# 4 Block Stalagmites

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## WHAT IS IT?

4-Block Stalagmites is part of the ProbLab middle-school curricular material for learning probability. Related materials are a random generator called a "marbles scooper" and a sample space called a "combinations tower". In classroom activities, students working with 4-Block Stalagmites will have interacted with the marbles box and built its sample space and thus would have inferred expectations as to outcome distributions in hypothetical experiments with the marbles box.

4-Block Stalagmites is designed to enable users to experience insights into the binomial phenomenon and in particular to witness and understand the emergence of experimental outcome distributions that, by and large, are both consistent and in accord with proportions in the sample space. The model includes an interactive simulation of a binomial experiment with a sample size of four, which is comparable to an experiment of tossing four coins over and over, only that the four "coins" can land on green or blue, not heads or tails, and each "coin" has a fixed position in a 2-by-2 table that we call a 4-Block, unlike a set of four coins that has not inherent structure and lands "all over the place" on the table.

A unique feature of the model is that the outcome distribution is composed of the actual experimental samples themselves that are stacked one above the other in their corresponding columns. This is different from classical histograms that do not record which specific samples were taken but only their aggregate properties. For example, the particular 4-Blocks sampled appear in the distribution, rather than as just a record of the fact that a 4-Block with 3 green and 1 blue squares (in any order) were sampled.

The outcome distribution is in the form of "stalagmites" of stacked samples that have "dripped" down into their correct column. This creates a picto-graph histogram that grows bottom-up like a stalagmite. When the probability in the model is set at 0.5, this stalagmite will grow to 1:4:6:4:1 proportions. For other p values, the stalagmite will be tailed.

## PEDAGOGICAL NOTE

There are four unique 4-blocks that each has exactly one green square, but there are six unique 4-blocks with exactly two green squares each. So, for a p value of .5 (when independent squares are equally likely to be green or blue), it is 1.5 times more likely to draw a two-green 4-block than a one-green 4-block (the ratio value of 6 to 4 is 1.5). This is worthy of attention, because students often need help in understanding how permutations are relevant to combinatorial analysis and, moreover, how combinatorial analysis is relevant to predicting the outcome distribution. When the sorting and coloring effects are activated as the simulation is running, the visual effect of the growing stalagmite is as though it is "stretching" the sample space of 16 elemental events into an outcome distribution of about 160 samples. Within columns we always expect all the elemental events to occur as frequently. However, this is true between columns only for a p value of .5. Note that we can change the p value and thus affect the overall shape of the stalagmites. For example, for the p value of .6, the two-green 4 Blocks will not occur as often as they would for a p value of .5, but the three-green 4 Blocks will occur more often than for a p value of .5. Due to the specifics of this change, the two-green and three-green columns are anticipated to be equally tall.

## HOW IT WORKS

4-Blocks are randomly generated by asking each square to choose a color with a 'probability-to-be-target-color' chance of being green. Each 4-Block sample "drips" down one of the five columns in accord to its number of green squares. The stalagmite distribution can be sorted by type, even as it grows. There are 16 unique outcomes, so sorting the experimental outcomes by type results in 16 groups. These 16 groups are typically of uneven size, even for the p value of .5, but most often their sizes revolve around the average. For example for 160 samples taken, most groups will contain roughly 10 outcomes. You can "paint" these groups to enhance their visual groupiness.

## HOW TO USE IT

Buttons: SETUP-initializes the View, essentially "emptying" the columns, and resets the variables and monitors. GO ONCE-generates a single 4-Block and sends it down its respective chute, whereas GO does so forever until one of the columns reaches the top of the display. GO-ORG-begins a run in which the samples sort themselves by type (see SORT OUTCOMES, below) ORGANIZE -rearranges outcomes within each column so that identical 4-blocks are grouped; DISORGANIZE -undoes this rearrangement. PAINT-colors outcomes by type so that identical 4-blocks appear of uniform color (the colors themselves are arbitrary-there is no inherent meaning or scaling); UNPAINT-returns the 4-blocks to their original appearance.

