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# Work out those angles – another worked exam question

Today we look at another exam question, where we are asked to work out the size on an angle.

I was working on this question with a student a couple of weeks ago, and he worked out the answer quickly, but wasn’t quite able to describe why….  so this one is for you, Joe.

For a question like this, we do need to show we understand the reasons why each stage of the answer works.  Relationships between angles often have names (Complimentary angles, Associate Angles) but we don’t need to know these for this question.

We do need to know.

• Angles in a triangle add up to 180
• Angles in a pentagon add up to 108 (That might not be as widely known, I’ll cover this in a later post)
• Sides in a regular pentagon are all the same
• Two angles in an isosceles triangle are the same.
• Angles in Right angle add up to 90.

So these are the steps that Joe took, in his head (He is an able student) and the reasons he, and we, need to give at each stage for full credit in an exam.

1. Angle ABC = 108  – Its an angle in a Pentagon
2. Triangle ABC is isosceles (Two of its sides are sides of a regular pentagon)
3. Angle ACB = 36 – Its one of the other two angles in triangle ABC so = (180-108)/2
4. Angle BCD = 108 – Its another angle in the pentagon
5. Angle ACD = 72 –  Angle ACD – ACB
6. The answer  DCF = 18 because its = 90 (Angle ACF, in the square) – 72( Angle ACD).And that is showing our working, with reasons. I think talented students sometime leave the rest of us behind with their logic, but explaining what we are ding is an important skill!

# Volume of a Prism – Exam worked example

Today’s post is a worked solution – of a question from straight off a GCSE paper.

With any question, the first ting to do is check if there are any words that are out of the ordinary.  Here, the word that stands out is ‘Prism’

I first remember using a ‘prism’ in Science lessons, because a ‘triangular’ prism can do beautiful things with a beam of light. But that is a special sort of prism – one with a triangle at each end.  Generally a prism is any solid shape that is the same all the way through.

And that is the clue to working out the volume. Work out the area of the shape of one end, and multiply that by the length.

That leaves us with the hard bit first – but at least we will know we’ve got that ‘out of the way’. How do we work out the area of the end, which is like an L lying on its side.

We need to split this up, but there are 3 ways of doing this. You might be able to see three ways of doing this; the third is a bit harder to spot.

On all of the splits, it will help you to fill in the missing sides. The short one we find by comparing the lengths on the left and right : 7cm – 4cm = 3cm

The longer missing length we can find by comparing the top and bottom. 11cm – 5cm = 6cm

I am showing the calculation for all three methods here  BUT YOU WOULD ONLY NEED TO CHOOSE ONE!

The first split is into rectangles that are 5cm x 3cm = 15cm and
11cm x 4cm = 44cm2.  Total is 59cm2.

The second alternative is 2 rectangles of 7cm x 5cm  = 35cmand
4cm x 6cm = 24cm2. Total is 59cm2 – And the result should be the same of course!

The third is harder to see but I think quite clever. The area of the shape without the missing part is 7cm x 11cm = 77cm2. The missing part is 6cm x 3cm = 18cm2

. This time we have to take the second area away – its ‘missing’  77-18 = 59cm.  Like I said, this had to be the same answer, but its a good check that it is!

[That’s a good hint with any problem solving. If you want to check your answer, find it in two different ways. If you check an answer by repeating the same steps, there is a chance you’ll make the same mistake, if you made one]

The final step is to calculate the volume by multiply the area of one end by the length. 59
cmx 20cm = 118cm3

There is a certain kind of question on Maths GCSE papers that confuse some students – What’s this question doing on a MATHS EXAM???

On the other hand, for the student nurses and midwives I taught to GCSE a few years ago, these were their favourite questions.

The reasons were the same – What’s this got to do with Maths!

Let’s look an example, taken from exam papers, so we can see what I mean.

So this is a question about language, right?  Yeah, its  also a Maths question; about how we can get meaningful data.

Can you see what’s wrong with the first survey?

Well consider you have my answer to the survey.  “I bought 1 CD last week for £7. Last year I spent £50”
So which answer do you take?  And how do you record £50?

The second problem is easier to spot.  £50 could go into £30-£50 OR £50-£70

First Problem Overlapping categories

We could make this answer better by noting that a spend less than £10 is not covered.

The second problem is the questions give no idea on time scale. Is Paula asking what I spent last week, or last year?

Second Problem Question is not time specific.

Although the question doesn’t ask for it, we can give an improved question here.

The survey is now clear on what it is looking for and what data we will get.  Do you think this will give an accurate picture of sales through the year?  OK maybe not. We could ask about the whole of 2017?  Why might that not work?

Now we have a good question, but does that mean we will get a good survey? The next part shows us a possible extra potential problem.  Who will we ask?  Can you think of a problem with only asking people in a CD shop? How will that skew our results?