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Text in italics shows examples from a single lesson, with associated videoclips and images. This is not intended to be a model lesson.
Aspect of a Rich TaskWhat pupils might doWhat teachers could dostep into a problem even when the route to a solution is unclear (see definition of a problem below), getting started and exploring is made accessible to pupils of wide ranging abilities. By starting with the challenge of making a Magic V pupils wide ranging abilities can get into this problem.
Sharing early findings can move the challenge on to finding all the solutions.Allow pupils to engage with the attributes of a Magic V by identifying similarities and differences between a Magic V and a nonmagic V.
Pupils are given a very open task to work on initially (what questions could we ask about this?) and the teacher funnels the suggestions to focus on particular ideas
Clips MagicV1.wmv, MagicV2.wmv, MagicV3.wmv and image MagicVA.jpg
Give pupils time to work on the problem on their own
Use Think pair share
Ideas about what you notice and then conjectures pupils might make
Share different ways of recordingpose as well as solve problems, make conjecturesThe task lends itself readily to pupils posing their own problems and making conjectures, for example:
Why is the number at the bottom always odd?
A pupil conjectures that this is the case if you have 3 odd numbers and 2 even numbers, but that if you have 3 even numbers and two odd ones then the bottom number will be even. Clip MagicV4.wmv
Will it always be odd?
I think the number at the bottom is odd because there are more odds in the numbers 15 than evens.
If I can find two pairs of numbers that add to the same total to go on the two arms of the V then it doesnt matter which number goes at the bottom.
A pupil tries to find an example that satisfies this conjecture:
Clip MagicV5.wmv
I think that opposite numbers in the arms of the V will have the same difference. See the clip MagicV6.wmv for further explanation of this. The pupils in this clip decide, wrongly, that this conjecture is incorrect.Ask questions such as:
What do you notice?
Can we justify that
Write conjectures on the board.
Encourage the whole group to work on an idea posed by one of their class.Aspect of a Rich TaskWhat pupils might doWhat teachers could dowork at a range of levelsPupils who do not readily work in a systematic way can gain insights into the value of being systematic and organised in their thinking. Some pupils are able to see why odd numbers must go at the bottom and the most able are able to develop convincing arguments for what will happen for any V.
A pupils explains that he tried putting an even number at the bottom but then found he was left with three odds and an even which dont make an even total and therefore cant be split equally between the two arms. This proof by contradiction is a higherorder skill that children rarely use. The clip MagicV7.wmv exemplifies this. [This clip also exemplifies the value of giving pupils thinking time.]
New knowledge can then be applied to different scenarios such as crosses or, more challenging Hs.
The clip MagicV8.wmv shows a pupil trying a Magic Cross.Encourage pupils to write down findings; the teacher could demonstrate good recording methods to the class, or could share ideas that the pupils have developed.
Provide materials (such as cards) for pupils to manipulate, so they can have greater confidence to try some ideas rather than aiming for an immediately correct solution. See the clip MagicV6.wmv for an example of this.
[Discussion point: should we allow all children to choose whether or not to use materials such as cards, or only issue them to certain pupils?]
Have ideas for extending the problem ready but try to encourage pupils to come up with ideas of their own with you helping them to select fruitful routes
Encourage able pupils to generalise be ready with counter examples to get them rethinking. For example always an odd at the bottom does not work with the numbers 2,3,4,5,6 so set them the problem with different numbers.
Ask:
What are the variables/what can we change?extend knowledge or apply knowledge in new contextsThis does not require the application of high level content knowledge but this means that proof and convincing arguments associated with the setting can be shared and understood. I have often seen generalisations produced (for example if there are more odds an odd goes at the bottom) that can be refuted. Refutation is a higher order thinking skill that pupils rarely employ rigorously. Ask pupils to prove it
Or ask
how do you know that will always be the case
When tackling problems in new contexts (such as larger Vs, crosses or, more challenging Hs). Ask pupils not only to solve the problems but to describe what strategies they reemployed.
