# Parallel lines

## Problem

Alter the positions of the line by moving the points.

How does the position of the line affect the equation of the line?

Now explore what happens when you have two lines.

Position the lines so that they are parallel to each other.

Try various arrangements.

What can you say about the equations of parallel lines?

You may want to take a look at Perpendicular Lines after this.

## Getting Started

It may help to keep one variable fixed and just change the other one:

you could keep the gradient fixed and change the intercept,

or you could keep the intercept fixed and change the gradient.

How do these changes affect the equation of the line?

## Student Solutions

Georgina from St. George's School correctly identified that it is the value of the number before the $x$ (the coefficient) that is important in determining how steep the line is, and how the number $x$ is multiplied or divided by relates to how it crosses the squares.

The steeper the line is, the larger the number before $x$ becomes. You can work this number out by counting how many squares up the line travels for every one across. The number up is the number before $x$.

The shallower the line, the larger the number x is divided by. You can work out this number by counting how many squares the line goes across for every one up.

Put the 2 together and you end up with an equation like the one below

$\frac{2x}{3}$ Simple!

Josh from Melbourn Village College pointed out that for two lines to be parallel, the formulae had to be the same except for the constant value added or subtracted and that this related to the distance of one line above/below the other. He also correctly came up with the rule for the formulae of vertical and horizontal lines.

When the red dot of the green line is on 0 to make the red line parallel below, the formula has to be the same but with minus however many squares there is difference.

If the red line is on 0 and the green is above then formula is the same but the green lines formula has plus the difference between them.

If both lines are horizontal the the formula for both is simple - its one number of how much the difference is from 0 e.g. $y=2$, $x=-7$

Both Daniel from AMVC and Beth, again from Melbourn Village College, correctly quoted the general equation for a straight line $y=mx+c$ and explained that providing the values of $m$ in the equations of two lines were the same, they would be parallel.

Daniel:

All straight lines can be expressed with the equation $y=mx+c$, where $m$ is the gradient (how steep the line is). This means that as long as the $m$ for both equations are equal the lines will be parallel.

Beth:

The equations of parallel lines are the same, except for the value of the constant (the value on the end of an equation). So, a line's equation will follow this structure: $y = mx$, but its parallel would follow this structure: $y = mx + c$.

Emma and Chloe from The Mount School noticed how the position of a line affects the equation:

When a line has a positive gradient, the equation never includes negative values of $x$.

When the line goes through zero there are no $x$ + or - somethings.

If the line does not go through zero then it will be $x$ + or - something else on the end of the equation.

The equations of parallel lines always have the same gradient but cross the $y$ axis at different points.

Boyang from Mountfields Lodge School added:

If the lines are parallel then it is always a $y = ax + b$ equation.

The $ax$ part is the same on both equations when the lines are parallel.

## Teachers' Resources

### Why do this problem?

This resource allows students to explore the connection between a straight line on a graph and the equation that represents or defines it - equations of the form $y = mx + c$. Students are encouraged to conjecture and test their conjectures.

### Possible approach

This problem could be used after How Steep is the Slope?

It is useful if students have also done some preliminary work plotting straight line graphs.

Working with the whole group demonstrate the interactivity, moving both points to allow the group to see the freedom the line has. Draw attention to the equation of the line showing beneath the graph.

Take time to discuss how an equation represents a line by defining the set of points that lie along it. Select points on the line to demonstrate how the coordinates of the point satisfy the equation. Ask questions to challenge this understanding, like: find the equation of a line that goes through (2,3); find the equation of a steeper line; find the equation of a line that slopes down instead of up.

The second interactivity allows the equations of two independent lines to be compared. Working in pairs at computers ask students to propose equations of parallel lines and use the interactivity to check their suggestions.

Encourage students to refine their earlier conjectures. Ask them to phrase what they have learned about equations of straight lines in exactly 25 words.

When students feel they understand this completely, suggest they play Diamond Collector in pairs or against the computer. Allow them to close the game and go back to the parallel lines interactivity if they find they need to learn a bit more to help them play the game better.

### Key questions

What are the connections between the properties of a line and its equation?

### Possible support

Concentrate on the first interactivity until the relationship between a line's properties and its equation are well understood. Ask students to predict what the graphs of different equations will look like and then use the interactivity to test their predictions.

### Possible extension