Mathematics, moon phases, and tides

Mathematics, moon phases, and tides

Aboriginal and Torres Strait Islander people have long observed the phases of the Moon and used it to understand the tides and the effect on the environment around them.

Lunar phases and tides

In many Aboriginal and Torres Strait Islander traditions, the Moon is generally viewed as masculine whilst the Sun is generally feminine1. In Yolngu traditions of coastal Arnhem Land, Ngalindi is the Moon-man. Yolngu traditions describe water filling Ngalindi as he rises, becoming full at high tide2. This causes him to grow as he rises becoming full at high tide. When the water drains out, there is a corresponding ebb in the tides. When full, he is viewed as fat and lazy. Anger drives him to kill his sons as punishment for not sharing their food. In retribution, his wives attack him, carving his body with their axes, causing him to wane away. He eventually dies for three days (New Moon) before resurrecting as a crescent (waxing Moon), growing again until full. The cycle then begins anew.

In Tasmania, Aboriginal people see this the other way around. A Palawa Creation story tells about about the Sun Man and Moon Woman. They rose into the sky together on the first day, but the Sun Man was faster than his wife, and she began to gradually fall behind. He illuminated more of her each night to encourage her to catch up (waxing Moon). Eventually she was on the opposite side of him in the sky, fully lit (Full Moon). This tradition not only explains why more of the Moon’s surface it lit as it waxes to Full Moon, but that the light we see on the Moon is reflected sunlight. Traditions describe how the cycle of the Moon is linked to time. The Moon completes one revolution around Earth with respect to the Sun every 29.5 days. Therefore, a “year” on the Moon is a month. In Many Aboriginal and Islander cultures, the words for ‘month’ and ‘Moon’ are the same, or closely linked. For example, the word “Meb” in the Meriam Mir language of the eastern Torres Strait means both Moon and month, explaining how the Moon completes a cycle over the course of a month3.

Coastal Aboriginal and Torres Strait Islander people linked lunar phases to the different tides and incorporated this into their knowledge systems. This knowledge informs hunting, fishing, and agricultural practices4. For example, Torres Strait Islanders observe the lunar phases to know the best times to go fishing. Elders explain that they prefer to fish during a neap tide rather than a spring tide. The main reason is that the stronger spring tides stir up the sediment, clouding the water, thus reducing the ability for fish to see the fisher’s lures, or the fishers to spot the fish. They explain that the best time to fish is during first and third quarter Moons, as these tides are foretold in the Moon5.

The Moon is always half lit by the Sun. How much of the lit side we see on Earth depends on where the Moon is in relation to the Sun and Earth - what we can call its phase angle. In terms of mathematics, as the Moon orbits the Earth, its angle between the Earth and the Sun changes. When the Moon appears close to the Sun in the sky, the side facing the Sun is illuminated, but not the side we see. This means the Moon rises with the Sun at dawn, which we call New Moon. Around this time, the phase angle is very low (reaching zero during eclipses). As the Moon slowly moves from West to East in the sky each day, a sliver of its lit side gradually becomes visible to us. We call this the crescent Moon. More of the Moon’s surface is illuminated each day (called “waxing”) until exactly half of the Moon is illuminated. This is the First Quarter Moon. On this day, the Moon rises at midday. From the New Moon to First Quarter, the phase angle is acute increasing in value until it reaches 90 degrees at First Quarter. When more than half of the Moon is illuminated, we call it “gibbous”. The phase angle of the Moon is now obtuse, increasing daily until it reaches 180 degrees at Full Moon. At this time, the Moon will rise at dusk, exactly opposite the Sun. As the days progress, the phase angle of the Moon decreases as less of the Moon is illuminated each night, which we call “waning”. When the Moon is again only half illuminated (opposite side from First Quarter), we call it Third Quarter (or sometimes Last Quarter) Moon. On this day, the Moon rises at midnight. It will gradually wane away until it is no longer visible, moving back to the New Moon phase, which occurs over roughly three days.

Meriam Mir people of the eastern Torres Strait have names for these lunar phases: The Moon is called meb (which is also the term for a month). The New Moon (thin crescent) is aketi meb. The First Quarter Moon is meb degemli. A nearly Full Moon (waxing or waning) is eip meb.Full Moon is giz meb. A Third Quarter Moon is meb zizimi. For reference, a lunar eclipse is meb dimdi.

