# Extending transformations beyond shapes on a grid

While I was teaching a higher GCSE class about  Reflections, Rotations and Translations  I wanted to explore extending transformations beyond shapes on a grid to include transforming straight line graphs.

About forty minutes into the lesson on reflections the majority of the students were quietly working their way through the activities.  The class were well behaved, attentive and on task.  However, were they being challenged?  I suspected not.

We had recently covered plotting and deriving the equation of straight line graphs.  This proved a challenging topic to the majority of the class as it always does.  Linking it to transformations as a plenary activity seemed the perfect mix of challenge and consolidation the students needed.

## Extending reflections on a grid

I presented the straight line graph y = 2x + 3 on a blank pair of axes and asked the students to sketch it on their mini-whiteboards.  I set two challenges.

Challenge One:  Draw a reflection of the line y = 2x + 3 in the line y = 1.

Challenge Two:  Derive the equation of the reflected line in the form y = mx + c.

When feeding back to the class we discussed the need to include the intercept on the sketched graph as this provided a reference point for the reflection.

Some students wrote y = -x/2 + 3 as they confused the gradient of the image as being perpendicular to the object line.   We discussed how each point on the image line must be the same distance from the mirror line to the object line.

## Extending rotations on a grid

Once I was confident all students could perform and describe a rotation of a shape on a grid we extended it to include a straight line graphs.

Challenge One:  Draw a rotation of 90° clockwise about the point (0, 3) of the line y = 2x + 3.

Challenge Two:   Derive the equation of the rotated line in the form y = mx + c.

All students had correctly marked the intercept value and used it as the center of rotation.  It was pleasing to hear a number of students discuss a rotation of 90° anti-clockwise would result in the same rotation as clockwise.  We discussed why this is true.

A few students wrote the gradient as -2 as they remembered what happened the previous lesson when reflecting the line.  We discussed the relationship between perpendicular gradients as the negative reciprocal.  I remember the class found this concept difficult when I first taught it.

## Extending translations on a grid

Translating shapes using a vector is often the easiest transformation for students to perform and describe.  However, translating a linear graph and deriving the new equation proved to be the most difficult activity out of the three.

Challenge One:  Draw a translation of the line y = 2x + 3 using the vector .

Challenge Two:  Derive the equation of the translated line in the form y = mx + c.

When performing the translation some students needed to label the numbers on each axis to translate the line as a whole.  We discussed the benefit of translating a point on the line rather than the line itself.  The easiest point to translate was the intercept.  Once the intercept has been translated it proved intuitive for the image to be parallel to the object.  All students realised this meant the two lines had the same gradient.

While all the class found the gradient of the new line only a quarter had sufficient algebra skills to find the intercept.  Using the translated intercept coordinate of (2, 2) we discussed x = 2 when y = 2.  Substituting these values into the equation y = 2x + c gave 2 = 4 + c.  The new intercept was therefore -2.

## Extending transformations beyond shapes on a grid

The point of extending transformations beyond shapes on a grid was not to teach the students something new but rather connect their understanding of transforming shapes on a grid, a topic they found relatively easy, to straight line graphs, a topic they found difficult.

Each activity was used a plenary of approximately 15 minutes with students working in pairs if they wanted to and on mini-whiteboards.

Most students were able to transform the lines but some did find deriving the equation of the images difficult .  I think it was important for the students to explore extending transformations beyond shapes on a grid to develop their algebra skills and recap finding the equation of straight line graphs.

My name is Jonathan Robinson and I am passionate about teaching mathematics. I am currently Head of Maths in the South East of England and have been teaching for over 15 years. I am proud to have helped teachers all over the world to continue to engage and inspire their students with my lessons.

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