Why some resized images look right and others look stretched

What to assume they know

• Multiplying and dividing whole numbers, and multiplying by a fraction such as one half or three halves
• Reading a fraction as an instruction to scale — "half of 60"
• Measuring lengths in millimetres and centimetres with a ruler
• Comparing two fractions or two quotients to see which is bigger
• That a rectangle has four right angles whatever its width and height

What they should be able to do

• Read a table of widths and heights and decide, from the numbers alone, which of several rectangles have the same shape
• Find the factor that carries one width to another, and check whether the same factor carries the first height to the second
• Show on an example that taking equal amounts off two unequal lengths does not preserve their comparison, and say which way the comparison moves
• Explain why a picture that keeps one side fixed and changes the other must look distorted
• Use the word proportional in the chapter's sense: both quantities changed by a single common factor
• Given a rectangle, construct a larger and a smaller one of the same shape, and justify the construction by naming the factor used
• Measure a real object — a blackboard, a classmate's limbs — and record the comparison as a pair of numbers rather than as two separate measurements

Where it usually goes wrong

• "The big one and the small one must look different." Size is not the variable being tested here. A and C differ by a factor of two and are the same shape; A and B differ by less and are not.
• "They look different because one is a square." The chapter raises this explanation itself and then destroys it: the other distorted picture is a rectangle. A wrong explanation that survives one example is the standard trap in this section.
• "Same difference, same change." Taking 20 mm off each side feels even-handed and is not. Twenty is half of forty and a third of sixty; the same subtraction is a different-sized event for the two measurements.
• "Subtracting from both makes it more square." It does the opposite, always — section 6 proves it. Students reliably guess the direction wrong, so make them predict before showing the picture.
• "Only whole-number factors count." The factor from A to D is three halves, and the factor from A to C is one half. A factor is any multiplier.
• "Every rectangle is similar to every other rectangle." Equal angles are not enough; the two side lengths have to be locked together as well.
• "Width is the horizontal one." True for the tigers, useless for the tilted rectangles on Part I p.166 and worse than useless for a limb. Pair the numbers by naming what each one measures, not by which way it points.

Questions to check understanding

• Given a table of widths and heights, group the rectangles that share a shape and name the factor inside each group
• Given one rectangle's measurements and a factor, produce the measurements of the resized rectangle
• Given two rectangles, decide whether one is a resize of the other, and if not, say which side is at fault
• Predict, without computing, whether removing the same length from both sides of a rectangle makes it look longer or squarer, then verify by measuring
• Draw a rectangle of a stated shape at a stated size, and a second one at a different size and the same shape
• Explain in words why two people's answers to "draw a bigger rectangle of the same shape" can differ and both be right
• Measure a real object and report the comparison, with units named

Examples worth working on the board

Values marked verified are worked out here or an added measurement on the printed page; the chapter states the table and leaves the comparisons to the reader.

