Recent solar eclipses provide relevant realworld contexts for learning about the scientific phenomena of the lunar phases. News coverage of the phenomenon may have raised questions such as, “Why does the Moon look different at different times, and sometimes as if it is not there?,” and “What patterns can be found in the lunar phases?” Teachers can use these recent events to launch investigations into the mathematics of the phases.
Connecting learning across the mathematics and science disciplines provides opportunities for students to deepen their conceptual understanding and apply what they have learned in new contexts (Vasquez et al., 2013). Our research team designed a set of simulations (link online) that can be used to weave together mathematics and science learning in Grades 5–7 (e.g., Basu et al., 2020). Here, we present how we used our Moon Pie simulation in a sixthgrade science classroom to bridge the mathematics of angle measurement, fractions, covariation, and cosplitting (multiplicative/proportional covariation) with the scientific phenomenon of lunar phases. For the simulation’s alignment to standards, see Table 1.
Mathematics and Science Standards
Standards  

NGSS Science Standards  MSESS11 Develop and use a model of the Earthsunmoon system to describe the cyclic patterns of lunar phases, eclipses of the sun and moon, and seasons. 
CCSS Mathematics Content Standards  4.MD.C.5.B An angle that turns through n onedegree angles is said to have an angle measure of n degrees. 
5.NF.B.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.  
5.NF.B.6 Solve realworld problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.  
6.RP.A.3 Use ratio and rate reasoning to solve realworld and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.  
7.RP.A.2 Recognize and represent proportional relationships between quantities.  
CCSS Mathematics Practice Standards  MP2 Reason abstractly and quantitatively. 
MP7 Look for and make use of structure. 
Note. Source: National Governors Association Center for Best Practices & Council of Chief State School Officers [NGA Center & CCSSO], 2010; Next Generation Science Statistics [NGSS] Lead States, 2013.
THE SCIENCE OF LUNAR PHASES
We chose the context of lunar phases because it is one of the main Earth and environmental content areas of middle school (NGSS Lead States, 2013; MSESS11). Children already have experiences with the lunar phases in informal learning situations throughout their lives and develop many alternative conceptions about how these phases are formed, such as believing that the Moon makes itself small, or the clouds or sky hide the Moon (e.g., Wilhelm, 2009).
To develop a more scientific conception of the lunar phases, the goal is for students to understand that each of the eight labels for describing phases depend on the positions of the Moon, Earth, and Sun. The New Moon occurs when the Moon is in between the Earth and the Sun and is often used as the starting point for the lunar cycle. It takes the Moon roughly 29.5 days to travel an apparent 360° around the Earth (known as the synodic period). Figure 1a shows the First Quarter phase, and Figure 1b shows the Full Moon phase, demonstrating how the Moon’s position in its orbit determines how much of the illuminated Moon is visible from Earth. The portion of the orbit the Moon has traveled can also be measured by fractions and degrees from the New Moon (Figure 2). To simplify the model for middle school students, we used a 28day month, a 360° apparent orbit, and the midpoints of the Waxing and Waning phases throughout.
