GCSE Maths - How to Find the Gradient of a Straight Line #65

Summary

This video explains the concept of gradient, illustrating how to calculate it from a graph using three methods. It demonstrates how the gradient measures the steepness of a line, with examples of positive, negative, and zero gradients. The methods include direct observation of line rise over one unit across, the rise over run equation, and the change in y over change in x, showing they are essentially the same. The video clarifies that gradients are consistent along a line and emphasizes the importance of directionality in interpreting gradients, concluding with practical examples for a clear understanding.

Takeaways

• 📈 The gradient represents the steepness of a line, indicating how quickly the height increases or decreases.
• 📊 A line with the highest gradient is the steepest, while a flat line has a gradient of zero, indicating no change in height.
• 🔽 A line sloping downwards has a negative gradient, with steeper descents corresponding to more negative values.
• 📐 Calculating the gradient can be done by determining the change in height for every unit moved horizontally.
• 📏 The 'rise over run' equation is a common method to find the gradient, which is the vertical change divided by the horizontal change.
• 📝 The change in y (rise) over the change in x (run) is another way to express the gradient, essentially the same as the rise over run.
• 🔢 The gradient can be calculated between any two points on a line by finding the difference in y-values divided by the difference in x-values.
• 📉 For lines that do not rise, the gradient remains zero regardless of the points chosen on the line.
• 🔻 When dealing with lines that slope downwards, the gradient is negative and can be found by the same methods as for upward sloping lines.
• 📚 Understanding the direction of the line is important; gradients are always considered from left to right on a graph.
• 👍 The video script effectively explains three methods to calculate gradients, emphasizing the consistency of results regardless of the method used.

Q & A

What does the term 'gradient' represent in the context of the video?

In the context of the video, 'gradient' represents a measure of the steepness of a line on a graph. It indicates how quickly the height of the line increases or decreases as you move across the graph.

How is the gradient of a line related to its steepness?

The gradient of a line is directly related to its steepness. A higher gradient indicates a steeper line, meaning it rises or falls more quickly, while a lower gradient indicates a less steep line.

What is the significance of a gradient of zero in the context of a graph?

A gradient of zero signifies a horizontal line on a graph, which means it neither rises nor falls. It is completely flat, indicating no change in the value of y for any change in x.

What does a negative gradient indicate about a line on a graph?

A negative gradient indicates that the line on the graph is sloping downwards. The more negative the gradient, the steeper the descent of the line.

How can one calculate the gradient of a line using the simplest technique mentioned in the video?

The simplest technique to calculate the gradient of a line is to determine how much the line goes up (or down) for every one unit it goes across. This can be visualized by drawing dashed lines perpendicular and parallel to the axes from any point on the line.

What is the 'rise over run' method for calculating the gradient of a line?

The 'rise over run' method involves calculating the gradient by dividing the vertical change (rise) by the horizontal change (run) between two points on the line.

How does the 'rise over run' method relate to the 'change in y over change in x' formula?

The 'rise over run' method is essentially the same as the 'change in y over change in x' formula. Both methods calculate the gradient by dividing the change in the y-coordinate by the change in the x-coordinate between two points on the line.

Can the 'rise over run' equation be applied to any two points on a line, regardless of their distance apart?

Yes, the 'rise over run' equation can be applied to any two points on a line, whether they are close together or far apart. It provides the gradient between those two specific points.

What is the gradient of a line that does not rise at all, as mentioned in the third graph of the video?

The gradient of a line that does not rise at all is zero. This is because the rise is zero, and any number divided by a nonzero number (the run) results in a gradient of zero.

How do you determine the gradient of a line that is sloping downwards, as shown in the last graph of the video?

To determine the gradient of a line that is sloping downwards, you can use the same techniques as for an upward sloping line. You calculate the change in y (which will be negative) and divide it by the change in x. The result will be a negative number, indicating a downward slope.

What is the importance of considering the direction of travel when calculating the gradient of a line?

The direction of travel is important because it determines whether the gradient is positive or negative. Lines that rise from left to right have a positive gradient, while lines that fall have a negative gradient.

