Although students saw how to find equations to fit data in an earlier course, this is the first time they fit polynomial equations to data. If students need a hint about how to find a polynomial equation that better fits the data, it may help them to know that polynomials of higher degree can fit data points more closely. A good strategy is to start with a polynomial of degree 1 or 2 and then add more terms. For example, students could start with an equation of the form \(y=nx^2\), and then move to \(y=nx^2+mx\) and \(y=nx^2+mx+k\). If the maximum number of terms (1 greater than the degree) is reached and the polynomial still is not a good fit, then the degree can be increased.

Graphing technology can be a helpful tool for finding models in this way, and students should be encouraged to use it. Graphing calculators automatically can find polynomials of specified forms that are the closest possible fit to a set of data. Desmos also has this feature. Calculators may report the value of \(R^2\) for the functions they produce. \(R^2\) measures how closely a function fits a set of data points. Tell students that the closer \(R^2\) is to 1, the better the function fits the data.

To conclude the task, invite students to share their models for the relationship between the orbital period and the distance, and for the relationship between the orbital period and the square root of the distance. Models should show a simpler relationship between the orbital period and the square root. If we cube the square root of the distance and multiply it by some constant, we get the period. This is exactly what Kepler found. Share with students the statement of Kepler’s third law: “The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit.” This means that the square of the orbital period is some constant times the cube of the distance. This can be expressed as \(p^2 = n \boldcdot d^3\), and then—substituting the proportionality constant, \(k\), for \(n\)—rearranged into the form \(p= k (d^{1/2})^3\), which is the relationship that students have been modeling.

Once Kepler saw this relationship, he could divide the square of any planet’s period by the cube of its distance in order to determine the proportionality constant. Students can try this for themselves to verify that all planets have the same proportionality constant. They also may want to compare what happens with Jupiter’s moons—do they have the same proportionality constant as the planets? Newton’s laws of gravity later explained why this relationship between the distance and the period is true.

Although students saw how to find equations to fit data in an earlier course, this is the first time they fit polynomial equations to data. If students need a hint about how to find a polynomial equation that better fits the data, it may help them to know that polynomials of higher degree can fit data points more closely. A good strategy is to start with a polynomial of degree 1 or 2 and then add more terms. For example, students could start with an equation of the form \(y=nx^2\), and then move to \(y=nx^2+mx\) and \(y=nx^2+mx+k\). If the maximum number of terms (1 greater than the degree) is reached and the polynomial still is not a good fit, then the degree can be increased.

Graphing technology can be a helpful tool for finding models in this way, and students should be encouraged to use it. Graphing calculators automatically can find polynomials of specified forms that are the closest possible fit to a set of data. Desmos also has this feature. Calculators may report the value of \(R^2\) for the functions they produce. \(R^2\) measures how closely a function fits a set of data points. Tell students that the closer to \(R^2\) is to 1, the better the function fits the data.

To conclude the task, invite students to share their models for the relationship between the orbital period and the distance, and for the relationship between the orbital period and the square root of the distance. Models should show a simpler relationship between the orbital period and the square root. If we cube the square root of the distance and multiply it by some constant, we get the period. This is exactly what Kepler found. Share with students the statement of Kepler’s third law: “The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit.” This means that the square of the orbital period is some constant times the cube of the distance, which can be expressed as \(p^2 = n \boldcdot d^3\), and then—substituting the proportionality constant, \(k\), for \(n\)—rearranged into the form \(p= k (d^{1/2})^3\), which is the relationship that students have been modeling.

Once Kepler saw this relationship, he could divide the square of any planet’s period by the cube of its distance in order to determine the proportionality constant. Students can try this for themselves to verify that all planets have the same proportionality constant. They also may want to compare what happens with Jupiter’s moons—do they have the same proportionality constant as the planets? Newton’s laws of gravity later explained why this relationship between the distance and the period is true.