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Four Difficult GCSE Maths Questions and How to Solve Them

Four Difficult GCSE Maths Questions and How to Solve Them

Difficult GCSE Level Maths Questions and How a Maths Teacher would Solve Them

Getting through the GCSE maths course requires working on as many practice questions and solutions as possible and learning the different methods for solving problems and equations.

Consequently, students can struggle with absorbing so much information, but the old saying ‘practice makes perfect’ applies as much to GCSE maths as to everything else.

Complete Questions for Practice

1. Jackson is trying to figure out the density of a woodblock in the shape of a cuboid; the density will then be deducted in g/cm. The length is 13.2 cm, the width is 16.0 cm, the height is 21.7 cm, and the mass is 1970 g, correct to the nearest 5 g.

Give your answer to the nearest degree, show all of your work, and explain.

Answer

Density equals mass divided by volume. Figure out the upper and lower bounds of the dimensions.

L: 13.15 ≤ 13.2 < 13.25

W: 15.95 ≤ 16.0 < 16.05

H: 21.65 ≤ 21.7 < 21.75

Mass: 1967.5 ≤ 1970 < 1972.5

Calculate the upper bound volume and the lower bound volume.

Upper bound volume: 13.25 × 16.05 × 21.75 = 4625.409375

Lower bound volume: 15 × 15.95 × 21.65 = 4540.925125

Now, we can work out the density, both the upper and the lower bounds.

The upper bound will be deducted by using the upper bound of the density divided by the lower bound of the volume: 1972.5 ÷ 4540.925125 = 0.434382

For the lower bound of the density, we’ll use the lower bound of the mass over the upper bound of the volume: 19675 ÷ 4625.409375 = 0.425367

Use these two numbers in two significant places. The upper bound will be 0.43, and the lower bound will also be 0.43, as the number in the third decimal place is (5); hence, it’ll increase the 2 to 3.

2. We have the following formula with (nth term) being repeated. You need to answer and simplify the formula to prioritise the (a term).

p=n2 + an + a

Answer:

Cross multiplication: P (n+a) = n2 +a

Expansion: Pn + Pa = n2 + a

Putting (a) on one side: Pa – a = n2 – Pn

Removing (a) factor: a (P-1) = n2 – Pn

Making (a) the subject: a = n2 – Pnp – 1

3.

Complete Questions for Practice

 

 

 

 

 

 

In front of you is a vector problem. You’ve got a quadrilateral (CDEF). These characters also represent the sides of this diagram, and most sides have another name.

CD = a

DE = B

FC = a – b

A) Name FE using the terms a or b. Give your solution in the simplest form.

Answer:

Primarily, students must understand that each side goes in one direction only (and not backwards). For example, CD starts at C and ends at D, meaning that DC is a separate side.

Another piece of information that you can prove without a calculator is that (M) is the mid-point of DE. As a result, DM equals 12b, and ME equals 12b as well.

To discover what the FE means in a and/or b language, you need to move from the angle F to the angle E, calculating all the sides involved.

That is a-b+a+b. The (-b) and the (+b) cancel each other out, leaving us with (2a), which is the solution for this maths equation.

B) We’ve already mentioned that M is the midpoint of DE. Now, if you look closely, you’ll see a dash stretching from F to M with a dot on it labelled X. This dot makes the ratio FX:XM equals n:1.

Now you need to work out what the (n) stands for. Keep in mind that the dash CXE, which isn’t drawn, is a straight line.

Answer:

First, we need to work out C to E, so, choosing the easier side, we’ll go C to D (a), then D to E, which is (b), leading to the equation CE (a+b).

Then, we should work out the distance FM. Again, we’ll choose the easiest side which is FC (a-b) + CD (a) + DM (12b). So, FM equals a – b + a + 12b. That will add up to 2a – 12b.

The next step will be figuring out FX which is in simple words n (2a – 12b) = 2an – 12b. The last step to knowing the value of (n) is to know CX, which involves CF + FX.

We’ve mentioned before that reading a side ‘backwards’ gives a different result from reading it forwards, so since FC equals (a-b), CF would equal the opposite of that (-a+b).

CX= -a+b + 2an – 12bn

Let’s facilitate that: -a + 2an -12bn

Factorization: a (-1 + 2n) + b (1 – 12n)

We need to place this entire equation on one line. So, whatever we do to the left side, we’ll do to the right side; hence, keeping the balance on both sides of the equation.

