Can You Really 

Count?

A WebQuest Created by 

Mrs. Larson

 Managing Impulsivity

Throughout your learning experiences you are about to endure, you are going to encounter several opportunities to collaborate with your learning partner and critical friends.  You will need to remember to resist the urge to act or react immediately.  Follow these helpful reminders:

• Think before you speak or act

• Do not interrupt your learning partner or your critical friends - listen as they express their ideas

• Do not blurt out the first answer that comes to mind - take time to think ideas through 

• Employ other habits of mind that we have been practicing such as questioning, to solve problems

Listening with Understanding and Empathy

As you and your learning partner and critical friends groups work together, you will need to follow our communication skills we are practicing:

• Use good body language such as eye contact, nodding, and facial expressions that show congruence with your partners words

• Paraphrase to put your partners' thoughts into your own words

• Ask probing questions, which we have been practicing, to clarify meaning

• Empathize by relating to the feelings, ideas, and perspectives of your partners

• Ask your partners to repeat something when your partners' minds start to wander

• Do not interrupt your partners

Metacognition

When working on your projects, remember to practice the following criteria:

• Generate a plan of action - brainstorm ideas and have discussions with your group

• Monitor to see if the plan is working - have two or more classmates check your work and provide feedback to you

• Reflect yourself; to see if the strategy worked

Introduction - A Sense of Wonderment

Mathematical questions often begin with the words how many  or how much.  As you know, the state of Colorado carries many automobile license plates that begin with three counting numbers and end with three letters of the alphabet.  A question might be, “I wonder how many different license plates are there in the state of Colorado that use this form?”  Being a student in middle school, you might ask another question as you try to open your locker for the next class.  A question such as, " I wonder how much time would it take to open a lock if I try all the possible combinations?”

To answer such questions, you must be able to count and to use the operations of addition, subtraction, multiplication, and division.  You can easily compute answers with a calculator, but only if you know which operations to apply and when to apply them.

The goal of this lesson is to assist your learning of new and applicable strategies for reasoning in situations that involve counting possibilities.  Throughout the lesson, you will journey through some experiences that will encourage you to review and apply previously learned skills such as mental-estimation, using problem-solving strategies, and discussing or writing about mathematical ideas and procedures.  If at any time you need to review these skills you could do so by using the webquest entitled, Probability, refer to your Math On Call, ask Dr. Math, or review your 7th and/or 8th grade journals with your learning partner and possibly with your critical friends group.

Start this review by using your KWL chart to record what you know about probability and then what you want to know more about this concept.  When you are finished reviewing, record what new learning you gained.

In this lesson, you will be provided with a diverse set of experiences, which will help you to develop a sense of the structure of problems involving counting. You will be able to consider the  special circumstances of various counting situations, such as whether or not order is important.

The Task

In this lesson, you will be introduced to combinations and permutations.  You will begin exploring through practice of various situations that will involve counting possibilities.  As a final outcome, you and your learning partner will need to choose one of the three project options:

Option A.  You will be responsible in assisting a case at South Valley Middle School to reason with those who are accusing their peers of trying all possible lock combinations until the lock in the girls' locker room was opened.

 

Option B.   You will brainstorm some ideas for a mystery case with your learning partner such as the lock combination case in the girls' locker room at South Valley Middle School.  You will use your ideas to write a story about a detective's investigation of a crime.

 

Option C.   In this project, you will make "trains" by joining "cars" of different lengths.  Each car is labeled with a number that indicates its length.  You will write a paper two or three pages in length summarizing your findings.

The Process

In order to reason the case with your peers at South Valley, write a justified story about a detective's crime investigation, or to explore patterns and reason with the differences and similarities of the "trains" and "cars" lengths, you will need to do the following to gain some background knowledge on combinations and permutations:

• Begin by gaining some background knowledge from the resources.  You may use any or all resources at any of the levels provided.

• As you are becoming familiar with combinations and permutations, try some of the interactive lessons offered in the resources until you feel comfortable with your competency in applying your newly acquired skills in our mystery case.  Check for accuracy by working with your learning partner and grouping with your critical friends to compare and contrast your strategies used.

 Completing the Task  

With all your newly acquired knowledge about combinations and permutations, you are going to apply this knowledge to one of the following project options:

Option A

You are going to now analyze the situation that has occurred at your school to assist your friends in solving this conflict reasonably.  Someone has opened a locker in the girls’ locker room at South Valley Middle School other than the owner of the lock that was on it. A CD player was stolen from the locker.  Your peers suspect that the culprit tried every possible lock combination until the lock opened.  The lock combinations used at South Valley consist of three random numbers from 0 to 39, with none of the numbers repeated. 

When working through this problem solving process, you and your learning partner should refer to the T.A.P.S. formula on the wall in the classroom to assist you.  You should also brainstorm your ideas together as well.  Refer to your metacognition, managing impulsivity, and listening with understanding and empathy skills. 

Based on your findings, write a letter to your student body explaining whether or not you agree with the accusations your peers have made.  Make sure in your letter, you are using examples to support your thoughts about the case.  You should not use: 

• Fuzzy wording such as "It was all, like...you know."  

