Download Permutations and Combinations

Primary Type: Lesson Plan
Status: Published
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Resource ID#: 71610
Permutations and Combinations
This is a seventh grade lesson that should follow a lesson on simple probability. This is a great introduction to compound probability and a fun,
hands-on activity that allows students to explore the differences between permutations and combinations. This activity leads into students
identifying situations involving combinations and permutations in a real-world context.
Subject(s): Mathematics
Grade Level(s): 7
Intended Audience: Educators
Suggested Technology: Scientific Calculator
Instructional Time: 2 Hour(s)
Freely Available: Yes
Keywords: permutations, combinations, compound events
Resource Collection: FCR-STEMLearn Algebra
ATTACHMENTS
Permutation Combination Activity Recording Sheet.docx
Permutations and Combinations guided practice.docx
Permutations and Combinations guided practice answer key.docx
Permutation and Combinations Activity Cards.docx
Permutation Combination Homework.docx
Permutation Combination Homework Key.docx
Permutations and Combinations Exit Quiz.docx
Permutation Combination Exit Quiz Key.docx
LESSON CONTENT
Lesson Plan Template: General Lesson Plan
Learning Objectives: What should students know and be able to do as a result of this lesson?
Students will be able to distinguish the similarities and differences between permutations and combinations.
Students will be able to correctly choose when to use permutations and combinations in order to determine the probability of compound events and to solve
problems.
Students will be able to calculate permutations and combinations to then find the probability of a compound event or to solve a problem.
Prior Knowledge: What prior knowledge should students have for this lesson?
MAFS.7.SP.3.8a - Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the
compound event occurs.
MAFS.7.SP.3.7 - Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement
is not good, explain possible sources of the discrepancy.
Guiding Questions: What are the guiding questions for this lesson?
What is a compound event?
How does a compound event compare to an independent event?
page 1 of 5 What is a permutation?
What do we do with combinations and permutations?
How can we use combinations and permutations to solve real world problems?
Teaching Phase: How will the teacher present the concept or skill to students?
Day 1:
The lesson will begin with the student activity Permutation and Combinations Activity Cards.docx. Have students answer the questions on the activity recording sheet
Permutation Combination Activity Recording Sheet.docx. When the activity is complete, the teacher will facilitate a discussion about the results of the activity with the
emphasis on deciding when order matters (permutations) and when order doesn't matter (combinations). Use the information provided below for each activity to
facilitate the whole class discussion and assist students in seeing the differences between permutations and combinations.
For the activity, students will be placed in groups of three, four or five depending on the number of students in your class. You can choose to group students based on
proximity of one another for ease or based on student ability. To help effectively manage the 4 activities, separate the activities onto separate card stock and have all
groups complete each part at the same time (e.g. each group does activity one at the same time and then moves through the other activities as they complete each
one). This allows the activities to proceed one at a time for the teacher’s pace. To teach the lesson as designed, all 4 activities require a fairly large flat surface to remain visible for the discussion. The discussion is the most important part of the
lesson as students explain and think through the patterns. Since the teacher is the only one who knows their classroom setup, please direct students to place the
activities in rows or corners, each group will complete a Permutation Combination Activity Group Sheet to record their findings.
Activity 1 should take less than 5 minutes. No two pairs of students will have the same color combination. The order does not matter because it is simply a pair
(group) - there is no need for students to greet each other a second time. The teacher will know when the activity is done correctly when there is [(number of
students in group) – 1] of each color. The illustration below reflects a group of 5 people:
Activity 2 should take between 5 - 10 minutes. First, the students (as a group) select any three colors. Students will make groups using all 3 colors in different
orders. When students expand their groupings to allow all patterns (order matters) the groupings will yield six permutations.
Activity 3 should take 10-15 minutes. Each group chooses all five colors and works together to create unique sets of three colors. No set can have the same colors
combined with one another, creating combinations. (It is much more difficult to complete Activity 3 than
Activity 4, because finding the duplicates can be difficult.)
See below:
Activity 4 should take 10-15 minutes. In this activity, each group chooses four different colors and work together to create all the sets possible using three colors at
a time. The colors in the set do not have to be unique but the positions of the colors must be different. It might be easier to find the 24 permutations using a
systematic list.
page 2 of 5 Some students will be able to form combinations quicker than permutations and vice-versa. The only help teachers should provide is encouraging students to develop
a pattern for Activity 3 and Activity 4 and to check if their totals are correct. The teacher should encourage students to look for patterns within the pattern. The teacher
can quickly tell if the sets are correct by counting the individual colors in each activity – they should all be equal. If they are struggling with a pattern, encourage them
to begin and then work in a methodical manner – there should be a 'method to their madness.' Suggest they start with one color first, then build combinations that start with that color. Use color 1 and color 2, then all the combinations that can be made. Then
color 1 and color 3; color 1 and color 4; and color 1 and color 5. Then they start with color 2 first, and so on.
