Scale and similar shapes

Activities to try

1. 1

   Enlargement studio: grab, scale, and compare with additive mode

   Why it works: Enlargement multiplies every corresponding side by the SAME factor, preserving shape and angles; adding the same amount to each side of a non-square rectangle distorts it. Because the shape grows in two dimensions, area scales by k², not by k.

   1. Work on a grid with a non-square rectangle or triangle whose aspect ratio is locked. Drag a corner to enlarge it and watch a live scale-factor readout while every side multiplies by the same factor and the angles stay unchanged; for a rectangle enlarged from a fixed corner, notice the dragged opposite corner moves along the original's diagonal, showing it is the same shape.
   2. Fill a ratio table alongside (old side | new side for each pair) and check that the same column multiplier appears for every pair of corresponding sides.
   3. Flip to an 'additive mode' that instead ADDS your chosen number to each side of the non-square rectangle; observe it stretch lopsided and fall off the original's diagonal — it is no longer similar. Compare directly with the multiplicative result.
   4. Turn on a grid-tiling overlay: with a scale factor of 2, see FOUR copies of the original tile the new shape; with a factor of 3, nine copies fit — connect this to area × k².
   5. If a scaling touches only one side, use the flag that highlights the unchanged side and correct it so every corresponding side is scaled.
2. 2

   Confront the additive error on paper

   Why it works: Adding a constant to every side changed the proportions of the 2×5 rectangle (as it does for any non-square rectangle); multiplying every side by a common factor always keeps corresponding sides in a single ratio and preserves the shape.

   1. Take a 2×5 rectangle. Add 4 to each side to get a 6×9 rectangle, and draw both. Check whether the side ratios match: 2:5 versus 6:9 — they do not, so the shapes are not similar.
   2. Now enlarge the same 2×5 rectangle by a scale factor, multiplying each side by 3 to get 6×15, and confirm 2:5 and 6:15 are the same ratio.
   3. State in your own words why 'add the same to each side' changes the shape while 'multiply each side by the same number' keeps it.
3. 3

   Solve and judge similar-shape problems

   Why it works: Problem solving means finding the scale factor from a known pair, applying it in the correct direction to every side, checking similarity by shared ratio, and using k² for area.

   1. Given a rectangle 6×9 enlarged so 6→18, find the scale factor (18 ÷ 6 = 3) and the missing side (9 × 3 = 27 cm). Beware of adding the difference 12 to get 21 — that is the additive error, not a valid answer.
   2. Find the scale factor between two similar triangles from a matched pair of side lengths, keeping new ÷ old consistent for the comparison.
   3. Judge whether two given shapes are similar by testing EVERY pair of corresponding sides for the same ratio, not by appearance, and by checking that corresponding angles are equal (for triangles, matching side ratios are enough; for other polygons they are not, e.g. a square and a non-square rhombus). One mismatched ratio means not similar.
   4. For a REDUCTION (e.g. 18 back to 6), decide first whether the answer should be bigger or smaller than the side you start from; then divide by the enlargement factor (÷3), which is the same as multiplying by the new ÷ old factor (6 ÷ 18 = 1/3).
   5. On an area follow-up for the 6×9→18×27 rectangle, reason that area scales by k² = 3² = 9 (54 cm² → 486 cm²), not by 3.