Switches: KEEP-REPEATS?-when Off, repeated outcomes are discarded from the Stalagmite. For example, say the simulation has already generated a 4-block with a single green square in the top-left corner. Any time later in the run, if the simulation generates another identical 4-block, it will descend the column and then disappear the moment it hits the stalagmite. But a 4-block with a single green square in the bottom-left corner would be kept, if it had not been generated. When On, repetitions are kept (as in standard outcome distributions). STOP-AT-ALL-FOUND?-when On, the run will end as soon as all 16 unique outcomes of the sample space have been randomly sampled. When Off, the run will continue until one of the columns reaches the top of the display. MAGNIFY?-when 'On,' a blown-up version of newly created 4-blocks is displayed to the side of the column. This helps, because the samples themselves are small and move fast. When Off, no blown-up sample is displayed.

Slider: PROBABILITY-TO-BE-TARGET-COLOR-determines the chance that each independent square in a 4-block will be green. For example, a value of 50 (50% or .5) means that each square has an equal chance of being green or blue, whereas a value of 80 means that each square has a 80% chance of being green and 20% chance of being blue. Monitors: EVENTS FOUND-keeps track of how many of the 16 possible 4-block outcomes have been randomly sampled.

Plot: EVENTS BY NUMBER OF OUTCOMES-shows how the sixteen elemental events are distributed by the number of outcomes sampled for each. When the first sample is taken, that event would be a '1' whereas all the other fifteen events are still at zero.

## THINGS TO NOTICE

Setup the model in its default settings (with the 'probability' slider set to the value of 0.5 and the 'magnify?' switch set 'On'), slow down the model, using the speed slider above the View, and press 'Go'. See how a random 4-block sample is generated at the top of the View, just to the left of the stalagmite columns. Count up the number of green squares in this 4-block and see that the 4-block descends down a column bearing the corresponding numeral at the bottom. For example, if there are exactly two green squares in the random 4-block, it will go down the column with a "2" at the base.

Keep running the model slowly. See how samples are stacked on top of each other in the columns. Look closely at these samples and see if you can locate repeated outcomes, for example see if the 4-block with exactly two green squares in a particular diagonal formation occurred at least twice.

## THINGS TO TRY

Set KEEP-REPEATS? to 'On' and STOP-AT-ALL-FOUND? to 'Off'. Press 'Go.' The columns will fill up until one of them hits the top, causing the run to stop. Compare the heights of the columns. What might you say about the relationship between these heights? Repeat this experiment and see whether any general pattern recurs.

Press SETUP then GO and wait until the run ends. Now press ORGANIZE. What happened? Press DISORGANIZE and then ORGANIZE, and watch the effect on the outcomes in the columns. Now press PAINT under each of the ORGANIZE and ORGANIZE conditions. When the outcomes are both organized and painted, what can you say about the relation among the sizes of the colored groups? That is, over repeated trials, is there any pattern in the relative sizes of these groups, or is it completely arbitrary?

Set KEEP-REPEATS? to 'Off' and STOP-AT-ALL-FOUND? to On. When you press GO the model will keep running until it has randomly sampled all of the unique outcomes in the sample space. How many samples, on average, are required in order to fill the entire sample space? Does this number change according to the settings of the probability? For example, if the probability is set at 80%, does it take as many trials to fill the sample space as compared to a setting of 50%? If not, why not? How about the extreme cases of 0% or 100%?

## EXTENDING THE MODEL

Add monitors and/or graphs to explore aspects of the experiments that are difficult to see in the current version. For instance: o How many trials does it take for the experiment to produce an all-green 4-block? How is this dependent on the various settings?

o Are there more samples with an even number of green squares as compared to those with an odd number of green squares?

o How symmetrical is the set of stalagmites? How would you define "symmetry?" How would you quantify and display its changes over time?

