# Exploring and Noticing Structure

Exploring and Noticing Structure is part of our Working Mathematically collection.

### Summing Consecutive Numbers

##### Age 11 to 14Challenge Level

15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?

### Magic Letters

##### Age 11 to 14Challenge Level

Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?

### Number Pyramids

##### Age 11 to 14Challenge Level

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

### Shifting Times Tables

##### Age 11 to 14Challenge Level

Can you find a way to identify times tables after they have been shifted up or down?

### Tilted Squares

##### Age 11 to 14Challenge Level

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

### Interactive Spinners

##### Age 11 to 14Challenge Level

This interactivity invites you to make conjectures and explore probabilities of outcomes related to two independent events.

### What Numbers Can We Make?

##### Age 11 to 14Challenge Level

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

### Calendar Capers

##### Age 11 to 14Challenge Level

Choose any three by three square of dates on a calendar page...

### Shape Products

##### Age 11 to 14Challenge Level

These eleven shapes each stand for a different number. Can you use the multiplication sums to work out what they are?

### Keep it Simple

##### Age 11 to 14Challenge Level

Can all unit fractions be written as the sum of two unit fractions?

### Have You Got It?

##### Age 11 to 14Challenge Level

Can you explain the strategy for winning this game with any target?

### Odds and Evens

##### Age 11 to 14Challenge Level

Are these games fair? How can you tell?

### Route to Infinity

##### Age 11 to 14Challenge Level

Can you describe this route to infinity? Where will the arrows take you next?

### Seven Squares

##### Age 11 to 14Challenge Level

Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?

### 1 Step 2 Step

##### Age 11 to 14Challenge Level

Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?

### Exploring and Noticing Structure - Short Problems

##### Age 11 to 16

A collection of short Stage 3 and 4 problems on Exploring and Noticing Structure.

##### Age 11 to 16Challenge Level

Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?

### Charlie's Delightful Machine

##### Age 11 to 16Challenge Level

Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?

### Arithmagons

##### Age 11 to 16Challenge Level

Can you find the values at the vertices when you know the values on the edges?

### Marbles in a Box

##### Age 11 to 16Challenge Level

How many winning lines can you make in a three-dimensional version of noughts and crosses?

### Steel Cables

##### Age 14 to 16Challenge Level

Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?

### Pair Products

##### Age 14 to 16Challenge Level

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

### What's Possible?

##### Age 14 to 16Challenge Level

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

### Attractive Tablecloths

##### Age 14 to 16Challenge Level

Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?

### Painted Cube

##### Age 14 to 16Challenge Level

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

### Pick's Theorem

##### Age 14 to 16Challenge Level

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

##### Age 14 to 18Challenge Level

Which of these roads will satisfy a Munchkin builder?

### Vector Journeys

##### Age 14 to 18Challenge Level

Charlie likes to go for walks around a square park, while Alison likes to cut across diagonally. Can you find relationships between the vectors they walk along?

### Simply Graphs

##### Age 16 to 18Challenge Level

Look for the common features in these graphs. Which graphs belong together?

### Polite Numbers

##### Age 16 to 18Challenge Level

A polite number can be written as the sum of two or more consecutive positive integers, for example 8+9+10=27 is a polite number. Can you find some more polite, and impolite, numbers?

### Fix Me or Crush Me

##### Age 16 to 18Challenge Level

Can you make matrices which will fix one lucky vector and crush another to zero?

### Impossible Triangles?

##### Age 16 to 18Challenge Level

Which of these triangular jigsaws are impossible to finish?