1 · Lesson 2 · Discovery tool 3 · Tier 1 · DOK 1 4 · Tier 2 · DOK 2 5 · Tier 3 · DOK 3
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One skill, start to finish.

This is exactly how a single concept moves through the hub: a projection-first lesson, a discovery tool that releases gradually, then practice that climbs in real cognitive demand. No account needed. Move through it yourself.

Today's concept: factoring x² + bx + c
1
The lesson · projected for the whole class

First, one idea made visible

The teacher projects this. One worked example, one big idea, paced by taps, never auto-advancing. The idea we are after has a name in the hub: the frozen product.

x² + 7x + 12
Layer 0·1List the number pairs that multiply to 12: (1,12), (2,6), (3,4). Smaller number first, always, so the answer matches every time.
Layer 2The click: the constant already froze the product at 12. So the only thing left to hunt is the sum. Which pair adds to 7? 3 and 4.
Layer 3Write it: (x + 3)(x + 4). The product was fixed; the sum is the part you steer.

Notice what the lesson did not do: it didn't hand over a rule to memorize. It made one structural idea visible, then stepped back. That is the on-ramp. The understanding itself forms next, and in practice.

2
The discovery tool · gradual release

Then the student does the thinking

I do, we do, you do. Here is a "you do" beat. The tool will not solve it for you and will not advance on its own. You place the pair yourself. (That is a rule we never break: the student completes every step.)

x² + 9x + 20  =  (x + ?)(x + ?)

Product is frozen at 20. Tap the pair whose sum is 9.

3
Practice · Tier 1 of 3

Tier 1 — Build the procedure

DOK 1–2 · Recall & skill Goal: fluent, accurate factoring of a clean trinomial.

Positive, friendly numbers. The student runs the method until it is automatic. This is the floor, and the floor has to be solid before anything else is honest.

x² + 8x + 15
Quick check
Which is the correct factored form?
4
Practice · Tier 2 of 3

Tier 2 — Manage the signs

DOK 2–3 · Strategic thinking Goal: flexibility when the constant is negative.

A genuinely different cognitive job, not just bigger numbers. A negative constant means the pair has opposite signs, and now the student has to reason about which sign goes where.

x² + 2x − 8
Quick check
The constant is −8 and the middle term is +2x. Which factored form is correct?
Coach Nive
This used to trip me up too. Before you write a single sign, check the sign of the constant. Negative constant means one plus and one minus. Then the middle term tells you which is bigger.
5
Practice · Tier 3 of 3

Tier 3 — Reason about the method itself

DOK 3 · Extended reasoning Goal: diagnose flawed thinking, not just execute steps.

The top tier asks for something procedure can't fake: catch why a method failed and explain it. A student who can only factor will get stuck here. A student who understands factoring will see it.

A student factored this and got it wrong:

x² + 11x + 24  →  (x + 5)(x + 6)
Quick check · find the thinking error
What did the student misunderstand?

5 + 6 = 11, so the sum looks right, which is exactly why this error is sneaky. But 5 × 6 = 30, not 24. The correct pair is 3 and 8: it multiplies to 24 and adds to 11. Seeing that is the whole point of the skill.

That was one skill. There are more than fifty.

Every concept moves through this same arc: made visible, practiced by the learner, then climbed tier by tier until the thinking is real. Nothing is auto-solved. Every step belongs to the student.

Built by a teacher to solve a real problem in a real classroom. Every concept connects.