All the Angles Inside

This island focuses on determining missing angles from existing angles but also provides opportunities to measure and draw lines at particular angles. It utilises pretty much all the rules that they will be taught for missing angles in primary schools. There is also a lot of scope for deeper exploration. I love the sense of awe that the children have that it is possible.

With an 8×8 square, let’s draw a line from each corner to the midpoint of the side opposite going in a clockwise manner. A Polypad of the first steps can be found here.

First of all, what do we notice?

  • The square has been split into nine shapes
  • A square inside our original square in the middle
  • Four triangles
  • Aside from the square, four other quadrilaterals
  • There are two pairs of parallel lines.

If we measure the angle of the line that we have drawn, we will find it to be about 27°. It is more like 26.565° but for the purposes of what we are doing, 27° is fine.

Without using a protractor, what are all the other missing angles in each shape?

The children can use a combination of each angle in a square being a right angle, angles in a triangle equalling 180°, angles in a quadrilateral equalling 360°, opposite angles within intersecting lines being equal, complimentary angles totalling 90° and supplementary angles totalling 180° to determine every other angle:

We can now see that we have the same angles repeated across the different shapes. A pattern immediately!

The question that presents itself is whether or not this is a special case or is this true for others? Can we explain what is going on?

What if we used a different sized square?

With this, we still draw a line to the midpoint of the side opposite. Being a square, and the resultant triangle created from the line being similar to what we drew originally, the results are equally the same.

There are two other ways that I have that we can deepen this inquiry easily:

What if we tried other rectangles?

Within this itself, you have two possible directions (here is a Polypad of each). If you continue to draw lines to the midpoint of the line opposite, we will get a parallelogram in the middle and the angles from each corner will not be equal. The children can then think about how many angles they need to measure to determine every angle inside each shape.

The alternative is that we could ensure that the angle from each corner remains the same after initially drawing a line to the middle of the opposite side. Like this:

Now we have a rectangle in the middle. It is perhaps more similar to the initial case that we looked at and the same repetitions of angles will be apparent. Hopefully you will see that we once again only have to measure one angle to determine every other one. Angles are repeated once again.

If they use extreme rectangles where one side is much longer than the other, a hexagon is produced in the middle rather than a rectangle:

However, it is still possible to determine the other angles from the initial angle. This is a nice example of using the Investigating skill of Expanding (see at the bottom of the page for all skills) as they will likely conjecture that it will always be a rectangle. They can also use their Arguing skills to explain why this might be. The first time it happens, you might encourage them to create another case that it also happens for which would promote them to think about why it is occurring.

What if we drew lines to different points?

Instead of going to the midpoint of the line opposite, they could go to a quarter of the way across or try different points. Or if they are ensuring that all four lines are at the same angle, they can choose a specific angle.

Above is an example where I have drawn the lines to a quarter of the way across the opposite side.

What if we used other shapes?

Here is an example of a regular hexagon. I would suggest that you stick to regular shapes. I think it is also easiest to stick to going to the midpoint of the next side when making this change. Straight away, we can see that a regular hexagon has been created in the middle in a similar way to the square being inside the square previously. I like how it isn’t at the same orientation though. Something to explore! How many angles need to be measured now?

What if we drew the line to the midpoint of the line it would otherwise intersect?

I’ve not tried this one out but I thought I’d throw it in here as the kind of ‘what if…?’ question that might occur in the Archipelago phase. Children come up with all sorts of interesting ideas when the culture is there. The whole point is that there are always creative ways to extend beyond the ideas we have as a teacher and this should be encouraged.

Children’s possible use of reasoning skills:

Search
Search inquiry reasoning skill in maths

This isn’t an inquiry that uses a lot of Seeking. However, Combing is important as we want the children to focus on making small changes or the same-sized shape repeatedly rather than going randomly.

Organise

You could encourage them to use a tally of the number of equal-sized angles or the number of different shapes. This could help lead them towards a conjecture.

Discover

There are so many similarities and patterns between cases and between different-sized rectangles and angles that you should expect plenty of conjectures without needing to lead them too much.

Investigate
Investigate inquiry reasoning skill in maths

Ensuring that they see that hexagons can be produced inside a rectangle is a really good example of why using the Expand skill is important. The peculiarity of the rectangle with one pair of sides being much longer than the other exemplifies this idea and might promote it in future inquiries.

Argue

A lot of discussion can come from why different angles have the same value. It’s important to give them time to consider and discuss this – we don’t just want them to think it is magical.

Explore

I have given a few different directions that this island can go in and it is one where you will have different children going in different directions. There is a lot of possibilities available to them. It is perhaps mostly Reorienting than Departing mostly but as always, who knows what they will come up with?

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