This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: Suppose we are given a circle of radius $r$. The goal of this task is to construct an equilateral triangle whose three vertices lie on the circle. Supp...

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: In quadrilateral $ABCD$ pictured below, $\overline{AB}$ is congruent to $\overline{CD}$ and $\overline{BC}$ is congruent to $\overline{AD}$. From the g...

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important aspects of the task and its potential use.

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: Jessica is working to construct an equilateral triangle with origami paper and uses the following steps. First she folds the paper in half and then unf...

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: Lisa makes an octagon by successively folding a square piece of paper as follows. First, she folds the square in half vertically and horizontally and a...

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: Suppose $\overline{AB}$ is a line segment and $D$ is a point not on $\overline {AB}$ as pictured below: Let $C$ be the point so that $|CD| = |AB|$, $\o...

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: Below is a picture of two triangles: Suppose there is a sequence of rigid motions which maps $\triangle ABC$ to $\triangle DEF$. Explain why correspond...

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important aspects of the task and its potential use.

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: Triangles $ABC$ and $PQR$ pictured below are congruent: Show the congruence using rigid motions of the plane. Can the congruence be shown with a single...

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: Suppose $\triangle ABC$ and $\triangle PQR$ are distinct, congruent triangles. Using the steps below, show that a congruence can always be shown with o...

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important aspects of the task and its potential use.

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: Suppose $ABCD$ is a quadrilateral for which there is exactly one rotation, through an angle larger than 0 degrees and less than 360 degrees, which maps...

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: There is exactly one reflection and no rotation that sends the convex quadrilateral ABCD onto itself. What shape(s) could quadrilateral ABCD be? Explai...

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: Jennifer draws the rectangle $ABCD$ below: Find all rotations and reflections that carry rectangle $ABCD$ onto itself. Lisa draws a different rectangle...

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: A satellite orbiting the earth in a circular path stays at a constant altitude of 100 kilometers throughout its orbit. Given that the radius of the ear...

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: For this problem, $(a,b)$ is a point in the $x$-$y$ coordinate plane and $r$ is a positive number. Using a translation and a dilation, show how to tran...

This question examines the algebraic equations for three different spheres. The intersections of each pair of spheres are then studied, both using the equations and thinking about the geometry of the spheres. For two spheres where one is not contained inside of the other there are three possibilities for how they intersect.

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: The goal of this task is to explain why the area enclosed by a circle $C$ of radius $r$ is $\pi r^2$. Recall that $\pi$ is the ratio of the circumferen...

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: Suppose we define $\pi$ to be the circumference of a circle whose diameter is 1: Explain why the circumference of a circle with radius $r \gt 0$ is $2\...

Reflective of the modernness of the technology involved, this is a challenging geometric modelling task in which students discover from scratch the geometric principles underlying the software used by GPS systems.