Today's schools introduce symbolic algebra near the end of elementary school. Students in the fourth and fifth grades should be able to understand algebraic relations mathematically, verbally and graphically. Topics in algebra should test and enforce students' understandings of algebraic concepts in these three areas. One possible project is to give students toothpicks with which to create squares. Let students record how many toothpicks are necessary to create a certain number of connected squares (4 toothpicks for 1 square, 7 toothpicks for 2 squares, and so on). Ask the students to explain the pattern in words and equations.
For fourth and fifth grade students, geometry is the exploration of different shapes and how these shapes relate to the real world and other mathematical fields. Projects in the field of geometry can focus on the words associated with shapes and characteristics of shapes, the dissection and arrangement of two- and three-dimensional shapes and relative positioning of shapes on a graph. One project example is to assign students to create a map on graph paper of their bedrooms. The students should attempt to map their bedrooms using their knowledge of shapes and ability to convert from large to small scales.
Fourth and fifth graders should learn how to measure and estimate certain quantities such as length, volume, weight, area and time. Students should also know how to order objects according to these quantities. One possible project in the field of measurement is to assign students a quantity, such as volume. Have students calculate the volume of various objects in their houses. The students then must write the names of these objects in a Venn diagram displaying "larger," "smaller" and "the same."
By the end of elementary school, students should have a good understanding of how to estimate the probability of certain events. They should have a good grasp of the probabilities associated with common objects like coins, dice and spinners. Projects of this sort should focus on making predictions and then justifying these predictions with data. One example is to have students create spinners of their own, using four different colors for the areas on which the needle can land. After creating a spinner, students should make predictions of how often the needle will land on the colored areas on the spinner. Then, students spin the needle, collecting data on where the needle lands. The students then compare their predictions with the actual results.