Moon Phases and Their Orbit Position from New Moon in Days, Fractions, and Degrees
Citation: Mathematics Teacher: Learning and Teaching PK12 117, 5; 10.5951/MTLT.2023.0160
Moon Phases and Their Orbit Position from New Moon in Days, Fractions, and Degrees
Citation: Mathematics Teacher: Learning and Teaching PK12 117, 5; 10.5951/MTLT.2023.0160
Moon Phases and Their Orbit Position from New Moon in Days, Fractions, and Degrees
Citation: Mathematics Teacher: Learning and Teaching PK12 117, 5; 10.5951/MTLT.2023.0160
THE MATHEMATICAL RELATIONSHIPS OF THE LUNAR PHASES
To define each named lunar phase, students need to analyze the situation into a structure of quantities and quantitative relationships (NGA Center & CCSSO, 2010; MP2). A quantity is a conceived attribute of an object or phenomenon that is measurable (Thompson, 1994). As mentioned in the previous section, the lunar phases can be mathematically described in terms of the number of days, the degrees (4.MD.C.5.B), and the fraction traveled by the Moon in orbit from the New Moon. Students can coordinate the changes in these quantities by reasoning covariationally, or mentally imagining two quantities’ values (i.e., magnitudes) changing simultaneously (Carlson et al., 2002). For instance, students can reason that as the days in orbit increase, the degrees traveled also increase. Additionally, students can engage in multiplicative (5.NF.B.4, 5.NF.B.6) and proportional reasoning (6.RP.A.3, 7.RP.A.2) through cosplitting (Corley et al., 2012), a specific form of covariation in which any multiplicative change in one quantity is coordinated with the same multiplicative change and thus establishes a ratio relationship. For example, they may observe that as the days in orbit double, the degrees traveled also double (Figure 3).
Cosplitting the Days and Degrees in Orbit
Citation: Mathematics Teacher: Learning and Teaching PK12 117, 5; 10.5951/MTLT.2023.0160
Cosplitting the Days and Degrees in Orbit
Citation: Mathematics Teacher: Learning and Teaching PK12 117, 5; 10.5951/MTLT.2023.0160
Cosplitting the Days and Degrees in Orbit
Citation: Mathematics Teacher: Learning and Teaching PK12 117, 5; 10.5951/MTLT.2023.0160
Measuring the orbit in terms of degrees involves a conceptualization of angles as rotations. However, angles as rotations are more difficult for students to quantify, and Devichi and Munier (2013) suggested a combination of static and dynamic representations to support students’ understanding (e.g., this GeoGebra application [link online]). Representing an angle as a wedge has shown potential in helping students recognize the attribute being measured by that angle (Browning & GarzaKling, 2009). Therefore, in our design, we added a wedge as a third quantity describing the area created by tracing the orbit. Our conjecture was that students would be able to use the part of the circle measured as a fraction of a full circle to define the orbit in degrees. For example, one full circle is 360°, half of a full circle is equal to 180°, and a quarter of a full circle is 90°.
MOON PIE SIMULATION
Several simulations have been designed to model the lunar phases (for example, this applet [link online]) that provide numerous features to support student learning (horizon diagram, threedimensional visualization of celestial positions). However, we have designed the Moon Pie simulation (Video 1) to be simplified and pedagogically accessible for middle school students (Weintrop et al., 2016) by using a 28day approximation of the 29.5day synodic lunar month and a limited number of variables to focus on specific quantitative relationships. The quantities highlighted in the simulation are the days in orbit, the fraction of the orbit traveled, and the arc length of the orbit traveled.
The Moon Pie Simulation
Citation: Mathematics Teacher: Learning and Teaching PK12 117, 5; 10.5951/MTLT.2023.0160
The Moon Pie Simulation (link online) can be used to identify fraction and degree values for each named Moon phase. The user can click on the Moon to drag it around in its orbital path, observing the resulting changes in its phase as displayed in the pictureinpicture view of the Moon as seen from Earth. The simulation supports student mathematical exploration by including representations of angles as wedges in addition to an arc traveled by the Moon in orbit. Students also have the ability to toggle the fraction and degree values and wedge overlays (see Video 1 for more detail on the simulation).
As students explore the Moon Pie simulation, teachers can use the guiding questions in Table 2 to help students make use of the mathematics practice standards to reason quantitatively and make use of structure (NGA Center & CCSSO, 2010; MP2, MP7). The questions can help students move from general noticings about the simulation to a more indepth focus on the patterns in the quantitative relationships between the lunar phases.
Guiding Questions
Questions about noticing.  • What do you notice? • What patterns do you see? 
Questions that led students to identify quantitative relationships in the simulation.  • What relationships do you see? • How are [quantities] changing together? • What is the relationship between the orbit measure in fractions and degrees? 
