Powers, Estimating, and Standard Form

• These notes cover key mathematical concepts and skills, broken down by topic.

STEM: Powers of 10

• Powers of 10 are used to represent very large or very small numbers in a concise way.
• Prefixes are associated with powers of 10 to denote specific multiples or submultiples (e.g., kilo-, milli-).
• Multiplying by an integer power of 10 shifts the decimal point to the right, while dividing shifts it to the left.

Calculating and Estimating

• Powers can be calculated using multiplication or exponents.
• Rounding to a certain number of significant figures simplifies a number while maintaining accuracy.

Indices

• Negative indices represent reciprocals (e.g., x^-1 = 1/x).
• Powers of fractions involve raising both the numerator and denominator to the given power.

Standard Form

• Standard form (scientific notation) expresses numbers as a x 10^b, where 1 ≤ a < 10 and b is an integer.
• To order numbers in standard form, first compare the powers of 10, then compare 'a' values if the powers are the same.

STEM: Calculating with Standard Form

• Calculations with numbers in standard form involve applying index laws and standard arithmetic operations.

Surface Area of Prisms

• A net of a 3D shape is a 2D representation that can be folded to form the shape.
• The surface area of a prism is the sum of the areas of all its faces, found by using its net.

Volume of Prisms

• The volume of a right prism is calculated by multiplying the area of its cross-section by its length.

Circumference of a Circle

• The circumference of a circle is calculated using the formula C = πd or C = 2πr (where d is diameter and r is radius).
• Problems involving circles or prisms can be solved using formulas for circumference, area, and volume.
• A diameter is a straight line passing from side to side through the center of a circle and radius is from the center to the circumference.
• The circumference is the perimeter of a circle, while an arc is a part of the circumference and a sector is part of a circle enclosed by two radii and an arc.

Area of a Circle

• The area of a circle is calculated using the formula A = πr^2 (where r is the radius).

Cylinders

• A cylinder's volume and surface area are calculated using formulas that incorporate the radius, height, and π.

Pythagoras' Theorem

• Pythagoras' theorem states that in a right-angled triangle, a^2 + b^2 = c^2, where c is the hypotenuse; it allows for determining the lengths of unknown sides.

Arithmetic and Quadratic Sequences

• The nth term formula generates terms in a linear or quadratic sequence.
• An arithmetic sequence has a constant difference between consecutive terms, and its nth term can be found using a formula.

Geometric Sequences

• Geometric sequences have a constant ratio between consecutive terms.

Expanding

• Expanding involves multiplying out brackets.
• Squaring a linear expression means multiplying it by itself.
• Quadratic identities can be used to simplify expressions.

Factorising

• Factorising quadratic expressions involves expressing them as the product of two binomials.

Solving Quadratic Equations

• Quadratic equations can be solved by factorising and setting each factor equal to zero.

Substitution

• Substitution involves replacing variables in expressions and formulas with given values.
• Formulas can involve multiple variables and can be used to solve problems.

Inequalities

• Linear inequalities can be solved using similar methods as equations, but with consideration for the direction of the inequality sign.
• Multiplying or dividing an inequality by a negative number reverses the inequality sign.

Using Index Laws

• Index laws apply to zero and negative powers where any number to the power of 0 is 1, and negative powers indicate reciprocals.

Expressions, Equations, Identities, and Formulae

• Expressions are mathematical phrases, identities are equations true for all values, equations are true for specific values, and formulas express relationships between variables.
• Expanding and factorising can be applied to expressions involving powers.

Solving Equations

• Complex equations can be constructed and solved using algebraic manipulation.

Changing the Subject

• Changing the subject of a formula involves rearranging it to isolate a different variable.

STEM: Planning a Survey

• Primary data is collected firsthand and secondary data is existing data.
• Bias in sampling and questionnaires should be reduced to ensure accurate results.

Collecting Data

• Questionnaires should be designed to be clear, unbiased, and effective.
• Data collection sheets and tables should be designed for efficient and accurate recording of data.

Calculating Averages and Range

• The mean from a grouped frequency table is estimated by using the midpoints of the intervals.
• The range from a grouped frequency table is estimated by finding the class width of the highest and lowest values.

Displaying and Analysing Data

• A line of best fit can be used to estimate missing values from a scatter plot.
• Outliers are data points that lie far from the general trend.
• Further lines of enquiry can be identified by analysing data and identifying patterns or anomalies.
• Frequency polygons visually represent frequency distribution.

Direct Proportion

• Quantities are in direct proportion when their ratio is constant.
• Equations can be set up to show direct proportion between two variables.

Solving Problems Using Direct Proportion

• Algebra is utilized to solve problems involving direct proportion by setting up and solving equations.

Translations and Enlargements

• Column vectors represent translations in the coordinate plane.
• The scale factor of an enlargement is the ratio of the lengths of corresponding sides of the image and object.
• Enlargements can be performed using positive scale factors about a center of enlargement.
• An enlargement on a coordinate grid changes the size but not the shape.

Negative and Fractional Scale Factors

• Enlarging with a negative scale factor results in an inversion.
• Enlarging with a fractional scale factor causes an image to be smaller than the original.
• The scale factor is the ratio of corresponding side lengths.

Percentage Change

• Inverse operations are used to find an original value after a percentage change.
• Percentage change is calculated as ((new value - original value) / original value) * 100.

Set Notation and Venn Diagrams

• Set notation is used to describe and manipulate sets.
• Venn diagrams visually represent relationships between sets.

Probability Diagrams

• Lists, tables, Venn diagrams, and sample space diagrams present possible outcomes of events.
• Mutually exclusive outcomes cannot occur at the same time.
• The probabilities of mutually exclusive outcomes/events sum to one.

Tree Diagrams

• Tree diagrams show the probabilities of two or more events occurring in sequence.

Experimental and Theoretical Probabilities

• Experimental probability is based on observed data.
• Theoretical probability is based on mathematical calculations.
• Probabilities can be compared to make predictions or draw conclusions.
• Probability can be applied to problem solving.

Explore powers of 10, prefixes, and scientific notation. Learn calculation techniques with powers and simplifying numbers using significant figures and rounding. Understand negative indices and powers of fractions, representing numbers as a x 10^b.