### All in the Mind

Imagine you are suspending a cube from one vertex (corner) and allowing it to hang freely. Now imagine you are lowering it into water until it is exactly half submerged. What shape does the surface of the water make around the cube?

### Rotating Triangle

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

### Instant Insanity

Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.

# Frogs

### Why do this problem?

This problem requires students to visualise, work systematically, and develop good representations. The engaging nature of the problem encourages students to persevere in order to get to a solution. Along the way they will notice patterns, make conjectures, and perceive structure to explain the patterns they notice.

### Possible approach

Show the video below or demonstrate the problem to the class. One way is to seat 4 students on 5 chairs, two wearing red hats/scarves/jumpers and two wearing blue, and act the problem out.

Give the class some time to play with the problem, using counters or the interactivity.

Full Screen and tablet version
Then bring the class together to share any initial noticings, and perhaps to demonstrate 3 red and 3 blue frogs, for example.
The interactive allows you to enter students' names after they have finished swapping the frogs, and these can be saved and replayed.

Next, set the challenge:
"In a while, I am going to choose a number of red and blue frogs, and challenge you to swap them over using as few moves as possible and be able to predict before you start how many moves it will take you."

Give the class some time to start thinking about the problem. When appropriate bring them together to move their thinking on:

"It may be helpful to look for patterns in the sequence of moves that it takes to swap the frogs over. You might find it easier to see those patterns if you have a good recording system."

While students are working, look for examples of clear thinking, explanations and recording strategies, ready to share these with the rest of the class.

Here are some possible methods students might employ - you may wish to suggest some of them to students who are struggling to express their thinking clearly on paper:

Bring the class together to share what they have found. Invite a few students to share their insights and any recording strategies that helped them.

Finally, use the interactive environment to perform a frog swap with, say, 7 red frogs and 6 blue frogs. You could invite one pair of students to start off, and then stop them part way through and ask another pair to take over.

### Key questions

How do you know whether you need to jump next, or slide next?
How many places will the frogs have moved by the end?
How many jumps will there be altogether?
Can you explain the patterns you notice?

### Possible extension

Without looking at the screen or your counters, and without drawing any diagrams, can you write out a sequence of moves to swap over some red frogs and blue frogs?

One approach might be to fix the number of blue frogs and explore what happens as you increase the number of red frogs.

### Possible support

Restrict the problem to exploring the number of moves when there are the same number of frogs on each side. Encourage students to record the number of jumps and the number of slides.