Math Term 2
Math Term 2
マインドマップ表示
マインドマップアウトライン
Math Term 2
1. Algebra
1.1. Inequalities
1.1.1. Meaning: compare values using <, >, ≤, ≥
1.1.1.1. < less than; > greater than; ≤/≥ include equality
1.1.1.2. Example: x > 3 means x can be 4, 5, 6…
1.1.2. Solve inequalities
1.1.2.1. Use inverse operations like equations
1.1.2.2. If multiplying or dividing by a negative, reverse the inequality sign
1.1.3. Show solutions
1.1.3.1. Number line: open circle for < or >
1.1.3.2. Filled circle for ≤ or ≥
1.1.4. Example: 2x + 3 < 11 → 2x < 8 → x < 4
1.2. Ratio and direct proportion
1.2.1. Ratio compares quantities, e.g. 2:3
1.2.1.1. Simplify by dividing both parts by the same number
1.2.1.2. Keep units consistent
1.2.2. Direct proportion
1.2.2.1. One quantity changes by the same multiplier as the other
1.2.2.2. Equation: y = kx; k is the constant of proportionality
1.2.3. Find missing values
1.2.3.1. Use a ratio table or scale factor
1.2.3.2. Example: 3 pens cost 6,so5penscost10
1.3. Solving equation and simplify
1.3.1. Simplify expressions
1.3.1.1. Collect like terms: 3x + 2x = 5x
1.3.1.2. Expand brackets: 3(x + 2) = 3x + 6
1.3.1.3. Use order of operations
1.3.2. Solve equations
1.3.2.1. Keep both sides balanced
1.3.2.2. Undo operations in reverse order
1.3.2.3. Example: 3x + 4 = 19 → x = 5
1.3.3. Check
1.3.3.1. Substitute your answer into the original equation
1.4. Sequences
1.4.1. Find the pattern
1.4.1.1. Term-to-term rule: how a term changes to the next
1.4.1.2. Look for adding, subtracting, multiplying or alternating patterns
1.4.2. Nth-term rule
1.4.2.1. For arithmetic sequences, use an + b
1.4.2.2. Find the difference to identify the coefficient of n
1.4.2.3. Example: 4, 7, 10, 13… has nth term 3n + 1
1.4.3. Continue a sequence
1.4.3.1. Apply the rule carefully to each term
1.5. Function
1.5.1. Function machine
1.5.1.1. Input → rule → output
1.5.1.2. Example: input x, rule ×2 + 3 gives 2x + 3
1.5.2. Substitution
1.5.2.1. Replace a letter with its given number
1.5.2.2. Use brackets for negative inputs
1.5.3. Function notation
1.5.3.1. f(x) means the output for input x
1.5.3.2. Example: f(x) = 2x + 1, so f(3) = 7
2. Statistic and probability
2.1. Data collection and sampling
2.1.1. Population and sample
2.1.1.1. Population: everyone or everything being studied
2.1.1.2. Sample: smaller group selected from the population
2.1.2. Sampling methods
2.1.2.1. Random sampling reduces selection bias
2.1.2.2. Convenience sampling is easy but may be biased
2.1.3. Good survey design
2.1.3.1. Use clear, neutral questions
2.1.3.2. Choose a sample that represents the population
2.2. Diagrams
2.2.1. Frequency diagram
2.2.1.1. Frequency diagrams
2.2.1.1.1. Show how often values or groups occur
2.2.1.1.2. Label both axes and use an even scale
2.2.1.1.3. Bar chart: bars have gaps for separate categories
2.2.2. Time series graph
2.2.2.1. Time series graph
2.2.2.1.1. Shows how data changes over time
2.2.2.1.2. Time goes on the horizontal axis
2.2.2.1.3. Look for trends, peaks, dips and unusual values
2.2.2.1.4. Do not assume a trend proves the cause
2.2.3. stem and leaf diagram
2.2.3.1. Stem-and-leaf diagram
2.2.3.1.1. Split each value into a stem and a leaf
2.2.3.1.2. Write leaves in ascending order
2.2.3.1.3. Include a key, e.g. 4 | 7 = 47
2.2.3.1.4. Keeps the original data visible
2.2.4. Pie charts
2.2.4.1. Pie charts
2.2.4.1.1. A full circle is 360°
2.2.4.1.2. Sector angle = frequency ÷ total × 360°
2.2.4.1.3. To find frequency: angle ÷ 360° × total
2.2.4.1.4. Check all sector angles add to 360°
2.3. Calculate using statistic
2.3.1. Mean
2.3.1.1. Add all values ÷ number of values
2.3.1.2. Sensitive to extreme values
2.3.2. Median
2.3.2.1. Middle value after sorting
2.3.2.2. For an even number of values, average the two middle values
2.3.3. Mode and range
2.3.3.1. Mode = most frequent value
2.3.3.2. Range = highest − lowest
2.3.4. Compare data
2.3.4.1. Use averages for typical value
2.3.4.2. Use range to describe spread
2.4. Experimental and Theoretical probability
2.4.1. Theoretical probability
2.4.1.1. Assumes outcomes are equally likely
2.4.1.2. P(event) = favourable outcomes ÷ total outcomes
2.4.2. Experimental probability
2.4.2.1. Uses results from an experiment
2.4.2.2. Relative frequency = successes ÷ trials
2.4.3. Probability scale
2.4.3.1. 0 = impossible; 1 = certain
2.4.3.2. All outcome probabilities add to 1
2.4.4. Compare
2.4.4.1. More trials often make experimental results closer to theoretical probability
3. Geometry and measures
3.1. 2D Shapes
