2. Differentiating work
One of the simplest transformations is translation. To translate a shape, we simply move its position on the page, up or down, left or right, but do not change the shape in any other way (see Resource 3: Translation).
Because translating a shape is simple, even very young pupils can grasp the idea, especially if they have physical shapes to manipulate. For older pupils, the activity can be made more challenging by using x-y coordinates and calculation, rather than manipulating physical shapes.
Case Study 2 and Activity 2 look at translation and how to differentiate tasks according to age and stage.
Case Study 2: Extending understanding of translation
Mrs Kiboa teaches a multigrade class in which she has a group of older children who are doing well at mathematics. Feeling their current work had not been stretching them enough, Mrs Kiboa took an opportunity to let them enjoy a real challenge. (For more information on teaching multigrade classes, see Key Resource: Working with multigrade classes.)
Mrs Kiboa had already introduced x-y coordinates to the whole class. One day, while most of the class were working on a triangle translation activity using cut-out shapes, Mrs Kiboa gave these four pupils extra support (see Resource 4: Translating and reflecting triangles).
Drawing a triangle with labelled x-y axes on some grid paper, she asked the pupils what the coordinates of the three corners (vertices) were – they answered easily, and wrote their answers down. Next, she asked them, ‘What would happen if I were to move the shape six spaces to the right? What would the new x-y coordinates be?’ When they had answered correctly, she went on: ‘And if I moved the shape 3 spaces down?’ Mrs Kiboa went on in this fashion until she felt the pupils clearly understood what was happening.
Next, she said to them, ‘Now, each of you set one another a problem – give coordinates for a triangle, and a translation to apply to the triangle. Write this down, then draw the triangle you have been set, calculate the translated coordinates, and draw the new position. If you do this correctly, you may then try shapes other than triangles to test each other with.’
The pupils enjoyed the respect of their teacher, as well as the opportunity to work more freely and to challenge each other mathematically.
Activity 2: Investigating translations practically
Make sure pupils understand how to give x-y coordinates, through whole-class teaching. To differentiate the task for older or younger pupils, see the notes on differentiation in Resource 4.
- Ask pupils to draw and cut out a triangle, square and rectangle from a piece of squared paper: emphasise that each corner (or vertex) of their shapes should be at one of the ‘crosses’ on their grid paper by drawing an example on the board. No side should be more than 10 squares long.
- On a second piece of grid paper, ask pupils to draw and label x-y axes at least 20 squares long (see Resource 4).
- Putting one of their cut-out shapes on the paper so that its corners are on the ‘crosses’ of the grid, they should mark the vertices (a, b, c & d as appropriate), then draw the shape and write down the coordinates of each vertex.
- Ask them to move their shape to a new position (keeping it the same way up) and repeat this process.
- Ask your pupils: ‘What happens to the x coordinates between the two positions? Does the same thing happen to each coordinate? What happens to the y coordinates?’
What parts of this activity caused difficulty for your pupils? How will you support them next time?