## NETLOGO FEATURES

## RELATED MODELS

Some of the other ProbLab (curricular) models, including SAMPLER-a HubNet Participatory Simulation-feature related visuals and activities. In Stochastic Patchwork and especially in Sample Stalagmite you will see larger blocks, such as an arrays of green and blue squares. In the Stochastic Patchwork model and especially in 9-Blocks model, we see frequency distribution histograms. These histograms compare in interesting ways with the shape of the stalagmites in this model.

## CREDITS AND REFERENCES

Thanks to Dor Abrahamson for the design and of this model as well as the implementation of the original model. Thanks to Josh Unterman for implementing the advanced procedures. This model is a part of the ProbLab Curriculum, originally under development at Northwestern's Center for Connected Learning and Computer-Based Modeling and now also at the Embodied Design Research Laboratory at UC Berkeley. For more information about ProbLab, please refer to http://ccl.northwestern.edu/curriculum/ProbLab/.

## HOW TO CITE

If you mention this model in a publication, we ask that you include these citations for the model itself and for the NetLogo software:

- Abrahamson, D. and Wilensky, U. (2006). NetLogo 4 Block Stalagmites model. http://ccl.northwestern.edu/netlogo/models/4BlockStalagmites. Center for Connected Learning and Computer-Based Modeling, Northwestern Institute on Complex Systems, Northwestern University, Evanston, IL.
- Wilensky, U. (1999). NetLogo. http://ccl.northwestern.edu/netlogo/. Center for Connected Learning and Computer-Based Modeling, Northwestern Institute on Complex Systems, Northwestern University, Evanston, IL.

## COPYRIGHT AND LICENSE

Copyright 2006 Uri Wilensky.

This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/3.0/ or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA.

Commercial licenses are also available. To inquire about commercial licenses, please contact Uri Wilensky at uri@northwestern.edu.

