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Section 4.1 Small Number and the Skateboard Park - Classroom Guide

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Suggested Grades: 4 - 7

Subsection 4.1.1 Mathematics

  1. geometry - shapes, patterns, angles
  2. geometrical construction
  3. mathematics and sports
  4. mathematical thinking
  5. mathematics and physics

Subsection 4.1.2 Mathematical Vocabulary

  • small, number, a lot of, big, every, math homework, geometry problem, full angle, walks through, against, geometry, around, shapes, geometric shapes and patterns, geometer, perfect number, 360掳, 720掳, two ramps, same height, different shapes, 铿俛t, curved, inside of a cylinder, a lot of speed, front, spins around fully two times, jumping up and down

Subsection 4.1.3 Cultural Components

  1. Indigenous:
    1. Learn more about a young skateboarder named of the who was chosen to represent his Nation at the 14th Annual All Nations Skate Jam.
    2. Learn more about .
    3. Totem poles: Draw and describe a totem pole that you鈥檝e recently seen. Learn more about traditional totem poles. You may start with the 丁香园AV Museum of Archeology: Building a totem pole.
    4. Learn more about the Indigenous tradition of weaving. For example. look at this 丁香园AV Museum of Anthropology website.
  2. General:
    1. Learn more about , the father of geometry.
    2. Learn more about .
    3. Do you have a skateboard? Do you know how to skateboard?
    4. Learn more about the different types of that skateboarders use.
    5. Learn more about .
    6. Learn about , a Canadian skateboarding olympian.
    7. Learn more why a full circle is associated with the .

Subsection 4.1.4 Mathematical Observations (Video)

Opening scene: Notice the various shapes, such as: buildings, mountain peaks, wooden structures. Notice the perspective.

0:09 - Notice the various shapes, parallel lines, a sphere, patterns on the totem pole, the symmetry of the totem pole. How many windows are there?

0:16 - Notice the shapes, patterns, perspective, parallel lines, right angles, tiling, shapes and patterns on the rug.

0:43 - Notice the image on Full Angle鈥檚 shirt and the skateboard; observe the central symmetry. Notice the shape of the skateboard, the shape of the helmet, and the perspective.

0:51 - Do the images on the rug and Full Angle鈥檚 shirt have anything in common?

1:16 - Notice the geometry of the rug.

1:48 - Think about the geometrical problem that Small Number is solving.

2:25 - A taste of the projective geometry: Are the lines formed by the tiles on the ground parallel or would they meet if they were extended?

2:31 - 鈥淚 can do a 360掳 and a 720掳.鈥�

2:46 - 鈥淔ull Angle points to two ramps that are the same height, but different shapes. One is 铿俛t, and the other is curved like the inside of a cylinder.鈥�

3:00 - 3:08 - 鈥淎s the front of his board comes off the ramp, he grabs it and spins around fully two times before letting go and landing perfectly!鈥�

3:20 - 鈥淲ell, Small Number, then you better be ready to learn some geometry! You will have to learn to do a 360掳 铿乺st.鈥�

Subsection 4.1.5 Answer: How did Full Angle know which ramp to choose in order to have enough time in the air to spin around twice before landing?

This should be treated as an open ended question. The teacher should moderate the discussion to make students aware of the following:

Point 1

The meaning of the phrases 鈥�360掳鈥� and 鈥�720掳鈥�.

Point 2

To make two rotations (720掳) requires more time than to make one rotation (360掳).

Point 3

To stay longer in the air, a skateboarder needs to launch themselves very high.

Point 4

The shape of the ramp is related to the height that the skateboarder will reach.

  • Point 1 - Understanding Angles:

    Here is a possible scenario.

    Teacher to the class: 鈥淗ow do you understand the phrases 360掳 and 720掳?鈥�

    \(\color{blue}{Script}\) \(\color{blue}{1:}\)

    Alice (correctly): 360掳 means that you turn a full circle. 720掳 means that you turn two full circles.

    Teacher to Alice: 鈥淭hank you, Alice!鈥�

    Teacher to the class: 鈥淒o you agree with Alice?鈥� [Teacher moderates the follow up discussion accordingly.]

