travel graphs lesson

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Distance-Time Graphs: Constant Speeds

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Ultimate Guide to Travel Graphs| Cambridge IGCSE Mathematics

Kulni gamage.

• Created on July 20, 2023

[Please watch the video attached at the end of this blog for a visual explanation on the Ultimate Guide to Travel Graphs]

Travel graphs fall under the bigger topic of Algebra, but unlike the other lessons under this topic, this is really simple and easy to understand… as long as you read the question carefully and pay attention to the readings of the graphs ?!

What is a travel graph?

A travel graph is a line graph that tells us how much distance has been covered by the object in question within a particular time. We can use travel graphs to represent the motion of cars, people, trains, buses, etc. This means travel graphs can tell us whether an object is still (stationary), moving at a constant speed, accelerating, or decelerating.  The y -axis represents the distance travelled while the x -axis shows the time it takes to travel a distance such as that.

James went on a cycling ride. The travel graph shows James’s distance from home on this cycle ride. Find James’s speed for the first 60 minutes.

We are told that James is going on a cycling ride and that the distance he travelled is what is shown here. What is required of us is to find James’s speed for the first 60 minutes. This means you need to mark the point at 60 minutes on the x -axis as shown in the picture below.

We have learnt when learning about Speed, Time and Distance that speed = distance/ time.

Therefore, in a distance-time graph, the slope that we see (also called the gradient) is what will give us the speed.

We can see that this particular cycle has travelled 12 km in 60 minutes.

When we substitute these values into the formula above, it would look like this:

When this formula is solved, then we get James’s speed for the first 60 minutes , which is 0.2 km/min .

A train journey takes one hour. The diagram shows the speed-time graph for this journey. Calculate the total distance of this journey.

We need to be careful here because this time what we have in this question is a speed-time graph and not a distance-time graph.

In a speed-time graph, the gradient of a slope represents either acceleration or deceleration, which will be concepts touched upon in later lessons. The distance travelled is found by calculating the area under a speed-time graph.

In order to calculate the distance travelled by the train therefore, we need to calculate the area under the graph that is drawn. To make things easier, we can separate this graph into shapes:

2 triangles and 1 rectangle

Triangle 1: Area of a triangle = ½ × base of triangle × height of triangle Area = ½ × 4 km × 3 km/min Area = 6 km2

Rectangle: Length of the rectangle can be found by finding out the difference between 50 and 4, which equals to 46 min. The height of the rectangle is 3 km/min. Area = length × base Area = 46 min × 3 km/min Area = 138 km

Triangle 2: Area of a triangle = ½ × base of triangle × height of triangle Area = ½ × 10 km × 3 km/min Area = 15 km

Total Area under the graph = Total distance travelled by the train Therefore, Total Distance = Triangle 1 + Rectangle + Triangle 2 Total Distance = 6 km + 138 km + 15 km Total Distance =  159 km

Those are the basic tips and tricks you would need to learn in this particular chapter!

Travel Graphs at Exams

Pay attention to what sort of graph is given to you. Is it a distance-time graph? A speed-time graph? Depending on the type of graph, remember, the value represented by the gradient changes, and so does the value represented by the area under the graph.

Practice as many questions as you can on this topic. They are quite simple once you get the hang of them. You can find some questions in this quiz to check where you stand!

If you are struggling with IGCSE revision or the Mathematics subject in particular, you can reach out to us at Tutopiya to join revision sessions or find yourself the right tutor for you.

Watch the video below for a visual explanation of the lesson on the ultimate guide to travel graphs!

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This is level 1; Reading information from distance-time graphs.

This is Travel Graphs level 1. You can also try: Level 2 Level 3 Level 4

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Description of Levels

Level 1 - Reading information from distance-time graphs

Level 2 - Matching distance-time graphs with their descriptions

Level 3 - Reading information from speed-time graphs

Level 4 - Draw a travel graph from the given description

Exam Style Questions - A collection of problems in the style of GCSE or IB/A-level exam paper questions (worked solutions are available for Transum subscribers).

More Graphs including lesson Starters, visual aids, investigations and self-marking exercises.

Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.

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Distance-Time Graphs

For a basic introduction to distance-time graphs see Hurdles Race . For more details play the video below.

Don't wait until you have finished the exercise before you click on the 'Check' button. Click it often as you work through the questions to see if you are answering them correctly. You can double-click the 'Check' button to make it float at the bottom of your screen.

Real Life Graphs

Four lessons in this section:

Conversion graphs is a complete lesson with a discussion starter and three visual explanations. Includes a differentiated main task and problem-solving questions.

Real life graphs features distance-time graphs and capacity-time graphs. Pair-work and differentiated main task, with literacy story-telling plenary.

Using gradients of line segments to calculate speed from distance-time graphs.

