A2 · Ramp Slope as Rise Over Run

Alberta Grade 11 Mathematics

Algebra

Understanding how one quantity changes compared with another

What you will learn

• Identify rise and run between two points.
• Interpret slope as rise divided by run.
• Explain slope as a constant rate and include its units.
• Check whether a slope makes sense in its context.

Start with a real situation

• Rise is the vertical change between two points.
• Run is the horizontal change between two points.
• Use changes between points, not just measurements from one point.

Read slope as a constant rate

• A constant rate gives the same change for equal runs.
• Slope units are vertical units divided by horizontal units.
• Read a graph’s slope as the change in its vertical value compared with its horizontal value.

Find and explain a slope

• Use axis scales to find actual changes.
• A straight line has a constant ratio of rise to run.
• Do not assume every graph has one constant slope.

Check units and reasonableness

• Check that the sign matches the direction of change.
• Check that units match rise divided by run.
• Use the rate to predict a change and see whether the result is sensible.

Worked example

A ramp’s steepness

1. Identify the changes
   The vertical change is the rise, 24 cm. The horizontal change is the run, 120 cm. Both are measured between the same two points.
2. Divide rise by run
   Slope compares rise with run. The centimetre units cancel because both measurements use the same unit.
   $$\frac{24}{120}=0.2$$
3. Interpret the result
   The ramp rises 0.2 cm for each 1 cm of horizontal distance. Equivalently, it rises 20 cm for each 100 cm of run.
   $$0.2\text{ cm/cm}=20\text{ cm}/100\text{ cm}$$

Worked example

A steady cycling rate

1. Calculate the rate
   Distance is the vertical quantity and time is the horizontal quantity. Divide the distance change by the time change.
   $$\frac{18\text{ km}}{45\text{ min}}=0.4\text{ km/min}$$
2. Explain the units
   The rate means the cyclist travels 0.4 km for each minute. This is the slope of the distance-versus-time graph.
3. Find the distance in 10 minutes
   At a constant rate, multiply the rate by the elapsed time. The minutes cancel, leaving kilometres.
   $$0.4\text{ km/min}\times10\text{ min}=4\text{ km}$$

Worked example

A decreasing amount

1. Find the changes
   The amount falls from 52 L to 37 L. Subtract the starting amount from the ending amount to find the change. Time increases by 10 min.
   $$37\text{ L}-52\text{ L}=-15\text{ L}$$
2. Calculate the slope
   Divide the change in amount by the change in time. The negative sign shows that the amount is decreasing.
   $$\frac{-15\text{ L}}{10\text{ min}}=-1.5\text{ L/min}$$
3. Predict at 18 minutes
   At the same constant rate, the amount changes by the rate multiplied by 18 min. Add this negative change to the starting amount.
   $$52\text{ L}+(-1.5\text{ L/min})(18\text{ min})=25\text{ L}$$

Common mistakes and how to avoid them

Lesson summary

• Slope is rise divided by run.
• Rise is the vertical change; run is the horizontal change.
• Slope can describe a constant rate, with units comparing vertical change to horizontal change.
• The sign shows whether the vertical quantity increases, decreases, or stays unchanged as the horizontal quantity increases.

Check your understanding

Key terms

Rise

The change in the vertical quantity between two points.

Run

The change in the horizontal quantity between two points.

Slope

The ratio of rise to run; it describes how a vertical quantity changes as a horizontal quantity changes.

Rate

A comparison of changes in two quantities, such as distance travelled for each unit of time.

Constant rate

A rate that stays the same as the horizontal quantity changes.

Continue through Math 20-3

View the complete Math 20-3 Alberta Grade 11 Mathematics curriculum and lessons

• N5 · Compare credit-card and loan options in context
• A3 · Solve proportional problems using unit analysis
• M2 · Solve volume and capacity problems with SI and imperial units
• G1 · Solve problems involving two or three right triangles
• G2 · Solve practical scale and scale-diagram problems
• G3 · Model 3-D objects and draw their top, front, and side views

Tutor support for A2 · Interpret slope as rise over run and as a constant rate

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Alberta Grade 11 Mathematics (Math 20-3), expectation A2. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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