Questions that led students to examine each phase.  • Where is the [Lunar Phase]? • How many degrees from the New Moon is the [Lunar Phase]? • How would you describe each phase in terms of the fraction of the Moon’s orbit? 
Questions that asked students to reason about the Moon traveling in the orbit.  • How many degrees of the orbit does it take for the Moon to get from [Lunar Phase] to [Lunar Phase]? • What fraction of the orbit is traveled by the Moon to get from [Lunar Phase] to [Lunar Phase]? • The moon has traveled [#degrees/#fraction] around the orbit from [Lunar Phase]. What phase is it in now? 
Questions that asked students to elaborate on their reasoning.  • How do you know that? • How did you figure that out? • Is there another way to do it? 
Questions about students’ mental picture of the phenomenon.  • What do you have in your mind that helps you calculate this? 
Prior to introducing students to the Moon Pie simulation, we asked them to draw the moon and discuss ways the moon can appear. We then introduced relevant vocabulary for the lunar phases (waxing, waning, crescent, gibbous, new, quarter, full) and the convention that the New Moon is considered the first phase. The teacher also conducted some discussion about the moon’s changing places in its orbit and the different amount of the illuminated side of the moon facing the Earth. Students began exploring the relationship between the lunar phases and the fraction of the month using our other simulation (link online). For example, students stated that when the Moon is in the First Quarter, 7/28ths of the 28day model cycle has passed. Teachers can also engage students in handson learning about the lunar phases by enacting a physical representation of the SunEarthMoon model using a lamp as the Sun and students acting out the roles of the Earth and the Moon (Ashmann, 2012).
Student Reasoning Using Moon Pie
Here, we describe the Moon Pie simulation lesson from our implementation with sixthgrade students, which took place over Google Meet during the COVID19 pandemic. We present on how a particularly expressive pair of students, Ali and Jaden, reasoned mathematically about the lunar phases to illustrate in depth the progression of reasoning that is possible when students engage with this simulation and our targeted questioning.
Discovering Relationships
The students had a few minutes to explore the simulation and then were asked to describe it. Ali explained the functioning of the simulation (Video 2) and identified three quantities in the phenomenon: the days in travel, the degrees, and the fraction of the orbit. Jaden added that the quantities were changing, “depending how, where the Moon is, changes, what the days, the degrees, and the fractions.” When asked about the relationship between these quantities Ali moved the Moon around the simulation and explained that, “As the degrees and the fraction become higher, the amount of days the Moon has been in orbit becomes higher.” Ali reasoned multivariationally about the degrees, the fraction of the circle, and the number of days the Moon has been in orbit.
Student Description of the Moon Pie Simulation
Citation: Mathematics Teacher: Learning and Teaching PK12 117, 5; 10.5951/MTLT.2023.0160
Fraction and Degree Values of Moon Phases
Using the guiding questions in Table 2, we asked students to reason about the fraction and degree values of the Moon throughout the named phases of the orbit. The first column of Tables 3–5 presents the named phases that students were asked to reason about, and the second column presents an illustration of students’ cosplitting strategies. In the illustration of student thinking, multiplication or division is used in the cosplitting relationships to align with student explanations (e.g., “half of 360,” represented as × 1/2 rather than ÷ 2). Students generated benchmark relationships of unit fractions (1/8, 1/4, 1/2) using cosplitting, which were then used to generate values for other phases.
Students’ CoSplitting Reasoning for Full Moon
Lunar Phase  Illustration of Students’ Thinking 

(a) Full Moon 
Student Thinking Jaden: “It’s half. . . . The degrees is 180. . . . because a whole cycle is 360°. And halfway through a cycle is the Full Moon. So half of 360 is 180.” 
