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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.
Use five steps to count forwards or backwards in 1s or 10s to get to 50. What strategies did you use?
Florence, Ethan and Alma have each added together two 'next-door' numbers. What is the same about their answers?
What do you see as you watch this video? Can you create a similar video for the number 12?
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?
Investigate which numbers make these lights come on. What is the smallest number you can find that lights up all the lights?
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?
Choose four of the numbers from 1 to 9 to put in the squares so that the differences between joined squares are odd.
Explore ways of colouring this set of triangles. Can you make symmetrical patterns?
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?
Follow the clues to find the mystery number.
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.
Try out this number trick. What happens with different starting numbers? What do you notice?
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?
Watch this animation. What do you see? Can you explain why this happens?
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?
How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.
These clocks have only one hand, but can you work out what time they are showing from the information?
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?
Start with a triangle. Can you cut it up to make a rectangle?
There are nasty versions of this dice game but we'll start with the nice ones...
Think of a number and follow the machine's instructions... I know what your number is! Can you explain how I know?
You'll need to know your number properties to win a game of Statement Snap...
The Number Jumbler can always work out your chosen symbol. Can you work out how?
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
Try out some calculations. Are you surprised by the results?
Take three consecutive numbers and add them together. What do you notice?
In this interactivity each fruit has a hidden value. Can you deduce what each one is worth?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
Who said that adding, subtracting, multiplying and dividing couldn't be fun?
Can you find any two-digit numbers that satisfy all of these statements?
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
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?
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?
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?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
Can you find a way to identify times tables after they have been shifted up or down?
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?
Which set of numbers that add to 100 have the largest product?
What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?
Where should you start, if you want to finish back where you started?
Play around with sets of five numbers and see what you can discover about different types of average...
Can you make a right-angled triangle on this peg-board by joining up three points round the edge?
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?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Seven balls are shaken. You win if the two blue balls end up touching. What is the probability of winning?
Can you work out what step size to take to ensure you visit all the dots on the circle?
Alison and Charlie are playing a game. Charlie wants to go first so Alison lets him. Was that such a good idea?
Six balls are shaken. You win if at least one red ball ends in a corner. What is the probability of winning?
Which countries have the most naturally athletic populations?
How well can you estimate 10 seconds? Investigate with our timing tool.
Can you recreate squares and rhombuses if you are only given a side or a diagonal?
Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?
If you move the tiles around, can you make squares with different coloured edges?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
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?
Imagine you were given the chance to win some money... and imagine you had nothing to lose...
An aluminium can contains 330 ml of cola. If the can's diameter is 6 cm what is the can's height?
Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?
Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?
Can you find the values at the vertices when you know the values on the edges?
There are lots of different methods to find out what the shapes are worth - how many can you find?
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?
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?
How many winning lines can you make in a three-dimensional version of noughts and crosses?
What's the largest volume of box you can make from a square of paper?
Can you do a little mathematical detective work to figure out which number has been wiped out?
Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.
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?
Imagine a room full of people who keep flipping coins until they get a tail. Will anyone get six heads in a row?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
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.
Chris is enjoying a swim but needs to get back for lunch. How far along the bank should she land?
Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?
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.
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".
You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?
The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?
An equilateral triangle rotates around regular polygons and produces an outline like a flower. What are the perimeters of the different flowers?
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?
If everyone in your class picked a number from 1 to 225, do you think any two people would pick the same number?
Can you work out the probability of winning the Mathsland National Lottery?
A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?
Explore the relationships between different paper sizes.
Can you work out which spinners were used to generate the frequency charts?
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?
Use your skill and judgement to match the sets of random data.
There are many different methods to solve this geometrical problem - how many can you find?