*How are they introduced?*

Very carefully. (Content coming)

*What do you mean by a “complete set”?*

A complete set of ProofBlocks for the congruent triangle unit usually contains the following:

- SSS, SAS, and ASA Postulates (1 of each)
- AAS Theorem (1)
- Given (3) and Picture (2)
- Reflexive Property (1)
- Vertical Angles Theorem (1)
- Definition of a Midpoint (1, reversible)
- Definition of a Segment Bisector (1, reversible)
- Definition of an Angle Bisector (1, reversible)
- Definition of Perpendicular Lines (1, reversible)
- Definition of a Perpendicular Bisector (1, reversible)
- CPCTC (1)

*Do you make a teacher set of ProofBlocks?*

Most teachers we know of just draw the blocks on the board or tape up the student blocks when necessary. Some teachers stick magnetic strips to the back of one set so that they can move the pieces around on the board easily. Other teachers create sets using pieces of overheads.

*Do you have any tips on making the manipulatives themselves?*

The process of ProofBlock creation is time intensive, but worth the effort. Here are a few things we’ve learned:

- The blocks can be printed on regular paper or cardstock. (Templates for the blocks are on the Resources page.)
- Laminate a class set (one for every two students). The lamination not only makes the blocks feel more substantial and protects them from wear, but it also protects them from the dry erase markers as students draw their proofs on the whiteboards.
- Lamination has been most effective when the edges of the ProofBlocks are sealed so that the paper cannot soak up liquids from the open sides. Usually we cut the blocks out of the paper, laminate each block separately, and then cut them all out again from the laminate material.
- Definition blocks are created to be double-sided with the inputs and outputs switched. Templates for both the fronts and the backs of these blocks are provided on the Resources page.
- Some teachers provide each student with a small set that they can take home and cut out themselves in their own time

*Why are they those shapes?*

Theorems, definitions, and postulates that can be expressed as biconditional statements are represented with rectangular blocks. All other conditional statements are D-shaped. Given and Picture are the only circular blocks.

Those with a background in engineering may have recognized the D-shapes as similar to “AND” gates. (For those who do not have this background but are curious, there is a pretty good description of logic gates here.) ProofBlocks were designed with logic gates in mind: the flow of the geometric information through the proof mimics the flow of electricity through a circuit, the format of the proof closely resembles a circuit diagram, and the concept of theorems as functions is strongly reinforced by the inputs and outputs of AND and OR gates. Generally a couple 15 minute lesson activities on logic gates is all it takes to give the majority of students a basic understanding of these systems, as well as a real-world connection for the logical reasoning required in the coming proofs.