Which rectangles break into identical squares, and which do not

These teaching notes are for members

What the video covers, what to say before it, where a class usually goes wrong, and what to set afterwards. An account is free, and it opens every chapter of every book.

Create a free account Sign in

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 — the square and rectangle conditions
Working out the order in which a figure has to be drawn — a rough diagram is drawn before the construction
Every curve in these figures is part of a circle — a compass holds a distance while it is carried across the page

What they should be able to do

Simplify a hard construction to an easier one and then build the hard one back up
Draw a rough diagram for a divided figure and mark the equalities on it
Chase a chain of equal sides through such a figure
Use the tick-mark convention to record which sides are equal
Given one side of a rectangle that splits into two identical squares, state the other, and do the same for three
Copy a length with a compass instead of measuring it, and say why that is enough here
Name two side lengths that rule out any split into two matching squares, and another pair that rules out three, and justify each choice

Where it usually goes wrong

"You need to be told a measurement before you can construct anything." Not here. The problem asks only for a rectangle with the property, so the size is yours; only the proportion is fixed.
"Two identical squares could also be stacked, or arranged some other way." Two squares of the same size joined into a rectangle can only sit side by side, which is why the long side comes out at exactly twice the short one.
"A rectangle twice as long as it is wide might split into three squares if you cut it cleverly." For two squares and for three, the number fixes the proportion completely: two means 2 to 1, three means 3 to 1, and nothing clever is available. Two and three are the only cases the Breaking Rectangles task asks for, and the argument works because neither number can be laid out in more than one row — four identical squares, by contrast, make a 4-to-1 rectangle in a row and a square in a two-by-two block.
"4 cm by 2.5 cm nearly works." Nearly is not a category here. Either the long side is exactly twice the short one or the squares are not identical.
"Measuring AF with a ruler and then measuring 4 cm again for AB is the same as copying it." It is not: the second measurement can disagree with the first. The compass never has to read a scale twice.
"The tick marks are decoration." They are how the rough diagram carries the result of the reasoning, so that the construction can be read off it.

Questions to check understanding

Build a rectangle that splits into two matching squares, no measurements given
Do the same for three (asked at p.201)
If AF = 4 cm in the two-square figure, what is AC? (asked at p.200)
Name side lengths that rule out a two-square split, and another pair that rules out three (asked at p.201)
Mark all the equal sides on the rough diagram using the tick convention
Copy a given length onto a new line using a compass only
A rectangle is 15 cm by 5 cm. Divided into the largest identical squares possible, how many squares is that, and how long is a side of each?

Examples worth working on the board

The target figure, p.199. A rectangle with two internal vertical lines cutting it into three matching squares in a row. No measurements are printed on it. Checked against p.199.
The rough diagram, p.200. The simplified two-square version: a rectangle with A, B and C along the top and F, E and D along the bottom, one vertical line BE cutting it into the squares ABEF and BCDE. Short tick marks are drawn on every one of the shorter sides. Checked against p.200 — the tick marks are artwork.
The chain of equalities, p.200. Since both squares are the same size, AB agrees with BC, and FE with ED. Because each piece is a square, all four of its own sides agree: AF = AB = BE = FE for the first, and BE = BC = CD = ED for the second. Every short line in the whole figure is therefore the same length. That one sentence is the entire solution, and it was reached with no number in sight.
The one number the chapter does supply, p.200. If AF is taken to be 4 cm, then AC — the long side of the whole rectangle — has to be 8 cm, because it is AB followed by BC and each of those is 4 cm. The chapter asks for this and does not answer it.
Carrying the length, pp.200–201. The chapter draws AF without measuring it, raises a perpendicular long enough to hold the top side, and then transfers the length of AF with the compass to place B, and again to place C. Two printed panels on p.201 show the compass first spanning AF and then set down to mark B. Checked against p.201.
Three squares, p.201. The chapter then hands the three-square case back to the student. The reasoning is unchanged: every short line is equal, and the long side is three of them. So the split into three works exactly when the long side is three times the short one.
Rectangles that resist the split, p.201. The chapter asks the student to name side lengths ruling out a two-square split, and another pair ruling out three. Any pair in the wrong proportion answers it. The chapter prints no examples of its own.

Figures to have open

The three-square rectangle and the two-square rectangle, drawn to the same short-side length so the difference is visible as one extra copy.
The rough diagram carrying all six corner labels, with tick marks on every short side. This is the topic's key image and the tick marks must be drawn.
The compass transferring AF, in at least two frames.
A rectangle that fails the test, with its two sides dimensioned so the failure can be checked by eye.
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.199–201 — the Breaking Rectangles task and the simplification on p.199, the rough diagram, the chain of equalities and the tick convention on p.200, and the compass transfer, the three-square extension and the cannot-be-divided question on p.201
§8.2, p.193, for the square condition that makes the chain of equalities run
Summary, p.216, for the line on rough diagrams as a planning device