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# Conjecturing and Generalising - Secondary Students

### More Less Is More

### Consecutive Numbers

### Subtraction Surprise

### Nice or Nasty

### The Remainders Game

### Method in Multiplying Madness?

### Remainders

### Number Lines in Disguise

### Got It for Two

### Factors and Multiples Game

### Elevenses

### Beach Huts

### Frogs

### Magic Letters

### Keep it Simple

### Mirror, Mirror...

### Picturing Triangular Numbers

### Can They Be Equal?

### Have You Got It?

### What Numbers Can We Make?

### Picturing Square Numbers

### Forwards Add Backwards

### Largest Product

### Tilted Squares

### Dicey Operations

### Interactive Spinners

### Summing Consecutive Numbers

### How Much Can We Spend?

### Number Pyramids

### Your Number Was...

### Crossed Ends

### Strange Bank Account

### Go Forth and Generalise

### Diminishing Returns

### Spaces for Exploration

### Shear Magic

### Farey Sequences

### Seven Squares - Group-worthy Task

### Seven Squares

### Coordinate Patterns

### ...on the Wall

### What Numbers Can We Make Now?

### Counting Factors

### Where Can We Visit?

### Power Mad!

### Same Answer

### Squares in Rectangles

### Tower of Hanoi

### Terminating or Not

### 1 Step 2 Step

### Route to Infinity

### More Number Pyramids

### Fibonacci Surprises

### Legs Eleven

### Stars

### Egyptian Fractions

### Differences

### Consecutive Negative Numbers

### Who is the fairest of them all ?

### Litov's Mean Value Theorem

### How Many Miles to Go?

### Conjecturing and Generalising - Short Problems

### Train Spotters' Paradise

### Arithmagons

### Charlie's Delightful MachineLive

### More Twisting and Turning

### Cuboid Challenge

### Take Three from Five

### Searching for Mean(ing)

### What Does it All Add up To?

### Speeding Boats

### Beelines

### Which Is Cheaper?

### Hollow Squares

### How Old Am I?

### A Little Light Thinking

### Pair Products

### Generating Triples

### Steel Cables

### Areas of Parallelograms

### Expenses

### Mystic Rose

### Square Number Surprises

### What's Possible?

### Which Is Bigger?

### Multiplication Square

### Partly Painted Cube

### Why 24?

### Multiplication Arithmagons

### For Richer for Poorer

### Difference of Two Squares

### Plus Minus

### Painted Cube

### Attractive Tablecloths

### Of All the Areas

### Pick's TheoremLive

### Fair Shares?

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*Conjecturing and Generalising is part of our Thinking Mathematically collection.*

Age 7 to 14

Challenge Level

In each of these games, you will need a little bit of luck, and your knowledge of place value to develop a winning strategy.

Age 7 to 14

Challenge Level

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

Age 7 to 14

Challenge Level

Try out some calculations. Are you surprised by the results?

Age 7 to 14

Challenge Level

There are nasty versions of this dice game but we'll start with the nice ones...

Age 7 to 14

Challenge Level

Play this game and see if you can figure out the computer's chosen number.

Age 7 to 14

Challenge Level

Watch our videos of multiplication methods that you may not have met before. Can you make sense of them?

Age 7 to 14

Challenge Level

I'm thinking of a number. My number is both a multiple of 5 and a multiple of 6. What could my number be?

Age 7 to 14

Challenge Level

Some of the numbers have fallen off Becky's number line. Can you figure out what they were?

Age 7 to 14

Challenge Level

Got It game for an adult and child. How can you play so that you know you will always win?

Age 7 to 16

Challenge Level

A game in which players take it in turns to choose a number. Can you block your opponent?

Age 11 to 14

Challenge Level

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

Age 11 to 14

Challenge Level

Can you figure out how sequences of beach huts are generated?

Age 11 to 14

Challenge Level

How many moves does it take to swap over some red and blue frogs? Do you have a method?

Age 11 to 14

Challenge Level

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

Age 11 to 14

Challenge Level

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

Age 11 to 14

Challenge Level

Explore the effect of reflecting in two parallel mirror lines.

Age 11 to 14

Challenge Level

Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

Age 11 to 14

Challenge Level

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

Age 11 to 14

Challenge Level

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

Age 11 to 14

Challenge Level

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

Age 11 to 14

Challenge Level

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

Age 11 to 14

Challenge Level

What happens when you add a three digit number to its reverse?

Age 11 to 14

Challenge Level

Which set of numbers that add to 100 have the largest product?

Age 11 to 14

Challenge Level

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

Age 11 to 14

Challenge Level

Who said that adding, subtracting, multiplying and dividing couldn't be fun?

Age 11 to 14

Challenge Level

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

Age 11 to 14

Challenge 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?

Age 11 to 14

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?

Age 11 to 14

Challenge Level

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

Age 11 to 14

Challenge Level

Think of a number and follow my instructions. Tell me your answer, and I'll tell you what you started with! Can you explain how I know?

Age 11 to 14

Challenge Level

Crosses can be drawn on number grids of various sizes. What do you notice when you add opposite ends?

Age 11 to 14

Challenge Level

Imagine a very strange bank account where you are only allowed to do two things...

