# Fresh formative checks · learner copy

**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.

1. Write a single expression `E(q)` for the count. Expand and simplify it. Show an equivalent **factorised** form and explain by expanding back.
2. 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.
3. 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.

1. 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.
2. 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.
3. 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.

Original SubjectNest fresh checks and fictional prompts © NeuroForgeIO Pty Ltd 2026, [CC BY 4.0](https://creativecommons.org/licenses/by/4.0/).