3.1.1. Quadrilaterals
3.1.1.1. Quadrilaterals have 4 sides
3.1.1.1.1. Square: 4 equal sides and 4 right angles
3.1.1.1.2. Rectangle: opposite sides equal, 4 right angles
3.1.1.1.3. Parallelogram: opposite sides parallel and equal
3.1.1.1.4. Trapezium: one pair of parallel sides
3.1.1.2. Angle facts
3.1.1.2.1. Interior angles add to 360°
3.1.1.2.2. Opposite angles in a parallelogram are equal
3.1.2. Polygons
3.1.2.1. Polygons are closed shapes with straight sides
3.1.2.1.1. Regular polygon: all sides and angles equal
3.1.2.1.2. Interior angle sum = (n − 2) × 180°
3.1.2.1.3. Each exterior angle of a regular polygon = 360° ÷ n
3.1.2.1.4. One exterior angle at each vertex adds to 360°
3.1.3. circle
3.1.3.1. Circle vocabulary
3.1.3.1.1. Radius: centre to circumference
3.1.3.1.2. Diameter: across the centre; d = 2r
3.1.3.1.3. Circumference: distance around the circle
3.1.3.2. Circle formulae
3.1.3.2.1. Circumference = 2πr = πd
3.1.3.2.2. Area = πr²
3.1.3.2.3. Use π or 3.14 if instructed
3.1.4. exterior angle of triangle
3.1.4.1. Exterior angle
3.1.4.1.1. Formed by extending one side of a triangle
3.1.4.1.2. Exterior angle + adjacent interior angle = 180°
3.1.4.1.3. Exterior angle equals the sum of the two opposite interior angles
3.1.4.2. Triangle facts
3.1.4.2.1. Interior angles add to 180°
3.1.4.2.2. Use known angles to find the missing angle
3.1.5. area of parallelogram and trapezium
3.1.5.1. Parallelogram area
3.1.5.1.1. Area = base × perpendicular height
3.1.5.1.2. Height meets the base at 90°; it is not the sloping side
3.1.5.2. Trapezium area
3.1.5.2.1. Area = ½ × (sum of parallel sides) × height
3.1.5.2.2. Use the perpendicular distance between parallel sides
3.1.5.3. Units
3.1.5.3.1. Area uses square units, e.g. cm²
3.2. 3D Shapes
3.2.1. Type of 3d shapes
3.2.1.1. Prisms
3.2.1.1.1. Same cross-section all along their length
3.2.1.1.2. Volume = cross-sectional area × length
3.2.1.2. Pyramids
3.2.1.2.1. One base and triangular faces meeting at an apex
3.2.1.3. Cylinder, cone and sphere
3.2.1.3.1. Recognise curved surfaces and circular bases
3.2.1.3.2. Identify faces, edges and vertices where appropriate
3.2.2. Surface area of triangular prism
3.2.2.1. Triangular prism
3.2.2.1.1. Two matching triangular ends and three rectangular faces
3.2.2.2. Surface area
3.2.2.2.1. Add the areas of all 5 faces
3.2.2.2.2. Area of each triangle = ½ × base × perpendicular height
3.2.2.2.3. Find rectangle areas using length × width
3.2.2.2.4. Check units are squared
3.2.3. Pyramids
3.2.3.1. Parts of a pyramid
3.2.3.1.1. Base can be triangular, square or another polygon
3.2.3.1.2. Side faces are triangles that meet at the apex
3.2.3.2. Volume
3.2.3.2.1. Volume = ⅓ × base area × perpendicular height
3.2.3.2.2. Height is measured at 90° to the base, not along a sloping edge
3.2.4. Volume
3.2.4.1. Cuboid
3.2.4.1.1. Volume = length × width × height
3.2.4.1.2. Surface area = add the areas of all 6 rectangular faces
3.2.4.2. Prism
3.2.4.2.1. Volume = cross-sectional area × length
3.2.4.3. Pyramid
3.2.4.3.1. Volume = ⅓ × base area × perpendicular height
3.2.4.4. Units
3.2.4.4.1. Volume uses cubic units such as cm³
3.2.4.4.2. 1 cm³ = 1 mL
3.3. Parallel lines and midpoint
3.3.1. Parallel lines
3.3.1.1. Corresponding angles are equal
3.3.1.2. Alternate angles are equal
3.3.1.3. Co-interior angles add to 180°
3.3.2. Midpoint
3.3.2.1. Halfway between two endpoints
3.3.2.2. Number line midpoint = (a + b) ÷ 2
3.3.2.3. Grid midpoint: average the x-values and average the y-values
3.4. Construction and Bearings
3.4.1. Geometric construction
3.4.1.1. Use ruler and compass for accurate lines and arcs
3.4.1.2. Angle bisector divides an angle into two equal parts
3.4.1.3. Perpendicular bisector crosses a line at 90° through its midpoint
3.4.2. Bearings
3.4.2.1. Measured clockwise from North
3.4.2.2. Write bearings using three digits, e.g. 065°
3.4.2.3. Draw a North line at the starting point and measure clockwise
3.5. Transformation of shapes
3.5.1. Reflecting
3.5.1.1. Reflection
3.5.1.1.1. Creates a mirror image across a mirror line
3.5.1.1.2. Each point is the same perpendicular distance from the mirror line
3.5.1.1.3. Size and angles stay the same
3.5.2. Rotating
3.5.2.1. Rotation
3.5.2.1.1. Needs a centre, angle and direction
3.5.2.1.2. Direction is clockwise or anticlockwise
3.5.2.1.3. Every point stays the same distance from the centre
3.5.2.1.4. Size and angles stay the same
3.5.3. Enlarging
4. Number
4.1. Percentage increase and decrease
4.2. Using a multiplier
