Lesson Title: Tree Diagrams
Creator: Marlene Addair
Grade Level: 4/5
Big Idea: Probability
Essential Question: How can I determine the number of combinations possible
to solve a problem?
CSOs, LS, TT:
Math CSOs:
M.O.5.5.1 Students will construct a sample space and make a hypothesis as
to the probability of a real life situation overtime, test the prediction
with experimentation, and present conclusions (with and without
technology).
M.O. 4.5.3
Students will design and conduct a simple probability experiment
using concrete objects, examine and list all possible combinations
using a tree diagram, represent the outcomes as a ratio and
present the results.
Learning Skills:
21C.O.5-8.2.LS.1
Student engages in a critical thinking process that supports
synthesis and conducts evaluations by applying
comprehensive criteria
21C.O.5-8.2.LS.2
Student draws conclusions from a variety of data sources to
analyze and interpret systems
21C.O.5-8.2.LS.3
Student engages in a problem solving process that divides
complex problems into simple parts in order to devise
solutions
21C.O.4.2.LS.3
Student engages in a problem solving process that promotes
questioning, planning investigations and finding answers and
solutions.
Technology Tools:
21C.O.5.8.3.TT.6 Student applies productivity/ multimedia tools and
peripherals to support personal productivity, group
collaboration, self-directed learning, lifelong learning, and
assistance for individuals with disabilities including
supplemental assistive technology tools
Launch/Introduction (suggested time 25 minutes)
Activating Prior Knowledge:
You have been hired to run a Smoothy Shop during your summer vacation. The
Smoothy Shop sells drinks in three different sizes: small, medium and large. The
flavors are: strawberry, banana, orange, and lemon-lime. Show all the possible
combinations for the drinks. (Students work individually to come up with
combinations). They can then volunteer to go to the board and explain their
methods. (example: make a chart, list, table or tree diagram) .
Teacher Note: These students have been previously introduced to using tree
diagrams to show combinations. This could be done as a teacher-led activity if
students are not familiar with this.
Specialized Vocabulary Development:
Probability, equally likely, unlikely, 50-50 chance, impossible, combinations, tree
diagrams
Discuss vocabulary words and place each word on a word wall for further
reference.
Whole Group: Remind students of the Essential Question they will try to answer:
How can I determine the number of combinations possible to solve a problem?
A stadium has three gates on the West side—A, B, and C. It has four gates on
the East side—W, X, Y, and Z.
Tell students that people can enter the stadium through any of the three West
gates. They can exit the stadium through any of the four East gates.
To support students who do not speak English as their first language, draw a
sketch of the stadium on the board or overhead.
Pose the problem: How many different ways can a person enter and leave the
stadium? Look for patterns as you work through these problems.
Suggest that one way to keep track of the ways is with a rectangular array in
which each row represents an entry gate, and each column represents an exit
gate. Each cell in the array contains an ordered pair of letters. The first letter
names the entry gate; the second letter names the exit gate.
A completed array is shown below:
Exit Gate (East)
Entry Gate (West)
W
X
Y
Z
A
(A,W)
(A,X)
(A,Y)
(A,Z)
B
(B,W)
(B,X)
(B,Y)
(B,Z)
C
(C,W)
(C,X)
(C,Y)
(C,Z)
Point out that the array has 3 rows and 4 columns, so there are 3 *4 = 12 ordered
pairs in the table.
There are 12 ways of entering through a West gate and exiting through an East
gate.
Show students the Multiplication Counting Principle on the overhead or ELMO.
Use the following problem: A school cafeteria offers these choices for lunch:
Main Course: chili or hamburger, Drink: milk or juice, Dessert: apple or cake
How many different ways can a student choose one main course, one drink, and
one dessert: Use the Multiplication Counting Principle.
2
*
2
*
2
=
8
Then draw a tree diagram to show all possible ways to select foods for lunch.
Main Course
Chili
Hamburger
Drink
milk
Dessert
apple
juice
cake
apple
milk
cake
apple cake
juice
apple
cake
Conclude by asking if all eight possible food choices are equally likely.
Ask students to discuss with a partner and report out why or why not the choices
are equally likely and be able to explain their answers.
(Example: No they are not equally likely because the foods are probably not equally popular. If
cake is more popular than an apple, lunches that include cake will be selected more often than
lunches that include an apple.)
Investigate/Explore (suggested time varies from 30-40 minutes):
Students now work in groups of four or five to complete combination problems.
Heterogeneous groups are established using benchmark scores.
Give each group a different problem from below. There may be more than one
group with the same problem depending on class size.
1. Sam has 3 clean shirts (red, blue, and yellow) and 2 clean pairs of pants
(tan and black). He grabs a shirt and a pair of pants without looking.
Make a tree diagram to show all possible ways that Sam can grab a shirt
and a pair of pants. List all possible combinations of shirts and pants.
Also tell if the combinations are equally likely.
2. Mr. Jackson travels to and from work by train. Trains to work leave at
6:00, 7:00, 8:00, and 9:00 a.m. Trains from work leave at 3:00, 4:00, and
5:00 p.m. Mr. Jackson is equally likely to select any 1 of the 4 morning
trains to go to work. He is equally likely to select any of the 3 afternoon
trains to go home from work. Make a tree diagram to show how many
different ways Mr. Jackson can take trains to and from work.
3. The ice cream shop has 3 flavors of ice cream and 5 different toppings.
How would you use the Multiplication Counting Principle to calculate the
total number of combinations of choosing a flavor of ice cream and a
topping? Draw a tree diagram to show the possible combinations.
4. There are 4 number cards—1 each of the numbers 1, 2, 3, and 4. If you
mix the cards, draw one card without looking, replace the card you drew,
mix the cards, and draw one card again, how many possible combinations
are there? Use a tree diagram and the Multiplication Counting Principle to
help solve the problem.
Summarize/Debrief the Lesson (suggested time varies from 30-40 minutes)
Each group reports their results and their process they used to get the answers.
They use the ELMO to show their tree diagrams to the entire class. Conclude
with a return to the essential question – How can I determine the number of
combinations possible to solve a problem? Students should be able to give
several answers. Discuss the advantages of each method and how you can be
sure that all combinations have been found.
Reflections:
Ticket Out the Door: Students use exit slips to explain in words how they can
determine all the possible combinations to solve problems.
Extension: Students may get on website below for practice with tree diagrams.
http://www.studyzone.org/mtestprep/math8/e/treediagram6p.cfm
http://www.pbskids.org/cyberchase/games/combinations/combinations.html
Materials: ELMO, Computers
Duration: 90 minutes