In physics terms, tides are caused by the gravitational effects of the Moon and Sun. When the Sun and Moon are aligned with the Earth, their combined gravity is stronger, creating the higher Spring tides. When the Moon is 90° to the Earth/Sun, their gravity counteracts the Earth-Sun pull, causing the lower Neap tides (Fig. 1).

There are usually two high-tides and two low-tides per day and they gradually increase in amplitude as the Moon goes from a Quarter Moon to either a Full or New Moon, then decrease as the Moon goes towards its Quarter phases. Since the Moon does not orbit the Earth in a circle, but rather a slight ellipse, the Moon is sometimes closer to us. Same applies with the Earth and Sun. If this occurs when the Earth reaches its closest point to the Sun, their combined gravity causes King Tides. The Earth reaches its closest point to the Sun around 2 January each year. If we get a Full or New Moon on or around this date, we will get a King Tide. In places like the Torres Strait, these can be very damaging as climate change causes the sea level to rise6.


Fig. 1: (top) Lunar phases. Image: NASA. (bottom) Lunar phases and their links to tides. Image: Janelle Wilson.7


Classroom activity - Mathematics Year 8

In this classroom activity, students will learn the relationship between lunar phases and tides, graphically plotting the tidal range versus Moon phase over an annual cycle for a selected Australian port, as well as determining the mean tidal range for each lunar phase. Students will then focus on determining the relationship between the lunar orbital phase angle (the angle the Moon makes with the Sun as seen from Earth) and percentage illumination of the Moon (i.e. how much of the Earth-facing lunar surface is illuminated by the Sun).

Curriculum connections

This resource addresses the following content descriptions from the Australian Curriculum:

  • Solve problems involving the use of percentages, including percentage increases and decreases, with and without digital technologies (ACMNA187)
  • Solve a range of problems involving rates and ratios, with and without digital technologies (ACMNA188)
  • Investigate the relationship between features of circles such as circumference, area, radius and diameter (ACMMG197)
  • Investigate techniques for collecting data, including census, sampling and observation (ACMSP284)

This resource addresses the following excerpts from the achievement standard for Year 8 in Mathematics:

  • students solve everyday problems involving rates, ratios, and percentages
  • name the features of circles and calculate the areas and circumferences of circles
  • determine the probabilities of complementary events

Inquiry-based learning questions

  • How do the phases of the Moon relate to the angle between the Earth/Sun/Moon system?
  • How do Aboriginal and Torres Strait Islander people measure the light-coverage of the Moon (as a percentage) to determine phase type?
  • How do observations of lunar phases inform datasets of tidal amplitude measurements?
  • How can knowledge of tides be applied to real world scenarios?

Activity 1 - Measuring lunar phase angle and percentage illumination

Suggested timing for activity: 30-45 minutes

Required resources: pen, graph paper, calculator, print-out of Moon phases

In this activity, students will use graph paper to estimate the area of the lunar surface illuminated every three days throughout a lunar cycle, as a ratio of the total surface area. Students then calculate the phase-angle of the Earth/Moon/Sun at that time.

  1. Give students a print-out of the Moon at seven different phases, along with a sheet of graph paper. Teachers can use images online or they can use a zoomed-in screenshot from the Stellarium planetarium software (www.stellarium.org). Suggested phases are three days before First Quarter, three days after First Quarter, Full Moon, three days before Third Quarter, Third Quarter, and four days after Third Quarter. In the following image, use days 6, 8, 11, 15, 18, 21, and 25. Teachers should keep in mind (and perhaps ask students if they realise) that the phases in this image are taken from the Northern Hemisphere. They are upside down when viewed from the Southern hemisphere.


