Denmark Anne Watson Denmark February 2011. Rods, tubes and sweets How many logs of length 60cm. can I cut from a long log of length 240 cm? How many bags.

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Presentation transcript:

Denmark Anne Watson Denmark February 2011

Rods, tubes and sweets How many logs of length 60cm. can I cut from a long log of length 240 cm? How many bags of 15 sweets can I make from a pile of 120 sweets? I have to cut 240 cm. of copper tubing to make 4 equal length tubes. How long is each tube? I have to share 120 sweets between 8 bags. How many per bag?

Three equal volume bottles of wine have to be shared equally between 5 people. How can you do this and how much will each get? Three equal sized sheets of gold leaf have to be shared equally between 5 art students, and larger sheets are more useful than small ones. How can you do this and how much will each get?

98 equal volume bottles of wine have to be shared equally between 140 people. How can you do this and how much will each get? 98 equal sized sheets of gold leaf have to be shared equally between 140 art students, and larger sheets are more useful than small ones. How can you do this and how much will each get?

A piece of elastic 10 cm. long with marks at each centimetre is stretched so that it is now 50 cm. long. Where are the marks now? A piece of elastic is already stretched so that it is 100 cm. long and marks are made at 10 cm. intervals. It is then allowed to shrink to 50 cm. Where are the marks now?

Division as problem solving

Example

When A varies, what changes and what stays the same?

Talk about.... In a right angled triangle, what happens to the ratio of the lengths of the side opposite the angle and the hypotenuse if the triangle is enlarged?

Give me an example of.... A rational number which gets bigger when you raise it to the power of a positive integer... and another generalise

Can you make a rational number which gets bigger when you raise it to the power of -0.5? What do all such numbers have in common? Convince somebody

When and why will Y happen?

Three people are sitting around a circular table; where can you put the coffee pot so they are all the same distance from it? Four people are sitting around on the grass for a picnic; when and why can you find a place to put the water bottle so they are all the same distance from it?

Do we meet? (from various ATM publications) On square grid paper Choose a position on the grid, label it A. Also choose nearby position B. Move from A horizontally or vertically n steps, then B has to move 2n steps in the same direction. The aim is for A to move in such a way that B catches up with it. What is the relation between A, B and the meeting point M? Why?

Number Cells (from Points of Departure 2, ATM, 1986)

What is problem-solving? Understand the problem Make a plan Act out the plan Reflect on the outcomes Adapt the plan or make a new one Act out the new plan......

Understand the problem Identify variables – same/different

Make a plan Change which variables and hold which ones constant

Act out the plan Variation

Reflect on the outcomes What varies and what stays the same – what are the relations, rules, covariation

Make a new plan Based on relations, covariation, properties

Act out the new plan Vary properties – same/different – what changes and what stays the same?

Etc.

What is missing? Naming new objects, drawing attention t o new features, providing the conventional notation, knowing what is mathematically important – taking it to a new level of mathematics