This activity is one in a series of tasks using rigid transformations of the plane to explore symmetries of classes of triangles, with this task in particular focusing on the class of equilaterial triangles. In particular, the task has students link their intuitive notions of symmetries of a triangle with statements proving that the said triangle is unmoved by applying certain rigid transformations.

This task examines some of the properties of reflections of the plane which preserve an equilateral triangle: these were introduced in ''Reflections and Isosceles Triangles'' and ''Reflection and Equilateral Triangles I''. The task gives students a chance to see the impact of these reflections on an explicit object and to see that the reflections do not always commute.

This activity is one in a series of tasks using rigid transformations of the plane to explore symmetries of classes of triangles, with this task in particular focussing on the class of isosceles triangles.

This interactive Flash applet gives students the opportunity to copy, create and extend repeating patterns utilizing two different triangles. Many patterns are possible since the angles in both are multiples of 30 degrees and the shorter sides are equal in length. The Teachers' Notes page includes suggestions for implementation, ideas for extension and support, and printable sheets (pdf).

The result here complements the fact, presented in the task ``Right triangles inscribed in circles I,'' that any triangle inscribed in a circle with one side being a diameter of the circle is a right triangle. A second common proof of this result rotates the triangle by 180 degrees about M and then shows that the quadrilateral, obtained by taking the union of these two triangles, is a rectangle.

This is the first version of a task asking students to find the areas of triangles that have the same base and height, and is the most concrete.

This is the second version of a task asking students to find the areas of triangles that have the same base and height. This presentation is more abstract as students are not using physical models.

In this activity students try to visualize 3-D shapes from given 2-D silhouettes (projections). Students can describe, draw, model or relate their ideas to objects in their environment. With several possible answers for each silhouette, students become more familiar with using the terms and describing the properties of solid figures. The Teachers' Notes page includes suggestions for implementation, discussion questions, ideas for extension and support. A warmup activity called "Skeleton Shapes" is offered (cataloged separately). [Note: "torch" in the UK = "flashlight" in the US.]

This activity gives students practice drawing straight lines with a ruler and looking for and categorizing shapes, for example, by the number of sides in polygons. The Teachers' Notes page includes suggestions for implementation, discussion questions and ideas for extension.

In this investigation students work systematically and keep organized records as they explore forming triangles from unit sticks. Learners look for patterns and trends in the number of triangles possible with a given integer perimeter. Ideas for implementation, extension and support are included along with a printable sheet of the problem.

Summer Reading: Standard 8.EE.6 (Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y=mx for a line through the origin and the equation y=mx+b for a line intercepting the vertical axis at b.)
This formative assessment exemplar was created by a team of Utah educators to be used as a resource in the classroom. It was reviewed for appropriateness by a Bias and Sensitivity/Special Education team and by state mathematics leaders. While no assessment is perfect, it is intended to be used as a formative tool that enables teachers to obtain evidence of student learning, identify assets and gaps in that learning, and adjust instruction for the two dimensions that are important for mathematical learning experiences (i.e., Standards for Mathematical Practice, Major Work of the Grade).

This 10-minute video lesson shows that three points uniquely define a circle and that the center of a circle is the circumcenter for any triangle that the circle is circumscribed about.

Watch the ZOOM cast build a chair out of newspaper by making good use of the strength of triangles.

In this video segment adapted from ZOOM, the cast tries to design and build a bridge made out of drinking straws that will support the weight of 200 pennies.

The triangle congruence criteria, SSS, SAS, ASA, all require three pieces of information. It is interesting, however, that not all three pieces of information about sides and angles are sufficient to determine a triangle up to congruence. In this problem, we considered SSA. Also insufficient is AAA, which determines a triangle up to similarity. Unlike SSA, AAS is sufficient because two pairs of congruent angles force the third pair of angles to also be congruent.

This activity asks students to visualize shapes, paying close attention to the definitions of special polygons. Learners are given a sheet of isometric grid paper and asked to find and sketch 12 specific shapes. Ideas for implementation, extension and support are included along with printable grids and shape definitions.

The two triangles in this problem share a side so that only one rigid transformation is required to exhibit the congruence between them. In general more transformations are required and the "Why does SSS work?'' and "Why does SAS work?'' problems show how this works.

For these particular triangles, three reflections were necessary to express how to move from ABC to DEF. Sometimes, however, one reflection or two reflections will suffice. Since any rigid motion will take triangle ABC to a congruent triangle DEF, this shows the remarkable fact that any rigid motion of the plane can be expressed as one reflection, a composition of two reflections, or a composition of three reflections.

This particular sequence of transformations which exhibits a congruency between triangles ABC and DEF used one translation, one rotation, and one reflection. There are many other ways in which to exhibit the congruency and students and teachers are encouraged to explore the different possibilities.

This is a youcubed favorite which comes from Mark Driscoll. The activity encourages students and teachers to engage in visual, creative thinking. We have coupled MarkÕs activity with asking students to reason and be convincing, two important mathematical practices.