Lesson 3
Probability
Lesson Plan
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Math
1
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Time: |
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One
(or two) 50-minute class period(s), or part of one 90-minute
block (or a whole one)
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Goals: |
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Students
will review or discover the Triangle Inequality.
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Students
will extend the Pythagorean Theorem to determine the relationship
between a2, b2, and c2 for acute and
obtuse triangles. |
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Students
will investigate probability, including conditional probability, as it
relates to the types of triangles
formed by given side lengths.
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Students
will learn more about the Law of Large Numbers.
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NCTM Content &
Process Standards
Addressed: |
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Number
and Operations (develop an understanding of permutations and
combinations as counting techniques)
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Data
Analysis and Probability (use simulations to explore the
variability
of sample statistics from a known population; use sampling
distributions as the basis for informal inference; use simulations to
construct empirical probability distributions; understand the concept
of conditional probability)
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Problem
Solving (build new mathematical knowledge through problem solving)
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Connections
(recognize and use connections among mathematical ideas; understand how
mathematical ideas interconnect and build on one another to produce a
coherent whole)
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Representation
(create and use representations to organize, record, and communicate
mathematical ideas)
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GPS Content &
Process Standards
Addressed: |
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MM1G3b
(understand and use the triangle inequality)
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MM1D1a,b
(determine the number of outcomes related to a given event)
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MM1D2c
(calculate
conditional probabilities)
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MM1P1b
(solve
problems that arise in mathematics)
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MM1P4a
(recognize
and use connections among mathematical ideas)
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MM1P5a,b
(represent mathematics in multiple ways)
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Supplies and Resources:
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3
sets of five balls or notecards
labeled 1 through 5
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3
bags to hold the balls (optional - could hold notecards upside down
instead)
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Uncooked
spaghetti, straws, compasses (optional - manipulatives for Triangle
Inequality activity)
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Rulers
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At
least 22 blank notecards
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Two
Fathom files (see files)
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Assessment: |
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Write-up
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Overview: |
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Introduction
(25 minutes)
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Let
a student draw one ball out of each of the three bags. Discuss:
Do we really need three bags? How could we do the same type of
thing with only one bag? (Answer: draw three times, with
replacement.)
Ask students whether the three numbers drawn could be the side lengths
of a triangle. (Students may use spaghetti or straws cut to the
correct length, compass constructions, or other types of
reasoning.) How can you tell?
Generalize: If a, b, and c are three numbers (where c is the largest of
the three), what must be
true in order for the three numbers to be the side lengths of a
triangle? (This
will go in students' write-ups.)
Have students determine all
possible triplets of the numbers in the bag that could be the side
lengths of a triangle. (This
will go in students' write-ups.) Have students compare
their lists.
Discuss: Why is it important to be systematic when making your list?
If students argue about whether 1-3-3 should be the same as 3-1-3, say
that for now it is convenient
to count them as the same.
Write
a "master list" on the board (see files
for correct master lists). Have students determine whether each
triplet forms an acute,
obtuse, or right triangle. Students can begin by working with
manipulatives, but then should be able to directly apply the following:
if a2 + b2 > c2, the triangle is
acute; <, the triangle is obtuse; =, the triangle is right.
Write these triplets on
notecards (e.g., one notecard will say 1-1-2) and place them in a
bag. Have a student draw a card from the bag.
Determine the theoretical probability of drawing an acute-, right-, or
obtuse-triangle-forming triplet from the bag. (This
will go in students' write-ups.)
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Triplets
Simulation (20 minutes)
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Run
the Fathom simulation once, sampling 10 pieces of data. Determine
experimental probabilities for acute, right, and obtuse. Compare
to theoretical probabilities. Repeat for 50, 100, 1000 pieces of
data.
Discuss: Why would the experimental probabilities be closer to the
theoretical probabilities when we sample more data? (Answer: Law
of Large Numbers)
Discuss: Do you think we calculated our theoretical probabilities
correctly?
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Wrap-Up (5
minutes)
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Why
wasn't our simulation data perfect? Discuss theoretical versus
experimental probability. (Students
will answer this question in their write-up.)
What happened as we collected larger data sets? Why? (Students will answer this question in their
write-up.)
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OneAtATime Simulation (45-50
minutes on the next day - optional) |
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Again,
have a student draw three numbers, with replacement. Using
yesterday's work, what is the probability that these three numbers form
any type of
triangle at all?
Do you think the probability of drawing an acute-, right-, or
obtuse-triangle-forming set of three numbers in this way would be the
same as the probabilities we found yesterday?
Run the OneAtATime Simulation. Ask students how many pieces of
data they want to collect and why. Calculate experimental
probabilities. Are these experimental probabilities close to the
theoretical probabilities we found yesterday? Why not?
This data includes non-triangles. Let's filter out all the
non-triangles. So given the
fact that the three numbers form a triangle, do you think the
probabilities would be the same?
Run the simulation again, with the filter on. Are these
experimental probabilities close to the theoretical probabilities we
found yesterday? Why not?
How many ways were there to draw, say, 2-3-4 yesterday? (Answer:
Only one.) How many ways are there to draw it today?
(Answer: 3! = 6.). So there are more
than 22 possible results of this experiment. How many are
there? (Answer: 65. Students need to calculate the
number of ways to get each triplet, then sum these numbers. See files for all possibilities.)
So what are our theoretical probabilities for this scenario (given that the three numbers form
the sides of a triangle)? (*) (This
will go in students' write-ups).
Now let's include all the non-triangles again. How many possible
outcomes are there now? What are the theoretical probabilities
for this scenario? What
is the relationship between your answer to this question and your
answer to (*)? (Answer: They are the probabilities in (*),
multiplied by the probability that a drawn triplet makes a
triangle.) Relate this fact to conditional probability. (This will go in students' write-ups).
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