Interpreting and evaluating - July 2009, All Stages

This month our problems involve aspects of the process of interpreting the information in a question and examining in various ways the solution or methods used to get the solution. For example, are the assumptions and arguments made correct or appropriate? How does the solution relate to the initial problem? Can we make sense of someone else's finding? Can we compare strategies used?

Problems

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Baked Bean Cans

Stage: 1 Challenge Level: Challenge Level:1

Is there a best way to stack cans? What do different supermarkets do? How high can you safely stack the cans?

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Domino Join Up

Stage: 1 Challenge Level: Challenge Level:2 Challenge Level:2

Can you arrange fifteen dominoes so that all the touching domino pieces add to 6 and the ends join up? Can you make all the joins add to 7?

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Hexpentas

Stage: 1 and 2 Challenge Level: Challenge Level:1

How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?

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The Mathemagician's Seven Spells

Stage: 2 Challenge Level: Challenge Level:1

"Tell me the next two numbers in each of these seven minor spells", chanted the Mathemagician, "And the great spell will crumble away!" Can you help Anna and David break the spell?

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Six Is the Sum

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

What do the digits in the number fifteen add up to? How many other numbers have digits with the same total but no zeros?

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Factor Lines

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

Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.

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Cycling Squares

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

Can you make a cycle of pairs that add to make a square number using all the numbers in the box below, once and once only?

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Inside Seven Squares

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

What is the total area of the four outside triangles which are outlined in red in this arrangement of squares inside each other?

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Parallel Lines

Stage: 3 Challenge Level: Challenge Level:1

How does the position of the line affect the equation of the line? What can you say about the equations of parallel lines?

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How Steep Is the Slope?

Stage: 3 Challenge Level: Challenge Level:1

On the grid provided, we can draw lines with different gradients. How many different gradients can you find? Can you arrange them in order of steepness?

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At Right Angles

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

Can you decide whether two lines are perpendicular or not? Can you do this without drawing them?

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Back Fitter

Stage: 4 Challenge Level: Challenge Level:1

10 graphs of experimental data are given. Can you use a spreadsheet to find algebraic graphs which match them closely, and thus discover the formulae most likely to govern the underlying processes?

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Four on the Road

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

Four vehicles travel along a road one afternoon. Can you make sense of the graphs showing their motion?

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Rationals Between

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

What fractions can you find between the square roots of 56 and 58?

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Calculus Countdown

Stage: 5 Challenge Level: Challenge Level:1

Can you hit the target functions using a set of input functions and a little calculus and algebra?

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Succession in Randomia

Stage: 5 Challenge Level: Challenge Level:1

By tossing a coin one of three princes is chosen to be the next King of Randomia. Does each prince have an equal chance of taking the throne?

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Tangled Trig Graphs

Stage: 5 Challenge Level: Challenge Level:1

Can you work out the equations of the trig graphs I used to make my pattern?

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Whose Line Graph Is it Anyway?

Stage: 5 Challenge Level: Challenge Level:1

Which line graph, equations and physical processes go together?

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Area L

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Draw the graph of a continuous increasing function in the first quadrant and horizontal and vertical lines through two points. The areas in your sketch lead to a useful formula for finding integrals.

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Operating Machines

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

What functions can you make using the function machines RECIPROCAL and PRODUCT and the operator machines DIFF and INT?