Reading a Diagram

This lesson is new to the sequence and addresses one of the most persistent problems from Year 1: students were reading tick marks, not geometry. When shared sides and vertical angles were not marked in the diagram, students did not see them. They were making assumptions from visual appearance rather than reasoning from the structure of the figure.

 

The opening task presents deliberately unmarked diagrams and asks students to list every geometric fact they can state with certainty. The most important diagrams to include are ones where shared sides and vertical angles are present but not marked — exactly the cases students missed. As groups work, ask: ‘How do you know that? What in the diagram forces it to be true?’ If they cannot cite a geometric reason, it does not belong in a proof.

 

This is also the right moment to connect back to Unit 2 logic explicitly. In formal argument, we cannot use the consequent of a conditional without first establishing the antecedent. The same principle applies here: we cannot use a geometric fact in an argument without first establishing that it is true — either from the diagram structure, or from a given, or from a previous step. The list of ‘free information’ is small and precise. Everything else must be earned.

 

A sharper framing worth naming for students: some facts can start a proof chain without needing to follow from anything prior. Vertical angles can be a starting point. A reflexive side can be a starting point. These do not need an antecedent. But they still need to be cited when used — they are not exempt from citation, only from the requirement to follow from an earlier step. This is a different thing from saying ‘vertical angles need no justification,’ which misleads students into thinking they can skip the citation. They still write ‘vertical angles’ below the congruence statement in a flow chart box. They just do not need a prior box pointing into it.

 

CPCTC is introduced here with the course policy: write the full definition before using the abbreviation, every time. The reasoning behind the policy is worth sharing directly with students: CPCTC is not a new idea. It is the definition of congruence they established in Lesson 3.2. When students defined congruence as ‘all six corresponding parts match,’ they were already saying what CPCTC says. Students were using it as a label without making that connection. The policy forces the connection every time.

 

The poster task (from mathmedic) should receive at least 55 minutes. If groups do not finish, allow them to complete it at the start of Lesson 3.4 — budget 10 minutes. Do not assign the poster as homework. The collaborative thinking is what the task is for, and the classroom is the only place we can guarantee students have the time and conditions to work through it together.

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