T870
Mathematics Success – Grade 7
LESSON 34: Uniform and Non-Uniform Probability Models
[OBJECTIVE]
The student will investigate chance processes and develop, use, and evaluate
probability models.
[PREREQUISITE SKILLS]
probability, relative frequency
[MATERIALS]
Student pages S444–S455
Number Cubes (1 per student pair)
Colored Pencils
Centimeter cubes (2 red, 2 blue, 3 green, 4 yellow per student pair)
[ESSENTIAL QUESTIONS]
1. Howcanauniformprobabilitymodelbeidentified?
2. Howcananon-uniformprobabilitymodelbeidentified?
3. Whydodiscrepanciesoccurbetweenprobabilitymodelsandrelativefrequencies?
[WORDS FOR WORD WALL]
probability model, uniform probability model, non-uniform probability model
[GROUPING]
Cooperative Pairs (CP), Whole Group (WG), Individual (I)
*For Cooperative Pairs (CP) activities, assign the roles of Partner A and Partner B to
students. This allows each student to be responsible for designated tasks within the
lesson.
[LEVELS OF TEACHER SUPPORT]
Modeling (M), Guided Practice (GP), Independent Practice (IP)
[MULTIPLE REPRESENTATIONS]
SOLVE, Verbal Description, Pictorial Representation, Concrete Representation,
Graphic Organizer
[WARM-UP](IP) S444 (Answers on T880.)
• HavestudentsturntoS444intheirbookstobegintheWarm-Up.Studentswill
be working with finding geometric probabilities. Have students complete the
problems and then review the answers as a whole group. {Verbal Description,
Pictorial Representation}
[HOMEWORK]
Take time to go over the homework from the previous night.
[LESSON][1–2 Days (1 day = 80 minutes) – M, GP, WG, CP, IP]
Mathematics Success – Grade 7
T871
LESSON 34: Uniform and Non-Uniform Probability Models
SOLVE Problem (CP, IP, WG)
S445 (Answers on T881.)
HavestudentsturntoS445intheirbooks.ThefirstproblemisaSOLVEproblem.
Students will complete the entire SOLVE problem as it relates to probability. The
SOLVE problem will then be extended to introduce uniform probability models.
Have students complete the SOLVE problem and then review the answers as a
whole group. {SOLVE, Verbal Description, Graphic Organizer}
Extend the SOLVE Problem – Uniform Probability Model
(M, GP, WG, CP) S445, S446 (Answers on T881, T882.)
M, WG, GP, CP:
HavestudentsturntoS446intheirbooks.Usethe
following activity to help students work with uniform
probability models to answer the questions. Students
will use information from the SOLVE problem on S445 to
complete the questions. {Verbal Description, SOLVE, Graphic
Organizer}
MODELING
Extend the SOLVE Problem – Uniform Probability Model
Step 1: Direct students’ attention to the question on S446 to extend the SOLVE
problem.
• PartnerA,whatconclusion can we make regarding the fairness of the
event that took place? (The event was fair because the probability of
1
choosing Michael’s name was 20, and the probability of choosing Maria’s
name was the same.) Record.
• Partner B, explain why you did not need to find the probability of
choosing each individual student in the classroom. (Each student’s
name was entered only once, therefore, their probabilities will all be
the same. They would all be formed with a numerator of 1 out of 20
total students’ names entered into the bag.) Record.
• PartnerA,whattermcanbeusedtodescribethemodelwhereyou
relyonchancetodecideanoutcomeusingprobability?(Aprobability
model) Record.
• Someschoolsrequirestudentstoadheretoastrictdresscodeupon
entering the building each day so that the staff can easily identify the
students by the similarities among their clothing.
• Partner B, do you know what these outfits are generally called?
(Uniforms)Record.
• Partner A, if the outcomes of a probability model are all the same,
whatdoyouthinkthistypeofmodeliscalled?(auniform probability
model) Record.
• PartnerB,doyouthinktheprobabilitymodeldiscussedintheSOLVE
problemisuniform?Explain.(Yes,themodelisuniformbecausethe
chance of choosing any individual student is exactly the same.) Record.
