Tag Archives: Making Thinking Visible

Circumscribed Angles

Circles: G-C. A. Understand and apply theorems about circles

  1. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

When I first googled circumscribed angle, I was led to Khan Academy. I’ve never specifically called an angle a circumscribed angle before, although of course it makes sense that an angle formed by two tangents to a circle from the same point outside of a circle can be called a circumscribed angle. I had always heard it called an angle formed by two tangents.

Suppose you are given that the measure of arc ADC is 260˚. What do you know?


How were you taught to calculate the measure of angle B?

How have you taught students to calculate the measure of angle B?

How have your students figured out to calculate the measure of angle B?

I’ve always thought of this angle as one other in the set of angles with vertices outside of the circle whose measure is half the difference of the outer intercepted arc and the inner intercepted arc.

Practicing look for and make use of structure gives my students and me flexibility in what I see and how I can calculate the measure of angle B.

I sent students a Khan Academy question on central, inscribed, and circumscribed angles to see what they could do. The auxiliary lines are drawn because of the central and inscribed angles, but students still practiced look for and make use of structure.

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Even though most got it correct, we still had a whole class discussion so that students could share how they got the angle measure.

Some worked it the way I had been taught.

Some saw the kite.

Even those who saw the kite didn’t think through calculating the angle measure exactly the same way.

At least one thought of the kite as a quadrilateral inscribed in a circle and said that the opposite angles were supplementary. Yikes! I’m glad that (incorrect) thinking was made visible.

Why are the opposite angles supplementary?

This item does give me hope that while I might not “cover” every standard (especially the next year or two), teaching mathematics using the Math Practices is worth our time. Even during an assessment, my students can figure out some of what they need to know by practicing look for and make use of structure and look for and express regularity in repeated reasoning. What #AskDontTell opportunities can you provide your leaners the next time you’re together?


Posted by on March 16, 2015 in Circles, Geometry


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Rigid Motions

We have started our CCSS geometry course with a unit on Rigid Motions. I’ve included the standards and “I can” statements that we are using at the bottom of the post.

We started last year with a unit called Tools of Geometry, where we did constructions by focusing on special segments in triangles, but we decided to try starting with Transformations this year, since our students have a bit more familiarity with transformations than medians, altitudes, angle bisectors, and perpendicular bisectors. We are hoping to ease them in to “making sense of problems and persevering in solving them” instead of bombarding them with it in the first unit….

We started the lesson with a routine called “Zoom In”, that I read about in Making Thinking Visible. In the routine, students look at a small part of an image. “What do you notice?” “What is your interpretation of what this might be based on what you are seeing?” As the teacher reveals more of the image, students think about the new things they notice & refine their interpretation of what the image might be based on the new information. We used a piece of fabric by Michael Miller.


The first image was part of the pair of flip flops – and then we zoomed out several times to reveal more of the fabric. Students noticed congruent figures – but they didn’t just say that the figures were congruent because they had the same size and shape. They began to discuss the congruence of the figures in terms of rigid motions. One brown and white cat is congruent to another because there is a translation that will carry one onto the other. One flip flop is congruent to the other in its pair because there is a reflection that will carry one onto the other. One fan is congruent to another because there is a translation that will carry one onto the other. One brown cat is congruent to another brown cat because there is a rotation that will carry one onto the other.

We used the Mathematics Assessment Project formative assessment lesson on Transforming 2D Figures as a guide for the rest of the lesson.

Students had a cutout L shape to transform as requested. We determined whether the figures were congruent based on whether there was a rigid motion (translation, reflection, rotation) that would carry one onto the other.


Then we asked questions from the MAP lesson such as …


Where will the L-shape be if it is translated by 1 horizontally and -4 vertically?

Where will the L-shape be if it is reflected over the x-axis?

Where will the L-shape be if it is reflected over the line x=2?

Where will the L-shape be if it is reflected over the line y=x?

Where will the L-shape be if it is rotated through 180° around the origin?

Oh – and just in case you are wondering whether students had remembered the graphs of x=2 and y=x over the summer, that was the topic of the bellringer for the lesson. I think that we often have students memorize that horizontal lines can be written in the form y=# and vertical lines can be written in the form x=# without having the students think about the actual points on the line. The point on the line in the TI-Nspire Graphs page shown below is dynamic, and so students moved the point and began to make sense of what was happening with the coordinates – and ultimately with the equation of the line. So that when we did ask them to reflect their L-shape over the line x=2, there was less discussion about where is the line x=2 and more discussion about the actual reflection.




The standards that we are using:

G-CO 3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

G-CO 2: Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

G-CO 4: Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

G-CO 5: Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

G-CO 6: Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

G-CO 7: Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

And our “I can” statements:

Unit 1 –Rigid Motions

Level 1: I can identify and define transformations and composite transformations.

Level 2: I can perform transformations and composite transformations.

Level 3: I can determine the congruence of two figures using rigid motions.

Level 4: I can apply transformations and composite transformations to figures in the coordinate plane.

Level 4: I can map a figure onto itself using transformations.

And so the journey continues….

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Posted by on August 25, 2013 in Coordinate Geometry, Geometry, Rigid Motions


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