The first image below is designed as a prompt for identifying equal corresponding angles. I chose to use this as it begins to get pupils filtering information, a key feature of the previous angles posts in this series.
In presenting this image to pupils, it was important to emphasise that the direction or placement of the arrows within a line is irrelevant to its parallel property (the first two images in the figure above show this). Similarly, in the bottom two images, extra lines or shortened line segments from what is typically offered in textbooks etc. puts some meat on the bones of an angle as a turn instead of an independent ‘thing’ separate from a vertex or series of lines.
The image above was then used to discuss and question in multiple ways. For example, “Which angle corresponds to y?” really assesses whether at this stage pupils can both ‘see’ the corresponding angle, but similarly ‘unsee’ angles u and v.
We then repeated this with alternate angles.
The same lettered angles from before were shown with and a mixture of questions were asked around alternate and corresponding angles. Here, the u and v caused problems as pupils had remembered z angles.
In terms of helping pupils identify which word is which, I used a language-based approach where we discussed the words corresponding and alternate in regular usage and then how that applies here. Similarly, with co-interior angles, pupils found that once they’d been shown examples of other words with the prefix co- (cooperate, cohabit, etc.) and told me about the word interior (interior design, the interior of a car) then displaying and discussing co-interior angles was easier to make sense of.
The prompt above was used as a primer for identifying parallel lines. Asking pupils which pair of angles were equal meant they had to filter the five lines in each diagram with the arrows and the relationships between them. This spatial reasoning is incredibly subtle, but the bread and butter of so much geometric understanding. This also providing a good lead in to whether or not there is anything mathematically significant in the differences between the top two images.
The next series of tasks provide a high level of challenge for identifying parallel lines from a figure that brings together previous angles work.
The following task is presented here as three images, but in the resources linked at the top of the post there are animations that provide a prompt for discussion of how we know what we can tell about the images. I’m trying to incorporate more tasks and discussions that don’t rely on numbers and measures as much as I’ve found pupils will latch on to the nearest number and assume something has to be done to it.
The task below is a minimally different exercise that combines the angles in parallel lines learning with some isosceles triangles reasoning.
Following a hefty dose of practice with angles in parallel lines, I included problems below which require extra parallel lines being added into figures.
This will become a key strategy for the bearings work coming up after this in the scheme of learning, so pupils need to get used to chucking in another North line. It will also sow the seeds for the idea of a plane and vectors later on.
It’s worth noting here, that while adding in extra parallel lines is the most efficient strategy for most of the questions above, that’s not always the case.
In the top right question, by extending the parallel line to make a pentagon, and using interior angles of polygons, this question becomes much simpler.
The final task in the resources are just some heavily involved questions that I used to provide more practice time for pupils that needed it on more simple cases of angles in parallel lines.
Addendum: I spent ages animating a proof for alternate angles being equal from the angles in a triangle, but then I realised that meant I’d need a proof for the angles in a triangle that didn’t rely on parallel lines.
I the went down a rabbit hole thinking about the parallel postulate.
I haven’t quite got my head around how rigourous and complete I’d like proofs articulated to pupils, and even how far into Euclid we go (Is it worth discussion axioms?, etc.) because there’s more pressing things going on, but I’ve bloody well made it so it’s going in the resources anyway.
[All the activities in this post are linked at the top with solutions, and while this post outlines a suggested sequence for use, it’s no way exhaustive or considered optimal, but an ongoing reflection and sharing of materials.]
This post is the seventh of a series on angles.
Part 1: Understanding Angle Labels
Part 2: Filtering With Basic Angle Facts
Part 3: Exterior Angles of Convex Polygons
Part 4: Concave Polygons
Part 5: Interior Angle Sum Derivations
Part 6: Reasoning With Interior Angles