### Is There a Theorem?

Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?

### Khun Phaen Escapes to Freedom

Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.

### Frieze Patterns in Cast Iron

A gallery of beautiful photos of cast ironwork friezes in Australia with a mathematical discussion of the classification of frieze patterns.

# Translating Lines

### Why do this problem?

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 complicated functions.

### Possible approach

It is useful if students have done some preliminary work plotting straight line graphs. The problems Parallel Lines and Diamond Collector offer good starting points. This problem works well alongside Reflecting Lines.

Working with the whole group demonstrate the interactivity by lining up the dots so the two lines are the same. Show how the line can be translated vertically by moving the blue dot. Draw attention to the equations of the lines showing next to the graph.

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 explanations/justifications.

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 blue 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 explanations/justifications.

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.

### Key questions

When we translate a graph, what changes? What stays the same?

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?

### Possible support

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.

### Possible extension

Instead of the card matching activity above, students could do the same activity with this 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?

Students could now try the problem Surprising Transformations.