TIMS Laboratory Investigations
Frequency Distribution
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(ISBN 0-7872-4025-7, primary, intermediate, & middle): This is a generic frequency distribution investigation that can be modified. It requires two quantitative variables and a bar graph.
Pockets
(ISBN 0-7872-4026-5, primary & intermediate): Pockets is our first
frequency distribution investigation. Consequently, the introduction
in the TLD contains an overview of frequency distribution’s multiple
sampling and probability. If a scientist wants to measure the
height of an adult howler monkey, he or she would not measure
just one. They would measure as many as possible and study the
distribution of heights. In this investigation we want to know
how many pockets a child has. Clearly, there is no one answer
for a class of 30 children. But by determining how many children
have two pockets, three pockets…ten pockets, we can find the most
likely number of pockets, the range or spread in the number of
pockets, the chance of having more or less than a certain number
of pockets, and predict how the distribution of pockets might
change if we had our coats on, or went to the beach, or had a
bigger class. Pockets involves collecting class data, making tallies,
doing lots of adding and subtracting, comparing numbers as to
greater or less than, and making and reading bar graphs. Probability
is a way of life and so are pockets. Pockets is a good way to
start on the road to understanding more about both.
Blocks From School
(ISBN 0-7872-4027-3, primary): Blocks is our second frequency
distribution investigation. It deals with small whole numbers,
basic addition and subtraction, and the fundamental vocabulary
and ideas of probability. For a classroom, we determine how many
blocks each child lives from school. We turn that raw data into
a frequency distribution where the number of blocks is the manipulated
variable and the number of children is the responding variable.
Clearly, all of the children do not live the same distance from
school, so we have a classic situation for a probability analysis
of the data to work with: What is the most likely number of blocks
from school? What is the range of blocks? What is the likelihood
of living more than five blocks from school? Some questions involve
more abstract thinking, like pretending a new student is coming
to class or imagining what the data would be like in a different
neighborhood. But it is just these abstractions, coupled with
the concrete operations like addition, which will develop the
children’s thinking skills.
Counting Kids
(ISBN 0-7872-4028-1, primary & intermediate): Throughout the curriculum
we have the children learn more and more about themselves and
how they fit into the world around them. We have already investigated
something about what they wear in Pockets and how far they live
from school in Blocks From School. In Counting Kids they will
look at the size of their families, and see how these sizes compare
among their classmates. Here, as in Pockets and Blocks, we will
stick to the basic ideas and vocabulary of probability, work with
small whole numbers, and use basic addition, subtraction, and
division facts. The data we want to collect are the number of
children in a family for each child in the classroom. Since all
of the children do not have the same size families, we have a
wonderful opportunity to use probability to analyze the data.
We ask such questions as: What is the most likely number of kids
in a family? What is the range of kids? Is a family likely to
have five or more kids? We conclude the lab with two class projects.
In the first we ask the children to find out what their bar graph
looks like when they include data from other classes at the same
grade, and in the second we ask the children to become real investigators
and determine if the frequency distribution depends upon another
grade, another location, or another generation. Being close to
home does not mean we cannot range far afield. Counting Kids shows
us how to do that.
High, Wide, and Handsome
(ISBN 0-7872-4029-X, primary & intermediate): One of the wonders
of nature is its diversity. No two people look exactly alike.
Some of us are tall or short; some of us are fat or thin. Indeed,
the range of height in an elementary school classroom is astonishing.
In this investigation we begin to explore the idea of diversity
in nature and tie it in with the concept of probability. We are
going to study the heights and arm spans of the children in your
class. There will be two frequency distribution graphs, one for
each variable. We are going to find the most likely height and
arm span, the range of each, and see how they compare to other
living things—like corn or sixth-graders. The children will have
the opportunity to make predictions based on the data sample.
We will also work on counting by fives, on addition and subtraction,
on rounding up and down, using the concept of half, and learning
about adding on and subtracting off. In order to keep the numbers
less than 100, we will use a chain of links as our measuring device.
But this will be the last time we use links. From now on it’s
centimeters all the way.
Pets I
(ISBN 0-7872-4030-3, primary & intermediate): To study animal
populations, biologists make frequency distributions of animals
around some natural unit of study such as small lakes and ponds
in the midwestern wetlands and water holes in Africa. In Pets
I and Pets II the unit of study is the children’s household, and
the animals studied are their pets. In Part I the children look
at the frequency distribution of the total number of pets in a
household. No distinction is made between the types of pets. We
ask the children about the most likely number of pets, the chance
of having a certain number of pets in a household; we ask how
the frequency distribution of pets might vary with grade, with
the size of their family, and from discussions with their grandparents,
how the distribution might change with time. We conclude with
an example of duck populations in a wetland area.