    \(\color{blue}{Script}\) \(\color{blue}{2:}\)

    Alice (incorrectly): 360掳 means that you turn two full circles. [Or any other incorrect answer.]

    Teacher to Alice: 鈥淭hank you, Alice!鈥�

    Teacher to the class: 鈥淒o you agree with Alice?鈥� [Teacher moderates the follow up discussion accordingly.]

    \(\color{blue}{Script}\) \(\color{blue}{3:}\)

    No student is replying or the discussion in Script 2 is not converging.

    Teacher to the class: 鈥淲hy don鈥檛 we start with the meaning of 90掳, i.e. with the measurement of a right angle?鈥� (贵颈驳耻谤别听4.1.2)

    Figure 4.1.2. 90掳

    Teacher to the class: 鈥淲hat about 180掳? Since \(180^o = 2 \times 90^o\text{,}\) this means two right angles which gives us a straight angle.鈥� (贵颈驳耻谤别听4.1.3)

    Figure 4.1.3. 180掳

    Teacher to the class: 鈥淣ow, what about 360掳? Since \(360^o = 2 \times 180^o\text{,}\) this means two straight angles which gives us a full angle.鈥� (贵颈驳耻谤别听4.1.4)

    Figure 4.1.4. 360掳

    Teacher concludes: 鈥淪o, 360掳 means that you turn full circle. Consequently, 720掳 means that you turn two full circles.鈥�

  • Point 2 - Time of Flight:

    Teacher to the class: 鈥淣ow, I鈥檇 like you to stand up and face the window.鈥�

    [After everyone is standing facing the window.] 鈥淭hink about a line that is perpendicular to the 铿俹or and passes through the top of your head.鈥�

    [After waiting for a moment] 鈥淒o you know what I mean? Do you have any questions?鈥�

    [Continues after possibly answering students鈥� questions] 鈥淣ow, I鈥檇 like you to slowly rotate around this imaginary line [that is perpendicular to the 铿俹or and passes through the top of your head] for 360掳, i.e. turn around till you face the window again. Is this OK? Do you understand what I鈥檇 like you to do?鈥�

    [Continues after possibly answering students鈥� questions] 鈥淎lso, I鈥檇 like to start counting once you start rotating. Remember the number that you reach once you stop rotating.鈥�

    鈥淎re you ready? Go!鈥�

    [After everyone is 铿乶ished] 鈥淏ob, What was the the number that you reached?鈥�

    Bob: [for example] 鈥淚 counted to 9.鈥�

    Teacher: 鈥淐arol, what did you get?鈥� [Teacher makes sure that students associate a number with the rotation.]

    [After a short break] 鈥淣ow, let us face the window again. This time I鈥檇 like you to do 720掳, i.e. to stop after two rotations. And, please remember to count the whole time. Try to keep the same speed as when you did 360掳.鈥�

    鈥淎re you ready? Go!鈥�

    [After everyone is 铿乶ished] 鈥淒ave, What was the the number that you reached?鈥� Dave [for example]: 鈥淚 counted to 17.鈥� 鈥淓rin, what did you get?鈥� [Teacher makes sure that students associate a number to this activity.]

    Teacher to the class: 鈥淎re you surprised that your count for 720掳 was higher than the count for 360掳?鈥�

    Class: 鈥淣辞!鈥�

    Teacher: 鈥淲ho would like to explain why the second count was higher than the 铿乺st one?鈥�

    Frank: 鈥淏ecause we needed more time to do 720掳 than 360掳.鈥�

    Teacher: 鈥淓xcellent thinking, Frank! Thank you!鈥�

    To the class: 鈥淒o you agree that Full Angle has to make sure that he stays longer in the air when he does a 720掳 than when he does a 360掳?鈥�

    Class: 鈥渊别蝉!鈥�

  • Point 3 - Air Time:

    For this activity you will need two quarters, two chairs and four volunteers. Place the chairs about 1.5 meters apart facing each other.

    Teacher to the class: 鈥淔or this experiment, I will need four volunteers.鈥� [Say that Gary, Heidi, Ivan and Judy volunteer to participate.]

