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 task examines the ways in which the plane can be covered by regular polygons in a very strict arrangement called a regular tessellation. These tessellations are studied here using algebra, which enters the picture via the formula for the measure of the interior angles of a regular polygon (which should therefore be introduced or reviewed before beginning the task). The goal of the task is to use algebra in order to understand which tessellations of the plane with regular polygons are possible.

Students play and record the “Mary Had a Little Lamb” song using musical instruments and analyze the intensity of the sound using free audio editing and recording software. Then they use hollow Styrofoam half-spheres as acoustic mirrors (devices that reflect and focus sound), determine the radius of curvature of the mirror and calculate its focal length. Students place a microphone at the acoustic mirror focal point, re-record their songs, and compare the sound intensity on plot spectrums generated from their recordings both with and without the acoustic mirrors. A worksheet and KWL chart are provided.

This is a textbook for first year Computer Science. Algorithms and Data Structures With Applications to Graphics and Geometry.

In this game activity students practice comparing shapes and naming something that is alike or different about them.

This lesson unit is intended to help teachers assess how well students are able to: work with concepts of congruency and similarity, including identifying corresponding sides and corresponding angles within and between triangles; Identify and understand the significance of a counter-example; Prove, and evaluate proofs in a geometric context.

This task provides a construction of the angle bisector of an angle by reducing it to the bisection of an angle to finding the midpoint of a line segment. It is worth observing the symmetry -- for both finding midpoints and bisecting angles, the goal is to cut an object into two equal parts. The conclusion of this task is that they are, in a sense, of exactly equivalent difficulty -- bisecting a segment allows us to bisect and angle (part a) and, conversely, bisecting an angle allows us to bisect a segment (part b). In addition to seeing how these two constructions are related, the task also provides an opportunity for students to use two different triangle congruence criteria: SSS and SAS.

In this activity, learners use a hand-made protractor to measure angles they find in playground equipment. Learners will observe that angle measurements do not change with distance, because they are distance invariant, or constant. Note: The "Pocket Protractor" activity should be done ahead as a separate activity (see related resource), but a standard protractor can be used as a substitute.

Find out how angles and symmetry come into play in the game of pool in this video adapted from Annenberg Learner’s Learning Math: Measurement.

We use the derivative to determine the maximum and minimum values of particular functions (e.g. cost, strength, amount of material used in a building, profit, loss, etc.).Differentiation is also used in analysis of finance and economics.

This lesson unit is intended to help you assess how well students are able to use geometric properties to solve problems. In particular, it will support you in identifying and helping students who have the following difficulties: Solving problems relating to using the measures of the interior angles of polygons; and solving problems relating to using the measures of the exterior angles of polygons.

In this math activity, learners explore the history of the Stomachion (an ancient tangram-type puzzle), use the pieces to create other figures, learn about symmetry and transformations, and investigate the areas of the pieces. The Stomachion, believed to have been created by Archimedes, consists of 14 pieces cut from a square, which can be rearranged to form other interesting shapes.

The famous story of Archimedes running through the streets of Syracuse (in Sicily during the third century bc) shouting ''Eureka!!!'' (I have found it) reportedly occurred after he solved this problem. The problem combines the ideas of ratio and proportion within the context of density of matter.

In this problem, students are given a picture of two triangles that appear to be similar, but whose similarity cannot be proven without further information. Asking students to provide a sequence of similarity transformations that maps one triangle to the other focuses them on the work of standard G-SRT.2, using the definition of similarity in terms of similarity transformations.

This problem is part of a very rich tradition of problems looking to maximize the area enclosed by a shape with fixed perimeter. Only three shapes are considered here because the problem is difficult for more irregular shapes.

The purpose of this task is primarily assessment-oriented, asking students to demonstrate knowledge of how to determine the congruency of triangles.

This course is an arithmetic course intended for college students, covering whole numbers, fractions, decimals, percents, ratios and proportions, geometry, measurement, statistics, and integers using an integrated geometry and statistics approach. The course uses the late integers model—integers are only introduced at the end of the course.

The purpose of this task is to provide students with a multi-step problem involving volume and to give them a chance to discuss the difference between exact calculations and their meaning in a context.

This task asks students to use similarity to solve a problem in a context that will be familiar to many, though most students are accustomed to using intuition rather than geometric reasoning to set up the shot.