Probability

Prodromou, T. and Pratt, D. (2005), ‘Harnessing the causal to illuminate the stochastic’, in D.Hewitt and A. Noyes (Eds), Proceedings of the sixth British Congress of Mathematics Education held at the University of Warwick, pp. 136-143.

This paper reports on the continuing design of a microworld that aspires to capture how students might use knowledge about the deterministic to explain probability distribution as an emergent phenomenon. This study builds on prior work, which identified that students of age 11 years had sound intuitions for short-term randomness but had few tools for articulating patterns in longer-term randomness.

Statistics

Sharmaa, S., Doyleb, P., Shandilc, V. and Talakia’atuc, S.(2012) ‘Developing statistical literacy with Year 9 students: A collaborative research project’, The University of Waikato; The University of Auckland and Marcellin College. BSRLM

In this collaborative research study, two cycles of teaching experiments were carried out in two year 9 classes. The data set consisted of audio and video-recordings of classroom sessions, copies of students’ written work, audio recorded interviews conducted with students, and field notes of the classroom sessions. The results shed light on tools and techniques which the research team used to help students develop critical statistical literacy skills.

Johnston-Wilder, P. (2008) ‘Children’s understanding of randomness as a model’, University of Warwick, BSRLM.

This paper presents two views of randomness from literature on philosophy of science. Each of these is discussed in relation to learning randomness, and they are related to a common restricted understanding of randomness.

These views are used to analyse extracts from interviews with secondary school pupils in which the pupils were invited to participate in simple experiments involving randomness, and to discuss situations using the idea of randomness.

Dave Pratt and Richard Noss, (2002) “The Microevolution of mathematical knowledge; the case of randomness”.

This is an accessible paper about randomness, which is a difficult idea for students. It describes how complex it is to move from meeting a new idea to recognising when it is useful in a new situation and how context affects learners’ understanding of a new abstract idea. It gives insight into how sequences of tasks might be designed that lead learners to apply a new idea.

Probability

Nines and tens;

This activity enables the pupils to:

• record, describe and analyse the frequency of outcomes of simple probability experiments involving randomness, fairness, equally and unequally likely outcomes, using appropriate language and the 0-1 probability scale
• understand that the probabilities of all possible outcomes sum to 1

Which team will win?

The dog ate my homework:

These activities will enable pupils to carry out experiments to compare theoretical and experimental outcomes that will enable them to:

• enumerate sets and unions/intersections of sets systematically, using tables, grids and Venn diagrams
• generate theoretical sample spaces for single and combined events with equally likely, mutually exclusive outcomes and use these to calculate theoretical probabilities.

Statistics

M, M and M:

The wisdom of the crowd:

These activities give pupils the opportunity to:

• describe, interpret and compare observed distributions of a single variable through: appropriate graphical representation involving discrete, continuous and grouped data; and appropriate measures of central tendency (mean, mode, median) and spread (range, consideration of outliers).

What’s the weather like?

Charting success:

These activities provide opportunities for pupils to:

• construct and interpret appropriate tables, charts, and diagrams, including frequency tables, bar charts, pie charts, and pictograms for categorical data, and vertical line (or bar) charts for ungrouped and grouped numerical data.

Going to school, from ‘Language of functions and graphs’, page 28 (from the Shell Centre).

(You must be logged in to the National STEM Centre website to access this document)

This activity provides opportunities for pupils to:

• describe simple mathematical relationships between two variables (bivariate data) in observational and experimental contexts and illustrate using scatter graphs

More Mind Games: from CensusAtSchool:

This activity uses data from CensusAtSchool 2005/2006 to provide opportunities for pupils to:

• work out averages and probabilities from a frequency graph. Contrasting mathematical and statistical thinking. (Interpreting charts and graphs, bar charts, mean, median, mode and probability for KS3.)

Probability

Example A

A box contains bricks which are orange or blue or brown or yellow.
Damion is going to choose one brick at random from the box.

The table shows each of the probabilities that Damion will choose an orange brick or a brown brick or a yellow brick.

Work out the probability that Damion will choose a blue brick.

Example B

Two boxes each contain a different number of black and blue ink pens.

Box 1 (Bikko) has 7 black and 11 blue ink pens.

Box 2 (Stylo) has 5 black and 7 blue ink pens.

Kylie picks a pen at random from a box. From which box is she more likely to select a blue pen? Why?

Example C

Trishane designs a game.
It costs £1.20 to play the game.

The probability of winning the game is 310

The prize for each win is £2.50
150 people play the game.

Work out an estimate of the profit that Trishane should expect to make.

Example D

Two coins are thrown at the same time. There are four possible outcomes:

HH HT TH TT

How many possible outcomes are there if:

1. three coins are used?
2. four coins are used
3. five coins are used?

Statistics

Example E

Three children have a mean age of 10.

The range of their ages is 6.

Give two sets of possible ages for the children.

Example G

20 students scored goals for the school hockey team last month. The table gives information about the number of goals they scored.

1. Write down the modal number of goals scored.
2. Work out the range of the number of goals scored.
3. Work out the mean number of goals scored.

Example H

In a survey, some students were asked what their favourite leisure activity was.

Their answers were used to draw this pie chart.

1. Write down the fraction of the students who answered “Television”.
   Write your answer in its simplest form.
   12 students answered “Reading”.
2. Work out the number of students who took part in the survey.
3. How many answered “Sport”?

Example I

Here is a scatter graph. One axis is labelled “Height”.

1. From the list below, choose the most appropriate label for the other axis.
   • length of hair
   • number of sisters
   • length of legs
   • GCSE French mark
2. Describe the correlation shown on this graph. Explain why you think this.

Probability

• The Monty Hall Problem is a famous (or rather infamous) probability puzzle. The puzzle and the counter-intuitive answer explained.
• What is probability? An animated guide to probability. Expressing probability as fractions and percentages based on the ratio of the number of ways an outcome can happen and the total number of outcomes is explained. Experimental probability and the importance of basing this on a large trial is also covered.
• Simple probability. An introduction to the concepts of Simple Theoretical Probability Part 1 of the Probability Series.

Statistics

• The Joy of Stats. Hans Rosling says there’s nothing boring about stats, and then goes on to prove it. A one-hour long documentary produced by Wingspan Productions and broadcast by BBC, 2010.
• What is descriptive statistics? Descriptive statistics is an effective way of analysing numbers and data. This video explains what is meant by descriptive statistics.