    Teacher to the four volunteers: 鈥淕ary, please take this coin, climb on the chair, and extend your hand with the coin. Heidi, here is a coin for you. You will sit down on the other chair, and extend your hand with the coin. When I say Go! you will release the coin so that it drops on the ground. Don鈥檛 throw the coin, just open your hand and let it go.鈥�

    鈥淚van and Judy, your job is to look and listen carefully and to tell the class which coin hits the 铿俹or 铿乺st.鈥� 鈥淒o you have any questions?鈥�

    Teacher to the class: 鈥淒o you understand what we will do? Do you have any predictions? Which coin will hit the ground 铿乺st, Grace鈥檚 or Heidi鈥檚?鈥� [Class, possibly: 鈥淗eidi鈥檚!鈥� 鈥淕ary鈥檚!鈥� 鈥淪ame time!鈥� 鈥淣ot sure!鈥漖

    Teacher: 鈥淟et us check!鈥� [To the four volunteers:] 鈥淎re you ready?鈥� 鈥淕o!鈥�

    Figure 4.1.7. Coin drop

    Teacher to Ivan and Judy: 鈥淲hat did you observe? Which coin did hit the 铿俹or 铿乺st?鈥�

    Ivan and Judy: 鈥淚t was Heidi鈥檚 coin!鈥�

    Teacher to the class: 鈥淒o you agree? Are you surprised with this outcome? Would you like to repeat the experiment?鈥� [Teacher moderates discussion.]

    Teacher to the class: 鈥淲ho would like to explain why Gary鈥檚 coin needed more time than Heidi鈥檚 coin to hit the 铿俹or?鈥�

    Kelly [correctly]: 鈥淏ecause Gary was standing on the chair, his coin was further away from the 铿俹or than Heidi鈥檚 coin.鈥�

    Teacher to the class: 鈥淭hank you, Kelly! You are absolutely right! Later in your science and math classes you will learn that if an object falls from the height of \(d\) meters then the objects needs about \(\sqrt{\frac{d}{5}}\) seconds to hit the ground.鈥�

    鈥淵ou will need to know a bit more of physics and mathematics to fully understand what鈥檚 going on here. What I would like you to know is that this mathematical expression con铿乺ms our observation that the larger height \(d\text{,}\) the longer the falling time \(t\) of the object is.鈥�

    Teacher [concludes]: 鈥淪o, if Full Angle wants to stay in the air long enough to do a 720掳, he must launch himself very high.鈥�

  • Point 4 - Ramp Geometry:

    Teacher to the class: 鈥淭here are three basic types of the skateboard ramps.鈥�

    Kicker ramp

    The ramp surface is 铿俛t.

    Launch Ramp

    The ramp surface is curved

    Quarter Pipe

    The ramp surface looks almost like the inside of a cylinder.

    Figure 4.1.9. Which of these ramps is a quarter pipe?

    Teacher to the class: 鈥淥ne of the reasons that every child in the world who is lucky to go to school studies mathematics, is that mathematics is useful. We are going to use drawings of lines of different shapes to answer Small Number鈥檚 question.鈥�

    Teacher continues: 鈥淟ook at this drawing and imagine that each blue line is a different ramp. Then, the red dotted lines represent the path of the skateboarder. Which of these three ramps in the skateboard park should Full Angle choose鈥�

    Figure 4.1.10. Which of these ramps should Full Number choose?

    Class: 鈥淔ull Angle should choose the quarter pipe ramp because the he would launch himself very high in the air.鈥�

Subsection 4.1.6 Small Number鈥檚 Homework Problem

Small Number: Full Angle, can you help me with my homework. This question is so hard that even Perfect Number couldnt solve it.

Full Angle: I can try. What is the question?

Small Number: I have no idea. This diagram is all I got: (贵颈驳耻谤别听4.1.11)

Figure 4.1.11. Small Number鈥檚 homework problem

I don鈥檛 even know where to start.

Full Angle: I agree, it is confusing. Can you tell me what you see in this 铿乬ure?

Small Number: I see this circle, a bunch of line segments, many points, two angles, and a question mark

Full Angle: (laughing) That鈥檚 very good, Small Number.

