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can/can't, have to / don't have tO 19 Short answers with will, won't and going to 38 . Short New Cutting Edge Pre-intermediate Students' Teacher's Resource.. New Cutting Edge Elementary Workbook With New Cutting Edge Pre-intermediate Teacher's Res.. New Cutting Edge. Find new research papers in: Physics · Chemistry · Biology · Health Sciences · Ecology · Earth Sciences · Cognitive Science · Mathematics · Computer Science.

Likewise, you know that each friend has a hobby — another way in which guests might find shared interests.

You figure the dance teacher who plays the cello might enjoy talking shop with the yoga instructor who plays tennis, or discussing music with the maple syrup farmer who plays the piano, but might be hard-pressed to slip into conversation with the political scientist who collects stamps. You could represent the relationships between different jobs with a graph whose nodes are the jobs, with links connecting any two jobs that are not conducive to shared shop talk.

Likewise, you could make a graph whose nodes are the different hobbies and connect two hobbies whenever they are incompatible.

The tensor product of these two graphs will have a node for each job-hobby pairing, and two nodes are connected if the two jobs and the two hobbies are both incompatible — exactly the situation you want to avoid at your weekend retreats.

The number of colors you used tells you how many weekends to block off in your calendar.

One thing to notice from this example is that any valid coloring of the job graph carries over to the tensor product — you can simply color each job-hobby combo the same color you used for its job. Such a coloring results in gatherings in which each pair of guests is compatible based purely on their shared professional interests, regardless of their hobbies.

Likewise, any valid coloring for the hobby graph will carry over to the tensor product.

Hedetniemi conjectured that whichever of the two carry-over colorings uses fewer colors is, in fact, the best possible way to color the tensor graph.

If, instead, you combined all your information about jobs and hobbies, maybe you could get away with using fewer colors and enjoy some well-earned guest-free weekends.

Yet for more than 50 years, mathematicians could not find a single tensor product that needed fewer colors than one of its two constituent graphs. The conjecture is true, they were able to show , whenever one of the two constituent graphs requires four or fewer colors.

These findings hinted that the conjecture might be true, but other findings suggested the opposite. In a paper whose main argument boils down to just over one page of dense mathematics, he shows how to construct a type of tensor product that requires fewer colors than either of its constituent graphs.

But mathematicians have long known that there is one graph that these five colors are sufficient to color: the tensor product made from G and its exponential graph.

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All exponential graphs, in fact, have this property: The tensor product that combines the exponential graph with the graph it was built from can be colored with exactly the same number of colors as in the original palette used to make the exponential graph. Why such an argument was overlooked for more than 50 years is a mystery to mathematicians.

The teacher chooses the order of when students will present their reports and may give the students some quick feedback on the content. At this stage the teacher may also play a recording of others doing the same task for the students to compare.

Analysis The teacher then highlights relevant parts from the text of the recording for the students to analyse. They may ask students to notice interesting features within this text.

The teacher can also highlight the language that the students used during the report phase for analysis. Practice Finally, the teacher selects language areas to practise based upon the needs of the students and what emerged from the task and report phases. The students then do practice activities to increase their confidence and make a note of useful language.

Cutting Edge is a series of a books that were created using the Task-Based Approach which teaching through meaningful tasks. The book is available in VK.

Skip to content Menu Englishpost. Task Based Learning.

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No cash or ATM access. Report Students then report back to the class orally or read the written report.

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The pre-task stage can also often include playing a recording of people doing the task. They then practise what they are going to say in their groups.

This enables them to improve their learning outcomes and have a sense of achievement.