Exploring and Noticing Structure
Consecutive Numbers
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
Calendar Capers
Choose any three by three square of dates on a calendar page...
Have You Got It?
Can you explain the strategy for winning this game with any target?
Summing Consecutive Numbers
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?
What's Possible?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
Marbles in a Box
How many winning lines can you make in a three-dimensional version of noughts and crosses?
Attractive Tablecloths
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?
1 Step 2 Step
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?
Pick's Theorem
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.
Pair Products
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Number Pyramids
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Tilted Squares
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
Painted Cube
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?
Arithmagons
Can you find the values at the vertices when you know the values on the edges?
Route to Infinity
Can you describe this route to infinity? Where will the arrows take you next?
Impossible Triangles?
Which of these triangular jigsaws are impossible to finish?
Interactive Spinners
This interactivity invites you to make conjectures and explore probabilities of outcomes related to two independent events.
Keep it Simple
Can all unit fractions be written as the sum of two unit fractions?
Cyclic Quadrilaterals
Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?
How Much Can We Spend?
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?
Shifting Times Tables
Can you find a way to identify times tables after they have been shifted up or down?
Charlie's Delightful Machine
Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?
What Numbers Can We Make?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
Vector Journeys
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?
Steel Cables
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
Magic Letters
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
Seven Squares
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
Exploring and Noticing Structure - Short Problems
A collection of short Stage 3 and 4 problems on Exploring and Noticing Structure.