Higher - Ratio, proportion and rates of change - GCSE Mathematics

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Higher - Ratio, proportion and rates of change

infoWhy this? We are teaching this unit so that pupils can use ratio, proportion and rates of change to solve complex, real-life problems, including working with compound measures and interpreting graphs that model motion, and at Higher level forming and analysing algebraic models of direct and inverse proportion

scheduleWhy now? We are teaching it now in Year 10 so that pupils can connect proportional reasoning with graphical and algebraic methods in their current work, strengthening their ability to interpret, model and solve multi-step contextual problems involving change over time

The difference between two values in a ratio can be used to determine the actual values when the total or one part is known.
Ratios can be combined by finding equivalent ratios or by expressing them in terms of a common variable.
Proportion means that two quantities increase or decrease at the same rate, and this can be used to scale recipes up or down.
On a direct proportion graph, the rate of change is constant and represented by the gradient of the straight line through the origin.
Average speed is calculated as total distance divided by total time.
Distance, speed, and time are related by the formula: distance = speed × time.
Density is calculated as mass divided by volume (density = mass/volume).
Mass, density, and volume are related by the formula: mass = density × volume.
Pressure is calculated as force divided by area (pressure = force/area).
Force, pressure, and area are related by the formula: force = pressure × area.
Direct proportion between two variables means as one increases, the other increases at a constant rate (y = kx).
Inverse proportion between two variables means as one increases, the other decreases such that their product is constant (y = k/x).
Distance-time graphs show how distance changes over time; the gradient represents speed.
Velocity-time graphs show how velocity changes over time; the gradient represents acceleration, and the area under the graph represents distance travelled.
The area under a distance-time graph does not have a physical meaning, but the area under a velocity-time graph represents distance travelled.
The tangent to a curve at a point gives the instantaneous rate of change (gradient) at that point.

rocket_launchYou must be able to

Use the difference between two values in a ratio to calculate the actual values, given the total or one part.
Combine two or more ratios to solve problems, including finding a common term or expressing ratios in terms of a single variable.
Use proportional reasoning to scale recipes up or down, including adjusting all ingredients in the correct ratio.
Deduce the rate of change from a direct proportion graph by calculating the gradient.
Calculate average speed, distance, or time using the formula: speed = distance/time.
Calculate density, mass, or volume using the formula: density = mass/volume.
Calculate pressure, area, or force using the formula: pressure = force/area.
Form and solve algebraic equations to represent and solve problems involving direct proportion (e.g., y = kx).
Form and solve algebraic equations to represent and solve problems involving inverse proportion (e.g., y = k/x).
Draw and interpret distance-time graphs, including identifying periods of rest and constant or changing speed.
Draw and interpret velocity-time graphs, including identifying periods of constant velocity, acceleration, or deceleration.
Calculate the area under a velocity-time graph to determine distance travelled, and interpret this in context.
Draw a tangent to a curve at a given point and use it to estimate the instantaneous rate of change (gradient) at that point.

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