# Full text and tactile routes for five A4 mats

All content of each original mat is stated here in reading order. The SVG/PDF is optional. Use this text in large type or a screen reader; for tactile work, use folded or raised paper strips. Colour is never needed to tell one result from another. Blank lines on the pages are writing space only.

## Exact value / approximate check

[SVG](exact-approx.svg) · [PDF](exact-approx.pdf). Subtitle: A radical can be exact without a decimal. “Square-number bounds give a fast reasonableness check. Use = for equal values and ≈ for rounded values.” Three boxes: (1) squares around n; (2) exact radical; (3) decimal check. Worked example: `9<10<16`, so `3<sqrt(10)<4`. `sqrt(10)` is exact; `sqrt(10)≈3.162` is rounded. Try it: What is wrong with `sqrt(10)=3.162`? **Tactile:** put 9, 10, 16 cards in order; connect them to root bounds 3 and 4.

## Find the square factor

[SVG](square-factor.svg) · [PDF](square-factor.pdf). Subtitle: Simplify a root and check by squaring. “Split a positive product under a square root. Do not split a sum using the same rule.” Three boxes: (1) n = square × rest; (2) √n = √square × √rest; (3) square the result. Worked example: `72=36×2`, so `sqrt(72)=6sqrt(2)`; `(6sqrt(2))²=36×2=72`. Try it: Why is `sqrt(36+2)` not `6+sqrt(2)`? **Tactile:** use 36 and 2 factor cards; fold the 36 card to show 6².

## Match the root part

[SVG](like-radicals.svg) · [PDF](like-radicals.pdf). Subtitle: Add or subtract only after simplifying. “Treat a shared √r like a shared variable. Keep unlike root parts separate.” Three boxes: (1) simplify each root; (2) match root parts; (3) combine coefficients. Worked example: `sqrt(12)+sqrt(27)=2sqrt(3)+3sqrt(3)=5sqrt(3)`; `5sqrt(3)≈8.66`, while `sqrt(39)` is only about 6.24. Try it: Can `2sqrt(3)+2sqrt(2)` combine into one like surd? **Tactile:** place 2 and 3 counters beside identical √3 strips.

## Multiply by one

[SVG](rationalise.svg) · [PDF](rationalise.pdf). Subtitle: A rational denominator, same value. “Choose a factor that removes the denominator root. Multiply the top and bottom by that same factor.” Three boxes: (1) start `7/sqrt(5)`; (2) multiply by `sqrt(5)/sqrt(5)`; (3) finish `7sqrt(5)/5`. Worked example: `(7/sqrt(5))(sqrt(5)/sqrt(5))=7sqrt(5)/5`; the multiplier equals 1. Try it: Why is `7sqrt(5)` alone not equivalent to `7/sqrt(5)`? **Tactile:** stack numerator and denominator strips; add √5 to both.

## Read the quadratic graph

[SVG](quadratic-map.svg) · [PDF](quadratic-map.pdf). Subtitle: `y=(x−2)²+1`, an abstract rule. Five plotted points on axes x=0–4, y=0–6: `(0,5)`, `(1,2)`, `(2,1)`, `(3,2)`, `(4,5)`. A solid upward parabola connects them and a dashed line shows the axis `x=2`; coloured point markers are decorative, with coordinates given here. Worked example: turning point `(2,1)`, axis `x=2`, y-intercept `(0,5)`; no x-intercept because `(x−2)²+1≥1`. Try it: Where is the matching point for x=1? **Tactile:** place five raised points on a grid and fold along x=2 to match pairs.