Students’ CoSplitting Reasoning for First and Third Quarter Phases
Lunar Phase  Illustration of Students’ Thinking 

(a) First Quarter 
Student Thinking Jaden: “90. Because it’s one fourth of 360°. . . . Because if you divide 360 by four, you will get 90. So one fourth would be 90°.” 
(b) Third Quarter 
Student Thinking Jaden: “So one fourth is 90°. If you multiply that by three, we’ll get 270. And one fourth times three is three fourths.” Ali [typed]: “because 360 × 3/4 = 270.” Q: “Did you do this multiplication of fractions?” Ali [typed]: “no. 360/4 = 90. 90 × 3/1 = 270.” 
Students’ CoSplitting Reasoning for Waxing and Waning Phases
Lunar Phase  Illustration of Students’ Thinking 

(a) Waxing Crescent 
Student Thinking Jaden: “An eighth, that would be half of 90 degrees, so that means 45 degrees for an eighth.” Ali: “So one fourth is equal to 90 degrees. So oneeighth is equal to 45 degrees.” 
(b) Waxing Gibbous 
Student Thinking Ali: “Because the Waxing Gibbous is threeeighths of the orbit. And three times 45, because 45 is equal to oneeighth, multiply three times 45, you get 135.” 
(c) Waning Crescent 
Student Thinking Jaden: “Because it’s the seventh phase. So I just multiplied 45 times seven.” 
Halves
Jaden used cosplitting to identify the fraction and angle values for the Full Moon by noting that it was halfway and 180° through a full orbit (Table 3).
Fourths
Jaden and Ali both identified 1/4 and 90° for the location of the First Quarter Moon (Table 4a) by cosplitting and dividing the fraction and degree relationship for a full orbit by four. After identifying the values for the First Quarter, the students were able to build on them by using cosplitting to identify the values for the Third Quarter Moon (Table 4b), which both students identified as 3/4 and 270°.
Eighths
Jaden and Ali also built on the benchmark values they had identified for the First Quarter, by using cosplitting to generate the benchmark values of 1/8 and 45° (Table 5a). Ali then went on to list the midpoint degrees that would correspond to the Waxing and Waning phases that are represented by the eighths (45°, 135°, 225°, 315°), with Jaden noting the pattern that “For every phase in between the quarters, it will be 45 degrees.” When asked to identify the midpoint values for the Waxing Gibbous phase (Table 5b), Ali used cosplitting to build on the benchmark relationship for oneeighth. Jaden also used the same strategy to explain that he got seveneighths and 315° for the midpoint of Waning Crescent (Table 5c).
Fraction and Degree Values of the Moon Traveling Between Phases
We also asked students to identify the values of the distance traveled from a nonNew Moon starting phase to another phase (such as from First Quarter to Full Moon). The first column of Table 6 presents the distance traveled that students were asked to measure, and the second column presents an illustration of students’ reasoning.
Students’ CoSplitting Reasoning for Moon Travel
Moon Travel  Illustration of Students’ Thinking 

(a) First Quarter to Third Quarter 

(b) Full Moon to next First Quarter 
When asked about the degrees traveled from the First Quarter Moon to the Third Quarter Moon (Video 3), Ali identified that it would be half of the Moon cycle and 180°. Jaden agreed and explained iterating by quarters from First Quarter to Third Quarter using the simulation. Ali shared similar reasoning and used cosplitting to double both the fraction and degrees traveled.
First Quarter to Third Quarter
Citation: Mathematics Teacher: Learning and Teaching PK12 117, 5; 10.5951/MTLT.2023.0160
When Jaden was asked to find the fraction of orbit traveled from the Full Moon to the next First Quarter Moon (Video 4), he answered threefourths and 270°. Jaden used the simulation to illustrate his thinking, showing he was iterating around the orbit by quarters. This allowed him to continue using cosplitting to provide both the fraction and degrees of the orbit traveled. While he noted the values of the phases in their static locations, such as Full Moon at 180°, he was more focused on the measures of the distance traveled by the Moon moving in orbit.
