compass and ruler constructions

not interactive, the MathOpenRef site has a variety of compass and ruler constructions that are demonstrated (videos)

these can be made full-screen

you can click an auto-repeat box so that the steps are shown repeatedly

boxplot plotting

you can change the number of data points (up to 16) and choose whether to have them sorted or not

from Braining Camp

see the effect of changing a single value incrementally

connect three

this game from NRich provides good practice in adding and subtracting directed numbers - to get three in a line
play against the computer or a partner

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

the eyeballing game

get students to use the eyeballing game to estimate the positions of various geometric objects

and find their average error (for either one, two or three goes)

there are various aspects to explore




e.g.
do people improve on their second/third goes?

which game are the class best/worst at?

fitting polygons together

octagon loops, produced using the Tyler applet

enables polygons to be drawn (and removed) easily


what is the relationship between the inside and outside perimeter of these octagon loops?

some interesting patterns can be formed











for example, the semi-regular tesselations could be explored

The illuminations (NCTM) polygon fitter is also good. Using this you can group blocks of polygons.

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

boxplots NLVM

the 'Old Faithful' geyser in Yellowstone Park has data for the intervals between eruptions that are sort of bi-modal















boxplots NLVM

enables a comparison to be made between a boxplot and the shape of a bar chart (of grouped data)

several data sets are provided - the 'Old Faithful' data set having 112 points

the group sizes can be changed with a slider

students can decide how many groups to have - to provide an overview of the data shape













students can enter their own data and compare a boxplot with a bar chart
they can also see how their data looks with easily changed class intervals (by means of a slider)

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)

UK statistics

a population pyramid for the UK enables you to choose a year (using a slider) or look at the variation over time (including projections). The display can be made full screen.

You can also look in more detail at a particular age group by hovering over it with a mouse.

histogram (bar chart) tool - Illuminations (NCTM)

Illuminations histogram tool

6 data sets given, or you can enter your own data

it is very helpful to be able to explore the shape of the distribution for various groups sizes (using a lovely slider)

students can consider the number of bars necessary to provide a view of the overall data shape

for example, in the 'horsepower' example above, a distribution with a further peak (as well as the modal class) gets smoothed out for fewer 'bins'

cumulative frequency - Waldomaths

Waldomaths














compare the cumulative frequency graph for various (10) standard and other data shapes

also compare these with the related boxplot

frequencies can be varied by dragging the red dots on the vertical axis - to see the effect on the cumulative frequency curve

questions can be asked about e.g. the frequency distribution from the cumulative frequency data and graph or the cumulative frequency graph from the boxplot (by clicking boxes on or off at the bottom)

scattergraphs - Waldomaths

waldomaths















useful to compare an estimated line of best fit with a computer calculated one. Plot some points using the mouse then drag a line (at the top of the screen) to where you think the line of best fit is located.

Then show the computer calculated (accurate) line of best fit.

averaging

what happens if you start with 2 numbers, produce a next number which is the average of these two and then iterate this process?
[sometimes this is know by the (fictitious) name 'Littov's' chain]

all number pairs reach a limit, which has a neat generalisation

students might explore this for their own two starting values, working on paper initially

it is also very easy to set up a formula that can be dragged down on a spreadsheet (using abs for the absolute value)

what happens for (4, 7)? for (10, 4)? for (2, 8)? plus some other ordered pairs....

mathsonline has a version of this task ('the mean series') available to non-members, either as an Excel sheet or online. The Excel sheet contains a graph, showing how the values home in (smooth) to their limit value.

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.

3D software: Kaleido tile

Kaleido tile enables various explorations of 3D shapes as well as tesselations (2D) and patterns wrapped around a sphere (as in the examples)

there are various options for colouring the faces - or adding in your own pictures

when named polyhedra are obtained their name pops up on the screen

this can be obtained from Jeff Weeks' collection

Perigal's dissection

The proof of the pythagoras rule by Perigal is animated here - you can drag the corners of the original triangle.

Henry Perigal, English, 1801 - 1898

The detail of the proof is not that simple.