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Section 2.1 Small Number and the Old Canoe - Classroom Guide

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Suggested Grades: 3 鈥� 7

Subsection 2.1.1 Mathematics

  1. counting
  2. patterns
  3. shapes
  4. mathematical thinking
  5. inclusion-exclusion formula

Subsection 2.1.2 Mathematical Vocabulary

  • small, number, 5-year-old, down, circle, the farthest, surface, quickly, to go far, smooth, flat, oval shaped, wanders far along the shore, tall, falling headfirst, into, stands up, looks around, stand around, along its smooth shape, very old, very big, How many?, How many generations ago?, long time, run back, huge, at least a hundred years old, the fastest, two of his brothers, all (the sons of my grandfather), three old totem poles, in front of the longhouse, each of them, by one of (my uncles), just before, two, three, four, five or more.

Subsection 2.1.3 Cultural Components

  1. Indigenous:
    1. Grandpa鈥檚 hat: Many First Nations along the Pacific Northwest Coast have tradition of wearing woven hats. Learn more about how these hats are made. You may start with the 丁香园AV Museum of Archeology: Haida Hats.
    2. Grandpa鈥檚 blanket: Maybe you鈥檝e noticed that some Indigenous people wear blankets during various ceremonies. Learn more about traditional button blankets. You may start here: .
    3. Canoe: Canoes have been one of the most important means of transportation along the Pacific Coast. Learn more about the traditional use of canoes and the ways they were built. You may wish to start with: the 丁香园AV Museum of Archeology (), the 丁香园AV Bill Read Centre (Northwest Coast Canoes), and the Canadian Encyclopedia ().
    4. Totem poles: Draw and describe a totem pole that you have recently seen. Learn more about traditional totem poles. You may start with the 丁香园AV Museum of Archeology: Building a totem pole.
    5. Feast dish: A potlatch is a feast practiced by Indigenous peoples of the Pacific Northwest Coast. Traditionally, during a potlatch food was served in so-called 鈥渇east dishes鈥�. Learn more about feast dishes. You may wish to start with the UBC Museum of Anthropology: .
  2. General:
    1. Have you ever tried skipping stones across the water? If yes, share your story. Do you think that the shape of the stone matters?
    2. Have you ever been in a canoe or a kayak? If yes, share your story.
    3. Have you ever made your grandparents worried by something that you did?
    4. What do you know about your great-grandparents?
    5. Have you ever encountered a mathematical question that had more than one possible answer?

Subsection 2.1.4 Mathematical Observations (Video)

Opening scene: Notice the shape of the canoe and the shapes of the carvings on its side.

0:15 - Notice the shapes of Grandpa鈥檚 hat, the mountain peaks, and the trees. How long is Small Number鈥檚 stick?

0:24 - How many boys are there? Notice their reflections in the water.

0:38 - 0:41 - What is the shape of the stone? Notice the pattern made by the skipping stone. How many times did it skip? Notice the perspective. 鈥淔or a stone to go far it needs to be smooth, flat, and oval shaped.鈥�

1:00 - What is the shape of the canoe?

1:17 - 鈥淗ow many people do you think [the canoe] could hold?鈥�

1:22 - 鈥淗ow many generations ago was [the canoe] built?鈥�

1:43 - Notice the geometrical shapes, the angles, and the perspective. Which objects are parallel to each other? Perpendicular?

1:47 - 鈥淕randpa is carving the surface of a huge wooden dish.鈥� What is the shape of the dish? Notice the patterns and shapes in the carving.

2:05 - 鈥淚 found an old canoe down on the beach! It must be at least a hundred years old!鈥�

2:10 - 2:16 - 鈥淚 know that canoe. It was once the fastest canoe in our village. It was built by my father and two of his brothers.鈥� How many bothers were building the canoe? What is the shape of the tools the brothers were using?

2:21 - 鈥淎ll the sons of my grandfather were known as great wood carvers.鈥�

2:23 - 2:29 鈥淵ou know those three old totem poles in front of the longhouse? Each of them was built by one of my uncles.鈥� Notice the shapes and heights of the totem poles.

2:37 - What is the shape of the log? The diameter of the log? The circumference of the log?

2:42 - Notice the shapes and the perspective.

Subsection 2.1.5 Answer: Why did Small Number think that his great-grandpa might have two, three, four, five or more brothers?

Grandpa: 鈥淚 know that canoe. It was once the fastest canoe in our village. It was built by my father and two of his brothers.鈥�

Small Number thought:

鈥淪o, my great-grandpa had at least two brothers!鈥� (See 贵颈驳耻谤别听2.1.1)

Figure 2.1.1. Great-Grandpa has at least two brothers!

Grandpa: 鈥淵ou know those three old totem poles in front of the longhouse? Each of them was built by one of my uncles.鈥� (See 贵颈驳耻谤别听2.1.2)

Figure 2.1.2. Each totem pole was built by one of Grandpa鈥檚 uncles!

