GeoGebra in Primary and Lower Secondary School Classrooms

1. www.geogebra.org , www.geogebra.org/IGI GeoGebra in Primary and Lower Secondary School Classrooms This looks at the ‘Shape’ requirements in the National Curriculum 2000 and the Primary National Strategy 2006 to investigate whether GeoGebra could be used to support/enhance teaching and learning in the Primary Years in English schools. It also considers the potential use of GeoGebra in Lower Secondary Mathematics.

2. In the initial stages the activities are more suited to a whiteboard or and adult working with a small group on a computer. Later (Year 3), learners can be shown the program and develop ideas themselves from prepared base files or creating their own examples. The pages can be printed, saved and copied (exported) to the clipboard. These files have been prepared using the freeware program GeoGebra. • These activities are also suitable for lower secondary and special needs classes.

3. Early Years • Use language such as ‘circle’ or ‘bigger’ to describe the shape of solids and flat shapes • Visualise and name common 2-D shapes … and describe their features; use them to make patterns, pictures • …identify shapes from pictures of them in different positions and orientations.

4. Match Shapes The size of the shapes can be changed and, as the drawing pad can be scrolled, there can be more shapes hidden from view.

5. Properties

6. Same shape, different properties

7. Reflection • Identify reflective symmetry in patterns and 2-D shapes and draw lines of symmetry in shapes • Draw and complete shapes with reflective symmetry; draw the reflection of a shape in a mirror line along one side

8. Show/hide Place points where the reflection is expected, then use show/hide.

9. Reflection in a line The shapes can be moved and the effect on their reflection can be shown using the tool bar.

10. Translations • Complete patterns with up to two lines of symmetry; draw the position after a reflection or translation. • Visualise 3-D objects from 2-D drawings; make nets of common solids.

11. Translations and Nets

12. Position on a grid (Y4 to Y6) • …..describe and identify the position of a square on a grid of squares • Read and plot co-ordinates in the first quadrant; recognise parallel lines in grids and shapes; … • Visualise and draw on grids of different types where a shape will be after reflection, after translations, or after rotation … about its centre or one of its vertices. • Use co-ordinates in the first quadrant to draw, locate and complete shapes that meet the given properties. • Use all four quadrants to find co-ordinates of points determined by geometric information.

13. Using Grids and Coordinates GeoGebra allowsthe option of having a grid as a background, with or without coordinates. The size and number intervals may also be set. In this shot all the lines and the square can be moved and the Reflect in a line used to show the consequences of so doing.

14. Angles Know that angles are measured in degrees and that one whole turn is 360⁰ compare and order angles less than 180 ⁰. • … calculate angles on a straight line. • ….calculate angles in a triangle or around a point. • Know the sum of angles on a straight line, in a triangle, and at a point, and recognise vertically opposite angles.

15. Angles

16. Another Idea: • This fraction activity can produce some interesting art work. • Take a regular shape and half the sides. Join them up so you get a smaller shape … • The questions might be what fraction of the whole shape is the new one? What fraction are the new lengths of the sides compared with the original? • What happens if I half it again, …… and again? • If you look at the triangle the first is ¼ of the first (can be checked by cutting out or folding inwards) and on the square it is half, again checking by folding • If it is a different triangle, such as a right angled triangle will this still be true? • Does this rule also work for other quadrilaterals?

17. Key Stage 3 • In Key Stage 3 GeoGebra has a place in supporting Algebra and well as the Shape area of the Mathematics Curriculum. • Y7: plot points on a coordinate grid from a table, choose an intermediate point and read off the values, including fractions and negatives. Recognise that all points on a line will fit a rule. • Y8: recognise y = mx + c, construct linear functions arising from real-life problems and interpret their corresponding graphs. • Y9: recognise ay + bx + c = 0 and consider their features, rearrange to give y in terms of x. Plot the graphs on paper and using ICT. Describe similarities and differences.

18. The Algebra Window

19. Examples of use in Shape • In Key Stage 3 the curriculum builds on the work in the Primary phase. For example: • Early stages of proof such as in angles in a triangle. • Relationship between lines. • Further work on translations and combinations of transformations. • Beginning to look at properties of circles.

20. Y8:enlargement by a scale factor, and on a grid link to ratio and proportion

21. Tessellation These are the same shape!

22. Work with lines

23. Finally • This is just a sample of what is possible, and sticks very much to the curriculum. • Geogebra offers many chances for students to investigate, if errors are made they can be quickly rectified, unlike with hand drawing. • Opens up opportunities for teachers and students to discover together. • Students are able to download program at home. • Enjoyment factor – students who are turned off might find the challenge of doing a drawing ‘that moves’ an engaging challenge. Whilst doing this they are extending their mathematical knowledge!