• The five-image plate (Part I, §7.1, p.159). One photograph of a walking tiger, reproduced five times at five sizes, captioned Image A to Image E. A sits top left, B and C to its right in one row, D and E side by side below. A, B, C and D are landscape rectangles; E is close to square. The chapter's verdict on the page is that A, C and D match one another.
• The measurement table (Part I, §7.1, p.160). Three columns — image, width in mm, height in mm. A: 60 and 40. B: 40 and 20. C: 30 and 20. D: 90 and 60. E: 60 and 60. These five pairs are the chapter's data and every argument in §7.1 runs on them.
• The factors between images. Verified: A to C is one half applied to both numbers (60→30, 40→20). A to D is three halves applied to both (60→90, 40→60) — the chapter asks for this factor and does not print it. A to B is one half on the height but two thirds on the width, which is the whole of B's problem. A to E is 1 on the width and three halves on the height.
• The subtraction that the chapter contrasts. A to B takes 20 mm off the width and 20 mm off the height. Verified: the resulting comparison is 40 to 20, which is a wider comparison than 60 to 40, not the same one.
• The general fact behind section 6. Verified algebraically: if the width w exceeds the height h, and the same positive k is removed from both — with ***k* smaller than h, so that both remaining lengths stay positive** — then (w − k) compared with (h − k) is a stronger mismatch than w compared with h. Cross-multiplying, (w − k)h − w(h − k) equals k(w − h), which is positive. So within that range equal subtraction never repairs a mismatch and never leaves it alone; it exaggerates it. The condition is not decoration: at k = h the second comparison has a zero term and says nothing, and past it both terms turn negative and the conclusion reverses. State the condition whenever the general rule is stated — the chapter's own instance (60 and 40, with k = 20) sits well inside the range, so nothing printed depends on it, but the explanation teaches the general rule. The chapter demonstrates this on that one example and does not state it in general.
• What the printed photographs actually measure. Measured by taking the bounding box of each photograph off the printed page, at two different thresholds with identical results: A prints 57.1 × 38.1 mm, B 38.1 × 19.3, C 26.7 × 19.1, D 85.1 × 56.4, E 52.1 × 57.1. So the plate reproduces the tabulated millimetres at about 95%, and the shapes it holds are A (1.50), B (1.97) and D (1.51), all within 1.5% of their tabulated ratios, while C prints 1.40 against a tabulated 1.50 and E 0.91 against 1.00 — both are narrow in width on the plate, and E is not square there. Note that B and C are bottom-aligned with their tops a single pixel apart, so their printed heights are necessarily equal. The consequence is the point: the two images whose plate ratios drift are C and E, so a class measuring the photographs will not reproduce the book's own A-C-D similarity grouping, which rests on the table. Treat the table as the data and the photographs as illustration.
• The blackboard task (Part I, §7.4, Example 5, p.163). Measure the classroom blackboard's width and height to the nearest centimetre, record the pair, then draw a rectangle of the same shape in a notebook and compare with classmates' rectangles. Two printed blanks, no values supplied.
• Five tilted rectangles (Part I, p.166, exercise item 4). Lettered A to E and set at different angles: A upright and narrow, B upright and wide, C large and tilted about 35° clockwise (its long axis running down to the right), D tilted about 18° anticlockwise (long axis running up to the right), E tilted about 45° clockwise. The senses matter: the item asks which are similar and suggests measuring with a scale, and the tilt has to survive the redraw. Verified by measuring the drawn outlines on the printed page: A is about 6 × 18 mm, B about 18 × 12, C about 51 × 20.5, D about 39 × 13, E about 17.5 × 7. So two of them are 3-to-1 shapes, two are 5-to-2 shapes, and one is 3-to-2 and matches nothing. Note as a check: because A stands upright and its partner lies down, a student who writes width-then-height gets 1 : 3 against 3 : 1 and may declare them different. That is the interesting part of the item, not a flaw in it.
• The rectangle to copy (Part I, p.166, exercise item 5). One upright rectangle. Verified: it prints about 35 × 21 mm, a 5-to-3 shape. The item asks for a smaller and a bigger rectangle of the same shape, then asks whether everyone's drawings agree and whether disagreement means someone is wrong.
• The human figure task (Part I, p.167, exercise item 7). Measure a friend's head, torso, arms and legs; record head : torso, torso : arms and torso : legs in three pairs of blanks; then draw a figure to those same comparisons. Three small drawn figures sit beside the blanks — a standing jointed figure, a running one, and a child holding a balloon. No lengths are printed.
• Note to the Teacher (Part I, p.163 and p.167). Both ask that students be made to give reasons, not just answers; the second asks specifically for a reason why a drawing is proportional.

Figures to have open

• The five-image plate, redrawn. Do not reproduce the printed photograph. Use one simple silhouette or a schematic animal and reproduce it at the five tabulated width-and-height pairs, since the argument needs the five shapes and nothing else. This is the chapter's own figure (Part I, §7.1, p.159).
• The width/height table as a five-row strip that stays in view through sections 3 to 7.
• A same-difference-versus-same-factor panel: 60 × 40 with 20 mm removed from each side, beside 60 × 40 halved. Standard schematic.
• The five tilted rectangles of Part I p.166, redrawn at their measured sizes and angles. The tilt is the point of the item and must survive the redraw.
• A jointed human figure with the four segments named. Standard schematic.
• No photograph is required anywhere in this topic.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part I, printed Chapter 7, "Proportional Reasoning-1", §7.1 "Observing Similarity in Change", Part I pp.159–160. The section carries one Math Talk marker (Part I p.160, beside the question of why B looks different when it too is a rectangle), four blue "?" question prompts across Part I pp.159–160 — a different device — and the five-row measurement table.
• Part I p.163, §7.4 Example 5 — the blackboard measurement task, and the Note to the Teacher below it.
• Part I p.166, exercise items 4 and 5, and Part I p.167 item 7, with the second Note to the Teacher. These items sit inside the "Figure it Out" block that follows §7.4 but they test §7.1's idea of similar shape, which is why they are carried here.
• Forward pointer: the word ratio arrives on the next page (Part I, §7.2, p.161) and is the subject of What a ratio claims, and why it is not a difference.
• The chapter's SUMMARY (Part I p.177) does not restate §7.1; it begins at the ratio. Read on the printed page.