This brief article describes the historical methods of marking and measuring time. Among the instruments and methods discussed are sundials, water clocks, celestial motions, and mechanical clocks. Included are descriptions of the sources of inaccuracies of these methods. Links to related resources and a separate page of pedagogical notes are provided.

This problem promotes logical thinking and introduces learners to the trial and error (guess and check) problem solving strategy, especially with the interactive provided. In this problem children need to understand the difference between having a certain number of brothers and the number of boys in a family to answer the question, "How many children are there in the Brown family?" The Teachers' Notes page offers suggestions for implementation, discussion questions, ideas for extension and support, and a link to a down loadable worksheet for students to table their trials.

This activity asks students to visualize and construct three-dimensional objects from the two-dimensional drawings. Students are shown four solids composed of cubes, and they must reproduce the objects with manipulatives or sketch them on isometric dot paper. Ideas for implementation, extension and support are included along with a printable sheet of dot paper.

This problem helps children begin to understand the various properties of common geometric solid shapes. It also promotes naming, discussion and experimentation concerning their features, and requires them to justify their ideas. It asks students to judge the stability of nine configurations made from six common solids. The Teachers' Notes page includes suggestions for implementation, discussion questions, ideas for extension and support, and printable sheets.

This activity gives students practice naming and using shape and color attributes to create patterned sequences. The first challenge asks students to use attribute differences to extend a sequence. A second, more open-ended challenge asks students to maximize the length of their sequences under a further constraint. An interactive applet is provided as an alternative to physical manipulatives. The Teachers' Notes page includes suggestions for implementation and discussion questions.

This activity gives students a chance to relate some common three-dimensional solids to their polygonal faces. The object is to put solids in a sequence in which adjacent solids have a polygonal face in common. Ideas for implementation, extension, hints and support are included along with printable cards of the polyhedra.

This hands-on activity helps students develop spatial sense and scaling concepts. Students use interlocking cubes to build first a chair and then a table of appropriate size for the chair. The student goes on to build two other sets of chairs and tables to make three different sizes in all. The Teachers' Notes page includes suggestions for implementation, discussion questions, and ideas for extension.

This brief article discusses the importance of young children creating their own informal graphical representations of their mathematical thinking and problem solving. As distinguished from formal recording of a completed process, these early markings and symbols enable children to develop understanding and make meaning as well as communicate their thinking. The article includes a list of references, including the authors' research on which the article is based.

In this activity students develop fraction concepts by reasoning about what choices they would make in order to get the most chocolate. Students determine how much candy they would receive as they enter one at a time and sit at one of three tables holding different amounts of candy which get shared equally. The activity includes the problem, tips for getting started, a teacher resource page, and a printable page of the problem.

This problem is an investigation into combinations of a number of cubes. It is a practical activity which involves working systematically, and visualizing and relating 3D shapes to their representation on paper. Children are asked how many different towers are possible using seven cubes on a base of two of them. The Teachers' Notes page includes suggestions for implementation, discussion questions, ideas for extension and support, and printable handouts (word/pdf).

This brief article relates the legend of young Gauss and the summing of consecutive numbers. Readers are asked to apply the method and they are shown a general solution. A link to a printable page is provided as well as links to related topics.

This activity helps students become familiar with positional words and develop systematic thinking. Students use the clues provided to arrange six colored squares in an array, either on paper or with the interactive Flash applet that is provided. The Teachers' Notes page includes suggestions for implementation, discussion questions, ideas for extension and support and a printable sheet (doc). Students may be asked to create a similar problem for others to solve.

This activity allows students to explore reflective symmetry. They are asked to color a given arrangement of triangles in symmetric patterns using specific numbers of colors. The Teachers' Notes page includes suggestions for implementation, discussion questions, ideas for extension and support, and printable sheets

This Flash game develops students' computational fluency and flexibility. It can be played alone or with others. Players choose 6 face-down number cards, and the applet provides a target number. Number cards include one each of 25, 50, 75, and 100, and multiple copies of 1 to 10. The goal is to use the selected numbers and the four basic operations (addition, subtraction, multiplication, division) to arrive at the target. Players can ask the applet to Show a Solution, although others may be possible. Several rule and scoring variations are suggested. A full screen option facilitates use on an interactive board. (This game is not self-checking.)

In this investigation students explore the concept of ratios and their relationship with factors and multiples. Given an assortment of cogs (gears) with different numbers of teeth, solvers are asked to find which pairs allow for each tooth of one to fit into every gap of the other, and to form conjectures regarding which pairs would work generally. The problem is well-suited for group work. The Teachers' Notes page offers suggestions for implementation, key questions, student sheets, and links to articles, videos, and an interactive Cog Environment (cataloged separately).

This problem gives children an opportunity to explore an increasing pattern and then generalize the results with a rule. Students begin with a single counter, surround it by a ring of other counters and then each new ring is surrounded with more counters. Solvers record results as they replicate the pattern and make predictions about many counters there will be in any given ring. The Teachers' Notes page offers suggestions for implementation, key discussion questions, ideas for extension and support.

This brief article discusses the Count the Beat project, which incorporates music into math instruction. The instructors in this project began each lesson with a brief math exercise followed by a more in-depth music task utilizing number lines, fractions, ratios, and/or probability. While music professionals carried out this project the article does offer some suggestions for application in the math classroom without music professional.

Solvers of this problem apply number sense and logical reasoning to determine the numbers of cows and sheep in each of five fields by using clues about how many cows and sheep can be seen by each animal. The problem includes questions for getting started, suggestions for implementation and differentiation, a printable student page, and sample solutions.

This problem requires some visualization and knowledge of 3D shapes. It gives children experience of identifying shapes from pictures of them in different positions and orientations. Ideas for implementation, extension and support are included along with a printable sheet of shape cards.

With this flexible Flash applet students can explore a wide range of concepts, including addition and subtraction, factors and multiples, fractions, ratios, and combinations. Users place, move and rotate virtual Cuisenaire Rods on a grid which can be enlarged, reduced, moved or printed. It contains a link to page of challenges which can take advantage of this tool.