Adventures of Small Number: Classroom Resources

Write:

Ask students, which number should replace the question mark?

Followed by:

Followed by:

Discuss the idea of counting by “skipping”. Ask students if anyone has heard about “even” and “odd” numbers. If yes, ask the student to explain. If not, briefly introduce the idea of even and odd numbers in the following way:

Odd numbers: Start with \(1\) and count by skipping \(2\text{:}\)

\begin{equation*} 1, 3, 5, 7... \end{equation*}

Even numbers: Start with \(2\) and count by skipping \(2\text{:}\)

\begin{equation*} 2, 4, 6, 8... \end{equation*}

You may wish to follow this by asking the following: Is \(15\) an even or an odd number? Is \(16\) an even or an odd number?

Watch the movie. After watching the movie invite students to explain how Small Number found his answer.

Explore Small Number’s strategy by discussing a smaller case, say with four tipis and looking for the number \(30\text{.}\)

You may wish to create the following play:

Divide the class into groups of five students. In each group, ask four students to form a circle. Make sure that you leave an empty space matching the east point of the circle, and make a point about the cultural connection with the story¹. Tell those four students to stay still and ask the reminding student to act as Small Number.

\(\color{red}{Teacher}\) \(\color{red}{(to}\) \(\color{red}{Small}\) \(\color{red}{Number):}\) “Small Number, go around the circle and count aloud by pointing to each “tipi”: \(1, 2, 3, 4\text{.}\) When you finish, stand in the middle of the circle and ask each of your friends: If I continue counting, what will be your number?”

[Small Number goes around the circle and counts aloud: “\(1, 2, 3, 4\)”. Then they go to the middle of the circle and ask the students: “If I continue counting, what will be your number?” The students answer: “\(5\)”, “\(6\)”, “\(7\)”, “\(8\)”.]

\(\color{red}{Teacher}\) \(\color{red}{(to}\) \(\color{red}{the}\) \(\color{red}{group):}\) “How many students are there in the circle?” [A: \(4\)] “Think about your two numbers. What happens when you add the number four to your first number?” [A: “I get my second number.”]

\(\color{red}{Teacher}\) \(\color{red}{(to}\) \(\color{red}{Small}\) \(\color{red}{Number):}\) “Pick one number from \(9, 10, 11, 12\text{.}\)”

\(\color{red}{Teacher}\) \(\color{red}{(to}\) \(\color{red}{Small}\) \(\color{red}{Number,}\) \(\color{red}{after}\) \(\color{red}{waiting}\) \(\color{red}{a}\) \(\color{red}{bit):}\) “Which number did you pick?”

\(\color{blue}{Small}\) \(\color{blue}{Number:}\) “\(10\)” [assume]

\(\color{red}{Teacher}\) \(\color{red}{(to}\) \(\color{red}{Small}\) \(\color{red}{Number):}\) “If you continue counting which of your friends will be counted as the number \(10\text{?}\)”

\(\color{blue}{(Script}\) \(\color{blue}{1)}\) \(\color{blue}{Small}\) \(\color{blue}{Number:}\) “I am not sure.” Teacher (to the class): “Who would like to help Small Number?” (and then follow up accordingly)

\(\color{blue}{(Script}\) \(\color{blue}{2)}\) \(\color{blue}{Small}\) \(\color{blue}{Number:}\) [correctly] “I think it’s Alice.” Teacher (to Alice): “Alice, do you agree?” (and then follow up accordingly)

\(\color{blue}{(Script}\) \(\color{blue}{3)}\) \(\color{blue}{Small}\) \(\color{blue}{Number:}\) [incorrectly] “I think it’s Bob.” Teacher (to Bob): “Bob, do you agree?” (and then follow up accordingly)

[In any case, make sure that students recognize that the number \(10\) is obtained as \(6 + 4 = 10\text{.}\) If you wish, you may follow up with the question: “What would happen if Small Number picked the number \(12\text{?}\)”]

\(\color{red}{Teacher}\) \(\color{red}{(to}\) \(\color{red}{the}\) \(\color{red}{group):}\) “Can you tell me your number if Small Number goes one more time around the circle?” [A: \(13, 14, 15, 16\)]

\(\color{red}{Teacher}\) \(\color{red}{(to}\) \(\color{red}{the}\) \(\color{red}{group):}\) “So we are counting by skipping by which number?” [A: \(4\)]

\(\color{red}{Teacher}\) \(\color{red}{(to}\) \(\color{red}{the}\) \(\color{red}{group):}\) “Can you tell me who will be the number \(30\text{?}\)”

If you feel that the students would like to continue playing, increase the number of tipis.

