Exit-ticket bank

20 items · Grade 7–8 linear equations · Every item is tagged by skill and misconception (M1–M5, the same taxonomy as the Socratic AI Tutor) · Each item includes the probing question the tutor reaches for when it’s missed — the ticket tells you where to dig; the question finds out why.

Items are drawn at 1–2 per session from the bank, chosen before the session (session plan template). Tags: SOLVE two-step solve · DIST distributing · MOVE moving terms · DIV dividing · BAL balance meaning of the equals sign · M1–M5 misconception most likely to surface.

Tier 1 — solve `ax + b = c` (weeks 1–2)

Item 1 · 3x + 5 = 20 · SOLVE · M1 risk The classic. Miss → probe: “You wrote 3x = 20. When you subtracted 5, what did you subtract from the right side?” (M1: operation applied to one side only.)

Item 2 · 2x − 7 = 9 · SOLVE · M2 risk Negative constant. Miss → probe: “The −7 moved across the equals sign. What operation is ‘moving’ it, and what happened on the right side?” (M2.)

Item 3 · x + 4 = 7 · SOLVE · M5 risk Deliberately trivial — and the sharpest M5 detector there is. A student who writes x = 7 (reading = as “the answer comes next”) or x + 7 exposes the operational equals sign. Miss → probe: “When you wrote the 7 after the x, what were you saying the whole line meant? What does the = promise about the two sides?”

Item 4 · 12 = 4x − 8 · SOLVE · M1 risk Reversed form (constant first) — students trained on ax + b = c sometimes “can’t start.” Miss → probe: “What’s the same about this one and 4x − 8 = 12? What does the = let you do?” (Tests whether the balance idea, not a left-to-right recipe, is doing the work.)

Item 5 · 6x + 2 = 2x + 22 · SOLVE/MOVE · M2 risk First x-terms-on-both-sides item. Miss → probe: “You moved the 2x — subtract or add, and from which sides?”

Tier 2 — distribute then solve (weeks 2–3)

Item 6 · 2(x + 3) = 10 · DIST · M3 risk Miss → probe: “What did the 2 multiply — say all of it out loud.”

Item 7 · 3(x − 2) = 9 · DIST · M3 + M2 risk Negative inside. Miss → probe: “What is 3 × −2? Where does that land on the right side?”

Item 8 · 2(x − 4) = 2x − 4 · DIST · M3 (classic) True/false format: “Amara wrote 2x − 4. Is she right? Why or why not?” Miss → probe: “What did the 2 multiply in her line?” (Explaining someone else’s error lowers the stakes of being wrong — keep this format for recurring errors.)

Item 9 · 4(x + 1) − 2 = 14 · DIST + SOLVE · M3 risk Two moves before the solve. Miss → probe: “Read your second line. What happened to the −2 — which terms did it hit?” (Distinguishes M3 recurrence from M1.)

Item 10 · 5(x + 2) = 5x + 10 · DIST/BAL · M3 + M5 risk Identity format — “is this true for every x? How do you know?” Tests whether “both terms” was learned as a rule or as a meaning. Miss → probe: “Pick a number for x. Test both sides. What does that tell you about the equation?”

Tier 3 — variables on both sides (weeks 3–5)

Item 11 · 5x + 3 = 3x + 11 · MOVE · M1/M2 risk Miss → probe: “When you subtracted 3x, which sides changed? What’s left on each?”

Item 12 · 3(x − 2) + 1 = 13 · DIST + MOVE · M3 risk The item used in the worked session plan. Miss → probe: “What did the 3 multiply? Then walk me through what you moved.”

Item 13 · 7x − 4 = 4x + 8 · MOVE · M2 risk The moving coefficient. Miss → probe: “The 4x moved from the right. What operation removed it — and what did that do to the left side?”

Item 14 · 9x + 4 = 4x + 4 · MOVE · M5 risk Solution x = 0 — the zero answer spooks students (“is that allowed?”). Miss → probe: “Test x = 0 in both sides. What do they give you? What does the = say about that?” (Converts the scare into the balance meaning.)

Item 15 · 6x − 3 = 2x + 5 → a student writes 6x − 3 = 5 after “moving 2x” · MOVE · M1 Error shown on the item (true/false again). Probe: “When 2x left the right side, what should have happened on the left? What does = promise?”

Tier 4 — dividing and special cases (weeks 5–8)

Item 16 · 4x = 12 → a student writes x = 12 · DIV · M4 (classic) Shown as an error to explain. Probe: “You divided the 4x by 4 — what did you divide on the other side of the equals sign?”

Item 17 · 4x + 2 = 14 → a student writes x + 2 = 3.5… then x = 1.5 · DIV · M4 The “divided everything except the +2” error. Probe: “If you divide ONE side by 4, what has to happen to the other side — every term on it?”

Item 18 · 2x = 2(x + 3) − 6 · BAL · M5 risk True for every x after simplifying — the identity case. Probe: “Simplify the right side. What’s left? Pick any x — test it. What kind of equation is true for ALL x?”

Item 19 · 3x + 1 = 3x + 5 · BAL · M5 risk No solution — another equals-sign-concept probe. Probe: “Simplify. What’s left on each side? Can any x make 1 = 5? What do we call an equation with no solution?”

Item 20 · −2x + 6 = −2(x − 3) · BAL · M5 + M3 risk Identity in disguise; the −2 distribution is the trap. Probe: “Distribute carefully, term by term. What does the right side become? Test x = 10 to be sure.”

Using the bank

Choose before the session (template rule). One item the group should mostly get (confidence), one at the edge (information).
Record with tags, not just scores — the tracker wants the tag (12·DIST), because “missed item 12” is trivia while “missed DIST twice” is a plan.
Escalation trigger: two consecutive tickets below mastery on the same tag → coordinator flag (D4). The tags make that automatic.
The probing question column is the bridge to practice: when an item is missed, the next session’s guided practice opens with its probe. Ticket → probe → plan is the whole data loop.
Items 3, 10, 14, 18, 19 are concept items (BAL/M5): use them when tickets say solves are fine but transfers keep failing — procedural success with conceptual collapse is exactly what these five are built to catch.

Fictional program, real content. Bridge Math Tutoring is fictional; the items are ordinary Grade 7–8 algebra constructed for training. The M1–M5 misconception taxonomy matches the misconception bank of the companion Socratic AI Tutor project.