Comparing two angles without measuring either of them

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

An angle is an amount of turn, not a pair of drawn arms: an angle is a turn about a vertex; vertex, arms and the three-letter naming convention
Line and ray: what changes when you refuse to stop: a ray keeps its identity however far along it you draw
Comparing two lengths by placing one beside the other, from earlier work on measurement

What they should be able to do

Rank a set of drawn angles from smallest to largest by inspection, and say when inspection is not enough
Superimpose two angles correctly — vertices together, one arm along one arm
State the condition for two angles to be equal in terms of superimposition
Explain equality a second way, in terms of equal amounts of turn
Show that changing the length of an arm changes nothing about the angle
Compare two angles that cannot be moved, using a circle placed at each vertex
Build a pair of rotating arms and use them to order several angles
Predict which of a set of rotating arms will pass through an angle-shaped slit, and justify the prediction
Explain why the slit test ignores the length of the arms

Where it usually goes wrong

"The angle with the longer arms is bigger." This is the chapter's declared target: §2.6, p.25 carries a note to teachers saying students often believe exactly this. Corrected twice over — by ∠XOB against ∠XOC on p.23, and by the slit, which the arms pass or fail regardless of their length.
"Superimposing means lining up the two drawings." It means putting the vertices together first. Corrected in §2.6, p.22, where the vertex condition is stated before anything else.
"Two angles are equal when they look the same size on the page." Corrected by ∠AOB and ∠XOY, whose arms are drawn to visibly different lengths and which are nonetheless equal, and by the undecidable pair, which look rankable and are not.
"If you cannot move the figures, you cannot compare them." Corrected by the transparent circle: the disc moves, the figures do not, and the disc carries the angle with it.
"The circle works because it measures the angle." It does not measure anything yet — there are no marks on it. It works because it converts both angles to marks on one rim. Keeping that distinction clean is what makes the next topic's move to degrees feel like a step rather than a jump.
"An angle can be reduced by trimming it." The snipping cartoon at §2.7, p.27 exists for this. It is the punchline of the explanation as well as of the previous one.

Questions to check understanding

Given two or three drawn angles, state which is greater and give the reason
Fold a rectangular sheet, name the angles the crease makes with the sides, and compare them
Identify a pair of angles in a figure that are equal because they share both arms, and say why the labelling differs
Explain why one of two angles cannot be ranked by inspection
Predict which rotating arm passes through a given slit, with justification
One-line reasoning items: does making the arms longer change the angle?
"Where else do we compare by superimposition?" — an open item the chapter asks at §2.6, p.23, which examiners echo as a transfer question

Examples worth working on the board

The animals (§2.6, p.21). Four creatures with their mouths open by different amounts — an eel, a fish, a chick and a crocodile. Each mouth is a pair of arms about a hinge. Use them to make ranking feel obvious before the section takes the obviousness away.
∠ABC and ∠PQR superimposed (§2.6, p.22). Three panels: the two angles apart, then laid together with Q on B and one arm shared. Once superimposed, the arm of one falls inside the other, and that settles the ranking. Show the movement of the lift; the still panels alone hide the argument.
∠AOB and ∠XOY (§2.6, p.22). Printed as three panels like the pair above: the two angles separately, then superimposed with A on X and B on Y. The two are drawn at the same tilt but with visibly different arm lengths — OA and OB are short, OX and OY run further out — so this is the worked case of equality and a second strike against arm length. State the matching arm by arm.
The three-part comparison (§2.6, p.23). One vertex O with rays to A, X, Y and, along one straight arm, B and then C further out. Three questions are asked of it: ∠AOB against ∠XOY, ∠AOB against ∠XOB, and ∠XOB against ∠XOC. The third is the important one — B and C sit on the same ray from O, so those two angles are the same angle written two ways, and nothing has changed except how far along the arm the second letter was chosen.
The undecidable pair (§2.6, p.23). A second figure at O with X, A, Y and B, where ∠XOY and ∠AOB genuinely cannot be ranked by looking. The honest answer is that superimposition or measurement is required.
Fig. 2.10, the two cranes (§2.6, p.23). Two birds with beaks open by different amounts, drawn in separate circular vignettes so that neither can be slid onto the other.
The transparent circle (§2.6, p.24). Photographs: a circular sheet laid on the first crane's angle with its centre on the vertex, the two rim points marked A and B; the same disc carried to the second crane's angle, its centre again on the vertex and one arm through OA. In the printed photograph the second crane's other arm falls inside the marked pair, between OA and OB, so the second crane's angle is the smaller.
The slit test (§2.7, p.26). Three panels: slit wider than the arms, slit narrower than the arms, slit equal to the arms. Only the third pair passes. The printed conclusion is a conditional worth keeping intact — passing depends on the angle alone, provided the arms are shorter than the slit.

Figures to have open

The superimposition triptych for ∠ABC and ∠PQR (§2.6, p.22). The printed stills are the compressed version of a motion.
The ∠AOB / ∠XOY equality figure (§2.6, p.22). Standard schematic.
The three-part comparison figure at O with A, X, Y, B, C (§2.6, p.23). Needed as drawn — B and C must lie on one ray from O, or the third question collapses.
The undecidable figure at O with X, A, Y, B (§2.6, p.23). Needed as drawn.
Fig. 2.10, the two cranes. Redraw; the printed art is the textbook's own.
The transparent-circle sequence (§2.6, p.24). These are photographs in the book. Redraw as clean schematics: circle, centre on vertex, two rim marks, then the same disc on the second angle.
The rotating arms and the slit panels (§2.7, pp.25–26). Redraw. The three slit outcomes must be shown as three, not summarised as one.
No table or dataset is required for this topic.

Where this sits in the book

NCERT Class 6 Mathematics (Ganita Prakash), Chapter 2 "Lines and Angles": §2.6 Comparing Angles, pp.21–25; §2.7 Making Rotating Arms, pp.25–27
Teacher's Note on arm length, §2.6, p.25
Figure it Out items used: §2.6, Q1–Q3 on p.23
The snipping cartoon, §2.7, p.27
Forward pointer: §2.9 Measuring Angles, p.32, which returns to the circle and puts marks on it