### Isosceles Triangles

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

### Route to Infinity

Can you describe this route to infinity? Where will the arrows take you next?

### Eight Hidden Squares

On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight hidden squares?

# Lost

### Why do this problem?

This problem offers students a chance to consolidate their understanding of coordinates whilst challenging them to think strategically.

### Possible approach

Demonstrate the problem to the class, either using the interactivity, or with a grid drawn on the board.

Give students about 10 minutes to work on the problem, either at computers, or on paper in pairs - taking it in turns to choose where the giraffe is and give the distances. Pairs can keep score of the number of guesses each student required to find the giraffe - the one with the lowest total wins.

Ask the class to share efficient strategies/useful ideas. Encourage the students to consider all the points that satisfy each condition, and to look at the shape of this locus. Re-emphasise that the problem is to develop a strategy to find the giraffe with the minimum number of guesses.

Return to the computers/pairs to work on the suggested strategies. Provide squared paper for rough jottings.

If students are familiar with coordinates in 4 quadrants, the game can be an excellent context for practising these - working on paper with suitable grids.

### Key questions

Which points satisfy the conditions given so far?
How can you narrow down the possibilities?

### Possible extension

Play the game on a grid with axes from -9 to 9. Restrict the guessing to the central square -5 to 5, but insist that the giraffe is lost outside this central square. Students are allowed one 'final answer' guess outside the square to locate the giraffe.

### Possible support

Encourage students to draw the situtation on squared paper, and colour code points that are possible/impossible; looking at the result of each new piece of information.