Why do this problem?
helps students to consolidate their understanding
of how equations of the form $y=mx+c$ describe the gradient and
position of lines. Students explore the effect of translating a
straight line on the equation that represents or defines it.
Students are encouraged to visualise the movement of the graphs in
order to conjecture and test their conjectures. This is good
preparation for future work on transforming the graphs of more
Working with the whole group demonstrate the interactivity by
lining up the yellow and green dots so the two lines are the same.
Show how the line can be translated vertically by moving the yellow
dot. Draw attention to the equations of the lines showing beneath
Choose a suitable line, and tell the students you are going to
translate it up or down by a number of units. Ask them to picture
what this will look like in order to predict what the equation of
the new line will be. Do this a number of times with different
lines and translations until students are able to predict the new
equation with confidence. Ask them to share insights and
The second part of this problem is perhaps a little more
challenging. Demonstrate, using the interactivity, how the line can
be translated horizontally by moving the yellow dot. Give students
plenty of time, perhaps working in pairs at computers, to picture
and sketch the effect these translations have on the equations of
lines. Clarify to the students that ultimately, the challenge is to
be able to predict the new equation whenever a straight line is
translated horizontally by a given number of units.
Later, bring the class together and use the interactivity to
test their ability to do this. Do this a number of times with
different lines and translations until students are able to predict
the new equation with confidence. Ask them to share insights and
Hand out this
card matching activity
and suggest the students work on this in
pairs, with the aim of producing a display of their results. This
could include sketches of the graphs and suggestions of other
combined translations which have the same outcome.
When we translate a graph, what changes? What stays the
How is this reflected in the equation of the graph?
What information do we need to predict what will happen to the
intercept when we translate horizontally?
For fractional gradients - how far do lines need to be translated
horizontally for the intercept to change by a whole number?
matching activity above, students could do the same activity with
set of cards
which has six additional lines (all with
fractional gradients) and three additional translation cards.
Students could create their own card matching activity for
their peers to complete.
An interesting line of enquiry is to look at pairs of
translations (one vertical, one horizontal) which link a pair of
parallel lines. Challenge students to explain why there are an
infinite number of possibilities.
Another option is to explore gradients of lines and the
corresponding horizontal translations which lead to a given change
in intercept. Can they explain any relationships they find?
Ensure that students are secure about the relationship between
a line's properties and its equation. Encourage students to sketch
the graphs of different equations and then use the interactivity to
test their predictions.
Then structure the activity above so that students start by
working on simpler cases.