Research is supported by NASA Astrophysics Research and Analysis Grant No. NNH18ZDA001N.

]]>This Demonstration shows initial steps in the iterative bracket method for creating the gold list of increasingly close rational approximants of a given real number target.

The initial bracket is the pair of fractions and with smallest possible denominators such that both approximate the target and lie to its left and right. The number line shows the sequence of repeated mediants of the bracket elements, monotonically approaching the target from the side, before crossing over to the side at the mediant. All repeated mediants between the and the are guaranteed to be in the gold list: the repeated mediants most closely bracketing the target become the bracket elements for the next iteration step.

For comparison, you can choose to see the continued fraction representation of the target.

]]>This Demonstration shows two simple methods for finding fractions that approximate a target real number (rational or irrational). The respective algorithms will "increment numerator () or denominator ()" or else will "increment denominator () only." Several target numbers are considered, with sliders selecting different parts of the sequences and graphs to clarify the algorithms and their results.

Using the method "increment or ," the red line on the numerator versus denominator graph represents the target; the dots give rational approximations. In each column of dots the *approximant* (the closest possible approximation to the target for that denominator) is shown in bold.

Using the method "increment only," only the approximants are shown, with errors and relative errors plotted. Here, bold points indicate members of the *gold list* of increasingly better approximations.

- Following the example in "Developing scientific decision making by structuring and supporting student agency" by N.G. Holmes, et al, students designed the research question in a three-week sequence and a written lab report was replaced with an online synchronous oral lab report, followed by time for questions from student colleagues. I will report on enhanced student engagement in class and in office hours and will share anecdotal comments. Students participated in a "belonging intervention". I will share practical lessons learned.

This work was supported by DARPA DETECT.

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