This task presents a context that leads students toward discovery of the formula for calculating the volume of a sphere. Students who are given this task must be familiar with the formula for the volume of a cylinder, the formula for the volume of a cone, and CavalieriŐs principle.

Unit: Area and Surface Area
Parallelograms: Lesson 5 Bases and Heights of Parallelograms

Students begin this lesson by comparing two strategies for finding the area of a parallelogram. This comparison sets the stage both for formally defining the terms base and height and for writing a general formula for the area of a parallelogram. Being able to correctly identify a base-height pair for a parallelogram requires looking for and making use of structure (MP7).

The terms base and height are potentially confusing because they are sometimes used to refer to particular line segments, and sometimes to the length of a line segment or the distance between parallel lines. Furthermore, there are always two base-height pairs for any parallelogram, so asking for the base and the height is not, technically, a well-posed question. Instead, asking for a base and its corresponding height is more appropriate. As students clarify their intended meaning when using these terms, they are attending to precision of language (MP6). In these materials, the words “base” and “height” mean the numbers unless it is clear from the context that they refer to a segment and that there is no potential confusion.

By the end of the lesson, students both look for a pattern they can generalize to the formula for the area of a rectangle (MP8) and make arguments that explain why this works for all parallelograms (MP3).

This lesson unit is intended to help you assess how well students are able to: recognize and use common 2D representations of 3D objects and identify and use the appropriate formula for finding the circumference of a circle.

This video is meant to be a fun, hands-on session that gets students to think hard about how machines work. It teaches them the connection between the geometry that they study and the kinematics that engineers use -- explaining that kinematics is simply geometry in motion. In this lesson, geometry will be used in a way that students are not used to. Materials necessary for the hands-on activities include two options: pegboard, nails/screws and a small saw; or colored construction paper, thumbtacks and scissors. Some in-class activities for the breaks between the video segments include: exploring the role of geometry in a slider-crank mechanism; determining at which point to locate a joint or bearing in a mechanism; recognizing useful mechanisms in the students' communities that employ the same guided motion they have been studying.

In this Cyberchase video segment, the CyberSquad must locate each other using a map's grid to construct a coordinate system using letters and numbers.

Addition of three Vectors, and displays resultant Vectors. Can drag the endPoints of the three different Vectors, but the resultant always starts at the origin.

This short video and interactive assessment activity is designed to teach second graders about visually evaluating parallel lines cut by a traversal.

Unit 4: Dividing Fractions
Lesson 15: Volume of Prisms

In this lesson, students complete their understanding of why the method of multiplying the edge lengths works for finding the volume of a prism with fractional edge lengths, just as it did for prisms with whole-number edge lengths. They use this understanding to find the volume of rectangular prisms given the edge lengths, and to find unknown edge lengths given the volume and other edge lengths.

Problems about rectangles and triangles in the previous two lessons involved three quantities: length, width, and area; or base, height, and area. Problems in this lesson involve four quantities: length, width, height, and volume. So finding an unknown quantity might involve an extra step, for example, multiplying two known lengths first and then dividing the volume by this product, or dividing the volume twice, once by each known length.

In tackling problems with increasing complexity and less scaffolding, students must make sense of problems and persevere in solving them (MP1).

This Nrich activity offers free exploration which will help learners develop a deep understanding of halves and halving. The task gives a context in which to discuss the importance of the part-whole relationship of fractions so that children realise halves can be different sizes, depending on the whole.

In this Nrich activity students look for connections in an array combining colors and shapes.

Unit 1: Scale Drawings
Lesson 1: What are Scaled Copies?

This lesson introduces students to the idea of a scaled copy of a picture or a figure. Students learn to distinguish scaled copies from those that are
not—first informally, and later, with increasing precision. They may start by saying that scaled copies have the same shape as the original figure, or that they do not appear to be distorted in any way, though they may have a different size. Next, they notice that the lengths of segments in a scaled copy vary from the lengths in the original figure in a uniform way. For instance, if a segment in a scaled copy is half the length of its counterpart in the original, then all other segments in the copy are also half the length of their original counterparts. Students work toward articulating the characteristics of scaled copies quantitatively (e.g., “all the segments are twice as long,” “all the lengths have shrunk by one third,” or “all the segments are one-fourth the size of the segments in the original”), articulating the relationships carefully (MP6) along the way.

The lesson is designed to be accessible to all students regardless of prior knowledge, and to encourage students to make sense of problems and persevere in solving them (MP1) from the very beginning of the course.

Unit: Area and Surface Area
Lesson 12: What is Surface Area?

This lesson introduces students to the concept of surface area. They use what they learned about area of rectangles to find the surface area of prisms with rectangular faces.

Students begin exploring surface area in concrete terms, by estimating and then calculating the number of square sticky notes it would take to cover a filing cabinet. Because students are not given specific techniques ahead of time, they need to make sense of the problem and persevere in solving it (MP1). The first activity is meant to be open and exploratory. In the second activity, they then learn that the surface area (in square units) is the number of unit squares it takes to cover all the surfaces of a three-dimensional figure without gaps or overlaps (MP6).

Later in the lesson, students use cubes to build rectangular prisms and then determine their surface areas.

At its most basic this Nrich task is an exercise in reading and recording information from a table. It also offers opportunities for children to do some elementary reasoning as they compare results with each other and work out why they differ.

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.

In this task students see different ways of partitioning a circle and a rectangle into two or more equal shares.

The Nrich interactivity offers an ideal context in which to observe the "messy" randomness of results after a small number of experiments, and the predictability of results after a large number of trials. The problem also offers a good starting point for considering different probability distributions and their features, which could be followed up with the tasks Which List is Which and Data Matching.

Beavers are generally known as the engineers of the animal world. In fact the beaver is MIT's mascot! But honeybees might be better engineers than beavers! And in this lesson involving geometry in interesting ways, you'll see why! Honeybees, over time, have optimized the design of their beehives. Mathematicians can do no better. In this lesson, students will learn how to find the areas of shapes (triangles, squares, hexagons) in terms of the radius of a circle drawn inside of these shapes. They will also learn to compare those shapes to see which one is the most efficient for beehives. This lesson also discusses the three-dimensional shape of the honeycomb and shows how bees have optimized that in multiple dimensions. During classroom breaks, students will do active learning around the mathematics involved in this engineering expertise of honeybees. Students should be conversant in geometry, and a little calculus and differential equations would help, but not mandatory.

Beavers are generally known as the engineers of the animal world. In fact the beaver is MIT's mascot! But honeybees might be better engineers than beavers! And in this lesson involving geometry in interesting ways, you'll see why! Honeybees, over time, have optimized the design of their beehives. Mathematicians can do no better. In this lesson, students will learn how to find the areas of shapes (triangles, squares, hexagons) in terms of the radius of a circle drawn inside of these shapes. They will also learn to compare those shapes to see which one is the most efficient for beehives. This lesson also discusses the three-dimensional shape of the honeycomb and shows how bees have optimized that in multiple dimensions. During classroom breaks, students will do active learning around the mathematics involved in this engineering expertise of honeybees. Students should be conversant in geometry, and a little calculus and differential equations would help, but not mandatory.

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.