This exploratory problem provides students a way to consolidate their understanding of halving and halves and gives students experience of mathematical proof. The students are given multiple images of squares split in half. The goal is to prove how they are correctly halved and to think of other ways to split a square into two halves. The Teachers' Notes page offers rationale, suggestions for implementation along with a PowerPoint presentation, discussion questions, ideas for extension and support, and printable (pdf) worksheets of the problem.

This problem consolidates children's understanding of halving in a spatial context and will help them to develop their powers of visualization. The students are given four different shapes and the goal is to divide them each into two parts that are exactly the same. The Teachers' Notes page offers rationale, suggestions for implementation, discussion questions, a link to an extension activity, Same Shapes (cataloged separately), and a link to a simpler activity, Halving (cataloged separately).

The goal of this task is to use geometry study the structure of beehives. Beehives have a tremendous simplicity as they are constructed entirely of small, equally sized walls. In order to as useful as possible for the hive, the goal should be to create the largest possible volume using the least amount of materials. In other words, the ratio of the volume of each cell to its surface area needs to be maximized. This then reduces to maximizing the ratio of the surface area of the cell shape to its perimeter.

In this investigation, students visualize and compare volumes in solids composed of unit cubes and look for patterns in the measurements. They work systematically to organize and analyze the results. Ideas for implementation, extension and support are included.

The purpose of this task is for students to apply the concepts of mass, volume, and density in a real-world context. There are several ways one might approach the problem, e.g., by estimating the volume of a person and dividing by the volume of a cell.

This is a mathematical modeling task aimed at making a reasonable estimate for something which is too large to count accurately, the number of leaves on a tree.

In this problem, the variables a,b,c, and d are introduced to represent important quantities for this esimate: students should all understand where the formula in the solution for the number of leaves comes from. Estimating the values of these variables is much trickier and the teacher should expect and allow a wide range of variation here.

As written, this problem gives students all of the information they need to estimate the thickness of a soda can.

his is a version of ''How thick is a soda can I'' which allows students to work independently and think about how they can determine how thick a soda can is. The teacher should explain clearly that the goal of this task is to come up with an ''indirect'' means of assessing how thick the can is, that is directly measuring its thickness is not allowed.

This activity provides an opportunity for students to work on spatial visualization and number sense together. Learners are presented with a hundred grid and told there is another printed on the reverse side. They must work out the unseen numbers for several positions on the grid. Ideas for implementation, extension and support are included along with a printable sheet of grids.

This rich task is an excellent example of geometric concepts in a modeling situation and is accessible to all students. In this task, students will provide a sketch of a paper ice cream cone wrapper, use the sketch to develop a formula for the surface area of the wrapper, and estimate the maximum number of wrappers that could be cut from a rectangular piece of paper.

This short video and interactive assessment activity is designed to give fourth graders an overview of quarter-circles and semi-circles.

This task shows how to inscribe a circle in a triangle using angle bisectors. A companion task, ``Inscribing a circle in a triangle II'' stresses the auxiliary remarkable fact that comes out of this task, namely that the three angle bisectors of triangle ABC all meet in the point O.

This task is primarily for instructive purposes but can be used for assessment as well. Parts (a) and (b) are good applications of geometric constructions using a compass and could be used for assessment purposes but the process is a bit long since there are six triangles which need to be constructed.

This task provides an opportunity for students to apply triangle congruence theorems in an explicit, interesting context.

This problem introduces the circumcenter of a triangle and shows how it can be used to inscribe the triangle in a circle. It also shows that there cannot be more than one circumcenter.

This task focuses on a remarkable fact which comes out of the construction of the inscribed circle in a triangle: the angle bisectors of the three angles of triangle ABC all meet in a point.

This task is closely related to very important material about similarity and ratios in geometry.

This brief article describes the number sequence of Leonardo of Pisa (Fibonacci), and its connecton to the golden ratio and rectangle. Links to related topics and a link to a printable page are included.

This activity allows students to investigate line symmetry and reflections. Using a mirror, students locate the lines of symmetry. in a square and then proceed to find other shapes by reflecting parts of the square. Ideas for implementation, extension and support are included along with a printable worksheet of squares (.doc)