The relationship between two variables is linear when they change together at a constant rate and form a straight line when plotted.

Features of linear relationships

Maths

Checking and securing understanding of plotting coordinates

Checking and securing understanding of plotting coordinates generated from a rule

Checking and securing understanding of plotting coordinates with technology

Checking and securing plotting a relationship

Defining features of linear relationships

Positive rate of change from a graph

Negative rate of change from a graph

Rate of change from a coordinate pair

The intercept point

The equation of a straight line

Finding the equation of the line y = mx + c

Finding the equation of the line ay + bx + c = 0

Using dynamic software to explore linear relationships

Problem solving with graphing linear relationships

In this unit, pupils saw that an arithmetic sequence can be represented graphically and that this is a linear relationship. Any point on the line satisfies this relationship. In the future unit, pupils learn how to find a value that satisfies the relationship when the second value is known, without the aid of the graph.

The sequences unit allowed pupils to explore different ways of generating sequences and then representing an arithmetic sequence graphically. This served as an introduction to this unit on graphical representations of linear equations.

In this unti we develop understanding of linear relationships and explore the key features of a linear equation such as the gradient, intercept and determining an equation from coordinate pairs.

Graphical representations of linear equations

The relationship between two variables is linear when they change together {{ }}.

No. The difference between $$-2$$ and $$3$$ in $$y$$ is different.

No. $$x$$ changes by $$+1$$ whereas $$y$$ changes by $$+5$$.

Yes. The $$y$$ values change constantly by $$+5$$ as $$x$$ changes by $$+1$$.

We can't know until we plot it and draw a line.

Does this table of values represent a linear relationship?

What is the constant rate of change in this relationship?

No. The $$x$$ values don't change by $$1$$.

No. The $$y$$ values are decreasing, not increasing.

Yes. The $$y$$ values change by $$-3$$ for every change of $$+2$$ in $$x$$.

Which of these rules will form a linear relationship?

Match the coordinate to the linear relationship it fits.

it has a common, constant difference between terms.

The sequence $$7,13,19,25,31, ...$$ is arithmetic (linear) because:

Which of these sequences is arithmetic (linear)?

Which of the below values will be a $$y$$ coordinate in this table of values for the relationship $$y=4x+5$$?

Here are the second and fifth terms of an arithmetic sequence. Which are the missing terms?

What is the $$10^{\text{th}}$$ term of the sequence $$2n^3$$?

What will the $$y$$ coordinate be when $$x=-2$$ for the relationship $$y=10-3x$$?

Features of linear relationships

Lesson outcome