Adventures of Small Number: Classroom Resources

The purpose of this activity is to introduce very young learners to the notions of permutations and combinations, i.e. the distinction between “ordering things” and “not-ordering things”.

Recall that finding the number of permutations and the number of combinations is related to counting in how many times we can choose \(r\) out of \(n\) given objects:

Permutation Rule:

Order matters. For example, \((1, 2, 3)\text{,}\) \((2,1,3)\text{,}\) and \((2,4,5)\) are three different 3-permutations of the set \(\{1, 2, 3, 4, 5\}\text{.}\)

Combination Rule:

Order does not matter. For example, \(\{1, 2, 3\}\) and \(\{2, 4, 5\}\) are two different 3–combinations of the set \(\{1, 2, 3, 4, 5\}\text{.}\) But, \(\{1, 2, 3\}\) and \(\{2, 1, 3\}\) represent the same 3-combinations of the set \(\{1, 2, 3, 4, 5\}\text{.}\)

Observe how round brackets \(( · , · , · )\) are used to indicate that order matters and how curly brackets \(\{ · , · , · \}\) are used to indicate that order does not matter.

• Part 1 - Permutations - Order Matters

  Put three chairs beside each other. Clearly label them: 1, 2, and 3. You may wish to use chairs of different colours.

  Prepare a chart like the one below (or draw it on the board) with 24 blank rows:

  \begin{equation*} \begin{array}{c|c|c|cc} \# \ \ &\ \ \text{Chair 1} \ \ &\ \ \text{Chair 2} \ \ &\ \ \text{Chair 3} \ \ \\ \hline &&&\\ \hline &&&\\ \hline &&&\\ \end{array} \end{equation*}

  Ask for nine volunteers:

  • Four students will perform the activity. (Let’s call these students Alice, Bob, Carol, and Dave.)
  • Two students will record the different stages of the activity into the chart. (Erin and Frank)
  • Three students will act as witnesses.⁴ (Grace, Heidi, and Ivan)

  Explain the activity:

  • We will count how many different ways Alice, Bob, Carol, and Dave can sit on the three chairs. Obviously, we have four people and three chairs, so one person will have to wait for their turn.
  • Erin and Frank will fill out the chart so that we do not lose count.
  • Grace, Heidi, and Ivan will be the witnesses of what is going on. They will observe everything carefully and later on they will tell us in their own words what happened.

  “Do you have any questions? Can we begin?”

  To Alice, Bob, Carol and Dave: “Alice, please sit on chair 1, Bob you sit on chair 2, and Carol you sit on chair 3. Dave, can you please stay standing.”

  To Erin and Frank: “Erin, please write ‘Alice’ under ‘chair 1’, ‘Bob’ under ‘chair 2’ and ‘Carol’ under ‘chair 3’. Frank, please write the number ‘1’ under the number sign.”

  \begin{equation*} \begin{array}{c|c|c|cc} \# \ \ &\ \ \text{Chair 1} \ \ &\ \ \text{Chair 2} \ \ &\ \ \text{Chair 3} \ \ \\ \hline 1&\text{Alice}&\text{Bob}&\text{Carol}\\ \hline &&&\\ \hline &&&\\ \end{array} \end{equation*}

  To the witnesses: “Remember that we started with Alice sitting on chair 1, Bob sitting on chair 2, and Carol sitting on chair 3.”

  

  To Alice Bob, and Carol: “Alice, please stay on chair 1. Bob and Carol, please exchange your chairs.” Bob and Carol switch their places.

  To Erin and Frank: “Erin, please write ‘Alice’ under ‘chair 1’, ‘Carol’ under ‘chair 2’ and ‘Bob’ under ‘chair 3’. Frank, please write the number ‘2’ under the number sign.”

  \begin{equation*} \begin{array}{c|c|c|cc} \# \ \ &\ \ \text{Chair 1} \ \ &\ \ \text{Chair 2} \ \ &\ \ \text{Chair 3} \ \ \\ \hline 1&\text{Alice}&\text{Bob}&\text{Carol}\\ \hline 2&\text{Alice}&\text{Carol}&\text{Bob}\\ \hline &&&\\ \end{array} \end{equation*}

  To the witnesses: “Remember that I asked Alice to stay on chair 1, and that I asked Bob and Carol to switch their places.”