Small Number continued:

鈥淢aybe the totem poles were built by the same two brothers that built the canoe together with my great-grandpa? Maybe my great-grandpa didn鈥檛 have any other brothers? So therefore, maybe my great-grandpa had only two brothers?鈥� (See 贵颈驳耻谤别听2.1.3)

Figure 2.1.3. Maybe Great-Grandpa only had two brothers?

Small Number scratched his head:

鈥淢aybe there was another brother? Maybe each totem pole was built by a different brother? It could be that my great-grandpa had three brothers.鈥� (See 贵颈驳耻谤别听2.1.4)

Figure 2.1.4. Maybe Great-Grandpa had three brothers?

Small Number had another idea:

鈥淢aybe my great-grandpa had four brothers? Maybe each totem pole was built by a different brother, but only one was built by one of Grandpa鈥檚 uncle who built the canoe?鈥� (See 贵颈驳耻谤别听2.1.5)

Figure 2.1.5. Maybe Grandpa had three brothers?

Small Number was getting sleepy:

鈥淏ut maybe each totem pole was built by a different brother and none of those brothers worked on the canoe? It could be that my great-grandpa had five brothers. Maybe there were other brothers? I have to ask Grandpa tomorrow how many brothers his father had. Two, three, four, five or more鈥︹��

Subsection 2.1.6 Discussion/Activities

  • Start with the following exercise:

    Prepare six cards: on three cards draw the same image of a canoe. On each of the remaining cards draw a (different) totem pole.

    Continue by asking: 鈥淲ho would like to be Small Number鈥檚 great-grandpa?鈥� (Hand over one of the 鈥渃anoe鈥� cards to the volunteer, Alice.)

    Next: 鈥淣ow I need two of you to be Great-Grandpa鈥檚 brothers who helped him build the canoe.鈥� (Hand over the other two 鈥渃anoe鈥� cards to Bob and Carol.)

    To the class: 鈥淟et us talk about the phrase at least two. For example, I watched each Small Number film at least two times. How do you understand that? How many times did I watch each Small Number film?鈥�

    [Moderate the conversation. Make sure that students understand that 鈥渁t least two鈥� means 鈥渢wo or more鈥�.]

    To the class: 鈥淒o you agree that Small Number鈥檚 great-grandpa had at least two brothers?鈥�

    To the class: 鈥淒o you remember when Grandpa said: 鈥榊ou know those three old totem poles in front of the longhouse? Each of them was built by one of my uncles.鈥�鈥�

    Show the three cards with the totem poles to the class: 鈥淚f Small Number鈥檚 great-grandpa had only two brothers, what should I do with these three cards?鈥�

    [Moderate the conversation. If needed lead the class to the conclusion that one of the already established brothers (Bob and Carol, those with a 鈥渃anoe鈥� card) should get two 鈥渢otem鈥� cards. Hand over one card to Bob and two to Carol.]

    Conclude: 鈥淚t is possible that Small Number鈥檚 great-grandpa had exactly two brothers?鈥� [The class should answer 鈥淵es!鈥漖

    Take one totem card from Carol and give it to Dave. 鈥淣ow, Bob is a brother who built a canoe and one of the totem poles, Carol is a brother who also built a canoe and one of the totem poles, and Dave is a brother who built the third totem pole. How many brothers are there now?鈥� [The class should answer 鈥淭hree!鈥漖

    To the class: 鈥淚s it possible that Great-Grandpa had four brothers?鈥� [This should lead to the conclusion that Bob鈥檚 totem card should be given to Erin.]

    To the class: 鈥淲hat about five brothers?鈥� [This should lead to the conclusion that Carol鈥檚 totem card should be given to Frank.]

    To the class: 鈥淲hat about more than five brothers?鈥� [This should lead to the conclusion that maybe there were other brothers, those who didn鈥檛 build either the canoe or one of the totem poles.]

  1. Start by reminding students how Venn diagrams are used to represent sets.

    You may wish to discuss that in mathematics we use the word set as a synonym for a 鈥渃ollection of objects鈥�. You may also wish to remind (or introduce to) the class some basic terminology related to sets.