Pets II
(ISBN 0-7872-4031-1, primary & intermediate): A biologist is not
only interested in the number of animals at a water hole, he or
she is also interested in the different kinds of animals. Pets
II tries to simulate this situation by having the children make
a frequency distribution of the number of different kinds of pets
in a household. Then they make separate frequency distributions
for each kind of pet (dogs, cats, fish, etc.). The children learn
about the most and least likely number of kinds of pets in a household,
the most popular kind of pet, the total number of dogs and cats,
etc.; and, by working with their grandparents, they see if the
frequency distribution of the number of kinds of pets in a household
has changed over time.
How Long Are Names?
(ISBN 0-7872-4032-X, primary & intermediate): What is more personal
than your name? What variables are related to your name? In this
investigation the children explore names by determining the frequency
distribution of the number of letters in their first and last
names. Since there will be a distribution in the number of letters,
the children can attach the idea of which number of letters is
most likely, look at the range of numbers of letters, and also
get involved with the addition and subtraction of whole numbers.
We continue to learn about graphing and develop the interpretive
skill of the children. To find out how close the class distribution
represents the real world, each child randomly collects names
from their local phone book and compares their frequency distribution.
The children then work together to combine all their individual
phone book data into a grand frequency distribution and compare
this with their more limited class data. Finally, we ask the children
to collect data from relatives. Will the distribution be narrower
or will it peak at the same spot as the class data? Their class
data, the phone book data, data from relatives—all give the children
a chance to work in a non-textbook environment and thus become
more like real scientists and mathematicians.
A Handful of Beans
(ISBN 0-7872-4033-8, intermediate): We introduce a new term, “median,”
to our discussion of probability and look more closely at how
we can effect the experimental probability of something happening
by controlling some of the “hidden” variables. In Part I each
child takes one handful of identical beans from a container, and
the class makes a frequency distribution of the number of beans
per handful, B, vs. the number of children. In Part II, each child
makes 32 grabs and plots a frequency distribution of B vs. the
number of grabs. The children compare the two investigations by
looking at the ranges and medians of the two distributions, and
see how controlling hand size affects their data. Using data from
Part II they are challenged to determine whether to get involved
with various “betting” situations. We conclude with the calculations
of experimental probabilities and an introduction of the idea
of normalizing their data to 100 trials and hence to the word
percent.
Rolling One Die
(ISBN 0-7872-4034-6, intermediate): Investigations that deal with
people, plants, and animals have many external variables that
are beyond our control. These variables interplay with chance
to produce a given frequency distribution. There is a class of
investigations, however, where we can control all the external
variables and leave the rest to chance. One such investigation
is Rolling One Die, the first in a series of labs studying the
concepts of probability theory. The children roll one die many
times and determine the frequency distribution of the face number
F. They calculate the theoretical probability of rolling a particular
value of F. By combining 3 sets of 20 rolls into 60 and ten sets
of 60 rolls to obtain a grand total of 600 rolls, the children
see that the experimental probability approaches the theoretical
probability as the number of rolls increases. The children calculate
the experimental probability for various combinations of face
numbers and even do some friendly “betting” on various outcomes
of rolling one die.
Spinners
(ISBN 0-7872-4035-4, intermediate): For a biologist, the chance
of finding a particular plant or animal is usually increased by
searching a larger area. Astronomers, hunting for supernovas,
can increase their chance of finding them by looking at a bigger
slice of the sky. In that spirit, this lab deals with the problems
of increasing or decreasing the chance of finding “something”
by increasing or decreasing the area one searches. In Experiment
1 a circle is divided into 4 equal areas; in Experiment 2 the
areas are in the ratio of 4:2:1:1. In each experiment the children
spin a spinner 100 times and determine how often it lands in each
“area.” They compare their initial predictions on how the 100
spins should be divided with the experimental results. Students
merge their data and compare theoretical predictions with experimental
results. The children are challenged to predict the results for
new area combinations and to study given results in order to determine
how the circle is divided. After determining the probability of
the spinner landing in a particular area, we conclude with an
example of biologists looking for spider monkeys in different
areas of the rain forest.
Flipping 3 Coins
(ISBN 0-7872-4036-2, intermediate): The children are again asked
to calculate the theoretical probability ahead of time, but with
a new twist. By flipping three coins, the children have to take
into account the possible permutations, or changes in order, of
the heads and tails. The children predict how many times they
will flip 0 heads, 1 head, 2 heads, or 3 heads in 80 tosses, and
then carry out the investigation where the manipulated variable
is the number of heads and the responding the number of tosses.
The children compare theory to experiment, and then collect data
from four other groups, and once again see that theory and experiment
converge as the data set increases in size. To make these comparisons,
the children calculate percent differences. In a take-home exercise
the children work out the probabilities and carry out the experiment
for Flipping 2 Coins.
