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Resources tagged with Manipulating algebraic expressions/formulae similar to Leonardo's Problem:

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Broad Topics > Algebra > Manipulating algebraic expressions/formulae

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Janine's Conjecture

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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. . . .

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Multiplication Square

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Always Perfect

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

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Never Prime

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.

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Sums of Pairs

Stage: 3 and 4 Challenge Level: Challenge Level:2 Challenge Level:2

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?”

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DOTS Division

Stage: 4 Challenge Level: Challenge Level:1

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

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There and Back

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Brian swims at twice the speed that a river is flowing, downstream from one moored boat to another and back again, taking 12 minutes altogether. How long would it have taken him in still water?

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Salinon

Stage: 4 Challenge Level: Challenge Level:1

This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?

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Perfectly Square

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Back to Basics

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

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Algebra from Geometry

Stage: 3 and 4 Challenge Level: Challenge Level:1

Account of an investigation which starts from the area of an annulus and leads to the formula for the difference of two squares.

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Magic Sums and Products

Stage: 3 and 4

How to build your own magic squares.

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Cosines Rule

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement.

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' Tis Whole

Stage: 4 and 5 Challenge Level: Challenge Level:2 Challenge Level:2

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?

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Simplifying Doughnut

Stage: 4 and 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

An algebra task which depends on members of the group noticing the needs of others and responding.

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Algebra Match

Stage: 3 and 4 Challenge Level: Challenge Level:1

A task which depends on members of the group noticing the needs of others and responding.

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Pair Products

Stage: 4 Challenge Level: Challenge Level:1

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

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Your Number Was...

Stage: 3 Challenge Level: Challenge Level:1

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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Robert's Spreadsheet

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Robert noticed some interesting patterns when he highlighted square numbers in a spreadsheet. Can you prove that the patterns will continue?

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Sitting Pretty

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?

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Matchless

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

There is a particular value of x, and a value of y to go with it, which make all five expressions equal in value, can you find that x, y pair ?

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Root to Poly

Stage: 4 Challenge Level: Challenge Level:1

Find the polynomial p(x) with integer coefficients such that one solution of the equation p(x)=0 is $1+\sqrt 2+\sqrt 3$.

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Dating Made Easier

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

If a sum invested gains 10% each year how long before it has doubled its value?

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Perimeter Expressions

Stage: 3 Challenge Level: Challenge Level:1

Create some shapes by combining two or more rectangles. What can you say about the areas and perimeters of the shapes you can make?

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Adding in Rows

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?

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Crossed Ends

Stage: 3 Challenge Level: Challenge Level:1

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

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Nicely Similar

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

If the hypotenuse (base) length is 100cm and if an extra line splits the base into 36cm and 64cm parts, what were the side lengths for the original right-angled triangle?

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Cubes Within Cubes Revisited

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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?

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Partitioning Revisited

Stage: 3 Challenge Level: Challenge Level:1

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

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Consecutive Squares

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?

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Triangles Within Squares

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Can you find a rule which relates triangular numbers to square numbers?

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Chocolate 2010

Stage: 4 Challenge Level: Challenge Level:1

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

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Always Two

Stage: 4 and 5 Challenge Level: Challenge Level:2 Challenge Level:2

Find all the triples of numbers a, b, c such that each one of them plus the product of the other two is always 2.

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Orbiting Billiard Balls

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

What angle is needed for a ball to do a circuit of the billiard table and then pass through its original position?

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Really Mr. Bond

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

115^2 = (110 x 120) + 25, that is 13225 895^2 = (890 x 900) + 25, that is 801025 Can you explain what is happening and generalise?

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The Medieval Octagon

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.

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Unusual Long Division - Square Roots Before Calculators

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

However did we manage before calculators? Is there an efficient way to do a square root if you have to do the work yourself?

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Temperature

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Water freezes at 0°Celsius (32°Fahrenheit) and boils at 100°C (212°Fahrenheit). Is there a temperature at which Celsius and Fahrenheit readings are the same?

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Lap Times

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Two cyclists, practising on a track, pass each other at the starting line and go at constant speeds... Can you find lap times that are such that the cyclists will meet exactly half way round the. . . .


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