Grade 6 · Math · Geometry
Two lines, four angles: which two are equal?
An interactive geometry lesson, 10 questions each asking for the reason
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An interactive lesson where the student turns one of two crossing lines, watches all four angles move together, and finds that two of them stay equal however far it goes.
This page is exactly what a teacher's published work will look like. Only the teacher is not real yet.
What the student does
- 1
Turn the line
The student takes one of the two lines and turns it about the crossing point, and the four angle measures change in front of them as it moves.
- 2
Notice what does not move
However far they turn it, the opposite pair stays equal and any two angles side by side still add to 180 degrees.
- 3
Predict before measuring
One angle's measure is given, and they are asked for the other three before any of them is revealed.
- 4
Justify instead of guessing
They pick the reason from three: «vertically opposite», «they add to 180», or «I judged it by eye». The third one is really there, and it is wrong.
What's inside
- Two crossing lines, one of which the student turns with a finger
- Four angle measures that update with every turn
- 10 questions marked the moment they are answered
- A «why?» question after every correct answer
Outcomes
Identifies the vertically opposite pair in any figure of two crossing lines.
Works out the other three angles from one that is given, and states the reason for each.
Separates an opposite pair, which is equal, from an adjacent pair, which adds to 180 degrees.
What the teacher focused on
Every year half the class writes that the two angles side by side are equal, because in the drawing they look it. So I made the drawing move until the resemblance falls apart in front of them.
Teaching strategies
- Predict, then reveal
- Keeping the misleading special case out of the opening
- Asking for the reason after every right answer
He opens on a slanted crossing, and turned down opening on two perpendicular lines. At a right angle all four angles are equal, so half the class leaves with a wrong rule that is hard to pull down later: «all four are equal». The figure that is easiest to draw is the figure that manufactures the mistake.
Questions from inside it
- 1Two lines cross and one of the angles measures 130 degrees. What is the angle vertically opposite it?
- 2Two angles sit side by side where two lines cross. The first measures 47 degrees. What does the second measure?
- 3Two lines cross and all four angles come out equal. What can you say about the two lines?
Curriculum
- Grade
- Grade 6
- Subject
- Math
- Unit
- Geometry
- Taught in
- Jordan
Where this lesson sits in the curriculum
The same lesson appears in these countries' curricula. One page, five paths to it.
When to use it
The explanation
When two lines cross, four angles form around one point and together they make 360 degrees. Any two of them that sit side by side lie on one straight line, so those two make 180. That is where the equality comes from: if a and b are side by side they make 180, and if b and c are side by side they make 180 as well, so a equals c. An opposite pair is equal because both share the same neighbour, not because they look alike in the drawing.
Common mistakes it catches
He used the adjacent rule where the opposite one belongs. 180 − 130 is the angle next door, not the one across the crossing point. Have him put a finger on the crossing point and travel straight through it to the angle he means.
True at 90 degrees and nowhere else. Turn the line to 60 in front of him: the angles become 60, 120, 60, 120, and the rule he built falls over in a second.
A drawing is not a reason. Ask for the justification before the answer rather than after it: which rule lets you say that? A student who cannot find one looks again before writing.