Area under graphs is a complete lesson on area under graphs and gradient, velocity-time graphs and acceleration/deceleration. Clear explanation, Bloom's Taxonomy questions and a differentiated main task.

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Distance-Time Graphs

Subject: Mathematics

Age range: 14-16

Resource type: Lesson (complete)

Last updated

25 April 2019

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A FULL LESSON on interpreting and drawing distance-time graphs.

We are learning about: Distance-time graphs We are learning to: Interpret and draw distance-time graphs in context.

Differentiated objectives:

• Developing learners will be able to interpret information from distance-time graphs.
• Secure learners will be able to identify the scale used on distance-time graphs.
• Excelling learners will be able to plot information onto a distance-time graph.

Starter: Clear demonstration of how the first example’s distance-time graph is constructed (Inspired by GeoGebra Traffic) Main: Walkthrough examples followed by practice questions on worksheets. All solutions given on PPT.

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Other graphs

Travel graphs: distance-time graphs, select lesson, probability, trigonometry, drawing shapes, lines and angles, properties of shapes, rates of change, straight line graphs, formulae and equations, algebraic manipulation, fractions, decimals and percentages, number calculations, types of numbers, explainer video, ​​in a nutshell.

Travel graphs show distance travelled on the  y y y -axis against time on the  x x x -axis. They can be used to determine the speed and direction of motion.

What do travel graphs look like?

Travel graphs have a series of line segments that join together to represent a journey. An example travel graph is given below. It shows a journey to a supermarket and back, with a small stop on the way to the shop.

The graph can be broken into its separate straight line segments:

• ( 0 0 0 ​ mins to 10 10 10 mins) In the first ten minutes, the walker has travelled one kilometre;
• ( 10 10 10 ​ mins to 15 15 15 ​ mins) then the person stops for five minutes. Perhaps they are chatting to someone on the way to the shop;
• ( 15 15 15 ​ mins to 20 20 20 ​ mins) the person continues on their journey to the shop, travelling  0.6 0.6 0.6  kilometres in five minutes;
• ( 20 20 20 ​ mins to 35 35 35 ​ mins) the person has stopped at the shop to do their shopping. They don't go any further for  15 15 15  minutes. With respect to their journey from home, they have stopped;
• ( 35 35 35  mins to 50 50 50  mins) the person has finished their shopping and goes directly home. They travel 1.6 1.6 1.6  kilometres in 15 15 15  minutes.

​​Direction

If the graph is going up, then the direction is  away  from the starting position. If the graph is going down, then the direction is  toward  the starting position. If the graph is flat, then motion has stopped.

Travel graphs give you enough information to calculate the speed of the motion. 

The speed of a segment is calculated by finding the gradient of the line segment. Gradient is calculated with the formula:

​ m = change in  y change in  x m=\frac{\text{change in }y}{\text{change in }x} m = change in  x change in  y ​

Note:  If a line segment has a negative gradient, ignore the negative sign for the speed. The negative just means that motion is back towards the starting position.

Given that each line segment is either a diagonal line or a flat line, finding the gradient is straightforward: for diagonal lines, just use the coordinates of the top and bottom of the line; for flat lines, the gradient, and hence speed, is zero. 

Note: The straight line means that the speed is constant.

Alternatively, use:

​ speed = distance time \text{speed}=\frac{\text{distance}}{\text{time}} speed = time distance ​ ​​

Note: The unit of speed will the unit of distance divided by the unit of time.

In the first ten minutes of the walker's journey, how fast do they travel?

In the first ten minutes, the walker has travelled one kilometre. Hence their speed is:

1  km 10  mins = 0.1  km per minute = 100  m per minute ‾ \frac{1\text{ km}}{10\text{ mins}}=0.1\text{ km per minute} = \underline{100\text{ m per minute}} 10  mins 1  km ​ = 0.1  km per minute = 100  m per minute ​

How fast did they walk on their way back home? Use the same graph above again.

Use the final  15 15 15  minutes of the journey; the graph is going down, hence the direction is back toward the starting location. In this time, the walker travels  1.6  km . 1.6\text{ km}. 1.6  km .  Hence their speed (rounded to three significant figures) was:

1.6  km 15  mins = 0.107  km per minute = 107  m per minute ‾ \frac{1.6\text{ km}}{15\text{ mins}}=0.107\text{ km per minute} = \underline{107\text{ m per minute}} 15  mins 1.6  km ​ = 0.107  km per minute = 107  m per minute ​ ​​

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Learning Goals

Faqs - frequently asked questions, what do the different directions of the graph mean in a travel graph.

If the graph is going up, then the direction is away from the starting position. If the graph is going down, then the direction is toward the starting position. If the graph is flat, then motion has stopped.

