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# Permuting Paper (and or) Permuting People Both lessons are described below.

Subjects:

Math, Science

Grades:

1, 2, 3, 4, 5, 6, 7, 8

**Permuting Paper**

To begin, I typically ask kids how many different ways 10 students can line up. I let them guess and write guesses on the board. Commonly, their guesses are off by a couple of orders of magnitude or more. I urge them not to feel bad about this, but I let them know the answer is much larger. Once in a while, someone may get pretty close in their guess.

If I am doing* Permuting Paper *using my YouTube video (see link), I ready the following video I created for projection in the classroom:

**IT IS ALSO POSSIBLE TO PERFORM THE ACTIVITY BY SINGING THE SONG YOURSELF OR PLAYING IT ON A DEVICE (WITHOUT PROJECTING IT) IF NO PROJECTOR IS AVAILABLE.**

1. Each student must cut and label three medium sized squares of paper.

2. The students label the squares **A**, **B**, and **C**.

3. They put them on their desks in **A**, **B**, **C** order.

4. Then they permute the squares to the YouTube song or live singing, trying to discover all the arrangements (permutations).

5. Then the audience is invited to guess how many arrangements or permutations were made. Pretty soon, the number 6 tends to start coming up regularly.

Commonly, at this point, I will provide some explanation as to why **6** is the result. There are **3** possible entries in the first slot, then **2** in the second slot, then **1**, in the third slot, and the results multiply to give **6**. Then, with the help of the students, we write out the 6 permutations of the letters **A**, **B**, and **C**.

**ABC**

**ACB**

**BAC**

**BCA**

**CAB**

**CBA**

6. Next, I ask how many permutations there would be of four letters (**A**, **B**, **C**, and **D**)? Pretty soon, but not always, students often come up with **24=4(3)(2)(1)**. Once students are getting a handle on the idea of a factorial for calculating permutations, I run through the numbers up to 10 letters to show how quickly these numbers get **VERY BIG!**

**5!=5(4)(3)(2)(1)=120**

**6!=6(120)=720**

**7!=7(720)=5,040**

**8!=8(5040)=40,320**

**9!=9(40320)=362,880**

**10!=10(362,880)=3,628,800**

There are many possible follow-ups to this lesson. At this point, I will leave them to your imagination. Many students begin to see connections to things like dealing cards from a deck and so forth.

**Permuting People **

What I typically do first off is ask kids how many different ways 10 students can line up. I let them guess. Commonly, their guesses are off by a couple of orders of magnitude or more. I urge them not to feel bad about this, but let them know the answer if much larger. Once in a while, someone may get pretty close in their guess.

Here are the steps for *Permuting People*:

1. I ask for three volunteers who will permute in front of the class.

2. The volunteers each receive a sheet of paper with the letter A, B, or C written large to hold in front of them while they “permute” to the song that goes with this activity.

3. The students **line up in A, B, C order**, and then permute (change orders, sometimes requiring a bit of guidance) while the teacher sings the song (or plays it through a device).

**As mentioned above, it is also possible to play the song from YouTube using a device rather than singing it yourself, although the song says *** *I hope to make another video with live students permuting at some point.

4. Then the audience is invited to guess how many arrangements or permutations they made. Sometimes, I will have 3 more students have a chance to permute in front of the class. Pretty soon, the number 6 tends to start coming up regularly as the number of arrangements.

5. Commonly, at this point, I will provide some explanation as to why 6 is the result. There are 3 possible enters in the first slot, then 2 in the second slow, then 1, and the results multiply to give 6. Then, with the help of the students, we write out the 6 permutations of the letters **A**, **B**, and **C**.

**ABC**

**ACB**

**BAC**

**BCA**

**CAB**

**CBA**

6. I ask how many permutations there would be of four letters (**A**, **B**, **C**, and **D**)? Pretty soon, but not always, students often come up with **24=4(3)(2)(1)**. Once students are getting a handle on the idea of a factorial for calculating permutations, I run through the numbers up to 10 letters to show how quickly these numbers get VERY BIG!

**5!=5(4)(3)(2)(1)=120**

**6!=6(120)=720**

**7!=7(720)=5,040**

**8!=8(5040)=40,320**

**9!=9(40320)=362,880**

**10!=10(362,880)=3,628,800**

There are many possible follow-ups to this lesson. At this point, I will leave them to your imagination. Many students begin to see connections to things like dealing cards from a deck and so forth.

These two lessons have proven themselves over and over again. I have used them with many different age groups. We have had a great deal of fun while exploring vital content. Good luck implementing these lessons should you wish to give one or both of them a try!