Age 11 to 14

Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.

Age 11 to 14

Challenge Level

How much of the square is coloured blue? How will the pattern continue?

Age 11 to 14

Alf Coles writes about how he tries to create 'spaces for exploration' for the students in his classrooms.

Age 11 to 14

Challenge Level

Explore the area of families of parallelograms and triangles. Can you find rules to work out the areas?

Age 11 to 14

Challenge Level

There are lots of ideas to explore in these sequences of ordered fractions.

Age 11 to 14

Challenge Level

Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?

Age 11 to 14

Challenge Level

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

Age 11 to 14

Challenge Level

Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?

Age 11 to 14

Challenge Level

Explore the effect of reflecting in two intersecting mirror lines.

Age 11 to 14

Challenge Level

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

Age 11 to 14

Challenge Level

Is there an efficient way to work out how many factors a large number has?

Age 11 to 14

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?

Age 11 to 14

Challenge Level

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

Age 11 to 14

Challenge Level

Aisha's division and subtraction calculations both gave the same answer! Can you find some more examples?

Age 11 to 14

Challenge Level

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

Age 11 to 14

Challenge Level

The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.

Age 11 to 14

Challenge Level

Is there a quick way to work out whether a fraction terminates or recurs when you write it as a decimal?

Age 11 to 14

Challenge 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?

Age 11 to 14

Challenge Level

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

Age 11 to 14

Challenge Level

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

Age 11 to 14

Challenge Level

Play around with the Fibonacci sequence and discover some surprising results!

Age 11 to 14

Challenge Level

Take any four digit number. Move the first digit to the end and move the rest along. Now add your two numbers. Did you get a multiple of 11?

Age 11 to 14

Challenge Level

Can you work out what step size to take to ensure you visit all the dots on the circle?

Age 11 to 14

Challenge Level

The Egyptians expressed all fractions as the sum of different unit fractions. Here is a chance to explore how they could have written different fractions.

Age 11 to 14

Challenge Level

Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number?

Age 11 to 14

Challenge Level

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

Age 11 to 14

Challenge Level

Explore the effect of combining enlargements.

Age 11 to 14

Challenge Level

Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...

Age 11 to 14

Challenge Level

How many more miles must the car travel before the numbers on the milometer and the trip meter contain the same digits in the same order?

Age 11 to 16

A collection of short Stage 3 and 4 problems on Conjecturing and Generalising

Age 11 to 16

Dave Hewitt suggests that there might be more to mathematics than looking at numerical results, finding patterns and generalising.

Age 11 to 16

Challenge Level

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

Age 11 to 16

Challenge Level

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

Age 11 to 16

Challenge Level

It would be nice to have a strategy for disentangling any tangled ropes...

Age 11 to 16

Challenge Level

What's the largest volume of box you can make from a square of paper?

Age 11 to 16

Challenge Level

Caroline and James pick sets of five numbers. Charlie tries to find three that add together to make a multiple of three. Can they stop him?

Age 11 to 16

Challenge Level

If you have a large supply of 3kg and 8kg weights, how many of each would you need for the average (mean) of the weights to be 6kg?

Age 11 to 18

Challenge Level

If you take four consecutive numbers and add them together, the answer will always be even. What else do you notice?

Age 14 to 16

Challenge Level

Two boats travel up and down a lake. Can you picture where they will cross if you know how fast each boat is travelling?

Age 14 to 16

Challenge Level

Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?

Age 14 to 16

Challenge Level

When I park my car in Mathstown, there are two car parks to choose from. Can you help me to decide which one to use?

Age 14 to 16

Challenge Level

Which armies can be arranged in hollow square fighting formations?

Age 14 to 16

Challenge Level

In 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays?

Age 14 to 16

Challenge Level

Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?

Age 14 to 16

Challenge Level

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

Age 14 to 16

Challenge Level

Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?

Age 14 to 16

Challenge Level

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

Age 14 to 16

Challenge Level

Can you find the area of a parallelogram defined by two vectors?

Age 14 to 16

Challenge Level

What is the largest number which, when divided into 1905, 2587, 3951, 7020 and 8725 in turn, leaves the same remainder each time?

Age 14 to 16

Challenge Level

Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.

Age 14 to 16

Challenge Level

There are unexpected discoveries to be made about square numbers...

Age 14 to 16

Challenge Level

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

Age 14 to 16

Challenge Level

Which is bigger, n+10 or 2n+3? Can you find a good method of answering similar questions?

Age 14 to 16

Challenge Level

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

Age 14 to 16

Challenge Level

Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?

Age 14 to 16

Challenge Level

Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.

Age 14 to 16

Challenge Level

Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?

Age 14 to 16

Challenge Level

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

Age 14 to 16

Challenge Level

What is special about the difference between squares of numbers adjacent to multiples of three?

Age 14 to 16

Challenge Level

Can you explain the surprising results Jo found when she calculated the difference between square numbers?

Age 14 to 16

Challenge 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?

Age 14 to 16

Challenge 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?

Age 14 to 16

Challenge Level

Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?

Age 14 to 16

Challenge 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 16

Challenge Level

A mother wants to share a sum of money by giving each of her children in turn a lump sum plus a fraction of the remainder. How can she do this in order to share the money out equally?