Ask students to explain the difference between “even” and “odd” numbers.

If needed, briefly introduce the idea of even and odd in the following way:

Odd numbers: Numbers that leave a remainder of \(1\) when divided by \(2\text{.}\)

Even numbers: Numbers that do not leave a remainder when divided by \(2\text{.}\)

Ask your students to pick a number between \(3\) and \(10\text{.}\) [Say they pick the number \(6\text{.}\)]

Ask students to explain in their own words how they understand the phrase: “a remainder when divided by \(6\)”.

[Answers should be something like: “The leftover when I divide the number by \(6\text{.}\)” Or, “The difference between the number and the biggest multiple of \(6\) that is less than or equal to that number.”]

Divide students in groups and ask each group to complete the following table:

Answer:

Ask students the following questions:

• What are the possible remainders when a number is divided by \(6\text{?}\)

  [A: \(0, 1, 2, 3, 4, 5\)]

• When you look at the table, do you see a pattern that the reminders follow?

  [A: They go in order from \(1\) to \(5\text{,}\) then \(0\text{,}\) and then the pattern repeats.]

Watch the movie. After watching the movie invite students to discuss how Small Number found his answer.

You may wish to create the following play:

Divide the class in groups of eight students. In each group, ask seven students to form a circle. Make sure that you leave an empty space matching the east point of the circle and make a point about the cultural connection with the story². Tell those seven students to stay still and ask the reminding student to act as Small Number.

\(\color{red}{Teacher}\) \(\color{red}{(to}\) \(\color{red}{Small}\) \(\color{red}{Number):}\) “Small Number, go around the circle and count aloud by pointing to each “tipi”: \(1, 2, 3, 4, 5, 6, 7\text{.}\) When you finish, stand in the middle of the circle and ask each of your friends: Please think carefully: Recall the number that I assigned to you. What is the remainder of your number when divided by \(7\text{?}\)”

[Small Number goes around the circle counting and assigns numbers \(1, 2, 3, 4, 5, 6, 7\) to each “tipi”. Then Small Number goes to the middle of the circle and asks the students: “Please think carefully: Recall the number that I assigned you. What is the remainder of your number when divided by \(7\text{?}\)” The students answer: “\(1\)”, “\(2\)”, “\(3\)”, “\(4\)”, “\(5\)”, “\(6\)”, “\(0\)”.]

\(\color{red}{Teacher}\) \(\color{red}{(to}\) \(\color{red}{the}\) \(\color{red}{group):}\) “How many students are there in the circle?” [A: \(7\)] “Think about the number that Small Number assigned you and the reminder of that number when divided by \(7\text{.}\) Now add \(7\) to your assigned number.” [Teacher waits for a moment to allow students to do their calculations.] “Which number did you get?” [A: “\(8\)”, “\(9\)”, “\(10\)”, “\(11\)”, “\(12\)”, “\(13\)”, “\(14\)”] “What is the remainder of your new number when divided by \(7\text{?}\)” [A: “\(1\)”, “\(2\)”, “\(3\)”, “\(4\)”, “\(5\)”, “\(6\)”, “\(0\)”] “What do you notice?” [A: “It is the same!”]

\(\color{red}{Teacher}\) \(\color{red}{(to}\) \(\color{red}{Small}\) \(\color{red}{Number):}\) “Pick one number between \(15\) and \(21\text{.}\)”

\(\color{red}{Teacher}\) \(\color{red}{(to}\) \(\color{red}{Small}\) \(\color{red}{Number,}\) \(\color{red}{after}\) \(\color{red}{waiting}\) \(\color{red}{a}\) \(\color{red}{bit):}\) “Which number did you pick?”