Full Moon to Next First Quarter
Citation: Mathematics Teacher: Learning and Teaching PK12 117, 5; 10.5951/MTLT.2023.0160
Benchmark Relationships and Mental Picture
At the end of the lesson, we asked the students to explain what they found useful in the simulation. Jaden summarized the important benchmark relationships: “I know the orbit is 360 degrees. And I know that a fourth is 90 degrees, an eighth is 45 degrees, and a half is 180 degrees. So those are the things that helped me.” Ali used his prior knowledge of fractions of circles to develop a mental picture of the orbit: “I look at the orbit as a circle. It’s cut into eighths. Sometimes I cut it into halves. Sometimes I cut it into fourths. And sometimes I cut it into eighths inside of my head. And that helps me see it.” Jaden also explained that he created a “picture in my head” that was “just the circle. So I imagine where each phase is at, and what, actually in what degrees that it traveled to. So like, Third Quarter is 270 degrees and it’s threefourths. Like I remember that.” Jaden’s mental picture shows that in his mind, the position of the moon in each phase around its orbit can be represented by both a degree measure and a fraction.
Tips For Implementation
Ali and Jaden’s reasoning demonstrates the potential for leveraging scientific contexts for meaningful mathematical learning. Teachers can use the Moon Pie simulation in either a mathematics or a science classroom to bridge the learning of these two disciplines. This also provides an opportunity for mathematics and science teachers to collaborate because there is much opportunity for proportional and quantitative reasoning when studying the celestial bodies of the solar system (NGSS Lead States, 2013).
While the ratios in Tables 3–6 are only illustrations of our attempt to depict students’ reasoning, teachers can have students create such tables themselves. Students may generate different forms of cosplitting to express relationships (e.g., multiplied by a 1/2 vs. divided by 2), which provides opportunity for meaningful discussion of multiplicative and proportional relationships. Additionally, the guiding questions in Table 2 can help teachers elicit different forms of students’ reasoning, providing opportunity for a discussion of different strategies. _
REFERENCES
Ashmann, S. (2012). A sunearthmoon activity to develop student understanding of lunar phases and frames of reference. Science Scope, 35(6), 32–36.
Basu, D., Panorkou, N., Zhu, M., Lal, P., & Samanthula, B. K. (2020). Exploring the mathematics of gravity. Mathematics Teacher: Learning and Teaching PK12, 113(1), 39–46. https://doi.org/10.5951/MTLT.2019.0130
Browning, C. A., & GarzaKling, G. (2009). Young children’s conceptions of angle and angle measure: Can technology facilitate? In C. Bardini, P. Fortin, A. Oldknow, & D. Vagost. (Eds.) Proceedings of the 9th International Conference on Technology in Mathematics Teaching (pp. 1–5), ICTMT 9.
Carlson, M., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002). Applying covariational reasoning while modeling dynamic events: A framework and a study. Journal for Research in Mathematics Education, 33(5), 352–378. https://doi.org/10.2307/4149958
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Thompson, P. W. (1994). The development of the concept of speed and its relationship to concepts of rate. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 181–234). SUNY Press.
Vasquez, J. A., Sneider, C., & Comer, M. (2013). STEM lesson essentials, grades 3–8: Integrating science, technology, engineering, and mathematics. Heinemann.
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ACKNOWLEDGMENTS
This research was supported by the National Science Foundation (#1742125). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the foundation. We thank Toni York, Erell Germia, and Anthony Cuviello for their work on the Moon Pie simulation and lesson.
Amanda Provost, she/they, provosta1@montclair.edu, is a doctoral candidate in mathematics education at Montclair State University in Montclair, NJ. She is interested in studying mathematical modeling, STEM education, and teacher professional development.
Nicole Panorkou, she/her, panorkoun@montclair.edu, is an associate professor of mathematics education at Montclair State University. She is interested in the design of transdisciplinary experiences for students that can strengthen their learning.