What things worked and what didnt?
What was the same and what different
Aspect of a Rich TaskWhat pupils might doWhat teachers could doallow for different methodsThis task opens up a wide range of methods for finding solutions and offers room for much discussion.
A pupil asks a friend to explain their idea with greater clarity: clip MagicV9.wmv.
An alternative method is described in clip MagicV10.wmv and image MagicVB.jpg. The five numbers add to a total of 15, so once one number is chosen to go at the bottom of the V (in this example, 5), the rest (10) must be split equally between the two arms of the V.Share different methods and discuss efficiency and effectiveness. An efficient method is only useful if you can use it.
For example: the sum of the numbers 1 5 is 15 to share equally in the two arms an odd goes at the bottom and the rest is shared so:
155 = 10, then 10/2 is 5. A total of 5 in each arm means 1+4 and 2+3.
153 = 12, then 12/2 is 6. A total of 6 in each arm means 1+5 and 2+4
Find all the solutions to V with 2,3,4,5,6 in your head
In the clip MagicV11.wmv the teacher draws attention to the efficient way that one group worked. They shared out the task so they all tried different possibilities.offer opportunities to broaden students problemsolving skills Being systematic is at the core of this problem.
This child demonstrates all the possible arrangements for a certain magic total: clip MagicV12.wmv and image MagicVC.jpg
Another child then explains how she uses the previous clip to work out how many Magic Vs there are altogether: clip MagicV13.wmv
Identifying pattern and generalisation then enables similar problems to be tackled more efficiently (Have you seen something like this before?)Share efficient and systematic recording methods and approaches to the problem.
Ask pupils if they would tackle a similar problem in the same or a different way next time. Why?
Where else has it been useful to be systematic in this way?deepen and broaden mathematical content knowledge In this task pupils are being asked to recognise and explain patterns and relationships, conjecture, generalise and predict.
At the highest levels they should justify their generalisations using convincing arguments and proofs. Less able pupils will be honing their number bond and mental calculation skills. They can be encouraged to look at different starting numbers and different sized Vs. Use pieces of paper to layout and try things out.
Establishing rules for adding odd and even numbers including simple proofs (picture proofs). For example odd+odd=even might look like:
: : : : : : . + . : : : : : : = : : : : : : : : : : : : :
More able pupils can be encouraged to generalise rules and assess peers on the rigour of their proofs.
Aspect of a Rich TaskWhat pupils might doWhat teachers could dohave potential to reveal underlying principles or make connections between areas of mathematics A powerful underlying concept here is the relationships between even and odd numbers and sums of consecutive numbers.See above re odds and evens.
That you can add, subtract, multiply or divide numbers in a Magic V and it will still work. Although a Magic T looks the same, if the trunk of the T is longer than the arms it does not work why?
Where else is it useful to be systematic? Where have we worked before where we have listed all possible outcomes?
Dipping games rely on odds and evens can you arrange to make sure that a particular person is left at the end.include intriguing contextsPupils are intrigued by identifying efficient and labour saving strategiesDiscussing efficient strategies
For example the method described above works because it is efficient and there is a clear structure. How about other methods, do they generalise?
Why do you like this method or someone elses method more?offer opportunities to observe other people being mathematical or the role of mathematics within cultural settingsAs a teacher you can model efficient techniques for solution to stimulate discussion
Now this is what I call efficient followed by modelling the process.
I have also found pupils seeing patterns in underpinning mathematics that I had not noticed and it is good for pupils to see you having to struggle to understand someone elses idea.When pupils suggest ideas and strategies try to take on the role of learner asking questions such as:
Why did you do that?
What should I do if
Would it work if I?
 even if you think you know.
In clip MagicV14.wmv and image MagicVD.jpg the teacher explicitly draws attention to the use of proof by contradiction as a powerful way to approach this problem.
Clip MagicV15.wmv shows the teacher highlighting how findings from Magic Vs can be applied to other letter shapes.
University of Cambridge
www.nrich.maths.org
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