\(\color{blue}{Small}\) \(\color{blue}{Number:}\) “\(18\)” [assume]

\(\color{red}{Teacher}\) \(\color{red}{(to}\) \(\color{red}{Small}\) \(\color{red}{Number):}\) “If you continue counting which of your friends will be counted as the number \(18\text{?}\)”

\(\color{blue}{(Script}\) \(\color{blue}{1)}\) \(\color{blue}{Small}\) \(\color{blue}{Number:}\) Small Number: “I am not sure.” Teacher (to the class): “Who would like to help Small Number?” (and then follow up accordingly)

\(\color{blue}{(Script}\) \(\color{blue}{2)}\) \(\color{blue}{Small}\) \(\color{blue}{Number:}\) [correctly] “I think it’s Alice.” Teacher (to Alice): “Alice, do you agree?” (and then follow up accordingly)

\(\color{blue}{(Script}\) \(\color{blue}{3)}\) \(\color{blue}{Small}\) \(\color{blue}{Number:}\) [incorrectly] “I think it’s Bob.” Teacher (to Bob): “Bob, do you agree?” (and then follow up accordingly)

[In any case, make sure that students recognize that the reminder of the number \(18\) when divided by \(7\) equals to \(4\text{.}\) If you wish you may follow up with the question: “What would happen if Small Number picked the number \(20\text{?}\)”]

\(\color{red}{Teacher}\) \(\color{red}{(to}\) \(\color{red}{the}\) \(\color{red}{group):}\) “Say that Small Number continues counting by going around the circle. What happens with the reminder of your newly assigned number when divided by \(7\text{?}\)” [A: “It stays the same!”]

\(\color{red}{Teacher}\) \(\color{red}{(to}\) \(\color{red}{the}\) \(\color{red}{group):}\) “Yes! isn’t that amazing? It does not matter how many times Small Number goes around the circle, each tipi will be assigned a number with the same reminder when divided by \(7\text{!}\)”

\(\color{red}{Teacher}\) \(\color{red}{(to}\) \(\color{red}{the}\) \(\color{red}{group):}\) “Can you tell me who will be the number \(100\text{?}\)”

You may wish to ask the following question:

\(\color{red}{Teacher}\) \(\color{red}{(to}\) \(\color{red}{the}\) \(\color{red}{class):}\) “Do you know why each day is divided into \(24\) hours?”

\(\color{blue}{(Script}\) \(\color{blue}{1)}\) \(\color{blue}{Alice:}\) “I know!” Teacher invites Alice to share her story with the class.

\(\color{blue}{(Script}\) \(\color{blue}{2)}\) \(\color{blue}{no}\) \(\color{blue}{student}\) \(\color{blue}{replies:}\) Teacher may say the following:

Our \(24\)-hour day comes from the ancient Egyptians who divided day-time into \(10\) hours. They measured time with devices such as shadow clocks, and added a twilight hour at the beginning and another one at the end of the day-time. Night-time was divided in \(12\) hours, based on observations of the stars.

Source and more details: Why are there \(24\) hours in a day?

\(\color{red}{Teacher}\) \(\color{red}{(to}\) \(\color{red}{the}\) \(\color{red}{class):}\) “So the reason that we have numbers \(1\) to \(12\) on the clock face is based on an ancient way of observing and measuring time. Now, I would like to show you some interesting mathematics that happens on the clock face.”

\(\color{red}{Teacher}\) \(\color{red}{(to}\) \(\color{red}{the}\) \(\color{red}{class):}\) “Here is app called: Analog Clock. What will happen if I click on the button - \(1\) Hour?”

\(\color{blue}{Bob:}\) “The clock will show \(11\text{!}\)”

[Teacher clicks and time changes to \(11\)] “Very good Bob.” To the class: “And you are not surprised because \(12 - 1 = 11\)”.

\(\color{red}{Teacher}\) \(\color{red}{(to}\) \(\color{red}{the}\) \(\color{red}{class):}\) “What will happen if I click on the button + \(1\) Hour three times?”