    For example, consider the following sets:

    \(A\) = {students who were born in June} = {Alice, Bob, Carol}

    \(B\) = {students who were born in July} = {Dave, Erin, Frank}

    \(C\) = {students with an older brother} = {Alice, Erin, Frank}

    \(D\) = {students with a younger sister} = {Dave, Frank}

    \(E\) = {students who play soccer} = {Alice, Erin, Frank}

    These five sets can be represented by Venn diagrams. (See 贵颈驳耻谤别听2.1.8)

    Figure 2.1.8. Five sets

    Continue by observing:

    1. Sets \(A\) and \(B\) do not have any elements in common. [If you wish you may introduce the terminology, 鈥淺(A\) and \(B\) are two disjoint 蝉别迟蝉鈥�.闭

    2. Sets \(A\) and \(C\) have only one element in common: Alice. See 贵颈驳耻谤别听2.1.9. [If you wish you may introduce the phrase 鈥淎lice is the only element in the intersection of \(A\) and \(C\text{.}\)鈥� You may introduce the notation \(A\) \(\cap\) \(C\text{.}\)]

      Figure 2.1.9. Alice is the only element in the intersection of \(A\) and \(C\)

    3. Observe that sets \(B\) and \(C\) and sets \(B\) and \(D\) have exactly two elements in common. See 贵颈驳耻谤别听2.1.10.

      Figure 2.1.10. Erin and Frank are in the intersection of \(B\) and \(C\text{,}\) Alice and Dave are not; Dave and Frank are in the intersection of \(B\) and \(D\text{,}\) Erin is not

      Observe the difference between these two cases: All elements of the set \(D\) belong to the set \(B\text{.}\) [If you wish you may introduce the phrase 鈥淺(D\) is a subset of \(B\)鈥�. Also, you may introduce the appropriate notation \(D\) \(\subset\) \(B\text{.}\)]

    4. Observe that sets \(C\) and \(E\) have all of their elements in common. We say that 鈥渟ets \(C\) and \(E\) are equal鈥�.

  2. Watch the movie.

    Repeat the question from the end of the movie: 鈥淲hy did Small Number think that his great-grandpa might have two, three, four, five or more brothers?鈥� Make sure that everyone understands the question. Then invite students to share their thoughts.

    Say: 鈥淲hat if we try to use Venn diagrams to help Small Number?鈥�

    Continue: 鈥淲hat are the sets of interest that may help in finding the answer?鈥� [Allow students to suggest 鈥渟ets of interest鈥�. You should end up with the following three sets. Students will probably need your guidance.]:

    \(B\) = {all of Great-Grandpa鈥檚 brothers}

    \(C\) = {Great-Grandpa鈥檚 brothers who built the canoe}

    \(T\) = {Great-Grandpa鈥檚 brothers who built the totem poles}

    Start by noticing that sets \(C\) and \(T\) are subsets of the set \(B\text{.}\)

    Ask: 鈥淒o we know how many elements the set \(C\) has?鈥� [The answer should be 鈥渢wo鈥�.]

    Next, ask: 鈥淒o we know how many elements the set \(T\) has?鈥� [The answer should be: 鈥淲e don鈥檛 know鈥�.]

    Remind students about Grandpa鈥檚 words: 鈥淵ou know those three old totem poles in front of the longhouse? Each of them was built by one of my uncles.鈥� Observe that Grandpa used the word 鈥渦ncles鈥� and ask: 鈥淒o you agree that there was more than one uncle involved in building the totem poles?鈥� [Students may agree with this or they may not. It does not matter. Go with the majority opinion.]

    Ask the class: 鈥淗ow do you understand Grandpa鈥檚 words: 鈥楨ach of them was built by one of my uncles鈥�? Is it possible that one brother built two totem poles?鈥� [Again, let students share their thinking. Lead students to the conclusion that there were at most three elements in set \(T\text{.}\)]

    Your strategy should be to discuss the following five cases:

    1. Sets \(C\) and \(B\) are equal. In this case the set \(T\) also must be equal to the set \(C\text{.}\) See 贵颈驳耻谤别听2.1.11.
      Figure 2.1.11. Two brothers: \(B\) = \(C\) = \(T\)

    2. Set \(C\) is a subset of the set \(T\text{,}\) but they are not equal. See 贵颈驳耻谤别听2.1.12.

      Figure 2.1.12. Three brothers: \(C\) is a subset of \(T\)

    3. The set \(T\) has three elements and sets \(C\) and \(T\) have one element in common. Make sure that you warn students about avoiding double-counting. 贵颈驳耻谤别听2.1.13.

      Figure 2.1.13. Four brothers: \(C\) and \(T\) have one element in common

    4. The set \(T\) has three elements and sets \(C\) and \(T\) have no elements in common. 贵颈驳耻谤别听2.1.14.

      Figure 2.1.14. Five brothers: \(C\) and \(T\) have no elements in common

    5. The set \(T\) has three elements and sets \(C\) and \(T\) have no elements in common, but there is another brother. 贵颈驳耻谤别听2.1.15.

      Figure 2.1.15. More than five brothers: \(C\) and \(T\) have one element in common

  • 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 the cultural, particularly Indigenous, aspects of the story. Ask students to make their own game that includes double counting.

Subsection 2.1.7 Challenge

Suppose there are 30 students, 10 who have a dog and 15 who have a cat. If there are 5 students that have both a dog and a cat, how many students have neither a dog nor a cat as a pet?