Published January 2010,February 2011.
A group of teachers were keen to embed rich tasks from the NRICH website http://nrich.maths.org into their curriculum for all KS3 and KS4 students. In this article, the teachers share the issues they needed to consider and what they are doing to address them.
The catalysts for change were our desire to:
These changes took some of us, and our colleagues, out of our comfort zone as we adapted the way in which we worked in our classrooms and tried out different roles for ourselves and our students.
We needed to take into account the different needs of individual teachers in our schools and their students as they began to use rich tasks.
In order to improve BOTH teachers' and students' abilities to work on the rich tasks, it was necessary to take account of, and find ways of supporting:
The challenges involved in learning mathematics through rich tasks
The use of a range of lesson structures
The use of effective questioning to promote thinking
In what follows, we have identified some problems from the NRICH website, closely linked to the curriculum, which are ideal for use by schools wishing to introduce rich tasks.
The aim is to be able to predict the area of any tilted square.
What areas are possible?
What areas are impossible?
Is there a connection between the "tilt" and the area of a square?
What observations, thoughts and conclusions can you offer?
The NRICH website offers Teachers' Notes to many problems. These include suggestions for how the problems might be used in the classroom.
Briefly, introduce the environment with an interactive whiteboard (IWB), or drawing diagrams on the board. Ask students to work in pairs to find one or more ways to calculate the exact area of the square displayed on the board.
Obtain suggestions, with students coming to the board to demonstrate their methods, until the group seem confident with both techniques i.e. 'boxing in' and dissection. Then pose the questions to learners.
Give them five minutes to work on their own/in pairs to think about, and make some brief notes on how they will start working on the problem and why they think such an approach would be useful.
Share ideas about strategies as a whole group. They will need to find ways of breaking the problem down working systematically and keeping track of their data. It may be useful to stop the group and share "good ideas" as work progresses. The aim is to mediate the problem by encouraging systematic approaches.
As conjectures emerge ask learners to write them on a poster/the board and ensure they have convincing arguments or proofs to hand that can justify them. Encourage students to find clear concise vocabulary to express their rules, and to express how certain they are about them (e.g. probable pattern/result from table/proved).
Choose four consecutive whole numbers, for example, 4 , 5 , 6 and 7.
Multiply the first and last numbers together.
Multiply the middle pair together.
Choose different sets of four consecutive whole numbers and do the same.
What do you notice?
Choose five consecutive whole numbers, for example, 3 , 4 , 5 , 6 and 7.
Multiply the second and fourth numbers together.
Choose different sets of five consecutive whole numbers and do the same.
What do you notice now?
What happens when you take 6, 7, 8,'?¦ n consecutive whole numbers and compare the product of the first and last numbers with the product of the second and penultimate numbers?
Explain your findings.
Establish that the group knows the meaning of 'consecutive' - consecutive days, consecutive letters in the alphabet...
Choose four consecutive numbers and tell your students that you will multiply the outer ones and the inner ones. Ask students to pick their own sets of four consecutive numbers and do the same. Record all the results on the board. What do they notice?
Will this always happen? Even with consecutive negative numbers?
How could we explain it? Encourage algebraic and geometric reasoning.
Much of what will follow will depend on the arithmetical and algebraic confidence of the group. Use the extension and support remarks below to indicate the best way to use this resource beyond the questions in the problem.
This problem only operated on the end numbers and the 'end but one' numbers. Could you make a more general statement and justify it?
If you have an odd number of consecutive numbers, what's the difference between the product of the end numbers and the square of the middle number?
This problem could also be approached purely numerically, as an exercise in developing fluency with multiplication tables while looking for pattern and structure.
A multiplication grid could be used for recording results, with the pair products highlighted according to how many consecutive numbers were being used.
Visualisation through blocks of dots or rectangle areas may help students explain why their pattern must work in every case.
Imagine you have a large supply of 3kg and 8kg weights.
Four 3kg weights and one 8kg weight have an average weight of 4kg.
How many of each weight would you need for the average (mean) of the weights to be 6kg?
If you had other combinations of the 3kg and 8kg weights, what other whole number averages could you make?
What's the smallest?
What's the largest?
Can you make all the whole number values in between?
What if you have a different pair of weights (for example 2kg and 7kg)?
What averages can you now make?
Try other different pairs of weights.
Do you notice anything about your results?
Do they have anything in common?
Can you use what you notice to find, for example, the combination of 17kg and 57kg weights that have an average of 44kg......of 52kg.......of 21kg.....?
Explain an efficient way of doing this.
Can you explain why your method works?
Given the original 3kg and 8kg weights, can you find combinations that produce averages of 4.5kg ... of 7.5kg ... of 4.2kg ...of 6.9kg ...? Convince yourself that all averages between 3kg and 8kg are possible.
What averages are possible if you are allowed a negative number of 3kg and 8kg weights?
You may initially wish to restrict the weights used to those which have a difference of 2kg, then 3kg, then 4kg, etc. in order to model working systematically, and to make the pattern of results more obvious.
Some students may find multilink cubes useful to support their visual images.
Here are two examples of students work on Searching for Meaning:
This is a game for two players.
You will need a 100 square which you can download or you can use the interactivity:
The first player chooses a positive even number that is less than 50, and crosses it out on the grid.
The second player chooses a number to cross out. The number must be a factor or multiple of the first number.
Players continue to take it in turns to cross out numbers, at each stage choosing a number that is a factor or multiple of the number just crossed out by the other player.
The first person who is unable to cross out a number loses.
Switch the challenge from winning the game to covering as many numbers as possible. Pupils can again work in pairs trying to find the longest sequence of numbers that can be crossed out. Can more than half the numbers be crossed out?
This challenge could run for an extended period: the longest sequence can be displayed on a noticeboard and pupils can be challenged to improve on it; any improved sequences can be added to the noticeboard.
Ask pupils to explain why their choice of numbers is good.
A semi-regular tessellation has two properties:
Can you find all the semi-regular tessellations?
Can you show that you have found them all?
To help you when you are working away from the computer, multiple copies of the different polygons are available to print and cut out.
Ask students to suggest shapes that could fit round a point. Test out their suggestions.
Record what works and what doesn't.
e.g. 1 pentagon, 2 squares and 1 triangle don't fit round a point
Why do some fit and some don't? Could you tell in advance, without using the interactivity?
Elicit from students the idea that the interior angles of regular polygons are a crucial factor in determining whether the regular polygons will fit together or not.
Before moving on to the computers ask students to work on paper finding combinations of regular polygons that will fit round a point. This may encourage students to work systematically when using the interactivity provided. Resources are provided at the end of the problem to support students working away from the computer.
Challenge students to find all the possible semi-regular tessellations and to provide convincing evidence that they have got the complete set.
Each school worked in different ways. Here are examples of what some of the schools did:
This is what the schools said about how their practice has changed:
This article is the result of the collaborative work of:
Susanne Mallett, Steve Wren, Mark Dawes and colleagues from Comberton Village College
Amy Blinco, Brett Haines and colleagues from Gable Hall School
Jenny Everton, Ellen Morgan and colleagues from Longsands Community College
Craig Barton, Debbie Breen, Geraldine Ellison and colleagues from The Range School
Danny Burgess, Jim Stavrou and colleagues from Sawston Village College
Catherine Carre and Fran Watson and colleagues from Sharnbrook Upper School
David Cherry, Chris Hawkins and Maria Stapenhill-Hunt and colleagues from The Thomas Deacon Academy.
Charlie Gilderdale, Alison Kiddle and Jennifer Piggott from the NRICH Project, Cambridge