Rolling 2 Dice
(ISBN 0-7872-4037-0, intermediate & middle): This lab introduces
the children to the idea of partitions in probability theory,
that is, that rolling a 4 can mean the dice may come up 3 + 1
or 2 + 2. That, along with the permutation of each partition,
allows the students to predict the relative frequency of rolling
a 5, 6, 10, etc. The children carry out the investigation for
108 rolls and then compare their experimental results with theory
by calculating percent differences. The children then collect
data from 4 other groups and see if theory and experiment converge.
They find ranges, medians, and calculate experimental probabilities.
We conclude the lab with two games that help develop the notion
of “odds.” The lab emphasizes ratios and proportional reasoning.
Sibling Pairs
(ISBN 0-7872-4038-9, intermediate & middle): The children collect
data on the gender of the two oldest siblings in their family.
They make a frequency distribution of sibling pairs, that is,
GG, BG, and BB, and calculate the chance that a sibling will be
a boy or girl or that both are girls, both are boys, or the combination
is boy-girl. After a discussion of X and Y chromosomes and their
role in determining the gender of a child, the students find the
theoretical probability a sibling will be a boy or girl or that
there will be GG, BB, or BG sibling pairs. The children compare
theory and experiment by calculating percent differences. A comparison
is made with the data from their take-home experiment Flipping
2 Coins.
What Are the Odds?
(ISBN 0-7872-4039-7, intermediate & middle): What Are the Odds?
is another in our series of frequency distribution investigations
that explore the concept of probability. Besides learning how
to set odds, we shall express probability as a decimal and continue
to work on ratios and equivalent fractions. The investigation
centers around drawing beans out of a container using a spoon.
The manipulated variable is the number of beans pulled each try,
and the responding variable is the number of tries. By making
many tries, the children establish the experimental probability
distribution, and then discuss how the fixed variables (size of
bean, type and size of utensil) might change the frequency distribution.
The children are challenged to answer questions on range, median,
and the probability of picking various combinations of beans.
Using the experimental probability distribution, we have the children
play a game. The “player” chooses how many beans he or she will
pull in one try. The “house” offers break-even odds (carefully
figured), and washers will be exchanged as currency. After 20
tries the winner is the person with the most washers. If each
child does the correct calculations, then everyone will be a winner.
Lives of Soap Bubbles and People
(ISBN 0-7872-4040-0, middle): An important problem in biology
is trying to understand how long plants and animals live and why
they die. For humans, the oldest records of life spans have been
gathered from birth and death records in European churches. We
can study more recent human populations by using local cemeteries.
Biologists look for patterns in life span data by making a frequency
distribution where the manipulated variable is the age at death,
and the responding variable is the number that die at that age.
In this investigation, the children substitute soap bubbles for
people and measure lifetimes of 20 soap bubbles using a stopwatch.
Because we deal with fractional lifetimes, the children learn
how to make a histogram where they lump the data between whole
numbers as one thick bar between numbers. From their data the
children determine the most likely lifetime of a bubble, the probability
that a bubble will live longer than a certain time, and what might
affect the lifetime of a soap bubble. 
The students then collect data for the whole class to obtain a
smoother distribution and better prediction. The real biology
begins when the children compare their data to cemetery data before
1910 and after 1950. There are striking differences between these
two distributions, but a stunning agreement with one of these
distributions and the soap bubble data. Which one and why? Try
Lives of Soap Bubbles and People and find out.
Germinating Seeds and Base Hits
(ISBN 0-7872-4041-9, middle): The children have seen that for
a single die the probability of rolling a 1 is the same as rolling
a 6. But what if the probability of rolling a 1 were different
from rolling a 6? It is this deeper understanding of probability
that we want to explore in this investigation. By carrying out
two experiments, we shall help the children discover that if we
have two independent events that are random, occurring with probability
P1 and P2, respectively, then the probability of both occurring simultaneously
is P1 x P2, and if 3 events, then P1 x P2 x P3, etc. In the first experiment each child plants 4 seeds. By collecting
class data, they determine the frequency distribution of germinating
seeds per container. In the second experiment, the children gather
data on the number of base hits that about 100 major league players
get in four at bats and determine the frequency distribution of
hits in four at bats. Although seemingly two very different situations,
with very different results, we shall, step by step, show the
children how they can predict these results by using the probability
ideas that they have previously learned along with our new idea
of P1 x P2 x P3 x P4. We think the investigation will be a big hit.
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Copyright © 1997 by Kendall/Hunt Publishing Company
Copyright © 1999 Institute for Mathematics and Science Education.
All rights reserved.
UIC—University of Illinois at Chicago