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Consecutive sums/Consecutive sums activity: Difference between revisions
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Consecutive sums/Consecutive sums activity (view source)
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The original NRICH version of this problem stated: | The original NRICH version of this problem stated: | ||
Many numbers can be expressed as the sum of two or more consecutive integers. | Many numbers can be expressed as the sum of two or more consecutive integers. | ||
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What can you say about numbers which can be expressed in this way? | What can you say about numbers which can be expressed in this way? | ||
Try to prove your statements | Try to prove your statements. | ||
The most recent version is now called ''Summing Consecutive Numbers'' and is at [http://nrich.maths.org/507 http://nrich.maths.org/507] | The most recent version is now called ''Summing Consecutive Numbers'' and is at [http://nrich.maths.org/507 http://nrich.maths.org/507] | ||
I have also presented this in a geometrical way as a problem I call ‘Steps’. A task-sheet is available [[file:Steps activity.doc]] | |||
Here is a diagram to show an example of steps: | |||
Here is a diagram to show an example of steps: | |||
[[File:Steps1.png]] | [[File:Steps1.png]] | ||
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This means that every consecutive sum answer can be considered to be a triangular number plus a rectangle that has the same height as the triangle. | This means that every consecutive sum answer can be considered to be a triangular number plus a rectangle that has the same height as the triangle. | ||
Depending on how the task is presented, some pupils think they have found a way of ‘breaking’ the problem. If you are allowed to include negative numbers you can make every single answer by summing consecutives. For example, 4 can be made as: 4 <nowiki>=</nowiki> -3 + -2 + -1 + 0 + 1 + 2 + 3 + 4. Either this can be excluded explicitly in the wording of the question (the Steps version clearly does not permit this!) or it can be left for the pupils to find, and the statement of the question altered if/when they do. | Depending on how the task is presented, some pupils think they have found a way of ‘breaking’ the problem. If you are allowed to include negative numbers you can make every single answer by summing consecutives. For example, 4 can be made as: 4 <nowiki>=</nowiki> -3 + -2 + -1 + 0 + 1 + 2 + 3 + 4. Either this can be excluded explicitly in the wording of the question (the Steps version clearly does not permit this!) or it can be left for the pupils to find, and the statement of the question altered if/when they do. | ||
'''Assessment''' | '''Assessment''' | ||
It is often difficult for pupils to assess their own investigative work. One way to encourage and support this is to use the answers that are available on NRICH. New problems on NRICH are provided with Teachers’ Notes and an opportunity to submit solutions. Pupils (or their teachers) can send in solutions either as attached documents or typed into a | It is often difficult for pupils to assess their own investigative work. One way to encourage and support this is to use the answers that are available on the NRICH website. New problems on the NRICH website are provided with Teachers’ Notes and an opportunity to submit solutions. Pupils (or their teachers) can send in solutions either as attached documents or typed into a form. These solutions are aggregated at NRICH and a summary is posted the following month, referencing the first name of the pupil who submitted each idea, with the ideas arranged roughly in ascending order of complexity. | ||
After pupils have completed their own work I sometimes use the NRICH solutions with them. | After pupils have completed their own work I sometimes use the NRICH solutions with them. | ||
One way to do this is to give them copies of the full solutions and to ask them to read them through. Then they need to position their own work with the solutions. Is it the same as someone else’s work? Is it more complex than some? What could they have done next to make their work even better? Are there any parts of the solutions they don’t understand? I print these off, often on A3 paper, and give them to pupils to work on in pairs. The older solutions from a previous version of the NRICH problem are available here as a Word document. The most recent solutions are on the NRICH site at: [http://nrich.maths.org/507/solution http://nrich.maths.org/507/solution] | One way to do this is to give them copies of the full solutions and to ask them to read them through. Then they need to position their own work with the solutions. Is it the same as someone else’s work? Is it more complex than some? What could they have done next to make their work even better? Are there any parts of the solutions they don’t understand? I print these off, often on A3 paper, and give them to pupils to work on in pairs. The older solutions from a previous version of the NRICH problem are available here as a Word document. The most recent solutions are on the NRICH site at: [http://nrich.maths.org/507/solution http://nrich.maths.org/507/solution] | ||
An alternative is to cut out the different parts of the NRICH solution and to provide them on individual cards. The pupils need to assemble them in order of complexity themselves. | An alternative is to cut out the different parts of the NRICH solution and to provide them on individual cards. The pupils need to assemble them in order of complexity themselves. | ||
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