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Year 8

Defining features of linear relationships

I can appreciate that there are two key elements to any linear relationship: rate of change and intercept point.

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New
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Year 8

Defining features of linear relationships

I can appreciate that there are two key elements to any linear relationship: rate of change and intercept point.

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Lesson details

Key learning points

  1. There is always a common difference in an arithmetic sequence.
  2. There is always a constant rate of change on a linear graph.
  3. There is always an initial value in an arithmetic sequence.
  4. There is always an intercept point with the y-axis on a linear graph.
  5. These two features uniquely describe any linear relationship.

Keywords

  • Rate of change - The rate of change is how quickly one variable changes with respect to another. If constant, there is a linear relationship.

  • Intercept - An intercept is the coordinate where a line or curve meets a given axis.

Common misconception

The arithmetic sequence 5n+2 starts at 7 so the linear equation y=5x+2 'starts' somewhere.

y=5x+2 has no start and no end; it goes on infinitely. However, all linear equations have a y-intercept and this helps define them.


To help you plan your year 8 maths lesson on: Defining features of linear relationships, download all teaching resources for free and adapt to suit your pupils' needs...

Open up a discussion about this misconception. "5n+2, 2n+5, 5-2n; where do they start?" Plot y = 5x+2, y = 2x+5, y = 5 -2x and ask, "Where do you see them starting?". Bring out the infinite nature of these relationships.
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This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

Lesson video

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6 Questions

Q1.
Two variables that change together at a constant rate when plotted on a graph will form a relationship.
Correct Answer: linear, Linear
Q2.
Match the arithmetic sequences to the descriptors.
Correct Answer:$$11,4,-3,-10, ...$$,Starts at $$11$$ and decreases by $$7$$.
tick

Starts at $$11$$ and decreases by $$7$$.

Correct Answer:$$11,7,3,-1, ...$$,Starts at $$11$$ and decreases by $$4$$.
tick

Starts at $$11$$ and decreases by $$4$$.

Correct Answer:$$7,11,15,19, ...$$,Starts at $$7$$ and increases by $$4$$.
tick

Starts at $$7$$ and increases by $$4$$.

Correct Answer:$$7,18,29,40, ...$$,Starts at $$7$$ and increases by $$11$$.
tick

Starts at $$7$$ and increases by $$11$$.

Correct Answer:$$11,15,19,23, ...$$,Starts at $$11$$ and increases by $$4$$.
tick

Starts at $$11$$ and increases by $$4$$.

Correct Answer:$$7,3,-1,-5, ...$$,Starts at $$7$$ and decreases by $$4$$.
tick

Starts at $$7$$ and decreases by $$4$$.

Q3.
Match the common differences to the arithmetic sequences.
Correct Answer:$$-8,-3,2,7, ...$$,$$+5$$
tick

$$+5$$

Correct Answer:$$7,2,-3,-8, ...$$,$$-5$$
tick

$$-5$$

Correct Answer:$$-8,-6,-4,-2, ...$$,$$+2$$
tick

$$+2$$

Correct Answer:$$-4,-6,-8,-10, ...$$,$$-2$$
tick

$$-2$$

Correct Answer:$$4,1,-2,-5, ...$$,$$-3$$
tick

$$-3$$

Correct Answer:$$-14,-11,-8,-5, ...$$,$$+3$$
tick

$$+3$$

Q4.
Which of the below coordinates can you see plotted here?
An image in a quiz
Correct answer: $$(-2,-2)$$
$$(-2,2)$$
Correct answer: $$(-1,2)$$
$$(6,0)$$
Correct answer: $$(0,6)$$
Q5.
Here is an arithmetic (linear) sequence for which we are given the second and fourth terms. Which of the below are the missing terms?
An image in a quiz
Correct answer: $$0.87$$
$$0.5$$
$$0.8$$
Correct answer: $$0.69$$
Correct answer: $$0.51$$
Q6.
For the linear relationship $$y=15-7x$$ what is the $$y$$ coordinate when the $$x$$ coordinate is $$1.5$$? State your answer as a decimal.
Correct Answer: 4.5, four point five

6 Questions

Q1.
The is how quickly one variable changes with respect to another. If the change is constant, there is a linear relationship between the variables.
Correct answer: rate of change
rate of increase
rate of decrease
increase
decrease
Q2.
Here is a table of values for the relationship $$y=2x-3$$. Which statements are true of the rate of change between the two variables?
An image in a quiz
For every change of $$+1$$ in $$x$$ there is a change of $$-2$$ in $$y$$.
Correct answer: For every change of $$+1$$ in $$x$$ there is a change of $$+2$$ in $$y$$.
For every change of $$+1$$ in $$y$$ there is a change of $$+2$$ in $$x$$.
Correct answer: For every change of $$+2$$ in $$x$$ there is a change of $$+4$$ in $$y$$.
Q3.
What constant rate of change is shown by this linear graph?
An image in a quiz
For every change of $$+1$$ in $$x$$ there is a change of $$+3$$ in $$y$$.
For every change of $$+1$$ in $$y$$ there is a change of $$+3$$ in $$x$$.
Correct answer: For every change of $$+1$$ in $$x$$ there is a change of $$-3$$ in $$y$$.
For every change of $$+1$$ in $$y$$ there is a change of $$-3$$ in $$x$$.
Q4.
An is the coordinate where a line or curve meets a given axis.
interval
inequality
Correct answer: intercept
axis
$$x$$ coordinate
Q5.
What is the $$y$$-intercept of this line?
An image in a quiz
$$2$$
$$(2,0)$$
$$-2$$
Correct answer: $$(0,4)$$
$$4$$
Q6.
Which unique statement describes this line?
An image in a quiz
Starts at $$4$$ and has a rate of change of $$-2$$
Starts at $$2$$ and has a rate of change of $$+2$$
$$y$$-intercept at $$(4,0)$$ and has a rate of change of $$+2$$
Correct answer: $$y$$-intercept at $$(0,4)$$ and has a rate of change of $$-2$$