Full Angle: (seriously) You know, when you have a problem like this, it is always helpful to annotate all of the signi铿乧ant points on your diagram. To avoid confusion, it is a good idea to use capital letters in the alphabetical order. To denote the centre of the circle people often use the capital letter \(O\text{.}\)

Small Number: Like this? (贵颈驳耻谤别听4.1.12)

Figure 4.1.12. Small Number uses capital letters to annotate all signi铿乧ant points on the diagram

Full Angle: Excellent work, Small Number! Do you observe any symmetry on your diagram?

Small Number: Yes, I see that the points \(A\) and \(B\) and the points \(D\) and \(E\) are symmetric to each other with respect to the line through the points \(O\) and \(C\text{.}\)

Full Angle: Very good! Remember that fact, Small Number. You will need it later. Now, can you tell me which angles are given?

Small Number: Oh, I see. I know that \(\angle AOB = 120\)掳 and \(\angle DCE = 40\)掳.

Full Angle: And what do you think that the red question mark stands for?

Small Number: For the [measure of] \(\angle DFE\text{?}\)

Small Number: (scratching his head) I still have no idea how to calculate this number.

Full Angle: (smiling) Patience, Small Number. We are almost there

Full Angle: What if you use your blue felt pen and colour the smaller of the two arcs with the endpoints A and B? Let us see if we can use that arc to calculate the measure of \(\angle ADB\text{.}\)

Small Number: Like this? (贵颈驳耻谤别听4.1.13)

Figure 4.1.13. Small Number colours \(\stackrel \frown {AB}\) and is trying to to calculate the measure of \(\angle ADB\)

Full Angle: Yes, very good. Now, do \(\angle ADB\) and \(\angle AOB\) have something in common?

Small Number: They kind of share \(\stackrel \frown {AB}\) .

Full Angle: Yes, they do! Next, look at the vertices of those two angles, the points \(O\) and \(D\text{.}\) What do you observe?

Small Number: The point \(O\) is the centre of the circle and the point \(D\) is a point on the circle.

Full Angle: Maybe you remember what those angles are called?

Small Number: I think that \(\angle AOB\) is called the central angle, but I don鈥檛 remember what we call \(\angle ADB\)

Full Angle: It is called an inscribed angle. Do you remember your teacher talking about the relationship between a central angle and an inscribed angle that share the same arc?

Small Number: I know that \(\angle ADB = 60\)掳! It was kind of funny how my teacher got excited when she was telling us that once when you choose the arc then it does not matter which point on the circle you pick as the vertex of the inscribed angle: That angle will be always exactly one-half of the corresponding central angle. So,

\begin{equation*} \angle ADB = \frac{1}{2} \times \angle AOB = \frac{1}{2} \times 120^o = 60^o \end{equation*}

Full Angle: I agree, it is kind of funny that something like that works for all points on the circle!

Full Angle: We need just one more thing and we are done. I鈥檇 like you to check if we can somehow connect \(\angle ADB\) and \(\triangle CDF\text{.}\) What if you colour in this triangle?

Small Number: Like this? (贵颈驳耻谤别听4.1.14)

Figure 4.1.14. Small Number writes \(\angle ADB = 600\) and colours \(\triangle CDF\)

Small Number: You should be a teacher, Full Angle! I know what to do next. Because \(\angle ADB\) is an exterior angle for \(\triangle CDF\) it must be:

\begin{equation*} \angle ADB = \angle DCF + \angle DFC \end{equation*}

I know this because of the symmetry:

\begin{equation*} \angle DCF = \angle ECF = \frac{1}{2} \times \angle DCE = \frac{1}{2} \times 40^o = 20^o \end{equation*}

Now it is easy:

\begin{equation*} \angle DFC = \angle ADB 鈭� \angle DCF = 60^o 鈭� 20^o = 40^o. \end{equation*}

Small Number: (very excited) Mom, I did it! The measure of the unknown angle is 80掳!

Small Number鈥檚 Mom and Full Angle look at each other, smiling.

Subsection 4.1.7 Challenge

Ask students to make up a problem that is related to circles and triangles and to share their solutions of the problem.

Prove that the measure of any inscribed angle is equal to one-half of the corresponding central angle.