Outlines

00:00

📈 Understanding Gradients and Calculation Methods

This paragraph introduces the concept of gradient, which is a measure of the steepness of a line. It explains three methods to calculate the gradient from a graph: the vertical rise over horizontal run, the rise over run equation, and the change in y over change in x equation, noting that the latter two are essentially the same. The paragraph uses illustrations of hills to demonstrate different gradients, including the steepest with the highest gradient, flatter lines with lower gradients, and a horizontal line with a gradient of zero. It also touches on the concept of negative gradients for lines that slope downwards. The paragraph concludes with an example of calculating the gradient by finding how much the line rises for every unit it moves across, using a point on the line and drawing dashed lines to measure the rise and run.

05:01

📉 Applying Gradient Calculation Techniques

Building on the understanding of gradients, this paragraph delves into the application of the calculation techniques. It emphasizes the importance of considering lines as moving from left to right, which affects the interpretation of the gradient's sign. The paragraph illustrates how to calculate the gradient for a line that does not rise, resulting in a gradient of zero, and for a line that slopes downwards, resulting in a negative gradient. It provides a step-by-step example using dashed lines to measure the rise and run between two points, and then uses the rise over run equation to calculate the gradient. The paragraph concludes by reiterating the process for finding the gradient between two distant points on a graph, using the change in y and x values to determine the gradient.

Keywords

💡Gradient

The term 'gradient' in the context of the video refers to the rate at which a line on a graph increases or decreases. It is a measure of steepness and is crucial for understanding the slope of a line. The video explains that a higher gradient indicates a steeper incline, while a lower gradient means a gentler slope. For example, the first hill in the video has the highest gradient because it rises quickly, making it the steepest.

💡Rise

In the script, 'rise' is used to describe the vertical change in a line's position on a graph. It is part of the equation for calculating the gradient, where the rise is the amount the line ascends from one point to another. The video illustrates this by showing how much a line goes up when it moves horizontally by a certain distance, such as going up by 0.5 for every unit it moves across.

💡Run

'Run' is the horizontal change in a line's position on a graph, which is used in conjunction with 'rise' to calculate the gradient. It represents the distance the line travels across before it rises. The video uses the term to explain the concept of the rise over run equation, where the run is the change in the x-value when calculating the gradient.

💡Change in Y

This term is used to denote the vertical distance or the difference in the y-values between two points on a graph. It is equivalent to the 'rise' and is used in the context of calculating the gradient. The video script mentions that the change in y is how much the y-value has changed by, which is essential for finding the gradient between two points.

💡Change in X

Refers to the horizontal distance or the difference in the x-values between two points on a graph. It is equivalent to the 'run' and is crucial for determining the gradient of a line. The video explains that the change in x is how much the x-value changes, which, when divided by the change in y, gives the gradient of the line.

💡Slope

The 'slope' of a line is another term for its gradient, indicating its steepness. The video script uses the term to describe the incline of different hills, with a steeper hill having a higher slope or gradient. The concept is fundamental to understanding how the gradient is used to measure the steepness of a line on a graph.

💡Flat Line

A 'flat line' in the video refers to a line on a graph that has no incline or decline, indicating a gradient of zero. The script explains that a flat line does not rise or fall, hence its gradient is zero, which is a key point in understanding that a zero gradient means no change in height or value.

💡Negative Gradient

A 'negative gradient' is used to describe a line that slopes downwards on a graph. The video script mentions that if a line is sloping downwards, it has a negative gradient, and the steeper the decline, the more negative the gradient becomes. This concept is important for understanding how the direction of a line's slope affects its gradient.

💡Steepness

'Steepness' is a descriptor of how quickly a line rises or falls on a graph. The video script uses steepness to compare the gradients of different lines, with steeper lines having higher gradients. The term helps viewers understand the relationship between a line's angle and its gradient.

💡Equation

In the context of the video, an 'equation' is used to calculate the gradient of a line. The script introduces two equations: rise over run and change in y over change in x, which are essentially the same and used to find the gradient. The equations are central to the video's explanation of how to determine the steepness of a line.

Highlights

The video explains the concept of 'gradient' and its calculation from a graph using three different methods.

Gradient measures the steepness of a line, with higher gradients indicating a quicker increase in height.

The simplest method to find the gradient is by determining how much the line rises for each unit it moves across.

A line with a gradient of one rises by the same amount it moves across, making it steep.

A line with a gradient of 0.5 is less steep than one with a gradient of one, indicating a gentler slope.

The rise over run equation is introduced as a method to calculate the gradient, equating to the change in y over the change in x.

The rise over run and change in y over change in x are essentially the same, allowing for flexible use in gradient calculation.

For longer stretches of a graph, dashed lines are drawn to determine the rise and run for gradient calculation.

