The shortest and longest crossing inside a rectangle

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What to assume they know

Constructing a square or rectangle from its side lengths — constructing a rectangle from its side lengths
The two properties that define a rectangle, and the one more a square needs — R1 and R2, and what makes a 4-sided figure a rectangle
Measuring a length in centimetres and millimetres and recording it
A point fixes a location; a segment is the shortest route between two points — the distance between two points is the length of the segment joining them

What they should be able to do

Set up the exploration: build the rectangle and identify the two sliding points
Predict where two sliding points will be closest and farthest, before measuring
Record many measurements compactly in a table with named columns
State the shortest possible XY and justify it without measuring
Recognise that a shortest distance can be reached at more than one position
State the longest possible XY and say which two positions produce it
Identify the 4-sided figure cut off when X and Y are level, and use it to explain the result

Where it usually goes wrong

"The closest position is the one at the top." Every level position is equally close. The chapter's own equal-distance table has three different level positions and the same answer three times.
"XY gets shorter as the points get higher." It gets shorter as they get level. Sliding both down together changes nothing at all.
"The shortest crossing is a new length you have to measure." It is AB, already drawn, already 7 cm. The rectangle ABYX is the reason.
"The farthest position is a corner, so there is one of them." There are two, and they are the two corner-to-corner lines of the rectangle.
"7 cm is the answer because 7 is the bigger side." No — it is the answer because it is the distance across, and X and Y are stuck on opposite 4 cm sides. Change BC to 40 cm and the shortest crossing is still 7 cm.
"Recording the trials is busywork." The table is where the pattern becomes visible; three sentences hide what three rows show.
"You need X and Y at the very ends for the maximum." True here, but say why: any tilt is longer than the level crossing, and the most tilt available is the full 4 cm.

Questions to check understanding

Where should X and Y be placed for XY to be shortest, and how long is it then?
How does the shortest XY compare with AB? (asked at p.198)
What shape is ABYX when X and Y are equally far from A and B? (asked at p.199)
How does the farthest XY compare with AC and with BD? (asked at p.199)
Fill in the three lengths for the positions given in the p.198 table
Repeat the exploration on a rectangle of different sides and state what changes
Explain why moving X and Y down by the same amount leaves XY unchanged

Examples worth working on the board

The set-up, p.197. A rectangle named ABCD, built with its long side AB = 7 cm and its short side BC = 4 cm. In the printed figures A is top-left, B is top-right, C is bottom-right and D is bottom-left, so AB is the long top side and AD and BC are the 4 cm sides. X slides along AD, Y along BC. Checked against p.197.
The four printed positions, p.197. Four copies of the rectangle with XY drawn as a dashed line: X 5 mm below A with Y marked 1 cm above C; X 1 cm below A with Y 2 cm below B; X sitting on D with Y sitting on B; and X sitting on A with Y sitting on B. Checked against p.197. These four are the whole range in miniature and are worth using in that order.
The three sample positions, p.198. X 5 mm from A with Y 3 cm from B; X 1 cm from A with Y 1 cm from B; X 2 cm from A with Y 4 cm from B. The chapter prints blanks for the three lengths.
The equal-distance table, p.199. Three rows in which X and Y are the same distance from A and from B: 5 mm and 5 mm, 1 cm and 1 cm, 1 cm 5 mm and 1 cm 5 mm. All three give XY = 7 cm. The chapter then asks two things about each row — how XY compares with AB, and what shape ABYX is. Both answers are the same answer: ABYX is a rectangle, so its two opposite sides XY and AB are equal.
The extremes. Shortest XY = 7 cm, reached at every level position, and there are as many of those as there are points on a 4 cm side. Longest XY is the corner-to-corner line, reached only when X is at D and Y is at B, or when X is at A and Y is at C. Its length is a shade over 8 cm — about 8 cm 1 mm for a 7 cm by 4 cm rectangle.
The comparison the chapter asks for, p.199. How the farthest XY compares with AC and with BD. Those two lines are equal to each other, which is exactly what §8.5 goes on to establish about the diagonals of a rectangle. This is the hinge between the two sections and should be named as such.

Figures to have open

The rectangle with the two sliding points, drawn so that X and Y can be shown moving along AD and BC. This is the topic's central image.
The four printed positions from p.197, redrawn as a strip.
ABYX shaded inside the rectangle when X and Y are level, with its four corners marked and its opposite sides ticked.
The two corner-to-corner placements with AC and BD both drawn, so their equality is visible before §8.5 argues for it.
A three-column table matching the chapter's own layout: distance of X from A, distance of Y from B, length of XY.
No photograph or dataset is required.

Where this sits in the book

NCERT Ganita Prakash, Class 6, Chapter 8 "Playing with Constructions", §8.4 "An Exploration in Rectangles", pp.197–199 — the set-up and the four positions on p.197, the Math Talk prediction and the shorthand table on p.198, and the equal-distance table with the ABYX question and the AC/BD question on p.199
§8.2, p.193, for R1, which is what makes ABYX settle the question
§8.5, p.203, where AC and BD acquire the name diagonals and their equality becomes the subject