Three ideas to kick-start enquiries in Grade 3 concerning geometry.
#1 Dotted paper polygons. Students explore how many unique polygons can be made with 1 dot in the middle. Further constraints could be added, e.g. how many quadrilaterals can be constructed?
#2 Perimeter 12 triangle problem
When solving this problem many students simply partition 12 into three side lengths, e.g. 3 + 4 + 5. However, 2 + 3 + 7, whilst summing to 12, is an impossible triangle. Why? How would you explain this mathematically? What can be generalized about the side lengths of any triangle? If we label the sides of a triangle ABC, then A+B must equal more than the length of C. Using compasses to construct the triangles helps students understand this relationship, particularly because their drawings will be accurate and arcs drawn with the compasses will only intersect with triangles that are possible. G3 students will generally use everyday language to discuss these side length relationships, however, as a teacher I like the idea of translating this spoken English into mathematical language (see above).
#3 12 cubes - an inquiry into cuboids (rectangular prisms)
Students are given 12 multilink cubes and invited to make all the different cuboids they can (students from North America are more used to calling these shapes rectangular prisms). For each cuboid the students make an isometric drawing (common in architecture and design engineering) using dotted isometric paper. The ITP above, isometric grid useful for modelling this. http://www.taw.org.uk/lic/itp/itps/isoGrid_16.swf
There are 4 uniques cuboids that can be constructed using 12 cubes. So, whilst all have the same volume, what about their nets? Encourage students to explore what the nets for these cuboids will look like. If we were to make these as boxes, which would be the most expensive to make/the cheapest? And how should we record our findings mathematically so that any generalizations will be revealed?
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