Using their observations
from Lesson 1, students will determine how many acute, right, and
obtuse triangles can be formed with whole number side lengths less than
or equal to 5. They will compare their own answers to the
experimental probability of making each type of triangle, as given by
two different Fathom simulations: one, where known
triangle-making triplets (e.g., 2-3-3) are placed in a single “bag”;
the other, where three random integers from
1 to 5 are chosen from three “bags”. These
two simulations will give quite different probabilities, since it is
more likely that 2-3-4 will be chosen than 2-2-2 in the second
simulation (there are 3! ways to choose 2-3-4 from three bags, but only
one way to choose 2-2-2), whereas all triplets have equal probability
in the first simulation. Students can learn more about
probability and sampling, while reviewing their observations from
Lesson 1 and recalling (or learning) the Triangle Inequality.
Students will perform actual bag draws a few times for each of the two
simulation types before asking Fathom to simulate numerous draws.
In this way, students can gain a better understanding of what exactly
Fathom is simulating, and can relate the computer’s data to their real
world experience, thus making it easier to understand why, for example,
2-3-4 is a more likely triple than 2-2-2 in the second
simulation.
Fathom is necessary, however, because it generates large amounts of
data that would be unfeasible for students to generate in a short
amount of time. In addition, Fathom allows students to filter out
non triangle triples (in the second simulation) with a single
click. Students can learn more about the Law of Large Numbers, as
they watch the experimental probability “settle down” and
approach their calculated
theoretical probability as Fathom’s data table gets longer and longer.