## Comments and Questions

globals [ ;; colors of the background of the view and of the two possible colors in samples background-color column-color target-color other-color num-columns ;; how many columns there are in the graphics-window histogram num-target-color ;; how many of the squares (patches) in the sample are of the favored color sample-right-xcor ;; the x-coordinate of the moving sample (not the magnified sample) sample-location-patch-agentset ;; patches where the moving sample will sprout token-sample-dude ;; bottom-left turtle in the moving sample stop-all? ;; Boolean variable for stopping the experiment side ;; tells how big the side of the block is--2 for 4-block popping? ] patches-own [ column ] breed [ column-kids column-kid ] column-kids-own [ binomial-coefficient sample-list ] breed [ sample-dudes sample-dude ] sample-dudes-own [ distance-for-jump ] breed [ baby-dudes baby-dude ] ;; jwu - instead of having the sample-dudes stamp, they're going to create ;; a sample-organizer. the sample-organizers are going to have a better idea ;; of which specific sample the sample-dudes represented. breed [ sample-organizers ] sample-organizers-own [ sample-values original-pycor ] to Go-org if stop-all? [stop] super-go organize-results end to super-go if stop-all? [stop] ifelse popping? [ no-display unpop go pop display ] [ go display ] end ;; the popping? global controls the popping visuals to pop set popping? true recolor-columns end to unpop set popping? false recolor-columns end ;; jwu - different color for each sample-summary-value to-report popping-color ; sample-organizers procedure report 15 + ((sample-summary-value * 10) mod 120) end to-report sample-summary-value ; sample-organizers reporter let result 0 let power-of-two 3 foreach sample-values [ if ? = 1 [ set result result + 2 ^ power-of-two ] set power-of-two power-of-two - 1 ] report result end to-report sample-patches ; sample-organizers procedure let result [] foreach n-values side [?][ let i ? foreach n-values side [?] [ set result lput patch-at ? (- i) result ] ] report result end to display-sample ; sample-organizers procedure let patch-popping-color popping-color (foreach sample-values sample-patches [ ask ?2 [ ifelse popping? [ set pcolor patch-popping-color ] [ ifelse ?1 = 1 [ set pcolor target-color ] [ set pcolor other-color ] ] ] ]) end to recolor-columns reset-column-colors ask sample-organizers [ display-sample ] end to reset-column-colors ; column-kids procedure ask column-kids [ ask patches with [ pcolor != black and (pxcor = [pxcor] of myself or pxcor = [pxcor] of myself - 1) ] [ set pcolor [pcolor] of myself ] ] end to make-a-sample-organizer ; sample-dudes procedure hatch-sample-organizers 1 [ ht set sample-values map [ ifelse-value ([color] of ? = target-color) [1] [0] ] sorted-sample-dudes display-sample ] end to organize-results ask sample-organizers [ if original-pycor = 0 [ set original-pycor pycor ] ] ask column-kids [ organize-column ] recolor-columns end to organize-column ; column-kids procedure let column-organizers sample-organizers with [ pxcor + 1 = [pxcor] of myself ] (foreach sort-by [ [ sample-summary-value ] of ?1 <= [ sample-summary-value ] of ?2 ] [self] of column-organizers sort [ pycor ] of column-organizers [ ask ?1 [set ycor ?2] ]) end to disorganize-results ask sample-organizers [ set ycor original-pycor ] recolor-columns end to startup ;set total-samples end ;; This procedure colors the view, divides patches into columns of equal length ( plus a single partition column), ;; and numbers these columns, beginning from the left, 0, 1, 2, 3, etc. to setup clear-all set-default-shape turtles "square big" clear-output set background-color white - 1;black set column-color grey set target-color green set other-color blue set side 2 set popping? false ;; num-columns is how many columns (bars) there are in the graphics-window histogram. ;; We need side ^ 2 + 1 columns in a histogram. For example, 3-by-3 samples (9 patches) ;; have 10 -- that is, 3 ^ 2 + 1 -- different possible counts of target-color (0, 1, 2, 3, ...9). set num-columns ( side ^ 2 + 1) ;; determines the location of the sample array beginning one column to the left of the histogram set sample-right-xcor -1 * round ( ( num-columns / 2 ) * ( side + 1 ) ) ;; assigns each patch with a column number. Each column is as wide as the value set in the 'side' slider ask patches [ set pcolor background-color ;; The following both centers the columns and assigns a column number to each patch ;; We use "side + 1" and not just "side" so as to create an empty column between samples set column floor ( ( pxcor + ( ( num-columns * ( side + 1 ) ) / 2 ) ) / ( side + 1 ) ) if column < 0 or column >= num-columns [ set column -100 ] ] ;; leave one-patch strips between the columns empty ask patches with [ [column] of patch-at -1 0 != column ] [ set column -100 ;; so that they do not take part