The two properties that define a rectangle, and the one more a square needs

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

Straight and right angles as the landmarks of a full turn — what a right angle is, and how a 90° corner is recognised
Reading and drawing an angle with a protractor — using a protractor, for the cases where the eye is not enough
Measuring a length in centimetres and comparing two lengths
The idea of a four-sided closed figure with straight sides

What they should be able to do

Name the corners, sides and angles of a four-sided figure, and identify which sides are opposite each other
State the two conditions the chapter gives for a rectangle
State the two conditions the chapter gives for a square
Decide, for a given figure, which conditions it passes and which it fails
Explain why every square passes the rectangle test
Produce a figure that satisfies one clause of a pair and fails the other, where such a figure exists
Test whether four right angles on their own already force the opposite sides to be equal

Where it usually goes wrong

"A square is a shape and a rectangle is a different shape." They are two tests, and the square test is the harder one. Anything that passes it passes the other. The two printed lists say so, even though the chapter does not spell out the conclusion.
"A rectangle is a square that got stretched." Backwards. Start from the conditions: a rectangle only promises that facing sides match; a square promises that all four do.
"Equal sides make a square." B and C in the p.194 collection have four equal sides and are not squares. S2 is doing real work.
"Four right angles make a square." D in the same collection has four right angles and is not a square. It is a rectangle.
"You can tell by looking." All four figures on p.194 are tilted precisely so that looking fails. The chapter's own Think prompt asks whether the corners' positions on the grid can settle it without any measuring at all — and they can.
"R1 and R2 are both needed, so they are independent." One of them is not independent: R2 forces R1, which is exactly what the p.197 question exposes. The chapter keeps both because both are useful to check.

Questions to check understanding

State the two properties of a rectangle and the two properties of a square
Identify which of these figures are squares, using measurements if needed (the chapter's own p.194 task)
Draw a 4-sided figure with all sides equal that is not a square
Draw a 4-sided figure with all angles 90° that is not a square
Is there a 4-sided figure with all angles 90° and unequal opposite sides? (asked at p.197)
Explain why every square is also a rectangle
Decide whether a figure on a dot grid is a square without using a ruler or a protractor, and say how you knew

Examples worth working on the board

Fig. 8.3, p.192. One rectangle in the middle with four equal squares set diagonally outward from its four corners, each clearly separated from the rectangle by a gap, the whole arrangement symmetric about the rectangle's centre. Checked against p.192: the squares nowhere meet the rectangle. No lengths are printed on it. It is the figure the chapter opens the section with and the one the p.194 Figure it Out asks students to reproduce on dot paper — where the placement is what the printed question is about.
Fig. 8.4, p.192. A rectangle labelled A top-left, B top-right, C bottom-right, D bottom-left. In the printed figure the pair AB and CD is drawn in one colour and the pair AD and BC in another, so the two pairs of opposite sides are distinguished by colour. Checked against p.192; the colours do not extract.
The four conditions, p.193. The chapter labels them, and the labels are reused later in the chapter: R1 — opposite sides equal in length; R2 — all angles 90°; S1 — all sides equal; S2 — all angles 90°. Note that R2 and S2 say the same thing, and that S1 is a stronger demand than R1.
The dot-grid collection, p.194. Four tilted 4-sided figures drawn with their corners on grid dots, labelled A, B, C and D. Checked against p.194 and reading the corners off the grid: A has diagonals of 8 dot-units each, crossing at right angles, so all four of its sides are 4 across and 4 up and all its angles are 90° — A passes both S1 and S2. B has diagonals of 6 and 8 units, giving four sides of 5 units each, so it passes S1 and fails S2. C has diagonals of 10 and 8 units, again four equal sides and no right angles — S1 yes, S2 no. D is the small tilted one: one side steps 2 dots across and 1 dot down, and the next steps 2 dots back and 4 dots down. Those two directions are at right angles, and the second side is exactly twice the first, so D passes R1 and R2 and fails S1. Only A is a square.
The 90°-but-unequal question, p.197. The chapter asks whether a 4-sided figure can have all its angles equal to 90° while its opposite sides are unequal. Work it as a genuine question in the explanation: walking the boundary, each corner turns a quarter turn, so after two sides the direction has reversed and the third side runs back parallel to the first. To close up, the third must match the first exactly. It cannot be done, so R2 already forces R1.
The rotated rectangle, p.194. A rectangle drawn on a slant, printed immediately before the Figure it Out. It is there to make the point that passing the test has nothing to do with sitting square to the page; that is the subject of Naming corners in order, and why a rotated square is still a square.

Figures to have open

Fig. 8.3 redrawn: a rectangle with four equal squares set diagonally outward from its four corners, each separated from the rectangle and the set symmetric about its centre. The squares must not be drawn touching the rectangle. Redraw rather than reproduce.
Fig. 8.4 redrawn with the two pairs of opposite sides in two colours, matching the printed colouring in structure.
The p.194 dot grid with all four figures A to D, drawn on a visible dot lattice so that side lengths can be counted rather than measured. This figure is essential and the grid must be accurate — the argument depends on the corners sitting on dots.
A two-column comparison card for the R and S lists.
No photograph or dataset is required.

Where this sits in the book

NCERT Ganita Prakash, Class 6, Chapter 8 "Playing with Constructions", §8.2 "Squares and Rectangles", pp.192–194 — Fig. 8.3 and Fig. 8.4 with the bold opposite-sides passage on p.192, the R1/R2 and S1/S2 lists on p.193, and the Figure it Out collection on p.194
§8.3 "Constructing Squares and Rectangles", p.197, Construct item 3, for the 90°-but-unequal question
§8.5, p.210, where R1 and R2 are cited again by their labels as the check on a finished construction
Backward links: Chapter 2 "Lines and Angles" for right angles and for measuring an angle