# Twenty optional transfer problems

Each prompt uses invented numbers/context. Use one after a lesson or for a home extension; students can always do it on paper. They should show an exact expression and a short check or reason. [Worked staff answers](teacher/KEY-AND-NEXT.md#optional-swaps--worked-results) are separate and public. No personal data, real workplace measurements or purchase is needed.

## Day 1 · Classification

- **Book-display squares:** areas 25 and 7 square units. Evaluate `sqrt(25)` and classify `sqrt(7)`; bound the latter between whole numbers.
- **Poster panels:** areas 49 and 15 square units. Which side is an exact whole number? Between which integers is the other?

## Day 2 · Factor transfer

- **Mosaic draft:** simplify `sqrt(108)` units using its largest square factor; square your result.
- **Fictional screen tile:** simplify `sqrt(200)` units and check the result by squaring.

## Day 3 · Comparison

- **Paper-track marks:** compare `sqrt(18)` and `sqrt(72)` cm; give the exact difference.
- **Gallery-cord samples:** compare `sqrt(27)` and `sqrt(75)` cm; give the exact difference.

## Day 4 · Addition

- **Kite-frame segments:** add `sqrt(20)+sqrt(45)` cm exactly; reject a radicand-addition shortcut.
- **Fictional map strips:** add `sqrt(28)+sqrt(63)` units exactly, showing common root parts.

## Day 5 · Subtraction

- **Prop-ribbon samples:** compute `sqrt(75)-sqrt(27)` cm exactly; check the sign.
- **Paper-lantern strips:** compute `sqrt(125)-sqrt(20)` cm exactly; explain the shared root.

## Day 6 · Products

- **Rectangle sketch:** sides `sqrt(8)` cm and `sqrt(18)` cm; find exact area with units.
- **Symbolic animation scale:** simplify `(3sqrt(2))(2sqrt(10))`; keep the whole-number coefficients visible.

## Day 7 · Rationalising

- **Pure design ratio:** rationalise `5/sqrt(3)` without changing its value.
- **Another pure ratio:** rationalise `6/sqrt(7)` and give a quick decimal equivalence check.

## Day 8 · Multi-step audit

- **Cue index:** simplify `(sqrt(50)+sqrt(8))/sqrt(2)` by two routes.
- **Route index:** simplify `(sqrt(75)-sqrt(12))/sqrt(3)` and locate the like terms.

## Day 9 · Vertex form

- **Abstract plotting card:** for `y=(x+1)^2+2`, find turning point, axis, y-intercept and number of real x-intercepts.
- **Digital-art plotting card:** for `y=2(x-1)^2+3`, find the same four features and show one mirrored x pair.

## Day 10 · Factor form

- **Fictional graph A:** `y=(x-2)(x-6)`. Find both zeros, axis, turning point and y-intercept.
- **Fictional graph B:** `y=-(x+2)(x-4)`. Find the same features and opening direction. Give one sign check between the zeros.