• Overgeneralizations such as "everyone is," "You never," and so on

• Vague descriptions such as "I cut a little here and measured some."

• Incomplete comparators such as, "This way was the best."

In completing this letter, you might want to consider some of the following thoughts:  

•       How many combinations are possible?  You could make organized lists or counting trees to help you find all the possibilities.  However, you may have a better method to count all the possible combinations. Remember that you need to assume that a number may not appear more than once in a combination.  You may want to take notes on the method you used.   

•     How long do you think it would take someone to try all the possible combinations?  Can you explain how you made your estimate?  What does the estimated time depend on?

Persistence

 Be sure to have two or more classmates (your critical friends group is the best selection) check your work with the checklist that the students choosing this project will create after reviewing the criteria of the project.  (These students need to have the checklist confirmed with Mrs. Larson when finished.) You will need to make appropriate corrections, deletions, or additions.  Your classmates should also be asking relevant questions that prompt critical thinking and enhance your understanding so your writing is clear to your audience.  Top of Page

Option B

Throughout the unit of study on combinations and permutations, you have been recording ideas for strategies to apply to your problem solving missions.  You will now use your ideas to write a story about a detective's investigation of a crime.  Your story should include some of the following:

• a cast of suspicious characters
• a detective with mathematical ability
• an assistant who always needs explanations
• a lock combination
• a getaway vehicle
• different getaway routes
• a partially legible telephone number found at the scene of the crime 

 Your story must show the detective doing the following:  

     ·        counting combinations  

     ·        figuring out how long it would take to make, do, or check something

     ·        using probability to show what is likely or unlikely to have happened 

     ·        using charts, graphs, or counting trees to explain his or her solution

You should not use: 

·         Fuzzy wording such as "It was all, like...you know." 

·         Overgeneralizations such as "everyone is," "You never," and so on

·         Vague descriptions such as "I cut a little here and measured some."

·         Incomplete comparators such as, "This way was the best."

Persistence

 Be sure to have two or more classmates (your critical friends group is the best selection) check your work with the checklist that the students choosing this project will create after reviewing the criteria of the project.  (These students need to have the checklist confirmed with Mrs. Larson when finished.) You will need to make appropriate corrections, deletions, or additions.  Your classmates should also be asking relevant questions that prompt critical thinking and enhance your understanding so your writing is clear to your audience.  Top of Page

Option C

 In this project, you will make "trains" by joining "cars" of different lengths.  A "car" 

is a rectangular strip of paper; a "train" is one car or two or more cars placed 

end-to-end.  You will find the cars on this Lab sheet.  

For each different car, find every possible train you can make with the same length as that car.  Consider trains made from the same cars, but in a different order, to be different.  This means that a 1-car followed by a 2-car is different from a 2-car followed by a 1-car.  For each different car on the lab sheet, record the length of the car and the number of trains with the same length as that car on a table, which you and your learning partner will create together.  Look for a pattern in your results.  Write an equation for the relationship between the length of a car, L, and the number of trains with that length, N.  Compare the pattern of change and the equation for this situation with the patterns and equations for other situations you have studied in this unit. (It would be wise to be making notes as you are making your observations.)  Note: You may cut the 1-car, 2-car, and 3-car rectangles apart from the lab sheet to create your train.

Write a paper two or three pages in length summarizing your findings.   

You should not use: 

• Fuzzy wording such as "It was all, like...you know."  

• Overgeneralizations such as "everyone is," "You never," and so on

• Vague descriptions such as "I cut a little here and measured some."

• Incomplete comparators such as, "This way was the best."

Persistence 

Be sure to have two or more classmates (your critical friends group is the best selection) check your work with the checklist that the students choosing this project will create after reviewing the criteria of the project.  (These students need to have the checklist confirmed with Mrs. Larson when finished.)  You will need to make appropriate corrections, deletions, or additions.  Your classmates should also be asking relevant questions that prompt critical thinking and enhance your understanding so your writing is clear to your audience.   Top of Page

Don't Forget to take a look at the rubric that will be used to score your work and behaviors on any of the three options you may choose.

Conclusion

You have now become an expert at computing combinations and permutations.  You are also quite skilled at knowing how to apply combinations and permutations to real life experiences.  Moreover, you have set a good example for your peers by role modeling positive conflict management.  It would be good for you to use your Second Step training and work on a new plan to improve the situation in which you worked on.  For instance, after solving the conflict with your peers in Option A, you could develop a plan to improve the situation in order to prevent this from happening again.  

Challenge

It is now a challenge to you, if you should decide to master your skills for you and your critical friends to meet with another critical friends group that worked on Option A, to encourage your student body to write a proposal to the principal of your school convincing her to purchase combination locks with more digits because of the increased security they offer.  (This request came from your student voices meeting as a possible solution to your locker theft issue at South Valley Middle School.)  A question to consider would be, “How does the number of combinations grow as the number of digits on the dial increase?”  You should also consider that our district is on a strict budget and the principal will not want to waste money. Your challenge is to help the student body make their case for purchasing locks with more digits.  Turn your proposal into Mrs. Larson for feedback after working on this with your groups.  We will then meet with Student Voices to discuss all possible proposals we come up with.  Student Voices can submit a final proposal to the principal based on our provided options.