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?
A 2-Digit number is squared. When this 2-digit number is reversed
and squared, the difference between the squares is also a square.
What is the 2-digit number?
Take a few whole numbers away from a triangle number. If you know
the mean of the remaining numbers can you find the triangle number
and which numbers were removed?
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
How to build your own magic squares.
The number 27 is special because it is three times the sum of its digits 27 = 3 (2 + 7). Find some two digit numbers that are SEVEN times the sum of their digits (seven-up numbers)?
The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for. . . .
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 box has faces with areas 3, 12 and 25 square centimetres. What is
the volume of the box?
A job needs three men but in fact six people do it. When it is
finished they are all paid the same. How much was paid in total,
and much does each man get if the money is shared as Fred suggests?
Triangle ABC is isosceles while triangle DEF is equilateral. Find
one angle in terms of the other two.
Think of a number... follow the machine's instructions. I know what
your number is! Can you explain how I know?
The well known Fibonacci sequence is 1 ,1, 2, 3, 5, 8, 13, 21....
How many Fibonacci sequences can you find containing the number 196
as one of the terms?
Find some examples of pairs of numbers such that their sum is a
factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and
16 is a factor of 48.
A car's milometer reads 4631 miles and the trip meter has 173.3 on
it. How many more miles must the car travel before the two numbers
contain the same digits in the same order?
We can show that (x + 1)² = x² + 2x + 1 by considering
the area of an (x + 1) by (x + 1) square. Show in a similar way
that (x + 2)² = x² + 4x + 4
Imagine starting with one yellow cube and covering it all over with
a single layer of red cubes, and then covering that cube with a
layer of blue cubes. How many red and blue cubes would you need?
Can you convince me of each of the following: If a square number is
multiplied by a square number the product is ALWAYS a square
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number.
Cross out the numbers on the same row and column. Repeat this
process. Add up you four numbers. Why do they always add up to 34?
Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?
Can you find the area of a parallelogram defined by two vectors?
Jo has three numbers which she adds together in pairs. When she
does this she has three different totals: 11, 17 and 22 What are
the three numbers Jo had to start with?”
A, B & C own a half, a third and a sixth of a coin collection.
Each grab some coins, return some, then share equally what they had
put back, finishing with their own share. How rich are they?
Crosses can be drawn on number grids of various sizes. What do you notice when you add opposite ends?
If the sides of the triangle in the diagram are 3, 4 and 5, what is
the area of the shaded square?
Janine noticed, while studying some cube numbers, that if you take
three consecutive whole numbers and multiply them together and then
add the middle number of the three, you get the middle number. . . .
A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .
What would you get if you continued this sequence of fraction sums?
1/2 + 2/1 =
2/3 + 3/2 =
3/4 + 4/3 =
A circle is inscribed in a triangle which has side lengths of 8, 15
and 17 cm. What is the radius of the circle?
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
Think of a two digit number, reverse the digits, and add the numbers together. Something special happens...
Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .
Semicircles are drawn on the sides of a rectangle ABCD. A circle passing through points ABCD carves out four crescent-shaped regions. Prove that the sum of the areas of the four crescents is equal to. . . .
Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten.
Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Many numbers can be expressed as the difference of two perfect
squares. What do you notice about the numbers you CANNOT make?
Find b where 3723(base 10) = 123(base b).
Two motorboats travelling up and down a lake at constant speeds
leave opposite ends A and B at the same instant, passing each
other, for the first time 600 metres from A, and on their return,
400. . . .
32 x 38 = 30 x 40 + 2 x 8; 34 x 36 = 30 x 40 + 4 x 6; 56 x 54 = 50
x 60 + 6 x 4; 73 x 77 = 70 x 80 + 3 x 7 Verify and generalise if
The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Use algebra to reason why 16 and 32 are impossible to create as the
sum of consecutive numbers.
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?
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?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?