T872
Mathematics Success – Grade 7
LESSON 34: Uniform and Non-Uniform Probability Models
Explore Uniform Probability Models
(M, GP, CP, WG) S447, S448 (Answers on T883, T884.)
WG, M, GP, CP:
HavestudentsturntoS447intheirbooks.Usethe
following activity to help students work with uniform
probability models. Students will explore the probability
model and identify reasons for discrepancies between
the model and the actual outcome. Distribute a number
cube to each partner pair before this section. {Concrete
Representation, Verbal Description, Graphic Organizers}
MODELING
Explore Uniform Probability Models
Step 1: Direct students’ attention to the paragraph at the top of S447. They
will be using the same SOLVE problem from S445 to work with uniform
probability models.
• HavestudentpairslookatthetablebelowtheSOLVEproblem.
Michael
Maria
Josh
Steven
Christina
Devon
Allison
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• P
artnerA,howmanytimesdidtheteacherpullanamefromthebag?
(10) Record.
• PartnerB,explainhowyouknowtherewere10trials.(Thereare10
tally marks.)
• PartnerA,wereallofthestudents’namesdrawnthesamenumberof
times?(No.)Record.
• Partner B, what happened during the 10 trials? (All of the students
listed were chosen once, except for Michael. There are a total of 20
students in the class, which means that 13 of the names were not
even chosen once.) Record.
• Partner A, did Michael have more entries in the bag than the other
students?(No,eachstudenthadoneentry.)Record.
• Partner B, what type of probability model is used in this situation?
(uniform) Record.
• PartnerA,whydidMichael’snameappearmorefrequentlythanthe
other students? (In this case, only 10 names were picked, so the
maximum number of students chosen could be ten. However, in the
actual experiment the results are most likely not going to be identical to
thetheoreticalprobabilitiesthatareidentifiedinthemodel.)Record.
Mathematics Success – Grade 7
T873
LESSON 34: Uniform and Non-Uniform Probability Models
Step 2: Direct students’ attention to the table at the bottom of S447.
Michael
Maria
Josh
Steven
Christina
Devon
Allison
Lisa
Carly
Jennifer
Matthew
Mia
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• P
artnerA,howmanytimesdidtheteacherpullanamefromthebag
accordingtothechart?(20)Record.
• PartnerB,howmanytimeswouldweexpecteachstudent’snameto
bepulled?(once)Record.
• PartnerA,iftheprobabilityforthismodelis1outof20,whydoesthe
datainthetablenotrepresentthisexactly?(Again,allstudentshave
a fair chance, having only entered each name in the bag once, but the
actual outcome can be very different from the model.) Record.
*Teacher Note: Students may have alternate answers. The point is that they
understand that the theory of what should happen will rarely match the actual
outcomes of an experiment.
Step 3: Direct students’ attention to page S448 and give each student pair a fair
number cube.
• Have student pairs identify the probability of rolling each of the
numbers on the number cube.
Number
Probability
1
1
6
1
6
1
6
1
6
1
6
1
6
2
3
4
5
6
T874
Mathematics Success – Grade 7
LESSON 34: Uniform and Non-Uniform Probability Models
• P
artnerA,whatdoyounoticeabouttheprobabilityofrollingeachof
thenumbersinthetable?(Eachprobabilityis1outof6).Record.
• PartnerB,whattypeofprobabilitymodeldoesthisexperimentdisplay?
(Uniformprobability,becausealltheprobabilitiesareequal.)Record.
• PartnerA,ifanumbercubeisrolled6times,howmanytimesdoyou
expectittolandoneachofthenumberslisted?(Intheory,thecube
would land on each number once.) Record.
Step 4: Have students roll the number cube 6 times and record the data in the
table provided. Give students time to complete the experiment and record
the data.
• PartnerA,whatdoyounoticeabouttheoutcomeofactuallyrollingthe
numbercube?(Theoutcomemostlikelyshowsthatnotallnumbers
were rolled exactly once.) Record.
