Year 10 Maths AT2 Cheat Sheet

This cheat sheet covers essential Year 10 Maths topics including graphing linear equations, calculating geometric properties like area, volume, and surface area, understanding error calculations, and providing key exam tips.

Core Principles

Linear equations have a consistent gradient and y-intercept.
Parallel lines share the same gradient.
Perpendicular lines have negative reciprocal gradients.
Area formulas apply to basic shapes; composite areas are sums/differences.
Volume formulas for prisms and cylinders use cross-sectional area or base area multiplied by height.
Advanced formulas exist for pyramids, cones, and spheres.
Surface area calculations require careful consideration of all visible faces.
Error calculations involve absolute, percentage, and rounding errors.

Action Steps

To graph a line: Put equation in y=mx+b form.
Plot the y-intercept (b).
Use the gradient (m) as rise/run to find another point.
Connect the points with a ruler.
To find a line's equation from gradient and a point: Substitute known values into y=mx+b.
Solve for the y-intercept (b).
Write the final equation.
For composite areas: Split the shape into simpler known shapes.
Calculate the area of each simple shape.
Add or subtract areas as needed.
For composite volumes: Split the solid into simpler shapes.
Calculate the volume of each simple shape.
Add or subtract volumes as needed.
For composite surface areas: Identify all visible faces.
Calculate the area of each visible face.
Sum the areas of the visible faces.

Formulas

$m = \frac{y_2 - y_1}{x_2 - x_1}$ (Gradient)
$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$ (Midpoint)
$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ (Distance)
$y = mx + b$ (Gradient-Intercept Form)
$A = lw$ (Rectangle Area)
$A = \frac{1}{2}bh$ (Triangle Area)
$A = \pi r^2$ (Circle Area)
$V = \text{Area of cross section} \times \text{length}$ (Prism Volume)
$V = lwh$ (Rectangular Prism Volume)
$V = \pi r^2h$ (Cylinder Volume)
$SA = 2lw + 2lh + 2wh$ (Rectangular Prism Surface Area)
$SA = 2\pi r^2 + 2\pi rh$ (Cylinder Surface Area)
$V = \frac{1}{3}Bh$ (Pyramid Volume)
$V = \frac{1}{3}\pi r^2h$ (Cone Volume)
$V = \frac{4}{3}\pi r^3$ (Sphere Volume)
$SA = \pi r^2 + \pi rl$ (Cone Surface Area)
$SA = 4\pi r^2$ (Sphere Surface Area)
Absolute Error = |measured - actual|
% Error = (absolute error / actual value) * 100

Key Terms

Gradient (m): The steepness of a line; rise over run.
Y-intercept (b): The point where the line crosses the y-axis.
Parallel Lines: Lines with the same gradient.
Perpendicular Lines: Lines whose gradients are negative reciprocals of each other.
Composite Area: The area of a shape made up of simpler geometric shapes.
Prism: A solid with a constant cross-sectional area along its length.
Slant Height (l): The distance from the apex of a cone or pyramid to a point on the edge of the base.
Absolute Error: The magnitude of the difference between a measured value and the actual value.
Percentage Error: The absolute error expressed as a percentage of the actual value.
Rounding Error: The uncertainty introduced by rounding a number to a specific decimal place.

Timeline

Ancient Greece: Development of basic geometric principles and formulas.
17th Century: Analytic geometry (Cartesian coordinates) developed by Descartes, linking algebra and geometry.
17th-18th Century: Calculus developed by Newton and Leibniz, providing tools for understanding rates of change (gradients) and areas under curves.
19th Century: Formalization of error analysis and measurement theory.
20th Century - Present: Widespread use of graphing calculators and software for visualizing and calculating properties of lines and shapes.

People

René Descartes: Pioneer of analytic geometry, linking algebra and geometry.
Isaac Newton: Co-developer of calculus, fundamental to understanding gradients and areas.
Gottfried Wilhelm Leibniz: Co-developer of calculus.

Quiz

What is the gradient of a line parallel to y = -3x + 5?: -3
What is the y-intercept of the line passing through (1, 5) with a gradient of 2?: 3
The volume formula for a cone includes which factor?: 1/3
If the actual value is 10 and the measured value is 12, what is the absolute error?: 2