Area of a triangle

I made these questions to hopefully reinforce the idea of area as the space inside a shape, rather than just the answer you get when you multiply numbers together.  Also I want them to see why the area of the triangle is half the area of the rectangle which it is enclosed within.  I made them using this Geogebra file but then pasted a few to make this worksheet, some with grids then some with just lengths.

A colleague of mine @DrPMaths made this impressive collection of triangles with 3 integer side lengths and an integer height.  Again, they are a great way to check that students are identifying the perpendicular height and multiplying that by the base rather than just multiplying the numbers they are presented with.  There are literally hundreds of them, this is a snapshot from somewhere near the beginning!

 

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Quarter the Cross

UPDATED POST. I used this task at my workshop at #mixedattainmentmaths on Saturday. I asked all teachers to have a go at this task but to do it in what they thought was the most obvious / simplest way.  An interesting experiment: what is obvious to some is not to others.  Of the solutions that I managed to take in, these were the choices:

This looks like a very useful open-ended task which provides an opportunity for creative solutions and rich discussion.  I have produce some printouts here.

In my view, the value in this activity is in representing each area as a fraction calculation.

According the Australian blog where I first read about this task, this is one of the most common first solutions

I’d be looking for some rationalising as to why the red area is a quarter. For example:

There are 100 solutions posted here!

And on a Prezi here enabling you to zoom into each one individually.

This is potentially very high ceiling. If students are struggling to come up with suitably challenging solutions of their own, you could always ask:

Show why this is a quarter:

Have a go first yourself.  I think this is a pretty mammoth task! This one caught my eye, but you might want to have a look at the 100 solutions to find something a bit easier!

Since this post was originally written back in January, I have used this task a couple of times at conferences and had some really good discussions on this example.  If you want a rather big hint, scroll down to see an animation.  Or you can find the Geogebra file here.

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Tethered Goat

No 4 in a series of posts based on Thinking Mathematically (1985) by Mason, Burton, Stacey

This is a classic task for working systematically:

This is a classic question that I usually use when teaching loci and constructions.  I usually scaffold it with a diagram, but I like the approach here that promotes students to introduce their own diagram as a way of getting closer to the problem. Next time I use this I will make sure I give it plenty of time, get students discussing the problem in pairs and really resist providing a diagram and thus removing some of the joy of discovering it!

A goat is tethered by a 6 metre rope to the outside corner of a shed measuring 4 metres by 5 metres in a grassy field. What area of grass can the goat graze?

If I felt the need to make this question easier,  I might start with the rope being 4 metres long, then try 5, then try 6.  In fact, this could be taken a lot further, what happens as the length of rope is increased further?  Try 10m, 50m, 100m.  What happens when the rope gets really long?

Or, as a way in to this problem I might ask:

A goat is tethered by a 6 metre rope to a simple post in an open field. What is the area of grass it can eat?

I will have this question ready written, but hidden on the same slide on the board so that I can show it to the whole class if I feel they need it. But alternatively I might just suggest this to individual students if they are really struggling and making no progress.  Either way, the point is that it is not another question that I want them to work on, it is a problem solving strategy that I am exposing them to: If the problem seems too difficult, try to find a simpler scenario as a way in to the problem. Solve that, and then think about what knowledge and methods you used and how they could be applied to the main problem.

Fundamentals of Area

Area is a topic where students are likely to have remembered a process (you multiply the lengths to get the area) but may not have a secure understanding of what area actually is.  It is worth spending time discussing what we mean by area.  I would prefer my students memorise “Area is the amount of space inside a 2D shape” rather that “Area is length times width”.

Counting the Squares is a good place to start even with students who feel like they know how to calculate area.   The concept of 1cm² as a quantity, an amount of something that bigger shapes have more of is powerful.  Estimation can be useful.  Physical pieces of 1cm² card and a series of rectangles, some with grids, some without is worth doing.

Reinforcing units is important.  Reminding students of the commutative law and rewriting 2cm x 3cm as 2 x 3 x cm x cm can help explain why we write cm² and link it back to algebra.  You could even play with 3cm x 2m.  Does make sense to say that this area would be 6 cmm?

A key step in the process is moving from:

…without losing sight of the underlying concept that the rectangle is made up of 12 little squares, each one 1cm².

Moving on to rectilinear shapes, I like this question as a way of challenging thinking.

 

The first misconception here would be multiplying 3 x 4 to get 12.  But of course the correct answer is “we don’t have enough information”, an answer which shows depth of understanding and one which I would like my students to be presented with more often.

Hippocrates First Theorem

Another one from the fabulous Don Steward:

You could of course just go straight for the algebraic proof but it does require a level of confidence with surds.  So you might want to scaffold this task. Maybe start by putting some numbers in for the radius of the smaller semi-circle, maybe 2.  You could then do it again with 4 and ask students if they are convinced by that.  (Here are some examples to warn against the dangers of extrapolating what appears to be a pattern).  If you do take the numbers approach it’s good calculator practice.  Can you type the whole expression for the area of the curved shape into the calculator to get an exact answer?

And here is a little GeoGebra drawing to go with it.

 

Generalising with Areas of Circles

So, at #MathsMeetGlyn last weekend Don Steward briefly put this picture up on the screen but then decided to talk about something else as time was pressing.

That’s a bit like a “Wet Paint” sign.  You have to touch it just to see if it really is wet…

So I had a little play with these. I couldn’t find anything about it on his blog but of course there is lots of other lovely stuff there.

A few different questions you could pose here:

• What is the ratio of red area to blue area?
• What is the red area as a fraction of the blue area?
• What is the red area as a fraction of the whole circle?
• What is the blue area as a fraction of the whole circle?

And then of course, can we generalise?  What about when there are n small circles.  What happens as n gets bigger? Why? Can we find a general expression for the area?

Have a go.  It actually turns out to be quite simple but depending on which question you start with, you can get into a lot of practice with ratio and dividing fractions.

Stumped by a Foundation GCSE Maths question

I have just returned from an inspiring morning at #mathsmeetglyn organised by @mathsjem watching Don Steward give a fantastic, brain-stretching whistle-stop tour of some of the great problems on his site.

The theme was “Generalising” – he started off by saying that his current mission is to get some generalising into every single lesson because without it, well, you’re not doing real maths.

I filled 10 pages of notes, some of which I’ll come back to, but wanted to share how he took this Edexcel Exemplar Foundation GCSE question and “generalised” it.

Now, first up, I don’t mind admitting that after working at it for about 3-4 minutes I was completely stumped.  Don had added “You can’t cut the tiles” to the question which was essential in my view, but didn’t actually help me. I was completely fixated on the fact that all tiles have to be in the same orientation.  They don’t.

I think it’s a pretty poor question, probably mainly because I couldn’t do it. But there is a serious point around what questions like this are really testing. If there is a simple “trick” you need to get, is that fair?

A debate to be had there, I’m sure, but more interesting was what Don did with it next.

What other areas can you fit 40 x 30cm “carpet tiles” into and how many do you need? Start with:

120cm x 60cm
120cm x 70cm
120cm x 80cm
120cm x 90cm

Do you need to go any further?  Can you write a general statement from this? i.e. can you prove that all multiples of 10 will work if the width is 120cm?  What other widths does this work for and why? And then, of course, what happens if you try different sized carpet tiles.

It feels a bit like one of those Maths GCSE coursework questions that were set in the days before I was a teacher.  But I really like the idea of taking what is a pretty bad question and turning into some interesting maths.