  

  To the class: “It is so easy to make a mistake when we count, so we always need to be very careful. What helps me to avoid mistakes is to count in some kind of order. I’d suggest that our strategy here be to first count all of the cases when Alice sits on chair 1. Do you agree?” [Class: “Yes, we agree!”]

  To the class: “Thanks! OK, what should we do next?” [The idea is that Dave takes Carol’s seat. Maybe a student will suggests this, but maybe the teacher will need to help.]

  To Dave and Carol: “Dave, please take Carol’s place. Carol, thank you, you will be standing for a little while.” [Recall that Alice is already sitting on chair 1 and that Bob is on chair 3.]

  To Erin and Frank: “What should you write on the chart now?”

  Frank: “I’ll write the number 3.”

  Erin writes:

  \begin{equation*} \begin{array}{c|c|c|cc} \# \ \ &\ \ \text{Chair 1} \ \ &\ \ \text{Chair 2} \ \ &\ \ \text{Chair 3} \ \ \\ \hline 1&\text{Alice}&\text{Bob}&\text{Carol}\\ \hline 2&\text{Alice}&\text{Carol}&\text{Bob}\\ \hline 3&\text{Alice}&\text{Dave}&\text{Bob}\\ \hline &&&\\ \end{array} \end{equation*}

  To Erin and Frank: “Excellent job, Erin and Frank!”

  To the class: “What should we do now? Don’t forget, our strategy is to keep Alice sitting on chair 1 for as long as possible.”

  A student: “Now, Bob should stand up and Carol should take his seat.” This will lead to two more cases and Erin and Frank would produce the following chart:

  \begin{equation*} \begin{array}{c|c|c|cc} \# \ \ &\ \ \text{Chair 1} \ \ &\ \ \text{Chair 2} \ \ &\ \ \text{Chair 3} \ \ \\ \hline 1&\text{Alice}&\text{Bob}&\text{Carol}\\ \hline 2&\text{Alice}&\text{Carol}&\text{Bob}\\ \hline 3&\text{Alice}&\text{Dave}&\text{Bob}\\ \hline 4&\text{Alice}&\text{Bob}&\text{Dave}\\ \hline 5&\text{Alice}&\text{Carol}&\text{Dave}\\ \hline 6&\text{Alice}&\text{Dave}&\text{Carol}\\ \hline &&&\\ \end{array} \end{equation*}

  This is the key moment of this activity. Students need to realize the following:

  1. The number of different cases with Alice sitting on chair 1 equals to six.
  2. Another person, say Bob, should sit on chair 1.

  Now, with Bob sitting on chair 1, ask the class: “What do you think, how many different cases will Erin and Frank complete with Bob sitting on chair 1?”

  \(\color{blue}{Script}\) \(\color{blue}{1:}\) A student, say, Jane, answers, “Six!”

  To Jane: “This is an excellent idea, Jane! But please explain to all of us your reasoning.”

  Jane points to the chart: “If we switch Alice’s name and Bob’s name on the chart, we will have all cases with Bob sitting on chair 1. So six cases all together!”

  OR

  Jane points to chair 1: “It does not matter who is sitting on chair 1. It will always the same number of cases: six!”

  OR

  Jane points to chairs 2 and 3: “We still have three people for chairs 2 and 3. Their names do not matter. So the number of cases will be: six!”

  \(\color{blue}{Script}\) \(\color{blue}{2:}\) No student comes up with the right answer. Encourage discussion. Lead students towards one of the “Jane’s answers” from Script 1.

  \(\color{blue}{Script}\) \(\color{blue}{3:}\) Repeat the activity with Bob sitting on chair 1 and Alice, Carol, and Dave taking turns on chairs 2 and 3. Erin and Frank should record all of the cases. Ask the class to observe that the number of cases is six. Ask: “Are you surprised?”

  Once when the class is convinced that, regardless who is sitting on chair 1, the number of cases is always six ask: “What do you think that the final answer should be?” If necessary, lead the class to conclusion that the answer is \(4\times6 = 24\text{.}\)

  Ask for questions. Thank Alice, Bob, Carol, Dave, Erin and Frank. Ask the witnesses to describe what happened during the activity.