A flat line, or one with no rise, has a gradient of zero, indicating no change in height.

Lines that slope downwards have negative gradients, with steeper descents corresponding to more negative gradients.

The direction of line travel, from left to right, is crucial when determining the sign of the gradient.

A simple graph can have its gradient calculated by picking any point and observing the change in y for a unit change in x.

Using two points on a line, the rise over run method can be applied to find the gradient, even for complex graphs.

The video concludes by summarizing the different techniques for calculating gradient and their applications.

Gradient calculation is essential for understanding the steepness and direction of lines on a graph.

The video provides a comprehensive guide to calculating gradients using various methods for different types of lines.

Transcripts

in this video we're going to look at

what the term gradient means and see how

we can calculate it from a graph using

three different methods

one where we find out how much the line

has risen by for each one that it goes

across

a second that uses the rise of a run

equation

and a third that uses the change in y

over change in x equation

although as we'll see these second two

methods are basically the same thing

now the gradient is basically just a

measure of how steep a particular line

is

so if we took these four hills

this first one in the top left has the

highest gradient because it's increasing

in height most quickly

or in other words it's the steepest

meanwhile the second one is less steep

and so has a lower gradient

the slope in the bottom left though

isn't rising at all it's completely flat

so this one has a gradient of zero

because it's not going up or down

and this last one is sloping downwards

so we say it has a negative gradient

and if it was sloping downwards even

more steeply then its gradient would be

even more negative

if we show these lines properly on

graphs instead though then we can

actually calculate the gradient of each

one

but for the sake of space let's move

them all over to the side for now and

look at them one by one starting with

the first

as we mentioned at the beginning there

are a few different ways that we can

find the gradients

but the most simple technique is just to

figure out how much the line goes up by

each time that it goes across by one

for example if we pick any point along

our line like this one here

and we draw little dashed lines going

across by one

and then up until we meet the line

we can see that for every one that it

goes across to the right

it also goes up by one

so the gradient of this line is one

and would have found the same gradient

no matter where we looked along our line

if we look at our next line though and

do the same thing

this time for every warner that it goes

across

it only goes up by 0.5

and so the gradient of this line is only

0.5

which means it's less steep than our

last line

another way to think about the gradient

is to use this equation here

which says that the gradient is equal to

the rise divided by the run

with the rise being how much the line

has gone up by

and the run being how much the line has

gone across by

you might also have seen it as change in

y divided by change in x

because the rise is basically how much

the y value has changed by

and the run is just how much the x value

has changed by

so these two equations are basically the

same thing which means you can use

whichever one of them you want

so if we use the equation with our

example here

we just figured out that it went up by

0.5

so our rise or change in y would be 0.5

and it went across by a 1.

so that's our run or change in x

which means that our gradient would be

0.5 divided by 1

which is just 0.5

just like we got before

importantly though we can also use this

equation for longer stretches of our

graph as well

for example if we wanted to find the

gradient between these two points which

are quite far apart

then we need to draw dashed lines

between them by going across and then up

and then figure out exactly how much we

want to cross and up by

so if we start with how much we went

across

we went from x equals negative four

all the way to where x equals two

so our x value has increased by six

then to figure out the rise we went from

where y equals negative one

up to where y equals two

which is an increase of three

then we can put these figures into our

equation

by doing the rise or change in y of 3

over the run or change in x of 6

which gives us 3 divided by 6

so 0.5 again

if we switch to our third graph now this

one doesn't rise at all

so no matter which points you pick along

the line the rise will always be zero

which means that our gradient will

always be zero as well

moving on to our last graph

one thing to point out is that you

always have to think of lines as

traveling from left to right

so this line is going down and will

therefore have a negative gradient

to find what that gradient is we can use

any of the techniques that we've looked

at so far

the easiest one for simple graphs like

this is to pick any point along the line

go across by one

and then see how many you have to go up

or down by

so because we had to go down by two

which is a change of negative two

we know that the gradient must be

negative two for this line

to use one of the equations instead we

pick any two points along the line

and find the rise over run

so if we draw dashed lines between these

two

we can see that it's gone down from

three to negative three on the y-axis

so a change of minus six

and along from negative one to two on

the x-axis

so a change of three

and if we then plug these values into

our equation we're going to get negative

6 divided by 3

which gives us a gradient of negative 2.

anyway that's everything for this video

so hope it all made sense and cheers for

watching