# Communicating and Reflecting - February 2009, All Stages

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

### Two-digit Targets

##### Stage: 1 Challenge Level:

You have a set of the digits from 0 – 9. Can you arrange these in the 5 boxes to make two-digit numbers as close to the targets as possible?

### Three Way Mix Up

##### Stage: 1 and 2 Challenge Level:

Jack has nine tiles. He put them together to make a square so that two tiles of the same colour were not beside each other. Can you find another way to do it?

### It's All about 64

##### Stage: 2 Challenge Level:

Write the numbers up to 64 in an interesting way so that the shape they make at the end is interesting, different, more exciting ... than just a square.

### Four-digit Targets

##### Stage: 2 Challenge Level:

You have two sets of the digits 0 – 9. Can you arrange these in the five boxes to make four-digit numbers as close to the target numbers as possible?

### Make 100

##### Stage: 2 Challenge Level:

Find at least one way to put in some operation signs (+ - x ÷) to make these digits come to 100.

### Bean Bags for Bernard's Bag

##### Stage: 2 Challenge Level:

How could you put eight beanbags in the hoops so that there are four in the blue hoop, five in the red and six in the yellow? Can you find all the ways of doing this?

### Ice Cream

##### Stage: 2 Challenge Level:

You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.

### 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.

### Can They Be Equal?

##### Stage: 3 Challenge Level:

Can you find rectangles where the value of the area is the same as the value of the perimeter?

### Days and Dates

##### Stage: 3 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: 3 Challenge Level:

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

### For Richer for Poorer

##### Stage: 3 Challenge Level:

Charlie has moved between countries and the average income of both has increased. How can this be so?

### Russian Cubes

##### Stage: 4 Challenge Level:

I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that?

##### 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?