Rich Mathemati

cal

Tasks

PMA • 7th June 2008Jim Hogan

Waikato SSShttp://schools.reap.org.nz/

advisor

Based on Chapter 11, Math Education Vol 1

By Jim Neyland, 1994

Problem Solving and

What could happen today. We…

• Try and decide what makes an RMT.• Explore some examples• Reflect on how rich they are• Change them to make them richer.• Solve problems

And so become a ‘Rich’ Mathematics Teacherhttp://schools.reap.org.nz/advisor/rmt.html

Problem #1• The Farmyard

There are some pigs and chickens in the farmyard. A worm counts there are 15 animals and 48 legs. How many pigs are there?

Your turn…

AcknowledgementsThis course refers to the detailed work in “Lighting

Mathematical Fires” 1999 by Prof Derek Holton (Otago) and Charles Lovitt (CDU Canberra).

and the path carving work in problem solving of George Polya (1945, Princeton).

Mathematics Educations Volume (1994) edited by jim Neyland.

What is a RMT?

• A silver bullet?• Something for bright kids?• Beyond me?• Difficult?• For last periods and end of term?• Something someone else does?

Have a chat and try and decide.

Let’s Go! Complete these two puzzles.

+ 4 6

3

7

+

4 10

7 13

Which caused more thinking?

+ 4 6

3

7

+

4 10

7 13

Your turn…x, -, +, /, fractions?

How can we make this easier? Harder? Can we use algebra?How many can we delete? Which ones?Can it be made using a bigger grid?

Constructivism

Mathematics is effectively learned only by experimenting, questioning, inventing, refuting, explaining and discussing.

Better Mathematics 1987

Learning is recognised to be an activity in which students construct knowledge for themselves.

Everybody Counts 1989

RMT

1. Most problems have the potential to be rich.2. It is not rich until you do it, get involved with

it…and then it is only rich for YOU.3. Has a lot to do with classroom environment

and teacher-student relationship.4. Is a rich source of learning when it happens.

Develops thinking and creates an opportunity to report.

Problem #4• 457457

Think of a three digit number and write it twice making a six digit number. Now divide it by 7, the answer by 11 and the answer by 13. What do you notice? Why does this happen?

Your turn…

RMT are…

• Accessible to everyone• Can be extended• Lets children do the thinking speculating,

conjecturing, proving, explaining, reflecting, reporting

• Is fun and enjoyable

See the reading for a more detailed list.

Modifying a task.

Original TaskWriting the date on the WB, say 25th

Modified taskWrite today’s date using any operations and only 2’s

and 5’s.

52 2 x 5 x 2 + 5 25 - 2 - 5 (5+2)x(5-2)+(5-2)+(5-2)/(5-2)

Ask the Answer!

I have an area of 24cm2. What does the shape look like?

This is 1 fifth. What does the whole shape look like?

Pentominoes

See my newsletter Term 1 2007. This activity was the first two weeks of my Year 9 class.

Why? No equipment. I wanted to destroy any preconceived ideas of what math is all about. It is fun, curious, hands-on, solveable, and connects to other ideas.

11 Squares…Thanks Andy Begg.

Break this square into 11 smaller squares.

Check with numbers.

Generalize your solution.

Farmer Brown

When Farmer Brown travels to town at 30km/hr he arrives an hour early. When he travels at 20km/hr he arrives an hour late.

What else do you now know?

Cards, reverse dealing

TaskArrange a the 1 to 10 playing cards so that when

you “deal one and tuck the next under” and so on we get:-

1 2 3 4 5 6 7 8 9 10

Extend to a suit, a pack, different patterns.

The Geoboard

Pic’s Formulae relates the boundary points B and the internal points I to the area of a shape.

What is the formulae?How can we use it?Why does it work?

Pure Logic

No box is labeled correctly. Select one sock from one box and re-label them

all correctly.

Black Socks

Black and White Sock Mixture

White Socks

• If I can give you some insights into how to make a task rich I will have succeeded.

NIM

Great for the fiddlers and to end a period.

Cross out 1,2 or 3. If you cross the last one out you loose.

Is there a strategy? How can I change this game?

Happy Birthday to You

How many people do you need in a group to be absolutely sure that two of them will have a birthday in the same month; on the same day?

How about a 50% chance?

How about a 30% chance?

The Broken Stick

If you break a stick into three pieces, what is the chance that it will form a triangle?

Happy Numbers

Think of a whole number. Square the digits and and add the results. This creates a new number. Repeat this process. If the sequence of numbers forms a circle then the original number is happy.

42 beomes 42 + 22 = 20, 4, 16, 37, 58, 89, 145, 42…aha a circle of happiness!

Are all numbers happy?

Computer Tasks, More Tasks…

Make a calendar, magic squaresSimulate a die, Make a Lotto ticket, Simulate the Monte Hall problemAnd then there is Prime Numbers, Chess and the

Knights Tour, Crossing Deserts, Optimizing a Job, Modeling a system or process…

The world is full of rich mathematical tasks!