# Clean learner cards · exact values first

These are fictional models and numbers. Give exact answers unless asked for an estimate. Show a square factor, algebraic step or graph feature that makes your reasoning visible. You may write, explain aloud with notes captured, or arrange labelled tiles/points and dictate the exact expression. No answer key appears in this file.

## Card A · Which roots stay irrational? (Day 1)

A paper display has separate square panels of area 9, 10 and 16 square units. Their side lengths are `sqrt(9)`, `sqrt(10)` and `sqrt(16)` units.

1. Evaluate the two perfect-square roots exactly. Which of the three side lengths is a surd, and why?
2. Place `sqrt(10)` between consecutive whole numbers without using a calculator. Explain using squares.
3. A classmate writes `sqrt(10) = 3.16`. Say what is useful about that statement and what symbol it needs.

## Card B · Find a square hiding inside (Day 2)

An invented mosaic tile has area 72 square units, so its side is `sqrt(72)` units.

1. Write 72 as a perfect square times another natural number. Simplify `sqrt(72)` exactly.
2. Verify by squaring your simplified expression. Say why `sqrt(72) = sqrt(36) + sqrt(2)` is not a valid shortcut.
3. Estimate the side between two whole numbers using `8^2` and `9^2`.

## Card C · Equivalent lengths, clearer comparison (Day 3)

Two fictional square stage marks have areas 50 and 98 square units. Their side lengths are `sqrt(50)` and `sqrt(98)`.

1. Simplify both exact lengths using their largest square factors.
2. Which side is longer? Give the exact difference in units.
3. Explain why changing both to decimals too soon can hide the shared `sqrt(2)` factor.

## Card D · Add like radicals (Day 4)

A fictional card route has two straight segments of lengths `sqrt(12)` cm and `sqrt(27)` cm. The total route length is their sum; this is a deliberately simple model, not a real navigation map.

1. Simplify both lengths, then add exactly.
2. A peer writes `sqrt(12) + sqrt(27) = sqrt(39)`. Test the claim by estimating both sides between whole numbers.
3. Write one sentence explaining what makes terms “like” after simplification.

## Card E · Subtract carefully (Day 5 practice)

Two ribbon lengths in a fictional prop kit are `sqrt(32)` cm and `sqrt(8)` cm.

1. Find the exact difference, longer minus shorter.
2. Find `sqrt(8) + sqrt(18)` exactly as a second case.
3. Point to the square factors used. The separate Day 5 check uses **new** expressions.

## Card F · Multiply surds and units (Day 6)

A fictional rectangle has sides `sqrt(6)` cm and `sqrt(15)` cm. A separate symbolic practice expression is `(2sqrt(3))(3sqrt(6))`.

1. Find the rectangle's exact area and simplify its radical. What units should the answer have?
2. Simplify the separate product. Explain where the whole-number coefficients go.
3. Estimate the rectangle's area to see whether your exact answer is reasonable.

## Card G · Move the radical out of the denominator (Day 7)

For a paper-model scale, an invented ratio is `7/sqrt(5)`. This is a pure number with no physical units.

1. Multiply numerator and denominator by a form of 1 to write the same number with a rational denominator.
2. Show that your exact new form is equivalent to the original.
3. Why is `7sqrt(5)/5`, not `7sqrt(5)`, the result? Try `4/sqrt(2)` as a quick second example.

## Card H · Audit a multi-step result (Day 8)

A stage-light designer has written the invented index `(sqrt(18) + sqrt(8))/sqrt(2)` on a planning sheet. The number is dimensionless; it is not a real lighting standard.

1. Simplify the numerator and then the whole expression exactly.
2. Find a second route by dividing each term by `sqrt(2)` first.
3. A draft says the answer is `sqrt(26)/sqrt(2)`. Locate the invalid step, and give an approximate check of your own answer.

## Card I · First quadratic graph (Day 9)

Study `y = (x - 2)^2 + 1` as an abstract rule. A plotted curve could be used in a fictional animation, but no physical fit is asserted.

1. Complete a table at `x = 0, 1, 2, 3, 4`. Plot or arrange the points.
2. State the turning point, axis of symmetry, y-intercept and number of x-intercepts, with reasons.
3. Explain what the square guarantees about the smallest possible y-value for real x.

## Card J · Factor form shows zeros (Day 10 practice)

For a different fictional graph, use `y = (x + 1)(x - 3)`.

1. Find both x-intercepts, the axis of symmetry, turning point and y-intercept.
2. Explain why the graph opens upward and where y is negative.
3. Sketch enough points to check your answers. The separate Day 10 check has a **new** graph and surd.