New equation: -1 + 2n = 1 – 12n

Add half: 212n = 2

Divide by 212 = n = 45

Ratio: 45: 15 = 4:1

N = 4

4.

Complete Questions for Practice

 

 

 

 

 

 

 

Here, we have a rectangle containing three identical circles. Each one of these circles is touching the others and the sides of the rectangle. The radii of these circles are equal, and each length is 24 mm.

Use this diagram to work out the area of the rectangle and provide your answer correct to THREE significant figures.

Answer:

Just like our previous problem, this one requires a little bit of imagination, and of course, trigonometry skills. Let’s start with the easy part first, which is working out the width of the rectangle.

If you look at the upper two circles, you’ll find that you can connect a line from the point where the circle touches the rectangle on the right side to where the other circle touches the rectangle on the left side. This line passes by four radii, two for each circle. Hence, the width of the rectangle is 24×4 = 96 mm.

Now, let’s try to find out the length of the rectangle. Trying to use the same method for the length as you did with the width won’t work. There will be an overlap of radii between the lower circle and the upper two.

Instead, you will need to draw a triangle connecting the three centres of the circles so that each centre represents one of the triangle angles. Since the three sides of this triangle are equal, as each one comprises two radii, the triangle will be equilateral, and each of its angles will add up to 60°.

When that happens, you can quickly figure out the triangle’s height by multiplying one side by Sin (60). Each side of this triangle equals two radii (48 mm).

As a result, Median = 48 x Sin (60) = 41.569 mm. The last step is to add the final two radii to get the full measurement. So, 41.569+24+24 = 89.569 mm.

Finally, we’ve got our length. We can multiply it by the width to get the area which is 9×96 = 8598.624 mm². Let’s correct that to three significant figures; the area is 8600 mm².

The Challenges of GCSE Maths

GCSE exams are tough – students are expected to learn multiple skills and absorb a considerable amount of information that they must then recall during the exam.  They need to be familiar withThe Challenges of GCSE Maths

trigonometry, geometry, algebra, and be able to apply them correctly when problem solving. They need to achieve the perfect balance of maths skills and calm reasoning. However, there are other aspects to consider.

The availability of materials and practice questions is important here; students need the chance to apply their knowledge and understanding of the curriculum and gain the confidence to excel in exams.

The language of the questions and expressions used are also critical.  A good maths teacher can explain to their pupils how to interpret question texts, but students must learn to simplify the problems for themselves so that they are not hindered during exams.

What Is the Hardest Topic in GCSE Level Maths?What Is the Hardest Topic in GCSE Level Maths?

The most arduous topic in the GCSE maths course depends on students’ individual strengths and weaknesses and problem-solving skills. Nevertheless, when asked this question, many students identify geometry as the most difficult topic.

Final Thoughts

Maths is an extraordinarily broad subject which covers a vast array of sub-topics, each exerting their own fascination. Most maths students develop a favourite topic, and many go onto study it in greater depth at sixth form college and university, but enthusiasm for maths is greatly dependent on having a teacher that sparks curiosity.

Our advice to students is to study hard, prepare well, make time for rest and relaxation, and take your exam with confidence!

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Suki

During her time at primary school, Sukirith struggled with maths finding it difficult to truly understand the more challenging topics.  Despite these initial struggles, Suki worked hard to overcome her low confidence in maths and achieved top grades in both GCSE and A-level maths!  Suki has shown that it is possible to turn a weakness into a strength and that lacking confidence in maths at an early age does not need to determine future success in the subject.  Suki has since helped students struggling in maths at her local primary school and she is currently completing a BSc in Economics at Aston University.  Suki is friendly, patient and understanding of the struggles students face in their learning and she takes great pride in the fact students love their online lessons with her.

Uswah

Uswah is an incredibly compassionate and enthusiastic tutor who embodies Tutoring Heroes’ core philosophy of helping others.  Uswah prides herself on helping students lose their fear of maths and English, replacing this fear with a newfound enjoyment of the subjects.  As an active member of her University’s Muslim Society, Uswah has played a leading role in raising over £6,000 for charity.  In addition to this, she volunteers weekly for the homeless charity, Strathclyde Street Care.  Uswah is able to build incredibly inspiring and productive relationships with her students, who benefit not only from her subject expertise and tutoring skills, but her ability to act as a role model.

