Scaffolded Proof Practice

This lesson was originally called ‘thin-sliced proof practice’ in the Year 1 design, borrowing Liljedahl’s language for a sequence of closely related problems that vary by one feature at a time. The term is close but not quite right. True thin-slicing in Liljedahl’s sense involves the teacher designing a finely graded sequence that students move through together at a pace the teacher controls, with each problem just beyond the previous one. What we are doing here is related but different: students choose their own entry level based on honest self-assessment, and the three levels within each problem are designed so that the mild result is always embedded inside the medium proof, which is always embedded inside the spicy proof. ‘Scaffolded proofs’ is the more accurate name for what this lesson is doing.

 

Open the lesson by returning the individual proofs from Lesson 3.5b without grades — as feedback only. Before handing them back, address the two or three most common errors on the board without identifying which students made them. Frame each as a question: ‘If someone writes “by ASA” without naming the specific parts, what is missing from the argument? What would a reader need to verify the claim?’

 

The key insight of this lesson is worth naming explicitly for students: the mild proof is always inside the medium proof, which is always inside the spicy proof. A student who completes spicy has actually written three connected proofs, not one hard one. When students get stuck at medium or spicy, they are almost always missing the mild result. Redirect them: ‘Have you proven the mild yet? Go do that, and then come back.’

 

The lesson can run as group board work or individual written work depending on where students are after 3.5b. If the individual proofs from 3.5b showed many students still struggling with basic structure, run the mild level as groups and medium/spicy as individual. If most students demonstrated solid structure, run everything individually with groups as a resource for getting unstuck. Either way, make the entry-point choice visible: students should tell you or write down which level they are starting at and why.

 

Close with 10–15 minutes of consolidation. Do not attempt to review all problems. Choose one. Ask: who moved from mild to medium? What did you have to do differently? Who reached spicy? What made it harder than medium — and what made it more achievable once you had the medium result? The goal is for students to articulate the embedded structure in their own words.