• Part 2 - Combinations - Order Does Not Matter

  Mark a spot in the classroom.

  Prepare a chart like the one below (or draw it on the board) with 4 blank rows:

  \begin{equation*} \begin{array}{c|c} \text{Team #} \ \ \ &\ \ \hspace{1cm}\text{Players}\hspace{1cm} \ \ \\ \hline & \\ \hline & \\ \hline & \\ \hline & \\ \end{array} \end{equation*}

  Ask for nine volunteers:

  • Four students will perform the activity. (Let’s call these students Alice, Bob, Carol, Dave,)
  • Two students will record the different stages of the activity in the chart. (Erin and Frank)
  • Three students will act as witnesses. (Grace, Heidi, and Ivan)

  Explain the activity:

  • We will count how many different ways Alice, Bob, Carol, and Dave can form a team of three players. Obviously, we have four people, so each time one person will not be in the team.
  • Erin and Frank will fill out this chart so that we do not lose count.
  • Grace, Heidi, and Ivan will be the witnesses of what is going on. They will observe everything carefully and later they will tell us in their own words what happened.

  “Do you have any questions? Can we begin?”

  To Alice, Bob, Carol and Dave: “Alice, Bob, and Carol you will be Team 1. Please come forward and stand around the marked spot. Dave, please stay where you are”

  To Erin and Frank: “Erin, please write ‘Alice, Bob, Carol’ under ‘Players’. Frank, please write the number ‘1’ under ‘Team’”

  \begin{equation*} \begin{array}{c|c} \text{Team #} \ \ \ &\ \ \hspace{1cm}\text{Players}\hspace{1cm} \ \ \\ \hline 1&\text{Alice, Bob, Carol} \\ \hline & \\ \hline & \\ \hline & \\ \end{array} \end{equation*}

  To the class: “Please listen carefully to this question: If Erin writes the names in a different order, would that be the same team or a different team?” [The class consensus should be: “Yes, that would be the same team!”]

  To the witnesses: “Remember that we agreed that Alice, Bob, and Carol form the same team regardless of which order we write or say their names.”

  

  To the class: “Thanks! OK, what should we do next?” [The idea is that Dave joins Alice and Bob as a team member and Carol walks away from the marked spot. Maybe a student will suggests this, but maybe the teacher will need to help.]

  To Dave and Carol: “Carol, please go back to your place. Dave, please go and join Alice and Bob.” [Recall that Alice and Bob are standing at the marked spot.]

  To Erin and Frank: “What should you write on the chart now?”

  Frank: “I’ll write the number 2.”

  Erin writes:

  \begin{equation*} \begin{array}{c|c} \text{Team #} \ \ \ &\ \ \hspace{1cm}\text{Players}\hspace{1cm} \ \ \\ \hline 1&\text{Alice, Bob, Carol} \\ \hline 2&\text{Alice, Bob, Dave} \\ \hline & \\ \hline & \\ \end{array} \end{equation*}

  To Erin and Frank: “Excellent job, Erin and Frank!”.

  To the class: “What should we do now?” [The class should come to the conclusion that two more teams are possible: Alice, Carol, Dave and Bob, Carol, Dave.]

  “So how many different teams can we make?” [The class agrees: Four.]

  \begin{equation*} \begin{array}{c|c} \text{Team #} \ \ \ &\ \ \hspace{1cm}\text{Players}\hspace{1cm} \ \ \\ \hline 1&\text{Alice, Bob, Carol} \\ \hline 2&\text{Alice, Bob, Dave} \\ \hline 3&\text{Alice, Carol, Dave} \\ \hline 4&\text{Bob, Carol, Dave} \\ \end{array} \end{equation*}

  Ask for questions. Thank Alice, Bob, Carol, Dave, Erin and Frank.

  Ask witnesses to describe what happened during the activity.

  To the class: “Can you think of another way to count how many teams there are?” [Maybe, “Each time we chose a team, one person was left over. So, for each of the four people, we have one team. So there are four different teams.”]

