# Communicating and Reflecting - February 2009, Stage 4&5

We are used to sharing findings and reflecting on what we've done when we arrive at the solution to a problem, but there is great value in doing this whilst working on problems. Good problem solvers reflect on their progress while they work, asking themselves "Is there a better way to proceed?". When recording their work they communicate their ideas (initially for themselves and then possibly for others); good ideas can come from sharing thoughts and reviewing progress.

## Problems

### 4 Dom

##### Stage: 1, 2, 3 and 4 Challenge Level:

Use these four dominoes to make a square that has the same number of dots on each side.

##### Stage: 3 and 4 Challenge Level:

Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.

### Russian Cubes

##### Stage: 4 Challenge Level:

How many different cubes can be painted with three blue faces and three red faces? A boy (using blue) and a girl (using red) paint the faces of a cube in turn so that the six faces are painted. . . .

### Days and Dates

##### Stage: 4 Challenge Level:

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

##### Stage: 4 Challenge Level:

A walk is made up of diagonal steps from left to right, starting at the origin and ending on the x-axis. How many paths are there for 4 steps, for 6 steps, for 8 steps?

### Angle Trisection

##### Stage: 4 Challenge Level:

It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.

### Iffy Logic

##### Stage: 4 and 5 Challenge Level:

Can you rearrange the cards to make a series of correct mathematical statements?

### Look Before You Leap

##### Stage: 5 Challenge Level:

Relate these algebraic expressions to geometrical diagrams.

### Parabella

##### Stage: 5 Challenge Level:

This is a beautiful result involving a parabola and parallels.

### Contrary Logic

##### Stage: 5 Challenge Level:

Can you invert the logic to prove these statements?

### Direct Logic

##### Stage: 5 Challenge Level:

Can you work through these direct proofs, using our interactive proof sorters?