Patterns appear when we begin to look for them. Let the students think it through and they will understand instead of simply showing them how to plug numbers into a
formula.
Guided Practice: What activities or exercises will the students complete with teacher guidance?
Day 2:
The guided practice/homework sheet, Permutation Combination Guided Practice, is designed to be duplexed and cut in half.
Once the activity is complete and the Permutation Combination Activity Group Sheet is collected the teacher will facilitate a discussion among the groups. Let the
students talk about the activity, the outcome and the process. What did they learn? What patterns did they see and how did they discover them? Have them walk
around the tables and look at the other teams' work. No two tables will look exactly alike.
The teacher will introduce and explain factorials. Work out a couple of quick scenarios. 5! = 5 X 4 X 3 X 2 X 1 = 120 etc. Next relate factorials to the counting principle
- they both multiply everything together to get the total possible outcomes. But with factorials you know what the numbers are based on a pattern. Make sure to note
that 0! = 1 since we can never divide by 0 and permutations where n = k always have zero factorial in the denominator.
If a factorial provides the answer for total outcomes, can we use that knowledge to determine how many outcomes there are when we only want to use some of the
outcomes? If factorial is everything and we don't use everything, we need to exclude what we don't use. Since we formed factorials with multiplication we will use
division to exclude the part we don't want. In simple terms, permutations are the total of all possible outcomes divided by the total of the outcomes not
selected. We use factorials to find those totals.
Have the students compare the even numbered activities with the odd numbered activities. When they discuss the difference between groups, introduce the idea that
order matters. When all arrangements are represented because order matters it is called a permutation. Show the formula
, then work out the
scenarios from the activity that are permutations (Activity 2 and 4).
Note for students that P(n,k) is the same as
Now introduce Combinations. Combinations are exactly like Permutations except we have to remove the non-unique sets because order does not matter, we are
simply forming groups. So how much do we remove? The space of the duplicates, the number of element in each set Factorial. Introduce the formula
, then work out the scenarios from the activity that are combinations (Activity 1 and 3). Again, note for students that C(n,k) is the same as
Discuss the definition for the word combination in general. Most students define combinations as all the ways different elements can be combined but that is actually
permutations! Remind them that Permutations > Combinations – but wait a minute – is it always? Make sure students know that When n and k = 1, Permutations
actually = Combinations.
What about using n or k = zero – is that possible? If there is nothing to choose from, can anything be chosen? Similarly, if something exists can groups of nothing be
created? Work out any permutation problem where k = n – 1 and k = n. Both yield the same number of outcomes. Why? It is easy to see that we are multiplying the
same numbers and dividing by 1 but where does the 1 come from? In one instance, the 1 comes from the difference and 1! = 1 and in the other instance, the
difference is 0 and 0! =1. So are they the same? If not, how are they different? A very quick visual is to redo Activity 2 to use the same 3 colors but create groups of 2.
Students should recognize that the number of outcomes are the same but both the arrangement of the pieces and the number of pieces used is different.
When beginning calculations, it is good to use factorials in expanded form for the students who may be struggling a bit and mark the numbers off before moving on to
using factorials and crossing the entire factorial off.
Once this class discussion is complete, allow students to work in pairs to complete the Guided Practice worksheet: Permutations and Combinations guided practice
answer key.docx Permutations and Combinations guided practice.docx
As students are working on the guided practice, the teacher should move around the room and assist students as they need help. Once all groups are finished, review
the solutions and discuss any questions that students have or clear up any misconceptions that you witnessed when moving around the room.
Independent Practice: What activities or exercises will students complete to reinforce the concepts and skills developed in the
lesson?
The students will complete the attached homework assignment, independently. Allow students to begin working on independent practice with the remaining time left in
the class period. If students do not finish, instruct them to finish for homework.Permutation Combination Homework.docx Permutation Combination Homework
Key.docx
While students are working independently, the teacher should be moving around the room checking on student understanding and answering any questions.
Closure: How will the teacher assist students in organizing the knowledge gained in the lesson?
The teacher will bring the whole class back together for a review of the lesson to reinforce what has been learned.
