The 2019 Year 6 SATs results indicate that mathematics performance has improved, with 79% of students reaching the expected standard. This is an increase of 3% from 2018: hardly significant you might think. (This and more data is freely available here.)
Regarding the eleven plus, one statistic is particularly important: 27% of pupils are reaching the high standard in mathematics (a scaled score of above 110 out of 120). This again represents a 3% increase from 2018.
While interpreting these statistics may not tell us much about the students' actual ability in mathematics, they indicate that students are improving in their test performance.
This is significant in regard to the eleven plus as the exams are designed to select from these top students. A greater pool of top students means that the pool of potential grammar and independent school students increases, lowering the probability of individual applicants achieving success.
In a nutshell, grammar and independent schools are getting more difficult to enter (though an expansion of grammar school places may eventually balance this out).
Foundations of Mathematics
Foundational skills of number and calculation skills need to be solid and is the first step for a student preparing for the 11 plus exam. This is because all mathematics questions are based around this core knowledge.
These basic skills are paramount for every year of a child’s school life. Even at ALevel, if you were to take on a differentiation problem, you still would rely upon the basic skills of subtraction and multiplication. The students who can understand and apply these foundational skills and practice them to mastery have the tools available to approach harder questions.
These foundational areas in maths are:
Place Value
Number Facts
Addition
Subtraction
Multiplication
Division
Parents often see the eleven plus exam as something that is separate from what is taught in school. Often they think endless practice papers is sufficient to give them entry to grammar or independent schools. However, true, solid preparation requires more than just practising exam papers. Timing is only part of the preparation.
At Redbridge Tuition, we ask that parents start preparing their child earlier than in year 5. The reason we encourage parents to start preparing for the 11 plus in year 4, or even at year 3, is to ensure that children have strong foundations moving into year 5 and 6.
What does it mean to master a skill such as addition?
Initially, students need to build their concept in the real world by putting groups of objects together in their early years. Secondly, they need to learn their number bonds so they do not need to rely so much on real objects and gain a more abstract concept of addition.
Then they need to practise their mental and written methods. Most students struggle with at least one of these steps, often because they do not practise enough. Mental methods are an especially valuable skill as students can benefit greatly from the act of juggling figures in their head.
An additional benefit of focusing on the foundations is that children can work on building their speed of calculation, which will give them precious time to spend on harder problems under exam conditions. No matter which exam your child takes, working with them on the foundational skills is unlikely to be a waste of time.
Recommended Resources
We recommend the Collins Basic Skills books as well as the Peter Robson series (especially Books 1, 2 and 3) as an initial resource:
Collins Practice in the Basic Skills
Book 1 

Book 2 

Book 3 

Book 4 

Book 5 
Peter Robson: Maths for Practice and Revision
Book 1 

Book 2 

Book 3 

Book 4 

Book 4 

Answers to Books 1  5 
Alongside these foundational resources, we recommend Schofield and Sims for practice in applying knowledge and problemsolving. We have generally found that students able to reach the end of book 5 with high speed and accuracy have performed well in the real exams.
Mental Arithmetic Key Stage 2 Student books
*Mental Arithmetic Teacher's Guide 

Mental Arithmetic Introductory 

Mental Arithmetic 1 

Mental Arithmetic 2 

Mental Arithmetic 3 

Mental Arithmetic 4 

Mental Arithmetic 5 

Mental Arithmetic 6 
Mental Arithmetic Key Stage 2 Answer Books
Mental Arithmetic Introductory Answers 

