CCSS.MATH.CONTENT.6.NS.C.6: Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.
CCSS.MATH.CONTENT.6.NS.C.6.A: Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., -(-3) = 3, and that 0 is its own opposite.
CCSS.MATH.CONTENT.6.NS.C.6.B: Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
CCSS.MATH.CONTENT.6.NS.C.6.C: Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.
CCSS.MATH.CONTENT.8.G.A.3: Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
Students enter this unit with a basic understanding of the Kūkuluokalani, the Hawaiian Star Compass. To begin, we will briefly review the quadrants, cardinal points, and house locations of the star compass.
Knowing precise direction is very important in navigation and in many aspects of modern life. Every location on Earth can be described by two numbers: how far you are east or west (longitude) and how far you are north or south (latitude). These measurements give your exact position on the globe. Technologies like GPS (Global Positioning System), satellite mapping, weather forecasting, shipping, and air travel all depend on these precise coordinates.
Imagine standing at the center of the star compass. If you take five steps toward the east and then two steps toward the north, you would find yourself in the house ʻĀina Koʻolau, which lies in the Koʻolau (northeast) quadrant. Now, if you instead take five steps west and two steps north, you end up in the house ʻĀina Hoʻolua, located in the Hoʻolua (northwest) quadrant. Similarly, taking five steps east and two steps south leads you to ʻĀina Malanai (southeast), while five steps west and two steps south places you in ʻĀina Kona (southwest). Notice that every movement can be described by how far you move in the east-west direction followed by how far you move in the north-south direction. Your combination of these two directions determines which house and quadrant you end up in.
In mathematics, we use a similar system to locate positions on a graph, called the Cartesian coordinate system. In this system, the east-west direction is referred to as the x-direction, while the north-south direction is called the y-direction. To distinguish between directions:
Moving east is represented by a positive x-value.
Moving west is represented by a negative x-value.
Moving north is represented by a positive y-value.
Moving south is represented by a negative y-value.
At the center is the origin, where both x and y are zero (0,0). The vertical line is called the y-axis(north-south direction), and the horizontal line is the x-axis (east-west direction).
We can now describe our earlier examples using ordered pairs (x, y), where x represents the east-west movement and y represents the north-south movement:
By using this system, we can describe any position on the plane. This entire system of plotting points using an east-west (x-axis) and north-south (y-axis) direction is called the Cartesian Coordinate System. The two crossing lines are called the coordinate axes, and together they form the coordinate plane. A coordinate is simply a pair of numbers, written as (x, y), that tells us how far to move along the x-axis (east or west) and the y-axis (north or south) to locate a point.
Draw a profile picture (side view) of Hōkūleʻa and of Hikianalia. Identify key features and assign coordinates to important points on your drawing. (Students can either free-hand draw these or we can provide a sketch to use)
Superimpose a simplified map of the Moananuiākea voyage route on a coordinate plane. Assign coordinates to key waypoints (e.g., Hawaiʻi, Tahiti, Rapa Nui, Aotearoa, Alaska). Use the coordinates to estimate distances between stops. Use either the distance formula or count units directly along the x- and y-axes.
Plot the point (5,3). For each reflection below:
Identify the coordinates of the reflected point.
Plot the reflected point on the coordinate plane.
Reflections:
Reflect the point across the x-axis.
Reflect the point across the y-axis.
Reflect the point first across the x-axis, then across the y-axis.
Reflect the point first across the y-axis, then across the x-axis.
The image below shows a sakman, a traditional single-outrigger canoe from the Northern Mariana Islands. These canoes were also known as flying proas for their speed. Use the Cartesian grid to identify and list all the ordered pairs that form the shape of the canoe.
