Start. Draw a bar in five equal parts: two lightly shaded, three dark. One group has total length 10 cm. Find the length of one part, then each colour: 2 cm, 4 cm and 6 cm. Double the total to 20 cm. Ask what doubles and what stays the same. Explain that actual length can be fractional, while a count of whole cards must stay whole.
Route 22A · A double number line
A paper pattern uses 4 dots for every 7 dashes. Make a double number line showing 0, 4, 8, 12 dots beside 0, 7, 14, 21 dashes. Add the point for 10 dots if partial lengths rather than whole marks were being modelled. Explain why 10 whole dots would not make an exact number of complete pattern groups.
Teacher next move: the continuous paired amount is 17½ dash-units; the scale factor is 10/4 = 2½. For whole marks in complete groups, use a multiple of four dots. Do not round 17½ to 18 and call the ratio unchanged.
Route 22B · Work backwards from a total
A set of paper counters must have stars:circles = 3:5 and contain 48 counters. Find each count using equal parts. Decide whether 50 counters can have this exact ratio with whole counters. Explain your answer without drawing all fifty.
Teacher next move: eight parts; 48/8 = 6, so 18 stars and 30 circles. Fifty would require 6¼ per part, yielding 18¾ and 31¼ counters; the exact whole-count condition is impossible. “Impossible under this condition” is a useful answer.
Route 22C · Compare methods, then choose
A length design has red:blue:grey = 1:2:3 and total 15 m on a fictional diagram. One learner finds one part first. Another scales the six-part base group by 2½. Show both methods and choose the one you would use for total 9 m. Explain why 1:2:3 and 2:4:6 describe the same mix but 2:3:4 does not.
Teacher next move: for 15 m, 2½ m, 5 m, 7½ m; for 9 m, 1½ m, 3 m, 4½ m. Each component must be multiplied by the same factor. All quantities here are continuous lengths, not whole objects.
Close: name the scale factor and the condition that permits or prevents a fractional answer.