How To Solve the GCSE Maths Question That's Leaving Parents Stumped
While most parents would love to be able to help their children with their maths homework, sometimes the questions are too hard even for them.
Here's an example of a GCSE geometry question that has been leaving many parents stumped. We will look at how the question is set up, what skills are needed to solve it and ultimately, how to find the answer.
Here's an example of a GCSE geometry question that has been leaving many parents stumped. We will look at how the question is set up, what skills are needed to solve it and ultimately, how to find the answer.
The Question
Take a look at the shape above. All measurements are in centimetres.
The area of the shape is A cm^2.
Show that A = 2x^2 + 24x + 46.
The area of the shape is A cm^2.
Show that A = 2x^2 + 24x + 46.
Why Is This Question So Difficult?
The problem with this question is that it brings so many different aspects of maths together. There's algebra and geometry in the setup of the question, and furthermore, it's asking us to 'show' something is true, rather than just asking for an answer. The geometry part of it is made trickier by this being an irregular shape with no obvious way of finding the area. It's not like finding the area of a rectangle or triangle where there are reasonably simple formulae to use.
To solve this problem, we first need to break the shape down into shapes with areas that can be found more easily.
To solve this problem, we first need to break the shape down into shapes with areas that can be found more easily.
The original shape is a compound shape, meaning it is constructed of simple shapes put together. By adding the red line above, we have split this shape into two rectangles. We know that the area of a rectangle is equal to its two side lengths multiplied together, so if we have these side lengths, we can find the separate areas.
This is easy enough for the top rectangle; it has dimensions of x + 1 and 4.
The bottom rectangle has length 2x + 6, but what about the height? We know that the overall height is x + 11 and we also have that the height of the small rectangle is 4. If we take this 4 from the x + 11 we will be left with the height of the large rectangle.
x + 11 − 4 = x + 7
Now we have all the required dimensions, we can find the separate areas by multiplying their heights and widths together.
This is easy enough for the top rectangle; it has dimensions of x + 1 and 4.
The bottom rectangle has length 2x + 6, but what about the height? We know that the overall height is x + 11 and we also have that the height of the small rectangle is 4. If we take this 4 from the x + 11 we will be left with the height of the large rectangle.
x + 11 − 4 = x + 7
Now we have all the required dimensions, we can find the separate areas by multiplying their heights and widths together.
^The area of the smaller rectangle is the easiest, so let's start with that one. As it's 4 multiplied by a bracket, this means we multiply everything inside the bracket by the 4.
Therefore area = 4(x + 1) = 4x + 4 cm^2.
The larger rectangle is a bit more difficult as this requires us to multiply two brackets by each other. To complete this we need to make sure we multiply each term in the first bracket by each term in the second bracket i.e. separately multiply the 2x in the first bracket by the x and the 7 in the second bracket and then do the same for the 6 in the first bracket.
Area = (2x + 6)(x + 7)
= 2x × x + 2x × 7 + 6 × x + 6 × 7
= 2x^2 + 14x + 6x + 42
= 2x^2 + 20x + 42
In the last step, we notice that we have two terms which are multiples of x (14x and 6x). We add these together to get 20x.
Therefore area = 4(x + 1) = 4x + 4 cm^2.
The larger rectangle is a bit more difficult as this requires us to multiply two brackets by each other. To complete this we need to make sure we multiply each term in the first bracket by each term in the second bracket i.e. separately multiply the 2x in the first bracket by the x and the 7 in the second bracket and then do the same for the 6 in the first bracket.
Area = (2x + 6)(x + 7)
= 2x × x + 2x × 7 + 6 × x + 6 × 7
= 2x^2 + 14x + 6x + 42
= 2x^2 + 20x + 42
In the last step, we notice that we have two terms which are multiples of x (14x and 6x). We add these together to get 20x.
This leaves us with two areas which we add together (again making sure to collect like terms) to get the total area of the original shape.
Total area = 2x^2 + 20x + 42 + 4x + 4
= 2x^2 + 24x + 46
The brilliant thing about 'show that' questions is that we can quickly see if we are correct or not. Our answer matches the one we are asked to show, hence we can be pretty certain that we've done it correctly. If it didn't match we would know to look back over our calculations to try to find any errors.
Another excellent thing about these questions is that there are plenty of marks available for giving it a go. Even if you make a mistake somewhere along the line, you could still gain marks for attempting to find a missing side length, or multiplying side lengths together etc. When faced with a long, trickier question like this one, the best bet is to give it a go and see what you can do. You never know, once you make a start, the rest of it may fall into place!
Total area = 2x^2 + 20x + 42 + 4x + 4
= 2x^2 + 24x + 46
The brilliant thing about 'show that' questions is that we can quickly see if we are correct or not. Our answer matches the one we are asked to show, hence we can be pretty certain that we've done it correctly. If it didn't match we would know to look back over our calculations to try to find any errors.
Another excellent thing about these questions is that there are plenty of marks available for giving it a go. Even if you make a mistake somewhere along the line, you could still gain marks for attempting to find a missing side length, or multiplying side lengths together etc. When faced with a long, trickier question like this one, the best bet is to give it a go and see what you can do. You never know, once you make a start, the rest of it may fall into place!
Comments
Could you solve this problem?
Don't forget to leave your comments below.
Don't forget to leave your comments below.