ws.
Adaptation
Texas Instrument’s TI 83 Plus programme application ProbSim will
allow a range of simulations to be explored, for example, rolling dices
or tossing coins. Choose a more appropriate context for pupils to
explore whilst focusing on the following objective.

If an experiment is repeated there may be, and usually will be,
different outcomes.
You could use the additional resource Straight dice. Ask questions
such as the examples given below.

If you roll a fair six-sided dice 60 times, how many times would you
expect to throw each number?
 What if you only rolled it six times?
Higher-attaining pupils could be asked to compare and contrast the
relative frequencies graphically by superimposing the line y = 1/6 over
This document is part of a range of materials designed to help teachers teach using ICT.
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STAT PLOTs of the relative frequency plotted against the dice score
for each dice. Ask the following questions.


Can you explain the significance of the line y = 1/6?
If the dice had been fair, what would you expect to see?
Evaluation
Lesson reflection
These prompts are designed to help you reflect on how the use of ICT
affected your teaching and pupils’ learning.
Prompts for reflection:
1. How did the use of ICT:
• help pupils to make better progress towards achieving the
learning objectives?
• affect the pace of learning?
• affect pupils’ motivation, interest and time spent on task?
• affect your ability to differentiate your teaching and personalise
pupils’ learning?
2. What knowledge or skills have you gained and extended in teaching
this lesson?
3. What adaptations would you make to the lesson and its resources
to suit the needs of your class?
You may wish to create a record of your evaluation and save it as
evidence of your professional development. If so, you can download a
template containing these prompts and spaces for your responses.
Materials evaluation
These prompts are designed to help you consider why, how and when
you would incorporate these lesson activities and resources into your
curriculum and teaching plans.
Prompts for evaluation:
1. What are the benefits of using these teaching and learning
approaches and resources to achieve the subject objectives?
2. How do the suggested activities fit with your existing curriculum and
teaching plans?
3. What adaptations would be required to the activities or resources to
suit the needs of your class?
4. Are there any requirements for ICT equipment, other resources,
space, etc. that might limit how and where the lesson is taught?
This document is part of a range of materials designed to help teachers teach using ICT.
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You may wish to make a note of your thinking for your own records
and to share with your colleagues. If so, you can download a template
containing these prompts and spaces for your notes.
Download module
Download module
You can download a module pack containing the lesson plan,
resources, supplementary information and extension activities using
the link below. The pack is in a zipped file to minimise file size, but
unless you have a broadband connection, the download may be slow.
To extract the files within the module pack you will need either Winzip
or Microsoft Windows Extraction Wizard. This software is freely
available and can be downloaded from the Software downloads
page.
This document is part of a range of materials designed to help teachers teach using ICT.
For more information, log on to www.teachernet.gov.uk/supportpack
Please note - Permission to reproduce, distribute, adapt and use this document is subject to the terms and conditions
of the Practical Support Pack website, available at www.teachernet.gov.uk/supportpack/termsandconditions.aspx.