Fresh formative checks… | Year 8 Maths · T1 W3–4 · Learner checks

Teacher: Give only the matching section on the day. Keep this page closed during the earlier lessons. Read the prompt neutrally if needed; a learner may use enlarged print, labelled tiles, a blank graph, tactile number line, speech/AAC or a scribe who records learner-chosen steps. Preserve the first independent response and note any mathematical hint separately from access support. These are internal checks, not standardised tests.

Day 15 · a new expression and its input

An invented display booth has four identical trays, each holding q+3 cards, plus two more groups of q blank cards. q is a non-negative whole number. The model counts cards only; it does not describe real stock, staffing or available tray sizes.

Write a single expression E(q) for the count. Expand and simplify it. Show an equivalent factorised form and explain by expanding back.
The invented total is E=42. Rearrange or solve to find q. Substitute your value in the original grouped expression and in the simplified expression to verify the same total.
Another invented total is E=39. Can it represent a whole-number q under this model? Show the calculation and give a one-sentence limit on what this classroom expression can tell us.

Day 20 · a new equation and a remaining-cap rule

Two invented display layouts use r whole batches, r≥0. Layout A uses 7+4r tokens. Layout B uses 19+2r tokens. A separate tray begins with 18 pins and loses 2 pins per batch; the tray model is meaningful only while pins remain non-negative. The display must have at least 6 pins remaining. These are teaching values, not real project costs or supplies.

Write an equation for A and B having the same total. Solve it algebraically and verify the crossing by substituting in both totals. State what a point (r,total) would mean on a graph.
Write and solve an inequality for “at least 6 pins remain”. Show why a negative division changes the inequality direction. Give the allowed whole batch counts and check the boundary plus one excluded count.
Compare A and B at r=5 and r=7. Write one bounded recommendation within this invented model that refers to the pin rule and one condition you would need before making any real display decision.