# Communicating and reflecting - September 2009, All Stages

This month NRICH gets involved in the conversational side of mathematics, with problems which naturally lead to discussion and sharing of ideas and methods. Which forms of communication and presentation of the mathematics will prove to be the most effective? Through discussion and explanation you will refine your thinking and perhaps make connections with other areas of mathematics.

## Problems

### Times of Day

##### Stage: 1 Challenge Level:

These pictures show some different activities that you may get up to during a day. What order would you do them in?

### Dienes' Logiblocs

##### Stage: 1 Challenge Level:

This problem focuses on Dienes' Logiblocs. What is the same and what is different about these pairs of shapes? Can you describe the shapes in the picture?

### Halving

##### Stage: 1 Challenge Level:

These pictures show squares split into halves. Can you find other ways?

### Tricky Track

##### Stage: 2 Challenge Level:

In this game you throw two dice and find their total, then move the appropriate counter to the right. Which counter reaches the purple box first? Is this what you would expect?

### A Square of Numbers

##### Stage: 2 Challenge Level:

Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?

### Coded Hundred Square

##### Stage: 2 Challenge Level:

This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?

### Sets of Numbers

##### Stage: 2 Challenge Level:

How many different sets of numbers with at least four members can you find in the numbers in this box?

### Buckets of Thinking

##### Stage: 2 Challenge Level:

There are three buckets each of which holds a maximum of 5 litres. Use the clues to work out how much liquid there is in each bucket.

### How Much Can We Spend?

##### Stage: 3 Challenge Level:

A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?

### Sticky Numbers

##### Stage: 3 Challenge Level:

Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?

### Where Can We Visit?

##### Stage: 3 Challenge Level:

Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?

### Chances Are

##### Stage: 4 Challenge Level:

Which of these games would you play to give yourself the best possible chance of winning a prize?

##### Stage: 4 Challenge Level:

Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.

### Carbon Footprints

##### Stage: 4 Challenge Level:

Is it really greener to go on the bus, or to buy local?

### Pythagoras Proofs

##### Stage: 4 Challenge Level:

Can you make sense of these three proofs of Pythagoras' Theorem?

##### Stage: 4 Challenge Level:

Investigate the properties of quadrilaterals which can be drawn with a circle just touching each side and another circle just touching each vertex.

### Model Solutions

##### Stage: 5 Challenge Level:

How do these modelling assumption affect the solutions?

### The Wrong Stats

##### Stage: 5 Challenge Level:

Why MUST these statistical statements probably be at least a little bit wrong?

### Dangerous Driver?

##### Stage: 5 Challenge Level:

Was it possible that this dangerous driving penalty was issued in error?

### Notty Logic

##### Stage: 5 Challenge Level:

Have a go at being mathematically negative, by negating these statements.