Math in Waʻa: Examples of math in waʻa and navigation

If a star rises in House Nālani Koʻolau, it will travel over the sky and set in the same House Nālani but in the quadrant Hoʻolua (symmetry in math). Similarly, if a wind or current enters a canoe in House Nālani Koʻolau, it will exit the canoe in House Nālani Kona (symmetry). Show this movement from one house, to the center (canoe), and continuing to the same house but different quadrant.

As we explore the Star Compass, it is important to know the relationship between Houses and their angles.

How many houses are in the Star Compass? Ans: 32 Houses.

How many degrees are in one cifcle? Ans: ${360}^{\circ }$

To find the angle associated with each house, we divide the total number of degrees in a circle (360) by the number of houses on the Star Compass (32):

If you prefer, you can obtain the same answer through halving.

Start with the total degrees in a circle, which is ${360}^{\circ }\text{,}$ and the total number of houses in the Star Compass, which is 32.

Halve ${360}^{\circ }$ to get ${180}^{\circ }\text{.}$

Halve the number of houses (32) to get 16 houses.

Continue halving:

Halve ${180}^{\circ }$ to get ${90}^{\circ }\text{.}$

Halve 16 houses to get 8 houses.

Halve ${90}^{\circ }$ to get ${45}^{\circ }\text{.}$

Halve 8 houses to get 4 houses.

Halve ${45}^{\circ }$ to get ${22.5}^{\circ }\text{.}$

Halve 4 houses to get 2 houses.

Halve ${22.5}^{\circ }$ to get ${11.25}^{\circ }\text{.}$

Each house on the Star Compass is ${11.25}^{\circ }\text{.}$

We will first construct the quadrant Koʻolau of the Star Compass. On the board, draw a blank quadrant Koʻolau. Label Hikina and ʻĀkau.

Now, focus on the names of the houses in each quadrant (specifically Koʻolau). Identify the name of the house in the middle (Manu) and the house halfway between Manu and Hikina (ʻĀina).

Continue the process of halving to locate all the houses in the quadrant.

For further practice, have students independently create the quadrant Hoʻolua using the same method.

By completing the construction of one quadrant of the Star Compass, students have gained a solid understanding of the interconnectedness and symmetry within the Hawaiian Star Compass. They can now comfortably locate houses by finding the midpoint between two other houses. This hands-on activity not only reinforces mathematical concepts but also provides a tangible connection to the cultural and navigational significance of the Star Compass

In this part, we will determine the angle for each house on the Star Compass.

On the board, with students following along on their papers, draw the the Koʻolau quadrant, setting the Baseline Angle: If we assume that Hikina is $0^{\circ }\text{,}$ what angle would ʻĀkau be? Answer: ${90}^{\circ }$

Now, recall when we first constructed the Star Compass, what house was located halfway between Hikina and ʻĀkau? Ans: Manu. If Manu is halfway between $0^{\circ }$ and ${90}^{\circ }\text{,}$ what is the angle for Manu? Ans: ${45}^{\circ }\text{.}$

What house was located halfway between Hikina and Manu? Ans: ʻĀina. If ʻĀina is halfway between $0^{\circ }$ and ${45}^{\circ }\text{,}$ what is the angle for ʻĀina? Ans: ${22.5}^{\circ }$

Repeat the Process: Continue applying these steps to determine the angles for all houses in the Koʻolau quadrant.

By following this method, students will not only grasp the angles associated with each house but also reinforce their understanding of angular relationships on the Star Compass. This activity covers the topic of dividing integers, providing a practical application of mathematical concepts in a real-world context.

What do you notice about the angles for the houses in the Star Compass? What is the angle between Hikina and Lā? What about between Lā and ʻĀina? Ans: Emphasize that all the spacing between each house is ${11.25}^{\circ }\text{.}$

Where have we seen ${11.25}^{\circ }\text{?}$ Ans: This is what we calculated earlier. Each house is ${11.25}^{\circ }\text{.}$

So if we know that Hikina is $0^{\circ }$ and each house is ${11.25}^{\circ }\text{,}$ what would the angle be for Lā? Ans: $0^{\circ }+{11.25}^{\circ }={11.25}^{\circ }\text{.}$

If Lā is at ${11.25}^{\circ }$ and each house is ${11.25}^{\circ }\text{,}$ where would ʻĀina be? Ans: ${11.25}^{\circ }+{11.25}^{\circ }={22.5}^{\circ }$

We can continue this process of adding ${11.25}^{\circ }$ to the previous house to find the angle of the next house.

This is an alternative way to finding the angles of the houses.