in commands that report relevant column numbers ] ;; colors the columns with two shades of some color, alternately ask patches [ if column != -100 [ ifelse int ( column / 2 ) = column / 2 [ set pcolor column-color ][ set pcolor column-color - 1 ] ] ] ;; This draws the base-line and creates a sample-kids turtle at the base of each column ask patches with [ ( pycor = -1 * max-pycor + side + 3 ) and ;; The base line is several patches above the column labels. ( column != -100 ) ] [ set pcolor black if [column] of patch-at -1 0 != column ;; find the leftmost patch in the column... [ ask patch (pxcor + side - 1) ;; ...then move over to the right of the column ( -1 * max-pycor + 1 ) [ set plabel column ] ask patch (pxcor + floor (side / 2)) ;; ...then move over to the middle of the column ( -1 * max-pycor + 1 ) [ sprout 1 [ hide-turtle set color pcolor set breed column-kids set sample-list [] ;; each column-kid knows how many different combinations his column has set binomial-coefficient item column binomrow (num-columns - 1) ] ] ] ] set stop-all? false set num-target-color false reset-ticks end to go if stop-all? [stop] ;; The model keeps track of which different combinations have been discovered. Each ;; column-kid reports whether or not its column has all the possible combinations. When bound? is true, ;; a report from ALL column-kids that their columns are full will stop the run. if stop-at-all-found? [if count column-kids with [length remove-duplicates sample-list = binomial-coefficient] = count column-kids [stop]] sample ifelse magnify? [ magnify-on-side ] [ ask baby-dudes [ die ] ] drop-in-bin tick if plot? [ histogram-blocks ] end ;; This procedure creates a square sample of dimensions side-times-side, e.g., 3-by-3, ;; located to the left of the columns. Each patch in this sample sprouts a turtle. ;; The color of the sample-dudes in this square are either target-color or other-color, ;; based on a random algorithm (see below) to sample ;; creates a square agentset of as many sample-dudes as determined by the 'side' slider, ;; positioning these sample sample-dudes at the top of the screen and to the left of the histogram columns set sample-location-patch-agentset patches with [ ( pxcor <= sample-right-xcor ) and ( pxcor > sample-right-xcor - side ) and ( pycor > ( max-pycor - side ) ) ] foreach sort sample-location-patch-agentset [ ask ? [ sprout 1 [ ht set breed sample-dudes setxy pxcor pycor ;; Each turtle in the sample area chooses randomly between the target-color and the other color. ;; The higher you have set the probability slider, the higher the chance the turtle will get the target color ifelse random 100 < probability-to-be-target-color [ set color target-color ] [ set color other-color ] st ] ] ] ;; num-target-color reports how many sample-dudes in the random sample are of the target color set num-target-color count sample-dudes with [ color = target-color ] end ;; procedure in which the sample turtles create an enlarged duplicate on the left side of the screen and this enlarged sample makes ;; a large duplicate of the sample. This helps users see the samples that may otherwise be too small to see comfortably. ;; Samples are small for side = 3 because we want the entire sample space to fit into the view. to magnify-on-side ask baby-dudes [die] ;; clears the way for new magnified sample ask sample-dudes [ hatch-a-big-baby ] end to hatch-a-big-baby ;; sample-dudes procedure hatch 1 [ set breed baby-dudes set size 12 * ( 8 - side ) / side ;; This is tricky. We want to center the new turtles vertically, and ;; put them to the right of all the columns, with a little space between. ;; The code is complicated, because it is supposed to work for other x-blocks, too setxy -1 * ( size * .35 ) + ( sample-right-xcor + ( .35 * size * ( xcor - sample-right-xcor ) ) ) ( ( side - 1 ) * size / 2 ) + ( .35 * size * ( ycor + min-pycor ) ) set size size * .33 ] end ;; This procedure moves the random sample sideways to its column and then down above other previous samples ;; in that column. to drop-in-bin find-your-column descend end ;; The random sample moves to the right until it is in its correct column, that is, until it is in the column ;; that collects samples which have exactly as many sample-dudes of the target color as this sample has. ;; The rationale is that the as long as the sample is not in its column, it keeps moving sideways. ;; So, if the sample has 9 sample-dudes (3-by-3) and is moving sideways, but 6 of them are not yet in their correct column, ;; the sample keeps moving. When all of the 9 sample-dudes are the sample's correct column, this procedure stops. to find-your-column ask sample-dudes [ set heading 90 ] while [ count sample-dudes with [ column = num-target-color ] != side ^ 2 ] [ ask sample-dudes [ fd 1 ] ] end ;; Moves the sample downwards along the column until it is either on the base line or ;; exactly over