• Givestudentsanotherfewminutestorollthenumbercube20more
times and add tally marks to the table. Provide students with colored
pencils or a different color of pen so they can record the additional 20
rolls in the table with another color. This will give students a reference
point when reviewing the data.
• PartnerB,whatdoyounoticeabouttheoutcomesinthetablenow?
(As students roll more, they should notice that the amount of times
each number is rolled is spread more evenly.) If students do not
experience this, encourage them to roll several more times to see if
this proves the point.
*Teacher Note: It will be helpful to have your own data ready to have an example.
• A
fter the experiment is complete, discuss with students that this is
again a situation where we would expect an even distribution, but
the actual outcome rarely results in data that matches the probability
model.
Mathematics Success – Grade 7
T875
LESSON 34: Uniform and Non-Uniform Probability Models
Extend the SOLVE Problem – Non-Uniform Probability Model
(M, GP, WG, CP) S449, S450 (Answers on T885, T886.)
WG, M, GP, CP:
Have students turn to S449 in their books and complete
the SOLVE problem. Students will be analyzing probability
models again and will be introduced to a scenario where
the model is not uniform. {Verbal Description, Graphic
Organizer, SOLVE}
MODELING
Extend the SOLVE Problem – Uniform Probability Model
Step 1: Direct students’ attention to the SOLVE problem on S449. Have students
work in pairs to complete the problem and then go over the steps of
SOLVE as a whole group.
Step 2: Direct students’ attention to the top of S450.
• PartnerA,whatissimilarabouttheprobabilitymodelsintheSOLVE
problems on S449 and S445? (Both of the models involve placing
names in a bag and choosing a name.) Record.
• PartnerB,whatisdifferentabouttheprobabilitymodels?(Answers
mayvary:Thefirstmodelhad20differentstudents,whilethesecond
onlyhad5.Thefirstmodelenteredeachstudent’snameonlyonce,
whilethesecondenteredthefivestudents’namesadifferentnumber
of times.) Record.
• Partner A, what about the second probability model makes it not
uniform?(Notallofthestudentsinthismodelhadthesamechances
of being chosen. Three of the students had the same probability, but
one of the students had a better chance than the others, and another
student had less of a chance than the others.) Record.
• Partner B, what type of probability model is this? (non-uniform
probability model) Record.
• Thegoalhereisforstudentstorecognizethatwithuniformprobability
models, each outcome has the same chance of occurring, whereas
with non-uniform models, outcomes can have different chances of
occurring.
T876
Mathematics Success – Grade 7
LESSON 34: Uniform and Non-Uniform Probability Models
Exploring Non-Uniform Probability Models
(M, WG, GP, CP) S450, S451 (Answers on T886, T887.)
WG, M, GP, CP:
Have students continue with the activity on S450. Students
will be working with a second experiment using the nonuniform probability model. {Verbal Description, Graphic
Organizer}
MODELING
Exploring Non-Uniform Probability Models
Step 1: Direct students to the paragraph below Question 4 on S450.
• Havestudentpairsreadthescenariotogether.
• PartnerA,howmanytimesdidtheteacherpullanamefromthebag?
(10) Record.
• PartnerB,explainhowyouknewtherewere10trials.(Thereare10
tally marks.)
• Partner A, did all of the students get chosen the same number of
times?(Yes.)Record.
• PartnerB,howdoyouknowthateachstudentwaschosenthesame
numberoftimes?(Eachnamehastwotallymarks.)Record.
• Partner B, do the frequencies in the table match what you would
expect based on the table created in the SOLVE problem on S449?
(No.) Record.
• Partner A, explain your thinking. (Since Michael had 3 entries, we
would expect his name to have been picked more than the others.
Also, Julia’s name was only entered once, so we would expect her
name to be picked less than the others.) Record. (*Answers may
vary.)
• Partner B, why do you think the actual results do not mirror the
probabilitytable?(Theexperimentonlyhad10trials.Asmallnumber
of trials will not always accurately display the expected probabilities
of the model.) Record.
Step 2: Direct students’ attention to the top of S451. The same names are in the
bag as in the experiment on S450.