Tiana

Tiana has volunteered as a primary school teaching assistant, been an Economics Subject Leader, and mentored children through her roles as House Prefect and Year 7 Prefect during her A-levels, and so she has extensive experience working with and teaching students.  Tiana has a natural ability and willingness to form genuine relationships with her students.  Due to her experiences, and having gone through the 11+, GCSEs and A-levels herself, Tiana understands exactly how challenging preparing for exams can be.  This is why she is so passionate about instilling perseverance into students, through her patience and enthusiasm, along with the belief that they can do anything they work hard at.  As well as her academic success, Tiana is an avid singer and enjoys playing netball for her university team.

Zaynab

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Jessica

 Jessica is currently studying Clinical Pharmacology at St George’s University of London.  She is an avid reader, netball player and Formula 1 enthusiast.  Jessica has always had an affinity for teaching and learning.  Throughout both secondary school and sixth form, Jessica took part in multiple tutoring, paired reading, and peer mentoring schemes to help support other students.  These experiences helped her to gain valuable skills which are essential for tutoring.  Jessica really understands the positive impact 1-to-1 tutoring can have on a student.  She believes that tutoring is not only about improving the student’s performance in a subject, but also about instilling self-belief in them.  Jessica is committed to being as patient and supportive as possible to ensure each student can achieve their goals.

Aliceah

Aliceah is currently studying for a degree in mathematics at the University of Sussex.  She loves sharing her passion for maths with her students and has a wide range of experience supporting young people to achieve their ambitions, including through her role as Widening Participation Ambassador at Sussex University.  Outside of her love for maths and helping younger students, Aliceah is a keen artist (achieving A* in A-level fine art), photographer and crochet enthusiast!

Arianna

Ariana is a second-year biochemistry student at the University of Nottingham and she loves all things maths and science.  In her spare time, she enjoys developing her artistic skills, such as: creative writing, photography and playing the cello.  Arianna is a strong believer in the value of maintaining a diverse skill set and she uses her creativity to personalise the learning for her different students.  She is acutely aware of the importance of demonstrating patience and compassion to bring the best out of her students, which ensures that they make excellent progress in both their subject skills and self-confidence.

Autumn

Autumn has a stellar academic background, holding a BSc in Biomedical Sciences, MSc in Psychology and MSc in Phytochemistry.  Autumn has a talent for instilling a passion for learning in her students and being able to explain challenging concepts in simple ways; we are very lucky to have Autumn as a member of the Tutoring Heroes team.  In her spare time Autumn plays the saxophone and is a keen diver and martial artist.

Ben

When it comes to tutoring, Ben has lots of experience from both the student and teacher perspective.  Currently completing a Masters in Engineering, Ben has practised maths for many years at a high level, enhancing his understanding of the foundations of mathematics, which he loves passing on to his students.  However, learning has not always come easy to Ben and he has first-hand experience of struggling in school and needing to catch up with his peers.  As a result, it is easy for Ben to empathise with students who are currently struggling with their learning.  In his spare time, he enjoys playing the guitar and a wide range of sports including football and table tennis.

Alexis

Alexis is currently a Year 2 Undergraduate at New York University, pursuing a double major in Economics and Sociology at the College of Arts and Science.  In secondary school, Alexis achieved stellar results in her A-levels, with straight As across the board.  In particular, Alexis enjoys mathematics and literature, and strives to share her enthusiasm with her students.   Alexis has significant experience tutoring both in a private and voluntary capacity, and is known to be exceptionally friendly and patient, often going the extra mile to ensure students have a deeper understanding of the subject.  Beyond academia, Alexis loves hiking across new terrain, trying her hand at embroidery, and watching the intensely thrilling sport that is competitive figure skating.  Needless to say, she looks forward to meeting and interacting with her students during their weekly lessons!

Chloe

Chloe recently completed A Levels in Maths, Psychology and Art, achieving grades A*, A, B. She is now at university studying Primary Education in the hope of becoming a Primary School teacher.  Chloe struggled with English in school, but with the right support was able to achieve a grade 8 (A*) in her GCSEs.  Therefore, Chloe is aware of the importance of extra tuition for children and strives to not only improve their understanding but also their confidence.  Throughout school, Chloe gained experience working with children, including SEND children and young adults, and has recently been teaching on placement for her degree.  Chloe is very patient, calm and encouraging and thoroughly enjoys seeing her pupils make progress.

