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• Author or Editor: Wayne Nirode
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## The First Days of Geometry

Students use movement and tracing paper to determine sets of points that are equidistant from two points, from two intersecting lines, and from a line and a point not on the line.

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## Proofs without Words in Geometry

Pictures and diagrams help high school geometry students develop reasoning and proof-writing skills.

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## Collecting Simulation Data with Google Forms

During their work with statistics, students should be able to compare two treatments from a randomized experiment and use a simulation to determine statistical significance informally (CCSSI 2010a; CCSSI 2010b; Franklin et al. 2007). To achieve these goals, I developed a method to collect student data in my classroom from hands-on simulations. The advantage of hands-on simulations over using formulas is that students can develop a conceptual understanding of statistical significance when they see the variation that occurs from sample to sample as the results of the experiment are rerandomized each time the simulation runs. I first explain a specific classroom experiment and the hands-on simulation. I then describe how to use Google Forms and Google Sheets to convert the simulation data that students submit using their cell phones into a single column of data that can then be displayed as a dot plot.

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## Doing Geometry with Geometry Software

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## Lines as “Foci” for Conic Sections

One of my goals, as a geometry teacher, is for my students to develop a deep and flexible understanding of the written definition of a geometric object and the corresponding prototypical diagram. Providing students with opportunities to explore analogous problems is an ideal way to help foster this understanding. Two ways to do this is either to change the surface from a plane to a sphere or change the metric from Pythagorean distance to taxicab distance (where distance is defined as the sum of the horizontal and vertical components between two points). Using a different surface or metric can have dramatic effects on the appearance of geometric objects. For example, in spherical geometry, triangles that are impossible in plane geometry (such as triangles with three right or three obtuse angles) are now possible. In taxicab geometry, a circle now looks like a Euclidean square that has been rotated 45 degrees.

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## Going Over the Test

To address student misconceptions and promote student learning, use discussion questions as an alternative to reviewing assessments.

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## Ferris Wheel Graphs

To introduce sinusoidal functions, I use an animation of a Ferris wheel rotating for 60 seconds, with one seat labeled You (see fig. 1). Students draw a graph of their height above ground as a function of time with appropriate units and scales on both axes. Next a volunteer shares his or her graph. I then ask someone to share a different graph. I choose one student with a curved graph (see fig. 2a) and another with a piece-wise linear (sawtooth) graph (see fig. 2b).

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## A Proof Progression for Geometry

A five-part proof progression can support diverse populations when engaging in proof-and-reasoning tasks.

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## Exploring New Geometric Worlds

Familiar sets of points defined by distances appear as unexpected shapes in larger-distance geometry.

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