Being Curious - Secondary Teachers

Being Curious is part of our Developing Mathematical Habits of Mind collection.

In Nurturing Students' Curiosity, we offer you support and advice on how to encourage your students to be curious mathematicians.

All humans are naturally curious, and good mathematicians get excited by new ideas and are keen to explore and investigate them. As teachers, we want to nurture our students' mathematical curiosity so they grow into creative, flexible problem-solvers. One way to nurture this curiosity is by providing the right hook to draw students in.

We hope that the problems below will exploit students' natural curiosity and provoke them to ask good mathematical questions.

You can browse through the Number, Algebra, Geometry or Statistics collections, or scroll down to see the full set of problems below.

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Being Curious - Number

Challenge level

Number problems for inquiring students.

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Being Curious - Algebra

Challenge level

Algebra problems for inquiring students.

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Being Curious - Geometry

Challenge level

Geometry problems for inquiring students.

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Being Curious - Statistics

Challenge level

Statistics problems for inquiring students.

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Next-door numbers

Age

5 to 7

Challenge level

Florence, Ethan and Alma have each added together two 'next-door' numbers. What is the same about their answers?

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Favourite

Eightness of eight

Age

5 to 7

Challenge level

What do you see as you watch this video? Can you create a similar video for the number 12?

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Favourite

Digit addition

Age

5 to 11

Challenge level

Try out this number trick. What happens with different starting numbers? What do you notice?

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Favourite

Number detective

Age

5 to 11

Challenge level

Follow the clues to find the mystery number.

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Favourite

Shaping it

Age

5 to 11

Challenge level

These pictures were made by starting with a square, finding the half-way point on each side and joining those points up. You could investigate your own starting shape.

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Favourite

If the world were a village

Age

5 to 11

Challenge level

This activity is based on data in the book 'If the World Were a Village'. How will you represent your chosen data for maximum effect?

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Favourite

Five steps to 50

Age

5 to 7

Challenge level

Use five steps to count forwards or backwards in 1s or 10s to get to 50. What strategies did you use?

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Light the lights

Age

5 to 7

Challenge level

Investigate which numbers make these lights come on. What is the smallest number you can find that lights up all the lights?

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Favourite

Ring a ring of numbers

Age

5 to 7

Challenge level

Choose four of the numbers from 1 to 9 to put in the squares so that the differences between joined squares are odd.

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Favourite

Colouring triangles

Age

5 to 7

Challenge level

Explore ways of colouring this set of triangles. Can you make symmetrical patterns?

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Chain of changes

Age

5 to 7

Challenge level

Arrange the shapes in a line so that you change either colour or shape in the next piece along. Can you find several ways to start with a blue triangle and end with a red circle?

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Favourite

Little man

Age

5 to 7

Challenge level

The Man is much smaller than us. Can you use the picture of him next to a mug to estimate his height and how much tea he drinks?

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Favourite

More numbers in the ring

Age

5 to 7

Challenge level

If there are 3 squares in the ring, can you place three different numbers in them so that their differences are odd? Try with different numbers of squares around the ring. What do you notice?

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Favourite

Statement snap

Age

7 to 14

Challenge level

You'll need to know your number properties to win a game of Statement Snap...

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Favourite

Tumbling down

Age

7 to 11

Challenge level

Watch this animation. What do you see? Can you explain why this happens?

game

Favourite

Dicey operations

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

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Triangle transformation

Age

7 to 14

Challenge level

Start with a triangle. Can you cut it up to make a rectangle?

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Favourite

Consecutive numbers

Age

7 to 14

Challenge level

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

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Favourite

Fruity totals

Age

7 to 16

Challenge level

In this interactivity each fruit has a hidden value. Can you deduce what each one is worth?

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Favourite

Number differences

Age

7 to 11

Challenge level

Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?

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Favourite

The Number Jumbler

Age

7 to 14

Challenge level

The Number Jumbler can always work out your chosen symbol. Can you work out how?

Age

7 to 11

Challenge level

Can you order the digits from 1-3 to make a number which is divisible by 3 so when the last digit is removed it becomes a 2-figure number divisible by 2, and so on?

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Satisfying statements

Age

11 to 14

Challenge level

Can you find any two-digit numbers that satisfy all of these statements?

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Favourite

Charlie's delightful machine

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?

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Favourite

Arithmagons

Age

11 to 16

Challenge level

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

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Favourite

Your number was...

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?

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Favourite

Summing consecutive numbers

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?

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Favourite

Semi-regular tessellations

Age

11 to 16

Challenge level

Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?

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Favourite

Blue and white

Age

11 to 14

Challenge level

Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?

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Favourite

What numbers can we make?

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?

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Favourite

What's it worth?

Age

11 to 16

Challenge level

There are lots of different methods to find out what the shapes are worth - how many can you find?

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Favourite

Largest product

Age

11 to 14

Challenge level

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

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Favourite

Elevenses

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?

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Favourite

Can they be equal?

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?

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Favourite

Perimeter possibilities

Age

11 to 14

Challenge level

I'm thinking of a rectangle with an area of 24. What could its perimeter be?

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Favourite

Special numbers

Age

11 to 14

Challenge level

My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?

