Developing spatial awareness

Q I am doing transformations, reflections, rotations and translations with my lower set Year 8. This of course is a subject that they have met before. I can hear the groans as we tackle the subject again. They don't seem to be able to visualise the movements and this is really frustrating....

A An area of development would be spatial awareness. Combine this with working to improve their memory and it becomes an effective activity.

You will need to see if you can get hold of some old bed sheets. Cut these into large squares, one per sheet. Paint a number on each corner, quite large so that it can easily be read. You'll need to paint the numbers on both sides so the sheets can still be read when turned upside down. Divide the class into groups of four, one group per square. They all stand by a corner; when an instruction is given they pick up the sheet and move as per the rule. For example:

A. The numbers in the top row of the square swap with the numbers in the bottom row of the square whilst keeping the order the same. Discuss with them how they interpreted this instruction. The correct solution is shown here. Ask them what single transformation would describe the movement of the original square to the new position (reflection about the horizontal line of symmetry). B. Rotate the shape clockwise through 180x.

After giving several one-step transformations, try giving them two, then three, instructions before they move the shape. Students of this ability will find it difficult at first to remember all the instructions, so it is important to make sure they are comfortable with the progress.

C. This could be, "reflect the square about its vertical axis of symmetry, then rotate through 90x clockwise, now swap numbers diagonally." What single transformation could have achieved the same result? A reflection about the diagonal that joins 1 3 in the original shape would do it.

These could be instructions for a dance! With some music they could create a transformation dance. I have the squares and instructions available as PowerPoint and Word documents at

Following this exercise, have a square on the board and ask them to imagine the square in their head. Clearly and slowly give an instruction and ask the pupils to move the square with their imagination. Then draw the position they think the new square is in. You could extend this activity to hexagons.

Having a grasp of the movements, you might like to take your groups on to the computers to play two excellent transformation games: Post the Shapes: reflect, rotate and translate shapes into holes.

Transformation Golf: reflect, rotate and translate the ball round the course.

The author of both games is David Womersley, who now creates resources for the Active Maths website. 2shapeindex.html.

Q This may be an obvioussilly question so I do apologise from the start. How do you explain the fact that - 3 x - 3 = 9 without going down the route of minus times a minus is a plus using the cross number pattern grid?For example, we can explain that 3 lots of minus 3 is minus 9 visually using the number line. Can you explain - 3 lots of - 3 = 9 using the number line and the correct language that pupils will understand?

Also, another question: is - 50 twice as great as - 25?

A I suggest you visit the 'Negator' page on and open up to full page. This is a circular number line. To demonstrate 3 x - 3, this is 'adding' - 3 three times; so, dial from the zero to - 3 three times, giving the answer of - 9. For - 3 x - 3 this is the same as dialling - 3 lots of - 3, a negative direction, so dialling from the - 3 to the zero three times this gives +9. Try it with - 1 lot of - 3. So in the case of - 3 x 3 this would be dialling from the positive three to the zero three times, giving an answer of - 9.

I would say - 50 is twice - 25, not twice as large. The debt is two times bigger, which would make sense in context. I would be interested in other reader's opinions.

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