Resources tagged with: Creating and manipulating expressions and formulae

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Broad Topics > Algebraic expressions, equations and formulae > Creating and manipulating expressions and formulae

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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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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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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?

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Christmas Chocolates

Age 11 to 14 Challenge Level:

How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

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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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Painted Cube

Age 14 to 16 Challenge Level:

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

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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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AMGM

Age 14 to 16 Challenge Level:

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

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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?

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Partitioning Revisited

Age 11 to 14 Challenge Level:

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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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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Triangles Within Triangles

Age 14 to 16 Challenge Level:

Can you find a rule which connects consecutive triangular numbers?

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Triangles Within Squares

Age 14 to 16 Challenge Level:

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

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Cubes Within Cubes Revisited

Age 11 to 14 Challenge Level:

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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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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Triangles Within Pentagons

Age 14 to 16 Challenge Level:

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

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Steel Cables

Age 14 to 16 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?

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Boxed In

Age 11 to 14 Challenge Level:

A box has faces with areas 3, 12 and 25 square centimetres. What is the volume of the box?

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Seven Squares

Age 11 to 14 Challenge Level:

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

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Diophantine N-tuples

Age 14 to 16 Challenge Level:

Can you explain why a sequence of operations always gives you perfect squares?

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Special Sums and Products

Age 11 to 14 Challenge Level:

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.

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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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DOTS Division

Age 14 to 16 Challenge Level:

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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' Tis Whole

Age 14 to 18 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?

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Sums of Pairs

Age 11 to 16 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?”

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Robert's Spreadsheet

Age 14 to 16 Challenge Level:

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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Multiplication Square

Age 14 to 16 Challenge Level:

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

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Leonardo's Problem

Age 14 to 18 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?

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Magic W

Age 14 to 16 Challenge Level:

Find all the ways of placing the numbers 1 to 9 on a W shape, with 3 numbers on each leg, so that each set of 3 numbers has the same total.

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One and Three

Age 14 to 16 Challenge Level:

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. . . .

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Plum Tree

Age 14 to 18 Challenge Level:

Label this plum tree graph to make it totally magic!

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Always a Multiple?

Age 11 to 14 Challenge Level:

Think of a two digit number, reverse the digits, and add the numbers together. Something special happens...

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Regular Hexagon Loops

Age 11 to 14 Challenge Level:

Make some loops out of regular hexagons. What rules can you discover?

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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?

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Around and Back

Age 14 to 16 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. . . .

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Adding in Rows

Age 11 to 14 Challenge Level:

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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Janine's Conjecture

Age 14 to 16 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. . . .

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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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Magic Squares for Special Occasions

Age 11 to 16

This article explains how to make your own magic square to mark a special occasion with the special date of your choice on the top line.

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Unit Interval

Age 14 to 18 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?

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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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Crossed Ends

Age 11 to 14 Challenge Level:

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

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Mediant Madness

Age 14 to 16 Challenge Level:

Kyle and his teacher disagree about his test score - who is right?

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Pythagoras Proofs

Age 14 to 16 Challenge Level:

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

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Chocolate Maths

Age 11 to 14 Challenge Level:

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. . . .

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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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Terminology

Age 14 to 16 Challenge Level:

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

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Attractive Tablecloths

Age 14 to 16 Challenge Level:

Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?

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Top-heavy Pyramids

Age 11 to 14 Challenge Level:

Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.

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Perfectly Square

Age 14 to 16 Challenge Level:

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