This page continues with more advanced factoring techniques and simplifying algebraic expressions. It covers factoring the difference of squares, factoring quadratic expressions, and simplifying algebraic fractions.

Example: Factoring x² - 36 = x + 6$$x - 6

Example: Factoring x² + 8x + 15 = x + 3$$x + 5

Highlight: The page emphasizes recognizing common factoring patterns, such as the difference of squares and perfect square trinomials.

Vocabulary: The difference of squares is an algebraic expression in the form a² - b², which can be factored as a + b$$a - b.

This page introduces the concept of surds and provides examples of simplifying and manipulating surd expressions. It covers simplifying square roots, multiplying and dividing surds, and rationalizing denominators.

Example: √200 = √(100 × 2) = 10√2

Example: $3 - √2$² = 9 - 6√2 + 2 = 11 - 6√2

Highlight: The page emphasizes the importance of recognizing perfect square factors when simplifying surds.

Vocabulary: A surd is an expression involving a root (usually a square root) that cannot be simplified to a whole number or fraction.

This page covers ratios and introduces basic counting principles. It provides examples of simplifying ratios and using the multiplication principle for counting possibilities.

Example: In a problem where Grace picks a 4-digit number with specific constraints, the total number of possibilities is calculated as 4 × 10 × 2 × 10 = 800.

Highlight: The page emphasizes breaking down complex counting problems into simpler steps using the multiplication principle.

Vocabulary: The multiplication principle states that if one event can occur in 'm' ways, and another independent event can occur in 'n' ways, then the two events can occur together in 'm × n' ways.

This page introduces the concepts of error intervals and bounds when rounding or truncating numbers. It provides examples of determining error intervals for rounded and truncated values.

Example: For a number rounded to 7.3 to one decimal place, the error interval is 7.25 ≤ x < 7.35.

Example: For a number truncated to 1.4 to one decimal place, the error interval is 1.4 ≤ w < 1.5.

Highlight: The page emphasizes the difference between rounding and truncation when determining error intervals.

Vocabulary: An error interval represents the range of possible values a number could have before being rounded or truncated.

This page continues the discussion on bounds, focusing on calculating upper and lower bounds for measurements and using them in calculations. It provides examples of finding bounds for areas and speeds.

Example: For a field with length 120m (to nearest 10m) and width 70m (to nearest meter), the lower bound for the area is 115 × 69.5 = 7992.5m².

Example: For a 100m run completed in 14 seconds (both to nearest unit), the greatest possible speed is 105 ÷ 13.5 = 7.778 m/s.

Highlight: The page emphasizes using the appropriate bounds (upper or lower) to calculate maximum or minimum possible values in applied problems.

Vocabulary: Upper and lower bounds represent the highest and lowest possible values for a measurement, given its level of accuracy.

This page introduces the concept of inverse proportion and provides examples of solving problems involving inverse relationships. It covers deriving formulas for inverse proportion and using them to calculate unknown values.

Example: If T is inversely proportional to the cube of L, and T = 5 when L = 0.2, the formula connecting T and L is T = 0.04 ÷ L³.

Highlight: The page emphasizes recognizing inverse relationships and setting up appropriate equations to solve problems.

Vocabulary: Inverse proportion describes a relationship where one quantity increases as another decreases in proportion so that their product is constant.

This page covers direct proportion and provides more examples of working with ratios. It includes problems on sharing quantities in given ratios and solving word problems involving proportions.

Example: To share £75 in the ratio 2:3, first calculate the value of one part $75 ÷ 5 = 15$, then multiply by the given ratio numbers $2 × 15 = 30 and 3 × 15 = 45$.

Highlight: The page emphasizes the importance of identifying the total number of parts in a ratio before calculating individual shares.

Vocabulary: Direct proportion describes a relationship where one quantity increases or decreases at the same rate as another, maintaining a constant ratio.

This page covers rounding numbers and introduces the concept of discrete data. It provides examples of finding the highest and lowest possible values for rounded numbers.

Example: For a population of 12,000 (to the nearest thousand), the lowest possible population is 11,500 and the highest is 12,499.

Highlight: The page emphasizes understanding the range of possible values when working with rounded numbers.

Vocabulary: Discrete data refers to data that can only take certain specific values, often whole numbers.

This page concludes with more examples of direct proportion problems and introduces writing equations to represent proportional relationships. It provides a complex example of solving a direct proportion problem involving squares.

Example: If C is directly proportional to the square of D, and C = 200 when D = 2, the equation linking C and D is C = 50D². Using this, when D = 5, C = 50 × 5² = 1250.

Highlight: The page emphasizes the importance of correctly identifying the type of proportion (direct or inverse) and setting up appropriate equations.

Vocabulary: In direct proportion, the general form of the equation is y = kx, where k is the constant of proportionality.

This page focuses on expanding and simplifying algebraic expressions. It provides several examples of expanding expressions with two or three brackets. The page also covers factoring quadratic expressions and simplifying algebraic fractions.

Example: x + 6$$x - 2 = x² - 2x + 6x - 12 = x² + 4x - 12

Example: x + 2$$x + 3$$x + 5 = x² + 5x + 6$$x + 5 = x³ + 5x² + 6x² + 30x + 5x + 30 = x³ + 11x² + 35x + 30

Highlight: The page emphasizes the importance of carefully distributing terms when expanding brackets and combining like terms when simplifying.

Vocabulary: Expanding refers to multiplying out brackets in algebraic expressions. Factoring is the reverse process of expanding, where an expression is written as a product of its factors.