### Hypotenuse Lattice Points

The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN?

### Lost

Can you locate the lost giraffe? Input coordinates to help you search and find the giraffe in the fewest guesses.

### 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

### Why do this problem?

This problem offers a good opportunity for students to discuss images and find convincing arguments for their solutions.

Reuben Hersh has written that:

"In the classroom, convincing is no problem. Students are too easily convinced. Two special cases will do it."

This problem offers an opportunity to ensure that students are justified in generalising from the particular cases that they have selected.

### Possible approach

This printable worksheet may be useful: Route To Infinity

You may be interested in our collection Dotty Grids - an Opportunity for Exploration, which offers a variety of starting points that can lead to geometric insights.

Show the diagram. It's available as a PowerPoint, or you could print out the poster.

"Have a look at this image. In a moment I'm going to remove it, and I want you to be able to describe the route that the arrows take to your partner."
Give students a short while to look at the image, then remove it.

"Without using paper or pencil, can you describe the route to each other?"
Once they have done this, show them the image again to check that what they have described is indeed what they saw.

"I'd like one person in each pair to turn their back to the screen and list the coordinates in the order in which they're visited, and your partner to look at the screen and check. When you make a mistake, swap over. See how far you can get."

Once students have spent some time listing the coordinates, bring the class together.
"I wonder if you can work out where the route will take you after visiting the point (18,17)? Spend a short while thinking about it on your own, then discuss it with your partner, and together develop a convincing explanation for your answer to share with the class."

As students are working, if they get stuck you could offer the following hint:
"What do you notice about the coordinates of the points visited when the arrows are sloping upwards/downwards?"

Students' explanations are likely to refer to specific examples on the visible grid. It is important to insist on clearly justified arguments that refer to the generality - a key question to ask is "How do you know it will always happen?".

Finally, introduce the last question: "I wonder if you can work out how many points the route will pass through before reaching (9,4)? Again, you may want to start by working on your own before discussing it with your partner, and then developing a convincing explanation to share with the class."

While pairs are talking, circulate and eavesdrop on discussions, correcting any misconceptions and making a mental note of any students with clear explanations.
Bring the class together and invite those students with interesting or elegant strategies to present their ideas to the rest of the class.

### Possible extension

Challenge students to design a route that will cover every grid point on an infinite coordinate grid in all 4 quadrants, and to create some questions (and answers), like those above, to go with their design. They could then swap with a partner.

The thinking involved in this problem could lead onto some investigation into countable infinity. This article by Katherine Korner would make a good starting point.

### Possible support

Before working on this problem, spend time on Cops and Robbers and Coordinate Patterns to develop students' fluency with coordinates.