sampling

this simulation, involves picking a marble at random from a bag: ten times, with replacement

this task from NRich is a good way of illustrating (on a good day) that longer runs usually lead to more accurate predictions

students are asked to predict the colours of the ten marbles in the bag

spinners

there are spinners available with most interactive whiteboards
for those that don't have these available - and with some benefits from other functionality - here are 3 examples of programmes:

experimenter (NRich 6033)

students could use a pencil and a paper clip to simulate using a spinner

using ICT you can have spinners with various sets of numbers and then add or subtract these

runs are single, in hundreds or 50 000s




one interest might be to compare e.g. experimental outcomes for subtracting numbers on two spinners, one with 5 numbers and the other with 4 and then theoretical probabilities (via a two-way 'outcomes' table)



spinning spinners (Actis)

has a restricted set of options and no speedy trialler

but the visuals are good and it can be left in the background, building up a bar chart

there is one difficult to caculate option (the bottom one) which is interesting...







Interactivate (Shodor) has two spinner programmes available:

adjustable spinner

 this automatically compares an experimental with a theoretical outcome









and experimental probability

which enables dice to be created with different numbers on their faces (only the same on both) as well as different spinners

capture - recapture

an interesting use of ratios is in the 'capture-recapture' method for estimating a population size

pull out a sample
mark it in some way
put the sample back
pull out another sample and use the fraction/ratio of marked things to estimate the full population size

suppose 40 butterflies are marked in some way and released in a particular location

then a further 20 are captured as shown:

how many butterflies would you estimate there to be from this information?

this program enables repeated capturing

repeatedly making second captures enables a comparison of different estimates, exploring the variation:
what are the extremes and what is the mean?

cereal packet problem

suppose a cereal packet manufacturer decides to put one of six Harry Potter figures inside each box:

Harry , Ron , Hermione , Dumbledore , Voldemort , Malfoy



how many boxes, on average, would it take to collect all six figures?

six is chosen because of the ease of simulation with a dice

on average it should need 14.7 'goes'

this is 6(1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6), a generalisable result

a simulation from Illinois enables various numbers of animal free gits to be (all) collected:

students can use this repeatedly to estimate the number of packets that need to be bought

if you don't mind promoting a commercial product (a breakfast cereal) then there is a good simulator that plots a histogram when all are collected - showing how many of each were obtained 

probability experiments - NRich

two probability experiments from NRich are based upon Geoff Giles' Dime probability kits and utilise the recording 'graph' paper that he developed

nrich 4311
 
'Two's company' win if the two blue balls touch each other

as well as the very helpful recording sheet, simulation results and a value for the experimental probability are given
you can keep accumulating 100s of runs

nrich 4313

'Cosy corner' win if at least one of the red balls is in any one of the three corner positions

students could be asked to estimate the probabilities after a short run experiment

theoretical probabilities for both of these simulation 'experiments' can be worked out but are not too easy to calculate (they are given, so could be quoted: 4/7 = 0.571 for 'two's company' and 57/60 = 0.95 for 'cosy corner')

NRich have a further probability experiment - equivalent to throwing two coins (4304) -a win is if both 'coins' show the same value. This can be a useful experiment to undertake since some students think there are three outcomes (both heads, both tails and one of each) so, mistakenly, think the probability of two the same is 2/3

it can also be interesting to explore the length of runs of wins or loses - maybe calculating an average length

NRich do a good copy of the (Geoff Giles) recording sheet - which has many benefits, especially in showing how results tend towards a certain value for a long run experiment

It's probably better to print the NRich student recording sheet (pdf) but here are two (ping) versions, for 50 trials:

probability - estarters

Matt Dunbar makes excellent use of Excel on 'estarters', some of which is freely downloadable

within 'probability theory' are the above two tasks involving a comparison of three probabilities (as fractions and decimals and positions on a probability scale)

Galton's quincunx

Sir Francis Galton - born in Sparkbrook, Birmingham (UK) in 1822

was a cousin of Charles Darwin  

made contributions to many branches of Science

used a 'quincunx' (also called a Galton board or bean machine) to show how a distribution tends to a 'normal' curve

(original is in the Science museum, Boston (USA))

The purpose of this work is for students to compare (i) a short run experimental probability with (ii) a longer run and (iii) a theoretical value. Hopefully the longer run value will be more accurate.... The task also involves a systematic consideration of all options and hence the theoretical values (which can lead to Pascal's triangle).

Resources: students will need two copies (front and back) of the 8 quincunx sheet and a coin. It can be helpful for a pair to work together (one copy of the sheet between them). Later they may need the 16 quincunx sheet, when they sort out the theoretical outcomes.

Instructions: throw a coin (or roll a dice and look for odds and evens).

If it is a head: go right at a junction.

If it is a tail: go left.

This can be usefully demonstrated on a board.

Four flips of a coin should result in the ball ending up in one of the five bins. Students are asked to record 16 experiments on their own sheets (front and back).

“Why do you think it ends up in the middle bin more than the others?”

After this it might be helpful to consider the purpose of the lesson :

to compare:

• A short run: your own experimental probabilities (their 16 trials)

• A long run: collect and sum the frequencies for each box for each pair of students. Label the bins (1 to 5) and work out the overall experimental probabilities from the class results.

• The theoretical probabilities: using another recording sheet ask students to find and/or draw all the different ways of getting from the top to the bottom: as a path or as combinations of 4 letters e.g. LLRR (following discussion). There are 16 ways altogether so they will need to be systematic. 
  
  
  
  “How can we use symmetry to simplify finding all the options?”

The theoretical results are a row of Pascal’s triangle: 1, 4, 6, 4, 1. These frequencies will be out of 16 and can be compared (e.g. as a decimal) with the ‘long run’ results by changing the fractions into decimals.

Utilising ICT: see (on a good day) how long-run experiments tend towards a theoretical value. 

There are several good versions of Galton’s quincunx on the net:

• Subtangent.com (Duncan Keith) (is good visually when made full size) as you approach a good number of trials, slow down the drop rate with the speed slider - then click on auto-drop to stop the flow. The frequencies just seem to keep on accumulating if you leave it to run. It can be enlarged (bottom left) for a whiteboard.  

• David Little (Penn State University) colourful (uses Java) can be made to fill a board and is downloadable - find the bin frequencies using the left and right arrow keys when the experiment has been halted, also displayed as percentages 

• maths is fun has a bar chart helpfully displayed alongside a frequency count as well as the pinboard - but is a little small for classroom display for 4 layers. Gave the following frequencies for 1000 goes:   58 , 261 , 358 , 270 , 53.

“Why do you think experimental results should be more accurate when more trials are involved (a longer run)?”

“What is the difference between an experimental (short run and long run) probability and a theoretical value?”

Extension: students could consider theoretical values for other layers (5 then 6 etc) leading to Pascal’s triangle.

subtangent.com Duncan Keith

subtangent.com

has the novel and engaging 'transformation golf' (go for the improved version: in resources, then in miscellaneous)


diagonal differences investigation on a number grid is probably helped by this programme

nrich resources

nrich helpfully give a 4 digit code to their resources
ou can find these resources by searching for nrich then the code










4348 cuisenaire rods

5618 dotty circle sine function, from a point going round a circle

2883 virtual geobaords (square, circle, isometric)

4901 large pinboard

2667 tilted square (coordinates of skew squares)

6703 mystic rose drawing tool, for various divisions of a circle, useful for creating a rule for the numbers of diagonals - with a focus on proving a generalisation
(you can also download an Excel tool for this from Bodmas.)

5528 tangram (square)