**Additional Materials**

**Materials Required**

**Estimated Time Needed**

*(Times are approximate and will depend on the needs of the students.)*

Lesson

- Whole-Class Interactive Discussion (15 minutes)
**Give each student the transparency***L-Shapes*, and a pin (to help find centers of rotation).

Using transparencies encourages students to test different transformations. Working dynamically should deepen students' understanding of transformations in a way that simply drawing shapes on a graph does not.

**Introduce the lesson by using slides P-1, P-2, and P-3 of the projector resource.****Ask the students where they think the image of the***L-Shape*will be after it has been translated, reflected, or rotated in different ways:*Where will the L-Shape be if it is translated −2 units horizontally and +1 units vertically?**Where will the L-Shape be if it is reflected over the line x = 2?**Where will the L-Shape be if it is rotated through 180° around the origin?*

**Ask volunteers to demonstrate their answers by placing their grid and***L-Shape*on the overhead projector.**Discuss these positions with the rest of the class, and encourage students to challenge their peers if they think the***L-Shape*has been positioned incorrectly.**Once the correct position has been agreed on, move on to the next transformation.****You may also want to move the***L-Shape*to a different position on the grid, and ask students:*What transformation will move the**L-shape to this new position? Show me.*

- Collaborative Work (30 minutes)
**Ask students to work in groups of two or three.**

**Give each group***Card Set A: Shapes*and*Card Set B: Words*and a copy of the transparency*Transformations*.**Introduce the activity:***You are now going to continue to transform L-shapes.**You've got six shape cards, each showing a different L-shape, and eight word cards, each of which describes a different transformation.**Take turns to link two shape cards with a word card. Make sure the arrow goes in the right direction! Each time you do this, explain your thinking clearly and carefully.**Your partner should then either explain their reasoning again in his or her own words or challenge the reasons you gave.**It is important that everyone in the group understands the placing of a word card between two shape cards.*

*Ultimately, you want to make as many links as possible. Use all the shape cards and all the word cards if possible.*

**You may wish to use Slide P-4 of the projector resource to display these instructions.****You have two tasks during the paired work: to make a note of student approaches to the task, and to support student reasoning.**

**Make a note of student approaches to the task.**- Listen and watch students carefully.
- In particular, listen to see whether students are addressing the difficulties they experienced in the assessment.
- For example, are students having difficulty rotating a shape around (2, 0) or reflecting a shape over the lines y = x and y = −x?
- You can use this information about particular difficulties as a focus for a whole-class discussion later in the lesson.

**Support student reasoning.**- Use the questions in the
*Common Issues*table to help address misconceptions. - Encourage students to explain carefully how they have made each connection.

*Lian, please explain why you've linked these two shapes with this transformation.**Laura, can you repeat Lian's explanation in your own words?*

Ask students:

*How does folding the L-Shape along the line of reflection help when reflecting the shape?**How does drawing a line from the center of rotation to a corner of the shape help when rotating the shape?*

**Students who are struggling should be encouraged to concentrate on linking Shape Cards A, B, C, and D.**

**Further Transformations**

**Once students have completed their arrangement of cards, give them a copy of***Card Set C: Additional Words*and a pair of scissors.**Ask students to add an appropriate transformation, where possible, between any shape cards that has not yet been connected.****On completion, students may then glue the cards on a poster. They will need a glue stick and a sheet of large poster paper to do this.**

**Extension Task**

**If a group of students successfully completes the task:***Can you find a combination of two transformations that could be replaced by a single one?**[For example, reflect B over the x-axis B onto A then reflect A over the y-axis onto C. These two transformations can be replaced by a single transformation: rotate B through 180° around the origin onto C. This can be seen on the example arrangement below.]*

**Students should be encouraged to investigate whether or not this is always the case.**

*For any shape, will this combination of transformations always replace this single one?*

A proof would involve considering what would happen to the general point (x, y). Under a reflection over the x-axis, this would go to (x, −y). After a further reflection over the y-axis, this would become (−x, −y). This is the same as the general point (x, y) being rotated through 180° around the origin.

**Students should be encouraged to look for other possible combinations in their card arrangements in the same way.**

- Whole-Class Discussion (15 minutes)
**Give each group of students either a mini-whiteboard, pen, and eraser, or a piece of square paper.****Use Slides P-5 and P-6 of the projector resource to support a whole-class discussion.****Ask students to do the following transformations using the coordinate grid on the transparency***Transformations*, then to write the new coordinate on their mini-whiteboard.*Use the transparency, Transformations. Mark the coordinate (1, 4)**on the coordinate grid.**Show me the new coordinates of the point (1, 4)**after it is:**Reflected over the x-axis.*(1, −4)*Reflected over the y-axis.*(−1, 4)*Rotated through 180° around the origin.*(−1, −4)*Reflected over the line y = x*. (4, −1)*Reflected over the line y = -x*. (−4, −1)*Rotated through 90° clockwise around the origin.*(4, −1)*Rotated through 90° counterclockwise around the origin.*(−4, 1)

**You may like to repeat this with a general starting point (x, y).***Show me the new coordinates of the general point (x, y) after it is:**Reflected over the x-axis.*(x, −y)*Reflected over the y-axis.*(−x, y)*Rotated through 180°**around the origin.*(−x, −y)*Reflected over the line y = x.*(y, x)*Reflected over the line y = −x*. (−y, −x)*Rotated through 90° clockwise around the origin.*(y, −x)*Rotated through 90° counterclockwise around the origin.*(−y, x)

**It may be helpful to write the new coordinates on the board to be able to extend discussions to include combinations of transformations.***What is the single transformation that will produce the same result as:**A rotation of 90° clockwise around the origin followed by a reflection in the y-axis?**[This is a reflection in the line y = −x.]*

*Show me two transformations that can be written as a single direction.**Show me two transformations that cannot be written as a single direction. Can you change the starting point of the shape so that it can be written as a single direction?*