another sample in that column. to descend let lowest-in-sample min [ pycor ] of sample-dudes ask sample-dudes [ set heading 180 ] ;; The lowest row in the square sample is in charge of checking whether or not the sample has arrived all the way down ;; In order to determine who this row is -- as the samples keeps moving down -- we find a turtle with the lowest y coordinate ;; checks whether the row directly below the sample's lowest row is available to keep moving down set token-sample-dude one-of sample-dudes with [ pycor = lowest-in-sample ] while [ ( [ [pcolor] of patch-at 0 -2 ] of token-sample-dude ) != black and ( [ [pcolor] of patch-at 0 -2 ] of token-sample-dude ) != target-color and ( [ [pcolor] of patch-at 0 -2 ] of token-sample-dude ) != other-color ] [ ;; As in find-your-column, shift the sample one row down ask sample-dudes [ fd 1 ] ;; Instead of establishing again the lowest row in the sample, the y coordinate of the row ;; gets smaller by 1 because the sample is now one row lower than when it started this 'while' procedure set lowest-in-sample ( lowest-in-sample - 1 ) ] ;; Once sample-dudes have reached as low down in the column as they can go (they are on top of either the base line ;; or a previous sample) they might color the patch with their own color before they "die." finish-off ;; If the column has been stacked up so far that it is near the top of the screen, the whole supra-procedure stops ;; and so the experiment ends if max-pycor - lowest-in-sample < ( side + 1 ) [ set stop-all? true ] end ;; we can't sort by who number, because who numbers get reused in weird ways, it seems. to-report sorted-sample-dudes ;report sort sample-dudes report sort-by [ (([pxcor] of ?1 < [pxcor] of ?2) and ([pycor] of ?1 = [pycor] of ?2)) or (([pycor] of ?1 > [pycor] of ?2)) ] sample-dudes end to finish-off ;; creates local list of the colors of this specific sample, for instance the color combination of a 9-square, ;; beginning from its top-left corner and running to the right and then taking the next row and so on ;; might be "green green red green red green" ;; jwu - need to use map and sort instead of values-from cause of ;; the new randomized agentsets in 3.1pre2 let sample-color-combination map [ [color] of ? ] sorted-sample-dudes ;; determines which turtle lives at the bottom of the column where the sample is let this-column-kid one-of column-kids with [ column = [ column ] of token-sample-dude ] ;; make the upper left sample-dude create a sample-organizer let the-sample-sample-dude max-one-of (sample-dudes with-min [ pxcor ]) [ pycor ] ;; accepts to list only new samples and makes a previously encountered sample if keep-duplicates? is on ifelse not member? sample-color-combination [sample-list] of this-column-kid [ ask the-sample-sample-dude [ make-a-sample-organizer ] ask sample-dudes [ die ] ] [ ifelse keep-repeats? [ ask the-sample-sample-dude [ make-a-sample-organizer ] ask sample-dudes [ die ] ] [ ask sample-dudes [ die ] ] ] ask this-column-kid [ set sample-list fput sample-color-combination sample-list ] end ;; procedure for calculating the row of coefficients ;; column-kids needs their coefficient so as to judge if their column has all the possible different combinations to-report binomrow [n] if n = 0 [ report [1] ] let prevrow binomrow (n - 1) report (map [?1 + ?2] fput 0 prevrow lput 0 prevrow) end ;;if the model has been run, report the number of patches ;;with the target-color -- otherwise, display "--" to-report #-target-color ifelse ticks != 0 [ report count patches with [ pcolor = target-color ] ] [ report "--" ] end ;;if has been run, report the number of patches ;;with the other-color -- otherwise, display "--" to-report #-other-color ifelse ticks != 0 [ report count patches with [ pcolor = other-color ] ] [ report "--" ] end ;; reports the proportion of the sample space that has been generated up to now to-report %-full ifelse samples-found = 0 [ report precision 0 0 ] [ report precision ( samples-found / ( 2 ^ ( side ^ 2 ) ) ) 3 ] end to-report samples-found report sum [ length remove-duplicates sample-list ] of column-kids end to-report total-samples-to-find report precision ( 2 ^ ( side ^ 2 ) ) 0 end to histogram-blocks let sample-value-summaries [ sample-summary-value ] of sample-organizers let possible-values n-values (2 ^ (side * side)) [?] let results [] foreach possible-values [ let i ? set results lput length filter [? = i] sample-value-summaries results ] set-current-plot "Events by Number of Outcomes" let max-results max results if mean results > 0 [ set-plot-x-range 0 (max-results + 1) ] set-current-plot-pen "Histogram" histogram results set-current-plot-pen "Mean" let mean-results mean results plot-pen-reset plotxy mean-results 0 plotxy mean-results plot-y-max end ; Copyright 2006 Uri Wilensky. ; See Info tab for full copyright and license.

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