GCSE Maths (9-1) H: Number » Foundation Number

GCSE (9-1) Higher Mathematics is broken down into six main areas:

  • Number (NH)
  • Algebra (AH)
  • Ratio, proportions, rates of change (RH)
  • Geometry and Measures (GH)
  • Statistics (SH)
  • Probability(PH)

NUMBER:H

1. Order positive and negative integers, decimals and fractions;

Order from smallest: 45%, 0.5, 5/9, −0.6

Use the symbols =, ≠, <, >, ≥, ≤

10 < x ≤ 12
Integer x can be 11 or 12.

2. Apply the 4 operations (+, –, ×, ÷), including formal written methods, to integers, decimals and simple fractions (proper and improper), and mixed numbers, positive or negative

Understand and use place value (e.g. when working with very large or very small numbers, and decimals.
0.002 × 3000 = 6 ;   0.2 × 0.4 = 0.08

Divide a decimal by a whole number or a decimal, including negative decimals, without a calculator.
e.g. 0.24 ÷ 6 ;   0.3 ÷ 0.6

Fractions:
11/2 + 3/4;    5/6 × 3/10;    –3 × 4/5

2/5 + 5/6;    2/5 ÷ 6/5;   2/3 + 1/2 × 3/5

3. Recognise and use relationships between operations, including inverse operations
e.g.  223 – 98 = 223 + 2 – 100 = 125
25 × 12 = 50 × 6 = 100 × 3 = 300

Use conventional notation for priority of operations, including brackets, powers, roots and reciprocals (BIDMAS)
e.g. 2 + 4 × (3 – 1) = 2 + 4 × 2 = 2 + 8

Cancellation to simplify calculations and expressions:
120 × 3 /  40 × 6

4. Use the concepts and vocabulary of prime numbers, factors (divisors), multiples, common factors, common multiples, highest common factor, lowest common multiple, prime factorisation, including using product (e.g. 2×2×3) notation

From 3, 6, 7, 8, 11, 14 , 16;
Find prime numbers, factors of 16, multiples of 7.

HCF and LCM of 54 and 84.

Bus A comes every 20 minutes, Bus B every 45 minutes. Come together @13pm when is next time come together.

Express a whole number as a product of its prime factors.
e.g. 24 = 2 × 2 × 2 × 3

Understand that every number can be expressed as a product of prime factors in only one way.

Find the HCF and LCM of two whole numbers from their prime factorisations

What are the factors of 20?

5. Apply systematic listing strategies including use of the product rule for counting

e.g. How many ways can we make 50 pence from a 2p, 5p, and 10p coins

NUMBER:H

6. Use positive integer powers and associated real roots (square, cube and higher), recognise powers of 2, 3, 4, 5;


Estimate powers and roots of any given positive number
e.g. 600 = 2³ × 3 × 5² is worked out from a Prime factor tree.
Estimate powers and roots. e.g. √51 to the nearest whole number

7. Calculate with roots, and with integer indices

e.g. 24=16 ;    √9 = 3;    3 8 = 2

Recognise simple powers of 2, 3, 4 and 5.  e.g. 27 = 33;   2–3 = 1/8

Law of indices:
am × am = a m+n  ;    am ÷ an = am–n
(am)n = amn

16 −¾ is the same as
(   1   )/ (∜16)³
or 1 / 8

8. Calculate exactly with fractions, surds and multiples of π
Add, subtract, multiply and divide simple fractions (proper and improper), including mixed numbers and negative fractions.

e.g. 11/2 + 3/4;    5/6 × 3/10;    –3 × 4/5

2/5 + 5/6;    2/3 + 1/2 × 3/5

Simplify surd expressions involving squares
(e.g. √12= √(4×3)= √4 × √3 = 2√3) and rationalise denominators)

 1  / 3 is the same as 3 / 3

9.Calculate with and interpret standard form A × 10n  where 1 ≤ A < 10, and n is an integer
1234 = 1.234 × 103
0.000123 = 1.23 × 10–4

Add, subtract, multiply and divide numbers in standard form, without a calculator.

10. Work interchangeably with terminating decimals and their corresponding fractions
(such as 3.5 and 7/2 or 0.375 or ⅜);

Use division to convert a simple fraction to a decimal. e.g. 1/6 = 0.16666..

Change recurring decimals into their corresponding fractions and vice vers
Recurring decimal:

Prove 0.44 = 44/99

11. Identify and work with fractions in ratio problems

Interpret a ratio of two parts as a fraction of a whole.
e.g. £9 split in the ratio 2 : 1 gives parts ⅔ × £9 and ⅓× £9

12. Interpret fractions and percentages as operators

Calculate a fraction of a quantity.
e.g. 2/5 of £3.50
Express one quantity as a fraction of another. e.g. 50p is 2/5 of £2.00

Add 10% to £2.50 by either finding 10% and adding, or by multiplying by 1.1 or 110/100
Calculate original price of an item costing £10 after a 50% discount.

13. Use standard units of mass, length, time, money and other measures (including standard compound measures) using decimal quantities where appropriate

14. Estimate answers; check calculations using approximation and estimation, including answers obtained using technology

Use the symbol ≈ appropriately

estimation

15. Round numbers and measures to an appropriate degree of accuracy (e.g. to a specified number of decimal places or significant figures);

Use inequality notation to specify simple error intervals due to truncation or rounding

e.g.
If x = 2.1 rounded to 1 dp, then 2.05 ≤ x < 2.15.
If x = 2.1 truncated to 1 dp,
then 2.1 ≤ x < 2.2.

 16. Apply and interpret limits of accuracy including upper and lower bounds

Calculate the upper and lower bounds of a calculation using numbers rounded to a known degree of accuracy.
e.g. Calculate the area of a rectangle with length and width given to 2 sf.
Understand the difference between bounds of discrete and continuous quantities.
e.g. If you have 200 cars to the nearest hundred then the number of cars n satisfies:
150 ≤ n < 250 and
150 ≤ n ≤ 249.