**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:

1^{1}/_{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

**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. 2^{4}=16 ; √9 = 3; ^{3}√` 8` = 2

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

Law of indices:

a^{m} × a^{m} = a ^{m+n} ; a^{m} ÷ a^{n} = a^{m–n}

(a^{m})^{n} = a^{mn}

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. 1^{1}/_{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 × 10 ^{n} where 1 ≤ A < 10, and n is an integer**

1234 = 1.234 × 10

^{3}

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

**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.