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#### Resources tagged with Creating expressions/formulae similar to Reciprocals:

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Broad Topics > Algebra > Creating expressions/formulae

### ' Tis Whole

##### Stage: 4 and 5 Challenge Level:

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?

### Magic Sums and Products

##### Stage: 3 and 4

How to build your own magic squares.

### Sums of Pairs

##### Stage: 3 and 4 Challenge Level:

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?”

### Back to Basics

##### Stage: 4 Challenge Level:

Find b where 3723(base 10) = 123(base b).

### Leonardo's Problem

##### Stage: 4 and 5 Challenge Level:

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?

### Pair Products

##### Stage: 4 Challenge Level:

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

### Janine's Conjecture

##### Stage: 4 Challenge Level:

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. . . .

### Pair Squares

##### Stage: 5 Challenge Level:

The sum of any two of the numbers 2, 34 and 47 is a perfect square. Choose three square numbers and find sets of three integers with this property. Generalise to four integers.

### Polynomial Interpolation

##### Stage: 5 Challenge Level:

Can you fit polynomials through these points?

### Inside Outside

##### Stage: 4 Challenge Level:

Balance the bar with the three weight on the inside.

### Balance Point

##### Stage: 4 Challenge Level:

Attach weights of 1, 2, 4, and 8 units to the four attachment points on the bar. Move the bar from side to side until you find a balance point. Is it possible to predict that position?

### Number Rules - OK

##### Stage: 4 Challenge Level:

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 number...

### Terminology

##### Stage: 4 Challenge Level:

Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?

### Perfectly Square

##### Stage: 4 Challenge Level:

The sums of the squares of three related numbers is also a perfect square - can you explain why?

### Pythagoras Proofs

##### Stage: 4 Challenge Level:

Can you make sense of these three proofs of Pythagoras' Theorem?

### And So on - and on -and On

##### Stage: 5 Challenge Level:

Can you find the value of this function involving algebraic fractions for x=2000?

### More Number Pyramids

##### Stage: 3 and 4 Challenge Level:

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

### Unit Interval

##### Stage: 4 and 5 Challenge Level:

Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?

### AMGM

##### Stage: 4 Challenge Level:

Can you use the diagram to prove the AM-GM inequality?

### 2-digit Square

##### Stage: 4 Challenge Level:

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?

### Mediant

##### Stage: 4 Challenge Level:

If you take two tests and get a marks out of a maximum b in the first and c marks out of d in the second, does the mediant (a+c)/(b+d)lie between the results for the two tests separately.

### Fair Shares?

##### Stage: 4 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?

### Lens Angle

##### Stage: 4 Challenge Level:

Find the missing angle between the two secants to the circle when the two angles at the centre subtended by the arcs created by the intersections of the secants and the circle are 50 and 120 degrees.

### Archimedes and Numerical Roots

##### Stage: 4 Challenge Level:

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?

### Hike and Hitch

##### Stage: 4 Challenge Level:

Fifteen students had to travel 60 miles. They could use a car, which could only carry 5 students. As the car left with the first 5 (at 40 miles per hour), the remaining 10 commenced hiking along the. . . .

### Look Before You Leap

##### Stage: 5 Challenge Level:

Relate these algebraic expressions to geometrical diagrams.

### What's Possible?

##### Stage: 4 Challenge Level:

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

### Plum Tree

##### Stage: 4 and 5 Challenge Level:

Label this plum tree graph to make it totally magic!

### Steel Cables

##### Stage: 4 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?

### Generating Triples

##### Stage: 4 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?

### Partly Painted Cube

##### Stage: 4 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?

### Little and Large

##### Stage: 5 Challenge Level:

A point moves around inside a rectangle. What are the least and the greatest values of the sum of the squares of the distances from the vertices?

### Pareq Calc

##### Stage: 4 Challenge Level:

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. . . .

### Circles in Circles

##### Stage: 5 Challenge Level:

This pattern of six circles contains three unit circles. Work out the radii of the other three circles and the relationship between them.

### Square Pizza

##### Stage: 4 Challenge Level:

Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal cuts through any point inside the square?

### W Mates

##### Stage: 5 Challenge Level:

Show there are exactly 12 magic labellings of the Magic W using the numbers 1 to 9. Prove that for every labelling with a magic total T there is a corresponding labelling with a magic total 30-T.

### Areas of Parallelograms

##### Stage: 4 Challenge Level:

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

### Chocolate 2010

##### Stage: 4 Challenge Level:

First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...

### Reasonable Algebra

##### Stage: 4 Challenge Level:

Use algebra to reason why 16 and 32 are impossible to create as the sum of consecutive numbers.

### Just Touching

##### Stage: 5 Challenge Level:

Three semi-circles have a common diameter, each touches the other two and two lie inside the biggest one. What is the radius of the circle that touches all three semi-circles?

### Days and Dates

##### Stage: 4 Challenge Level:

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

### Old Nuts

##### Stage: 5 Challenge Level:

In turn 4 people throw away three nuts from a pile and hide a quarter of the remainder finally leaving a multiple of 4 nuts. How many nuts were at the start?

### Marbles in a Box

##### Stage: 3 and 4 Challenge Level:

In a three-dimensional version of noughts and crosses, how many winning lines can you make?

### Seven Squares

##### Stage: 3 and 4 Challenge Level:

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

### Semi-square

##### Stage: 4 Challenge Level:

What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?

### A Tilted Square

##### Stage: 4 Challenge Level:

The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?

### Screen Shot

##### Stage: 4 Challenge Level:

A moveable screen slides along a mirrored corridor towards a centrally placed light source. A ray of light from that source is directed towards a wall of the corridor, which it strikes at 45 degrees. . . .

### Triangles Within Pentagons

##### Stage: 4 Challenge Level:

Show that all pentagonal numbers are one third of a triangular number.

### Around and Back

##### Stage: 4 Challenge Level:

A cyclist and a runner start off simultaneously around a race track each going at a constant speed. The cyclist goes all the way around and then catches up with the runner. He then instantly turns. . . .

### Three Four Five

##### Stage: 4 Challenge Level:

Two semi-circles (each of radius 1/2) touch each other, and a semi-circle of radius 1 touches both of them. Find the radius of the circle which touches all three semi-circles.