How do you work out speed on a travel graph?

The speed of a segment is calculated by finding the gradient of the line segment.

What does a travel graphs look like?

Travel graphs are have a series of line segments that join together to represent some journey.

Travel Graphs including Average Speed Lesson

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Description

This is a whole lesson on Travel Graphs . This comes in a fantastic sequence of lessons and it is recommended that you teach speed, distance, and time first before looking at travel graphs. This lesson is ready to go, with no prep required. It is great for home learning too. Great as well for flip learning. 25-slide presentation + ORIGINAL VIDEO CONTENT + lots of supplementary resources.

The lesson comes with:

+ a Starter (quick worksheet - describing travel graphs)

+ Learning Objectives (differentiated)

+ superb teaching slides (with LOTS of custom animation)

+ Lots of examples

+ Helpful time conversion slide

+ FULL ORIGINAL VIDEO CONTENT (10 MINS)

+ MWB activity for conversion between minutes and hours (great for AFL)

+ Fantastic Worksheet (with answer key)

+ Matching Activity (great for Homework)

ALL LESSONS on Geometry in one MEGA BIG Bundle:

Geometry - ALL Lessons

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Travel Graphs

This section covers travel graphs, speed, distance, time, trapezium rule and velocity.

Speed, Distance and Time

The following is a basic but important formula which applies when speed is constant (in other words the speed doesn't change):

Remember, when using any formula, the units must all be consistent. For example speed could be measured in m/s, distance in metres and time in seconds.

If speed does change, the average (mean) speed can be calculated: Average speed = total distance travelled                              total time taken

In calculations, units must be consistent, so if the units in the question are not all the same (e.g. m/s, m and s or km/h, km and h), change the units before starting, as above.

The following is an example of how to change the units:

Change 15km/h into m/s. 15km/h = 15/60 km/min               (1) = 15/3600 km/s = 1/240 km/s      (2) = 1000/240 m/s = 4.167 m/s        (3)

In line (1), we divide by 60 because there are 60 minutes in an hour. Often people have problems working out whether they need to divide or multiply by a certain number to change the units. If you think about it, in 1 minute, the object is going to travel less distance than in an hour. So we divide by 60, not multiply to get a smaller number.

If a car travels at a speed of 10m/s for 3 minutes, how far will it travel? Firstly, change the 3 minutes into 180 seconds, so that the units are consistent. Now rearrange the first equation to get distance = speed × time. Therefore distance travelled = 10 × 180 = 1800m = 1.8km

Velocity and Acceleration

Velocity is the speed of a particle and its direction of motion (therefore velocity is a vector quantity, whereas speed is a scalar quantity).

When the velocity (speed) of a moving object is increasing we say that the object is accelerating . If the velocity decreases it is said to be decelerating. Acceleration is therefore the rate of change of velocity (change in velocity / time) and is measured in m/s².

A car starts from rest and within 10 seconds is travelling at 10m/s. What is its acceleration?

Distance-Time Graphs

These have the distance from a certain point on the vertical axis and the time on the horizontal axis. The velocity can be calculated by finding the gradient of the graph. If the graph is curved, this can be done by drawing a chord and finding its gradient (this will give average velocity) or by finding the gradient of a tangent to the graph (this will give the velocity at the instant where the tangent is drawn).

Velocity-Time Graphs/ Speed-Time Graphs

A velocity-time graph has the velocity or speed of an object on the vertical axis and time on the horizontal axis. The distance travelled can be calculated by finding the area under a velocity-time graph. If the graph is curved, there are a number of ways of estimating the area (see trapezium rule below). Acceleration is the gradient of a velocity-time graph and on curves can be calculated using chords or tangents, as above.

The distance travelled is the area under the graph. The acceleration and deceleration can be found by finding the gradient of the lines.

On travel graphs, time always goes on the horizontal axis (because it is the independent variable).

Trapezium Rule

This is a useful method of estimating the area under a graph. You often need to find the area under a velocity-time graph since this is the distance travelled.

Area under a curved graph = ½ × d × (first + last + 2(sum of rest))

d is the distance between the values from where you will take your readings. In the above example, d = 1. Every 1 unit on the horizontal axis, we draw a line to the graph and across to the y axis. 'first' refers to the first value on the vertical axis, which is about 4 here. 'last' refers to the last value, which is about 5 (green line).] 'sum of rest' refers to the sum of the values on the vertical axis  where the yellow lines meet it. Therefore area is roughly: ½ × 1 × (4 + 5 + 2(8 + 8.8 + 10.1 + 10.8 + 11.9 + 12 + 12.7 + 12.9 + 13 + 13.2 + 13.4)) = ½ × (9 + 2(126.8)) = ½ × 262.6 = 131.3 units²