## A Note on using Angle Find! for Educators

Angle sense, or the ability to estimate angle measurements by sight, is an important foundational skill for students who will be facing angle related problems throughout their mathematical studies. The NCTM Principles and Standards for School Mathematics states:

Estimating that an angle is less than 90 degrees should prevent a student from misreading a measurement of 150 degrees for a 30-degree angle. Students can develop a repertoire of benchmark angles, including right angles, straight angles, and 45-degree angles. They should be able to offer reasonable estimates for the measurement of any angle between 0 degrees and 180 degrees. Checking the reasonableness of a measurement should be a part of the process.

Angle Find! is a game that helps students to develop both angle sense and their understanding of angle relationships. Players of the game are given an angle measurement and a screen full of lines that meet at various angles. The game contains three sets of levels. The levels of the first set contain three lines and their intersections. In the second set of levels, parallel lines are introduced. The final set of levels contains "challenge" levels that include more different angles and smaller differences between angle measurements.

I suggest having students play through at least the first two sets of levels. If you are in a setting with multiple students, try to have as many of them as possible get through the first two sets of levels while allowing those who have already made it through the first two sets to continue on to the third set. Total playing time should be about 5 to 10 minutes.

In addition to encouraging angle related discoveries, this game may also have a side benefit of training working memory. Remembering where one put (or tried to put) a given angle the last time it came up can be quite helpful when it appears again. Conversely, if a student does poorly at, for example, matching an angle to a previously correctly placed vertical angle, this may indicate trouble with keeping angles in working memory rather than a failure to notice the congruence of vertical angles.

After you have students play the game, I would suggest leading a conversation about the discoveries they have made. Here are some questions you might want to ask students:

• What tricks did you discover to help you figure out which angles were correct?
• What kinds of situations produced angles of the same size?
• How did the parallel lines affect the situation?

This conversation may also be a good time to introduce vocabulary:

Student: I noticed that the angles that were across from each other were the same.

You: That's right. By the way, those angles that are "across from each other" are called "vertical angles."

Level scores are not intended to be used as assessments of student knowledge. They are meant to serve more as motivational checkpoints: getting a good score or a score that has improved over the previous level score feels good and may encourage someone to keep playing. However, a student whose score are consistently low and don't show improvement,

Common Core standards that could be addressed in a program of study that uses Angle Find! include the following:

7.G.5. Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.
8.G.5.Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.

For both of the above standards, the game should expose students to some concepts and encourage discovery, but it is not intended to be a substitute for formal instruction.

If you use Angle Find! with your students, I would love to know how it went, what problems if any you may have, and what suggestions you may have for its improvement. Please contact me at pz@puzzlezapper.com. If you'd like to discuss #anglefind on Twitter, I'm @two_star there.