 classroom activities data warmups warmup 1 heads or tails  the goal of this activity is to demonstrate that there is natural variability when we make measurements and observations the research question for this activity is  is there a difference between canadian and american coins in a coin toss  materials for each group or student you will need 1 canadian and 1 us coin nickels work great paper and pencils or overhead transparencies and markers doing the activity have each group flip the canadian coin 10 times then the us coin 10 times record the number of heads resulting from the tosses the teacher compiles the data on an overhead or white board go around the room asking each group how many heads where thrown for each type of coin calculate the mean for each coin type an example of the data compilation follows group canadian heads us heads a 6 5 b 4 6 c 5 5 d 7 8 e 3 4 mean 50 56 pose the question is 56 enough of a difference from 50 that we would think it s a meaningful difference due for example to differences in how the coins are made or just from random variability or chance to delve into the question repeat the coin toss a second time record the results and calculate the means are they the same or different if you tossed coins again would you get the same or different values why the differences between the two types of coins and among the different trials are most likely due to chance there are just natural differences in coin tosses that make the tosses different each time discuss some sources of variability with your students tossing style if a coin is worn dirt on one side etc to look at the variability more closely plot the data on the board or overhead there are two types of plots we ll use to look at variability among groups of things a dotdensity plot a type of histogram and a box and whisker plot dot density this type of graph shows how many times a certain value appears in your data set use different symbols or colors to denote the us versus canadian toss results for the example coin toss data above it would look like this where open circles represent canadian coin tosses and filled circles are us coin tosses this graph is like a histogram it s counting how many times you got each value so for this data set 3 heads came up once 4 heads came up twice etc you can also do this more kinetically drawing the number line scale on the board and giving students postits again colored by coin type so they can stick their values on the board and make a stack above each number imagine that each dot above is a postit use this type of graph to begin seeing the mean or balance point where are most of the points is there a big bump near a certain value in the dot density plot above most of the data are centered between 4 and 6 with the greatest number of points at 5 the 3 and the 8 are less common values if you calculate the mean see above it s near 5 which makes sense when you see where the data mostly lie talk about variability do the two sets of measurements canadian and us overlap a little or a lot if you were to do another trial what would you predict a second useful type of graph is a box and whisker plot at right sometimes just called a boxplot the boxplot groups the canadian and us coins on the x horizontal axis and the height of the various parts of the box represent the y values number of times heads came up more specifically the line across the middle of each box here represents the median or middle value no calculator needed just put the measurements in order and see which one is in the middle for canadian coin tosses you had 34567 so 5 is in the middle then you want to plot the middle 50 of the data called the interquartile range between the two middle quarters if the middle median is 5 then 25 of the values are from 45 and 25 are from 56 the whiskers represent the outer quarters of data technically most statistics folks will tell you it s the 95th and 5th percentiles so in our canadian toss example the whiskers go to 7 and 3 in the case of the us coins plotted above the median and the lower interquartile range happen to be the same because 5 appears twice so you just can t see the line across the middle of the box because it s the same as the line at the bottom of the box we use a boxplot because we can compare the data overlap again in order to talk about variability and whether the two groups have meaningfully different data if half of the canadian data the interquartile range are shifted and don t overlap much at all with the interquartile range of the us data then we d think we might have a meaningful difference if as in this case the boxes mostly overlap then we may not have a difference that means much warmup 2 point up or point down  the goal of this activity is to build on warmup 1 with a more datarich experiment materials for each group or student you will need 20 thumbtacks not pushpins but flathead thumbtacks 20 upholstery tacks get them at the hardware store 2 paper cups paper and pencils or overhead transparencies and markers doing the activity first have teams inspect the tacks and make observations what s the same about them what s different then ask them when tossed like dice which ones do you think are more likely to land with the point facing up have them tie their hypotheses to an observation eg we think the upholstery tacks will land pointup more often because they have a heavier head each group then throws a cupful of tacks they can decide on the method whether each type should be thrown separately or not and counts the number of points facing upward repeat 10 times for each type of tack each group should record the number of each pointup tacks for each throw and each tack type have each group plot their data as a box plot ask was there a difference between upholstery tacks and thumbtacks compile all the data into one graph for the class were the results the same for all the groups what does the variability look like is the difference meaningful look at the interquartile range and the overlap compared between the two types of tacks  warmup 3 it s a stretch the goal of this activity is to become familiar with scatterplots the graph type used when looking at correlation are two things related  materials for each group or student you will need a stick 1x1 lumber about 3 long with two cut tied together rubber bands attached to one end like a fishing pole and a binder clip attached to the loose end of the rubber band a ruler 8 washers of different sizes and weights about 3060 grams each it s helpful to paint washers a few different colors reds greens blues and number them 18 a balance or other way to weigh the washers paper and pencils or overhead transparencies and markers doing the activity have each group create a data table with washer color and a space for washer weight and stretch washer color weight grams stretch cm 1 green 45 12 2 blue 30 8 3 blue 55 18 etc weigh a washer then clip it to the rubber band and measure the amount the band stretches record the weight and stretch length in the table continue to weigh and measure stretch of all the washers have students make a graph of their data you can leave this fairly open ended to see if they ll get tripped up by the washer numbers or colors the goal is to put weight on the xaxis and stretch on the yaxis to see if weight and stretch are correlated once graphs are made have each group report out or present to the class was washer weight and stretch related how do the points fall on a straight line linear or are they curved they should be linear did the washer color or number matter they do not point out that it s important to figure out which pieces of data are useful here when the question is whether or not washer weight and stretch are related the other data don t matter  mercury in watersheds  