\(\color{blue}{Candice:}\) “The clock will show \(2\text{!}\)”

[Teacher clicks on the button three times and time changes to \(2\)] “Excellent thinking, Candice.” To the class: “But what kind of mathematics is this: \(11 + 3 = 2\text{?}\)”

\(\color{red}{Teacher}\) \(\color{red}{(to}\) \(\color{red}{the}\) \(\color{red}{class):}\) “Help me to complete the following table:”

[Solution:]

\(\color{red}{Teacher}\) \(\color{red}{(to}\) \(\color{red}{the}\) \(\color{red}{class):}\) “What do you observe from this table?” [Possible answers: “The number is always between \(1\) and \(12\text{.}\)” “If we add \(12\) then we get the same number.” “We got the same number when we added \(3\text{,}\) \(15\text{,}\) and \(27\) to \(6\text{.}\)” “When we add \(10\) hours to \(6\) we get \(4\) and when we add \(6\) hours to \(10\text{,}\) we get \(4\) again. It looks like that the order in which we add hours does not matter.”]

\(\color{blue}{Script}\) \(\color{blue}{1:}\) Dave: “I noticed that \(5 + 8 = 13 = 12 + 1\text{.}\) Also, \(6 + 10 = 16 = 12 + 4\text{.}\) It looks like we are writing down the remainder when the number is divided by 12.” Teacher congratulates Dave: “This is a great observation, Dave. Congratulations!” Then, to the class: “Do you agree with Dave? Should we check if this works for other numbers? And does \(12 + 12\) follow Dave’s pattern?” After moderating the class discussion: “Dave, what if we replace \(12\) by \(0\text{?}\) In this case, would we always get the remainder when divided by \(12\text{?}\)” (And then the teacher continues accordingly.)

\(\color{blue}{Script}\) \(\color{blue}{2:}\) No student mentions the remainders. Teacher may follow Dave’s words from Script 1: “Notice that \(5 + 8 = 13 = 12 + 1\text{.}\) Also, \(6 + 10 = 16 = 12 + 4\text{.}\) It looks like that we are writing down the remainder when the number is divided by \(12\text{.}\) Do you agree? Is there an exception to this rule?” (Students notice that \(9 + 3\) gives \(12\text{,}\) not the remainder \(0\text{.}\))

What if we take a clock that has the number \(0\) instead of the number \(12\text{?}\) See Figure 1.1.5.

\(\color{red}{Teacher}\) \(\color{red}{(to}\) \(\color{red}{the}\) \(\color{red}{class):}\) “Use the new clock to complete the following table:”

“For example, what happens if the current time is 5:00 and we add \(8\) hours?” (See Figure 1.1.6.)

[Solution:]

“Believe it or not, this kind of addition has inspired a part of mathematics called modular arithmetic. Just imagine a clock that has any number of hours on its face. For example, see Figure 1.1.7 for a clock with seven hours.”

“Let us see how this works! Who will volunteer to complete this table?”

[Solution:]

“In this world of modular arithmetic we can also do multiplication. For example, if we use the clock with seven hours, then \(2 \times 5 = 5 + 5 = 10 = 7 + 3\) and we say that (on this seven hour clock) \(2 + 5\) equals to \(3\text{.}\)”

“Let us complete this table:”

For example:

[Solution:]

(Reference: Modular Arithmetic)

Watch the movie. After watching the movie invite students to discuss how Small Number found his answer.

You may wish to help them by using a diagram such as Figure 1.1.8.

Answer: Since there are seven tipis in the circle, we think about this circle the same way we think about a clock with seven hours on its face. We take that Grandma’s tipi, the one that is just south form the entrance, is counted as \(1\text{.}\)

Observe that \(100 = 1 + 99\) and that \(99 = 14 \times 7 + 1\) Hence, if we think in the way of modular arithmetic, i.e. the way we add numbers on a clock face, then \(1 + 99\) is the same as \(1 + 1 = 2\text{.}\) Therefore, the hundredth tipi will be the one that is south from Grandma’s tipi.

You may wish to discuss the following:

Ask students to pay attention to mathematics (words, geometrical shapes, patterns, numbers). Ask students to pay attention to cultural, particularly Indigenous, aspects of the story. Ask students to make their own game that includes skip-counting/remainders/modular arithmetic.