The teacher should review the answers of the guided practice and answer any lingering student questions.
page 3 of 5 Summative Assessment
The teacher will determine if students have reached the learning target by checking the self-assessment section of the guided practice and having students complete
an exit quiz. Permutation Combination Exit Quiz Key.docx Permutations and Combinations Exit Quiz.docx
Formative Assessment
During the Activity, the teacher can monitor student progress by walking around the room and observing students. Try to offer encouragement rather than answers
and let students work through each part as a group.
After the Activity, the teacher will be able to check for understanding through whole class discussion, in the self assessment section of the guided practice, and by
checking the exit quiz.
Scale:
The Self-Assessment section of the guided practice is where each student rates their knowledge of the concepts listed in the learning goal from the beginning of the
lesson until the guided practice is complete. This lesson will use a generic scale. Post a permanent display in the classroom in a prominent spot. The only item the
teacher will need to change is the daily learning goal. If unit goals are needed, for this lesson simply add a unit goal about Compound Probability.
Since this is the first time factorials, combinations and permutations have been taught:
1 for Beginning, "I have no prior knowledge of the learning goal, I need help." would be appropriate.
Just before the student turns the guided practice over assess student understanding. Most students should now be able to enter a 2 for Developing, "I can solve
simple problems with the learning goal, I may need help” or a 3 for Applying, "I can solve complex problems with the learning goal, I don’t need help.”
Remind them to assess one last time before the guided practice is collected. If they are struggling, encourage them to fill in the part that says I don’t understand …. At
the end of the lesson, many students will enter a 4 for Expert, "I can solve complex problems with the learning goal, I can teach another.”
Feedback to Students
Students will receive feedback as a group during the activity and discussion, then individually through guided practice and homework
ACCOMMODATIONS & RECOMMENDATIONS
Accommodations:
The teacher may choose to place struggling students in pre-determined groups so that they are grouped with a high ability student and at least one average ability
student.
The teacher should be conscientious of students with color-blindness, as these activities require students to differentiate by color. A possible accommodation might be
to label each colored slip with a distinct letter designation to remove the need to see color (e.g. the blue paper have a B, the red paper an R).
Students with auditory processing issues may be provided with hard copies of any information the teacher presents.
Extensions:
Students can create their own problem situations where permutations and combinations are appropriate tools. Students can create their own and then share with a
partner. Each person can then complete the partners problems and compare solutions and rationale.
Suggested Technology: Scientific Calculator
Special Materials Needed:
brightly colored card stock in 5 colors pre-cut (see Further Recommendations for specific dimensions)
glue stick
timer
Further Recommendations:
Calculator Note:
Save the Scientific Calculators until a later date. This lesson focuses on students learning how to solve combinations and permutations with paper and pencil providing
a concrete foundation of the concepts.
Later, once they have a firm understanding, they can be shown the calculator functions.
Students get excited when solving combinations and permutations – it feels good to cross those numbers off! Combinations and permutations are easy to set up.
When using a calculator, students are not able to see the patterns.
Materials:
Cut each sheet lengthwise in 1/2 inch strips and then cut those strips into 1 inch pieces. Use SMALL binder clips to cut several sheets of cardstock at the same time.
Put three clips on the left side of the sheets, oriented lengthwise. Cut ½ inch strips. Take several of the ½ inch strips and put a binder clip on the end and then cut 1
inch pieces. There will be 187 pieces per sheet so 3 pieces of cardstock per color is the minimum quantity needed for 6 groups. Cut out the activity cards and glue to
1/6 of a sheet of cardstock. Use a different color for each of the four activities. Perfection is not required but try to make the pieces as uniform as possible.
Recommendation:
Select card stock colors that vary in hue. For some students, it is difficult to distinguish between yellow, orange and green.
page 4 of 5 SOURCE AND ACCESS INFORMATION
Contributed by: Laura Quinton
Name of Author/Source: Laura Quinton
District/Organization of Contributor(s): Orange
Is this Resource freely Available? Yes
Access Privileges: Public
License: CPALMS License - no distribution - non commercial
Related Standards
Name
MAFS.7.SP.3.8:
Description
Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.
a. Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the
sample space for which the compound event occurs.
b. Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams.
For an event described in everyday language (e.g., “rolling double sixes”), identify the outcomes in the sample
space which compose the event.
c. Design and use a simulation to generate frequencies for compound events. For example, use random digits as a
simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the
probability that it will take at least 4 donors to find one with type A blood?
page 5 of 5