    Fig. 2: Tony Honchar (2013), retrieved from: http://rooftopweather.info/moonphases.htm

  2. Have students label each Moon phase with their Meriam Mir names.
  3. Have the students place clear graph laminate over the Moon phase image, marking out the illuminated area. They then count the lit bozex to estimate the surface area of the Moon lit during each of these phases. (The smaller the boxes, the more accurate their measurements).
  4. Students then calculate the ratio of the lit area of the Moon to the total area of the Moon.
  5. What is the percentage change from the previous phase they calculated?
  6. Students then calculate the phase-angle based on the Moon phase, using the data they calculated before. This can be plotted as a Sine graph, where 0 and 180 degrees are the x-axis, and the y-axis shows +90 to -90 degrees, where +90 degrees is First Quarter and -90 degrees ii Third Quarter.

If the teacher uses screenshots from Stellarium (from a given place and time), the students’ work can be checked by clicking on the Moon during that phase. A drop down list of information is provided. One of them gives the percentage of Moon illuminated.


Fig. 3 : (Left) Relationship between lunar orbital phase angle (angle between the Moon & Sun) and lunar phase as seen from Earth. (Right) Determining % illumination of the Moon as seen from Earth. Retrieved from: https://www.astronomynotes.com/nakedeye/s13.htm

Activity 2 - Establishing the link between moon phases and tides

Suggested timing for activity: 30 minutes

Required resources: paper, pen, handout

Using data from Activity 1 (or alternatively simply getting the information online), students will determine the correlation between lunar phase angle and tide amplitude.

  1. Access the tide data from Mer (aka Murray Island, as we will base this on Torres Strait Islander knowledge)8https://tides.willyweather.com.au/qld/far-north/murray-island.html

    Students will calculate the difference between high tide and low tide on the day of each lunar phase throughout the year. Teachers can develop their own spreadsheet in the following manner (using additional columns for additional phases, depending on their time constraints). Use the Meriam Mir names for the lunar phases so the students are encouraged to remember them.

     

    Tidal Range/Amplitude (High Tide minus Low Tide)

    Month

    New Moon

    First Quarter

    Full Moon

    Last Quarter

    January

        

    February

        

    March

        

    etc....

        
  2. Students then plot their data on a curve to show the correlation between lunar phases and tides. Since two tides normally occur each day, students can average the two values for that day.
  3. Students then calculate the amplitude of high tides throughout the year, which peak at King tide around 2 January and being lowest at neap tide around 2 July.

Based on the calculated information, have student discuss what days are best for fishing, using the basic Meriam knowledge about fishing, Moon phases, and tides.

Notes

1 Clarke, P.A. (2007/2008). An Overview of Australian Aboriginal Ethnoastronomy. Archaeoastronomy, 21, 39-58.

2 Hamacher, D.W. and Norris, R.P. (2011). “Bridging the Gap” through Australian Cultural Astronomy. In Archaeoastronomy & Ethnoastronomy – Building Bridges Between Cultures, edited by Clive Ruggles. Cambridge University Press, pp. 282-290.

3 Hamacher, D.W., Tapim, A., Passi, S., and Barsa, J. (2018). Dancing with the stars – astronomy and music in the Torres Strait. In Imagining Other Worlds: Explorations in Astronomy and Culture, edited by Nicholas Campion and Chris Impey. Lampeter: Sophia Centre Press, pp. 151-161.

4 Hamacher, D.W. and Norris, R.P. (2011). Eclipses in Australian Aboriginal Astronomy. Journal of Astronomical History and Heritage, 14(2), 103–114.

5 Hamacher, D.W., Tapim, A., Passi, S., and Barsa, J. (2018). Dancing with the stars – astronomy and music in the Torres Strait. In Imagining Other Worlds: Explorations in Astronomy and Culture, edited by Nicholas Campion and Chris Impey. Lampeter: Sophia Centre Press, pp. 151-161.

https://www.sbs.com.au/news/torres-strait-pleads-for-help-as-super-moon-king-tides-hit

https://www.slideshare.net/janellewilson/seasons-phases-eclipses-and-tides

https://tides.willyweather.com.au/qld/far-north/murray-island.html

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The development of these resources was funded through an Australian Government initiative delivered by the University of Melbourne's Indigenous Studies Unit. The resources include the views, opinions and representations of third parties, and do not represent the views of the Australian Government. They have been developed as a proof of concept to progress the inclusion of Aboriginal and Torres Strait Islander content in Australian classrooms. In drawing on the material, users should consider the relevance and suitability to their particular circumstances and purposes.