• Conclusion.

  To the class: “Remember that in both activities we started by asking four people to do something similar: One person stayed aside and the remaining three participated in some way.”

  • Sitting on one of the three chairs (permutations): You remember that Erin had to be very careful in which order she entered the names in the chart and that we had many different cases.
  • Staying around a marked spot (combinations): In this activity, Erin could write the team members’ names in any order she wished. It didn’t matter for our count.

  “Can you think about something where it is important to do things in order?” [Possible answers: “entering a password”, “playing music”, “telling a story”, or maybe, “the order of mathematical operations, like addition and multiplication”.]

  “Can you think about something where it is not important to do things in order?” [Possible answers: “putting socks on”, “making a fruit salad”, “getting on the school bus” or maybe, “the order in which we add two numbers”.]

  “Later in school you will learn words like permutations and combinations and how to calculate the number of permutations and number of combinations very quickly. You will also learn that the part of mathematics that studies permutations and combinations is called combinatorics.”

• “Let me show you how you can count the number of teams quickly, without writing anything down.”

  “This is why I like mathematics: Once you understand what is going on, you can answer many questions with not too much effort.”

  “Think about making a team with three players as a task of filling each of three boxes with a name, selected from the six options.” See Figure 3.1.3

  

  We just need to keep in mind that the order in which we list the players’ names does not matter. So we need to make sure that we count each team only once. For example, see Figure 3.1.4. Now we think in this way:

  

  

• Step 1: In how many different ways can we put six names into three boxes so that each box contains exactly one name and that each name is used only once? Think for the moment that the boxes are coloured, \(\color{blue}{blue}\text{,}\) \(\color{green}{green}\text{,}\) and \(\color{orange}{yellow}\text{.}\) See Figure 3.1.5. For the \(\color{blue}{first}\) \(\color{blue}{(blue)}\) \(\color{blue}{box}\) we have six choices, for the \(\color{green}{second}\) \(\color{green}{(green)}\) \(\color{green}{box}\) there are five choices (because one name is in the \(\color{blue}{blue}\) \(\color{blue}{box}\) already) and for the \(\color{orange}{third}\) \(\color{orange}{(yellow)}\) \(\color{orange}{box}\) we have four choices (because two out of six names are already taken). So, there are

  \begin{equation*} 6\times5\times4 = 120 \end{equation*}

  different ways to put six names into three coloured boxes!

  We know that, in this count, each team appears more than one time. For example, see Figure 3.1.6

  

• Step 2. Now we count how many times each team of three appeared in the number 120 that we obtained in Step 1.

  In how many different ways can we put three names into three boxes so that each box contains exactly one name and that each name is used only once? Think for the moment that the boxes are coloured, \(\color{blue}{blue}\text{,}\) \(\color{green}{green}\text{,}\) and \(\color{orange}{yellow}\text{.}\) For the \(\color{blue}{first}\) \(\color{blue}{(blue)}\) \(\color{blue}{box}\) we have three choices, for the \(\color{green}{second}\) \(\color{green}{(green)}\) \(\color{green}{box}\) there are two choices (because one name is in the \(\color{blue}{blue}\) \(\color{blue}{box}\) already) and for the \(\color{orange}{third}\) \(\color{orange}{(yellow)}\) \(\color{orange}{box}\) we have only one name remaining (because two out of three names are already taken). See Figure 3.1.7.

  

  So, there are

  \begin{equation*} 3\times2\times1 = 6 \end{equation*}

  different ways to put three names into three coloured boxes!

• Step 3: From step 1, we know that there are 120 different ways the six players can be organized into three boxes. But, this method counts the same team multiple times (as in Figure 3.1.6). From step 2, we know that each team of three players can appear in 6 different arrangements within the three boxes. Since these arrangements all represent the same team, the number of unique teams can be calculated as:

  \begin{equation*} 120\div6=20 \end{equation*}

  Therefore, there are 20 different teams!

• Ask students to pay attention to mathematics (words, geometrical shapes, games, patterns, numbers). Ask students to pay attention to cultural, particularly Indigenous, aspects of the story. Ask students to make their own puzzle that includes double counting.