Farah

Farah has significant experience helping children improve their skills and understanding across a range of subjects (English, Maths, Coding and TOEFL), as well as being a young leader at her local Girl Guiding group.  A lover of the English language, Farah is currently studying for a Masters in Speech and Language Therapy.  Farah is an incredibly friendly, calm and patient tutor, who takes pride in the fact her students love their online sessions with her.

Grace

Grace is currently studying Architecture using her talents in art and design, combining them with her love and aptitude in maths, English and history to help to create her projects.  Her university career follows on from her successful school and college years where she gained the highest grades in her GCSEs and A-levels.  Grace has a wealth of experience working with younger students to help instil confidence and enjoyment - from peer mentoring, sports coaching and English and maths tutoring.  Having recently worked through exam processes herself, Grace understands the difficulties and pressure on young people, and is passionate about supporting students to achieve their full potential.  Grace strives to create a positive and reassuring learning environment where she can support her students with the most appropriate learning strategies and techniques for each individual student.  Grace is very proud of the partnerships she has built with her students, and how the subjects she supports them with often become her students’ stronger subjects.

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Mel

Mel is currently studying for her Masters in Theoretical Physics, and enjoys being able to pass her enthusiasm for maths onto her students.  From experience, Mel appreciates the importance of having confidence in your own abilities in order to succeed.  This inspires Mel to focus her teaching on building her students’ confidence, knowing that it will in turn improve their maths.  In her spare time, Mel loves to read and crochet, as well as being a gym enthusiast.

Prisha

Prisha has a fantastic record of academic achievement in not only English and maths, but across the board, which makes her a very well-rounded tutor.  Prisha is currently pursuing a BSc in Product Design, which deepens her understanding of maths (and how it is applied) which enables her to bring maths to life for her students.  As an avid reader, English language and literature has always held a special place for Prisha, and this passion helped her to score full marks in the IELTS test.  Having worked on educational projects with young people, she is experienced at how to interact with them and explain concepts in a manner that is not only fun, but also does not overwhelm them.  Prisha works hard to ensure lessons with her students are not considered a chore and strongly believes that with the right support, every child is capable of great achievement.  Outside of education, Prisha plays for her university basketball team and is also a big fan of baking and dancing.

Rabia

Rabia is a highly organised individual who, throughout her GCSEs and A-levels, had been a peer-mentor to younger students, aiding them to build confidence with their reading and writing skills.  Through her Audiology degree at university, in which she supports patients of varying ages, Rabia has attained an appreciation of how to modify and adapt her communication style, in order to facilitate effective communication with different individuals, something that she also utilises in her tutoring sessions.  Being an avid reader, Rabia spends her free time reading books of different genres and time periods, and enjoys discussing with others the themes and messages in a particular novel/text.

Shair

Shair has a notable range of English tutoring experience across the student spectrum, teaching from KS2 and up to GCSE level.  Her aspirations to make a difference in the field of education continue to grow.  As well as through Tutoring Heroes, she teaches a range of students at her local tuition centre, ensuring they also receive the 1-to-1 support they need in order to fuel their progression in English.  An avid reader with an immense love to learn about different cultures and writings, Shair is currently in her final year of studying BA Comparative literature.  Shair is an approachable, friendly and considerate tutor, who ensures her students leave their lessons feeling confident and ready to take on new academic challenges.

Sipeng

Sipeng was originally educated in China where she developed a very strong understanding of maths.  After moving to the UK, she achieved the highest grade (A*) in GCSE and A-level maths.  Crucially, in addition to her strong maths ability, Sipeng has significant experience of tutoring and inspiring younger students to achieve beyond their expectations.  Sipeng recognises the positive impact that several teachers have had on her own education, and is driven by the desire to help her students in a similar way, so that they too develop confidence in their maths ability and achieve higher grades as a result.

Shatha

On moving to the UK aged 11, Shatha found that her maths skills were behind the standard expected, so Shatha has first-hand experience of what it feels like to struggle with maths.  Despite this initial struggle, Shatha achieved a Grade 8 in GCSE Maths and Further Maths, and an A grade in A-level Maths, proving how it is possible to turn a weakness into a strength.  Shatha has a range of experience helping children, including two years’ tutoring at a Kumon centre and volunteering at Sheffield Children’s Hospital.  A keen tennis player and trilinguist (English, Arabic and Spanish), Shatha particularly loves watching her students overcome their fear of maths, just as she did.