Mental Arithmetic Answers 1 

Mental Arithmetic Answers 2 

Mental Arithmetic Answers 3 

Mental Arithmetic Answers 4 

Mental Arithmetic Answers 5 

Mental Arithmetic Answers 6 
*I recommend the teacher’s guide for the Mental Arithmetic Books as it helps narrow down which books to purchase.
Find out what Exam Board your Child’s School uses
Although the skills required across the exam boards in mathematics are quite similar, there is a pronounced difference between the style of questions.
CEM and GL style questions are designed to be tackled quickly and rely mainly upon agility of thought. This means they usually require students to spot connections between numbers and take shortcuts. CSSE maths problems require a little bit more thought. CEM and GL mathematics questions are not difficult, however, under time pressure it can be difficult for some students to adapt to the situation. Misreading questions, as well as the options provided, can also be a source of mistakes so taking enough time to think about each question is crucial. I think that an important skill that many students need to acquire is estimation. Estimating an answer allows students to eliminate less logical options.
Independent school exams are quite similar to CSSE (Essex) papers in this regard and can have extremely difficult questions near the end of the paper to test the strongest candidates. These can be quite enjoyable problems and most students would not be able to solve them simply by applying their knowledge. Most of them require experience in dealing with similar types of problems. In some Independent School exams, they may even provide students with new mathematical knowledge that they have never seen in school and then expect students to solve a problem using that new knowledge.
Whatever the board, using practice papers is a crucial part of the preparation. However, taking many practice papers does not substitute for learning each topic. Furthermore, paying close attention to a student’s score rather than concentrating on weak elements is more likely to result in a worse performance.
Free Maths Questions with Answers
(Answers are at the end of the article)
A good way to understand what students are up against is to consider some 11 plus style problems. I have split them into beginner, intermediate, and advanced problems; our booklets are organised in a similar manner.
Beginner:
1. What is the sum of four hundred and eightytwo and three hundred and seventyeight?
2. What is the product of 1.2 and 3.4?
3. How many prime numbers are there from 1 to 20?
4. If there are 300 sweets and they are shared amongst 4 students, how many sweets does each student receive?
5. What is ⅜ of 200?
Intermediate:
1. If the ratio of boys to girls at a school is 4 : 3, and there are 200 boys, how many girls attend the school?
2. If Piper is 5 years older than Hailey, Hailey is twice the age of Scott, Scott is 3 years younger than Magnus, and Magnus is 14, how old is Piper?
3. What is the next number in the following sequence:
1 1 2 3 5 _
4. Stan did five mathematics tests and achieved scores of 75, 80, 67 and 70 on the first four tests. If his mean score after taking five tests was 78, what was his final score?
5. If a bag is on sale at £63 and the original price was £70, what was the percentage offered in the sale?
Advanced:
1. A farmer has a rectangular plot of land. It has a width of twice its length and its perimeter is 120m. What is the length of the plot?
2. Fill the gaps in the following sequence:
1 _ _ 10 15 21
3. The sides lengths of a rightangled triangle are labelled as follows:
If Pythagoras’ Theorem states a^2 + b^2 = c^2, what is the value of c when a = 5 cm and b = 12 cm?
4. If a caterpillar starts doubling in mass every day and reaches 32g by Friday, how heavy was the caterpillar on Monday of the same week?
5. Fill in the missing numbers for the sum given below:
If:
B = 2A
C = 3B
What are the values of:
A =
B =
C =
D =
Resources for further exam practice
I think that the best additional resources on the market for further exam practice in mathematics are the CGP 10minute tests on worded problems as they are wellwritten short papers and have simple explanations provided with the answers.
CGP 10Minute Tests
11+ Word Problems Book 1 (CEM) 

11+ Word Problems Book 2 (CEM) 

11+ Word Problems Book 1 (GL) 
Maths Practice Papers for Senior School Entry
Peter Robson’s Maths Practice Papers for Senior School Entry is an excellent resource to use to prepare for independent school exams. I like the way the questions are written and presented. The answer book also explains concepts clearly.
Maths Practice Papers for Senior School Entry 

Maths Practice Papers for Senior School Entry: Answers 
Conclusion
In order to reach the standard required for 11 Plus and independent school exams, students need to go far beyond the material they ordinarily study in school. Extra preparation with a tutor can be beneficial in this regard and is practically essential for scholarship applicants.
While mathematics in the UK seems to have a bright future, as there seem to be many talented youngsters with huge potential, the school system will have to adapt greatly to nurture this swell of talent. Tuition centres and other supplementary education will have to continue to play their part to make up for any shortfalls in the education system.
ANSWERS with explanations
Beginner:
1. 860  Sum of numbers is another way of saying add the numbers together
2. 2.88  Product means multiply
3. Primes are 2, 3, 5, 7, 11, 13, 17, 19
There are 8 prime numbers between 1 and 20
4. 300 ÷ 4 = 75 sweets
5. 200 x ⅜ = (200 ÷ 8) x 3 = 75
Intermediate:
1. Girls = 200 x 3/4 = 150 girls
2. Scott’s age = 14  3 = 11, Hailey’s age = 11 x 2 = 22, Piper’s age= 22 + 5 = 27
3. This is a Fibonacci sequence. The next term is found by adding the previous two terms together. 3 + 5 = 8
4. As we know the mean is 78, we can calculate the total 5 scores by doing 78 x 5 = 390. Then we can add up the known scores and subtract them from the total to find the missing score.
Final score = 390  (75 +80 + 67 + 70) = 88
5. First, find the percentage difference between the original price and the sale price:
£70  £63 = £7
Then express the difference over the original price as a fraction:
7/70 = 1/10
Then convert the fraction into a percentage by multiplying by 100:
1/10 x 100 = 10%
Advanced:
1. First, think about the way the shape appears visually and label the sides.
Modelling the sides algebraically is a logical way to start:
Length = x
Width = 2x
Then we can express the perimeter of the shape in terms of x:
x + x + 2x + 2x = 6x = 120 metres
Then solve for x by dividing both sides by 6:
6x/6= 120/6= 20 metres
2. This is a triangular number sequence. Imagine the following:
If each row accumulates you end up with the following pattern:
1 1 + 2 = 3 1 + 2 + 3 = 6 1 + 2 + 3 + 4 = 10
1 + 2+ 3 + 4 + 5 = 15 1 + 2 + 3 + 4 + 5 + 6 = 21
3. This question involves a student being provided a new mathematical fact and then applying this fact:
5^2 + 12^2 = C^2
25 + 144 = 169
C = √169
C = 13 cm
4. First, think of the problem as a geometric sequence:
To find Monday, you can either keep dividing the last term by two, or you can state that:
x × 2^4 = 32
x = 2 grams
5.
A good way to approach this is to use substitution and elimination. If the result of the sum is a 3 digit number, then the highest possible result is 999. This, in turn, means that B equals 3 or less as C = 3B.
The only possible combination is:
A = 1
B = 2
C = 6
D = 9
Students are likely to use trial and error to solve this type of problem, but sometimes it is possible to use more technical methods.