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Favourite

Number pyramids

Age

11 to 14

Challenge level

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

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Favourite

How much can we spend?

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?

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Favourite

Shifting times tables

Age

11 to 14

Challenge level

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

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Favourite

Tilted squares

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?

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Favourite

On the edge

Age

11 to 14

Challenge level

If you move the tiles around, can you make squares with different coloured edges?

problem

Favourite

Sending a parcel

Age

11 to 14

Challenge level

What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?

problem

Favourite

Square coordinates

Age

11 to 14

Challenge level

A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides?

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Stars

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?

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Favourite

Right angles

Age

11 to 14

Challenge level

Can you make a right-angled triangle on this peg-board by joining up three points round the edge?

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Favourite

Two's company

Age

11 to 14

Challenge level

Seven balls are shaken. You win if the two blue balls end up touching. What is the probability of winning?

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Favourite

Cosy corner

Age

11 to 14

Challenge level

Six balls are shaken. You win if at least one red ball ends in a corner. What is the probability of winning?

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Favourite

Who's the best?

Age

11 to 14

Challenge level

Which countries have the most naturally athletic populations?

problem

Favourite

Opposite vertices

Age

11 to 14

Challenge level

Can you recreate squares and rhombuses if you are only given a side or a diagonal?

problem

Favourite

Marbles in a box

Age

11 to 16

Challenge level

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

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Favourite

Think of two numbers

Age

11 to 14

Challenge level

Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?

problem

Favourite

Non-transitive dice

Age

11 to 14

Challenge level

Alison and Charlie are playing a game. Charlie wants to go first so Alison lets him. Was that such a good idea?

problem

Favourite

Take three from five

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?

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Favourite

Searching for mean(ing)

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?

problem

Favourite

What numbers can we make now?

Age

11 to 14

Challenge level

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

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Favourite

Cuboid challenge

Age

11 to 16

Challenge level

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

problem

Favourite

Estimating time

Age

11 to 14

Challenge level

How well can you estimate 10 seconds? Investigate with our timing tool.

problem

Favourite

Wipeout

Age

11 to 16

Challenge level

Can you do a little mathematical detective work to figure out which number has been wiped out?

problem

Favourite

More number pyramids

Age

11 to 14

Challenge level

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

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Favourite

Reversals

Age

11 to 14

Challenge level

Where should you start, if you want to finish back where you started?

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Favourite

Unequal averages

Age

11 to 14

Challenge level

Play around with sets of five numbers and see what you can discover about different types of average...

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Favourite

Which solids can we make?

Age

11 to 14

Challenge level

Interior angles can help us to work out which polygons will tessellate. Can we use similar ideas to predict which polygons combine to create semi-regular solids?

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Favourite

Litov's mean value theorem

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...

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Favourite

A chance to win?

Age

11 to 14

Challenge level

Imagine you were given the chance to win some money... and imagine you had nothing to lose...

problem

Favourite

Cola can

Age

11 to 14

Challenge level

An aluminium can contains 330 ml of cola. If the can's diameter is 6 cm what is the can's height?

problem

Favourite

A little light thinking

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?

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Curve hunter

Age

14 to 18

Challenge level

This problem challenges you to sketch curves with different properties.

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Favourite

Last one standing

Age

14 to 16

Challenge level

Imagine a room full of people who keep flipping coins until they get a tail. Will anyone get six heads in a row?

problem

Favourite

How old am I?

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?

problem

Favourite

Beelines

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?

problem

Favourite

Vector journeys

Age

14 to 18

Challenge 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?

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Data matching

Age

14 to 18

Challenge level

Use your skill and judgement to match the sets of random data.

problem

Favourite

Which spinners?

Age

14 to 18

Challenge level

Can you work out which spinners were used to generate the frequency charts?

problem

Favourite

Curvy areas

Age

14 to 16

Challenge level

Have a go at creating these images based on circles. What do you notice about the areas of the different sections?

problem

Favourite

Pair products

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?

problem

Favourite

Circles in quadrilaterals

Age

14 to 16

Challenge level

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

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Favourite

Where to land

Age

14 to 16

Challenge level

Chris is enjoying a swim but needs to get back for lunch. How far along the bank should she land?

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Favourite

Same number!

Age

14 to 16

Challenge level

If everyone in your class picked a number from 1 to 225, do you think any two people would pick the same number?

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Favourite

Triangle midpoints

Age

14 to 16

Challenge level

You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?

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Favourite

What's possible?

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?

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Favourite

Arclets

Age

14 to 16

Challenge level

Each of the following shapes is made from arcs of a circle of radius r. What is the perimeter of a shape with 3, 4, 5 and n "nodes".

problem

Favourite

Trapezium four

Age

14 to 16

Challenge level

The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?

problem

Favourite

Multiplication arithmagons

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?

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Favourite

Pick's theorem

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.

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Favourite

Odds and evens made fair

Age

14 to 16

Challenge level

In this follow-up to the problem Odds and Evens, we invite you to analyse a probability situation in order to find the general solution for a fair game.

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Favourite

Triangles and petals

Age

14 to 16

Challenge level

An equilateral triangle rotates around regular polygons and produces an outline like a flower. What are the perimeters of the different flowers?

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Favourite

Partly painted cube

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?