Name
Frequency
Maria
Michael
Jake
Josh
Julia
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Mathematics Success – Grade 7
T877
LESSON 34: Uniform and Non-Uniform Probability Models
• P
artnerA,howmanytimesdidtheteacherpullanamefromthebag
accordingtothechart?(20)Record.
• PartnerB,explainhowyouknowthis.(Thereare20tallymarks.)
• PartnerA,dotheresultsmatchtheexactprobabilitiesfromthemodel?
(No, the results do not match the exact probabilities because we can
immediately see that Josh, Jake, and Maria have different outcomes.)
Record.
• PartnerB,dotheresultslookmoreorlesslikethemodelcreatedinthe
SOLVEproblemonS449?(Theresultslookmoreliketheprobability
model. We expect Michael’s name to be chosen more than the other
names, and Julia’s to be chosen less than the other names. We also
expect Josh, Jake, and Maria’s names to be chosen about the same
number of times.) Record.
• Partner A, in both SOLVE problems and the number cube example,
what did you notice about the outcomes of the experiments with a
smallernumberoftrials?(Theresultsoftheactualoutcomesdidnot
match the expectation from the probability models.) Record.
• Partner B, in all of the experiments, what did you notice about the
outcomes with a larger number of trials? (While the results of the
actual outcomes are not identical to the probability model, when the
number of trials is increased, the actual outcomes begin to look more
like the probability model.) Record.
• PartnerA,whatconclusioncanyoudrawregardingtheseexperiments
andthenumberoftrials?(Withbothuniformandnon-uniformmodels,
the actual outcomes will rarely match the probability models. As the
number of trials increase, the actual outcomes will most likely begin
to show results that are similar to the original probability models.)
Record.
Creating Probability Models
(CP, IP, M, WG, GP) S452, S453 (Answers on T888, T889.)
WG, M, GP, CP:
Have students turn in their books to S452. Students will be
conducting an experiment with a uniform probability model
using centimeter cubes. Pass out the centimeter cubes
to each set of student pairs. {Verbal Description, Graphic
Organizer, Concrete Representation}
T878
Mathematics Success – Grade 7
LESSON 34: Uniform and Non-Uniform Probability Models
MODELING
Creating Probability Models
Step 1: Students will be working in student pairs as the teacher models completing
the experiment with a uniform probability model. For the experiment on
page S452, each student pair will need 2 red, 2 blue, 2 green, and 2
yellow cubes.
• Havestudentpairsdeterminetheprobabilityofselectingeachcolor
and write the probability in the table. Review the answers as a whole
group so that each student pair has the correct information.
• Selectoneormorestudentstocometothefronttoassistinselecting
a cube from a bag without looking.
• Students should place a tally mark in the appropriate color box for
each selection.
Step 2: Usingthedataalreadycollected,modelforthestudentshowtodetermine
the relative frequency of each event.
• Selectoneormorestudentstocometothefronttoassistinselecting
a cube from a bag without looking.
• Studentsshouldmarkwithatallymarkintheappropriatecolorbox
for each selection.
• Afteranadditional10trials,determinethetotalrelativefrequencyof
the 20 trials and then compare them to the probability of each color.
Step 3: Partner A, how did the actual outcomes compare to the probability model
after10trials?20trials?(Answerswillvarybasedonactualoutcomes.)
Record.
• PartnerB,werethereanydiscrepanciesbetweenyouractualresults
andthemodel?Ifso,howcanyouaccountforthem?(Answerswill
vary based on actual outcomes. Students should refer to the fact
that actual outcomes rarely mirror the theoretical outcomes from the
probability models.) Record.
CP, IP, WG:
Students will work together in partners to complete the
experiment on S453 with a non-uniform probability model.
They will need 4 yellow cubes, 1 red cube, 2 blue cubes,
and 3 green cubes. Students will collect data and record
information from 10 and 20 trials. Students should follow
the same procedure as the experiment with the uniform
probability model. Have students complete the questions
about their experiment at the bottom of S453 and then
come back together as a class to discuss the results of
their experiments. {Verbal Description, Graphic Organizer,
Concrete Representation}