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We have two circles, touching each other externally. Makes sense?

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In the previous case, what if we move one of the circles slightly away from the other? The two circles will not touch each other anymore, and lie outside each other. In this case, the distance between the centres equals the difference of the radii of the circles, i. Samantha speaks up first.

She suggests a mountain trail in West Virginia. It is one she knows very well. In fact, everyone in her family circle knows it really well. As a child her family spent summer vacations in the area. For 15 minutes, he talks in circles. He gives reasons that do not make any sense about why they should not hike this trail.

He adds that the trail is much too easy for him. This leads the others to wonder if the trail is worthwhile. So, they go around in circles for hours, talking about many trails but deciding on none. They decide to hike the trail in West Virginia after all.

Who's in Your Inner Circle?

Samantha rolls her eyes but she is happy the selection process has come full circle. When the time comes, the group hikes the trail and everyone has a wonderful time -- well, again, everyone except Alex. He hiked the trail far ahead of everyone else, took a wrong turn and got lost. After a two-hour search-and-rescue effort, the park police found him shivering under a lot of leaves and sticks in the dark, cold night. Alex recovered. The last the group heard, he quit hiking and joined a competitive yoga group. That is one circle, Samantha thought, that I want nothing to do with.

And that is some of our expressions using this simple shape that goes around in degrees — the circle. We began our show with a simple definition of a circle.

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And now, we have come full circle. Join us again next week when we will circle back with new expressions to teach. Ashley Thompson was the editor. Load more comments.

Search Search. Audio menu. Learning English Broadcast. Previous Next. As usual they should come up with a conjecture first and then try to find a proof of that conjecture. When a proof has been found get a student to present it to the rest of the class. Let them criticise it. If no one is able to find a proof, even with your help, give a proof yourself. Let them criticise that. It might be useful if you are doing this to out in an odd error to keep them on their toes.

Let them all write up a proof. There are many answers to this question. You might want to restrict the different sums to being consecutive. Your student will tell you what this is. Can the numbers 1, 2, 3, 4, 5, 6 be put into the six circles below, so that the sums of the numbers on each side of the triangles are different? If they can, find all solutions.

How do I create another Circle?

Log in or register to create plans from your planning space that include this resource. Use the resource finder. Home Resource Finder. The point of this unit is to give students a chance to see how mathematicians operate display ingenuity and creativity practice arithmetic in context learn what generalisations, extensions, conjectures, theorems, and proofs are work through a completely novel situation and try to develop a mathematical theory around it. Specific Learning Outcomes. Description of Mathematics. Required Resource Materials. Lesson Sequence Session 1 Discover all four answers to the Six Circle Problem and be reasonably convinced that there are no others.

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Background First we start off with the problem on which this whole unit is based. The important thing for them to do at the start is to use their intuition. What that really means is to encourage them to try anything that comes into their minds. Then, when they start to get answers, you will need to get them to think about which answers differ from each other. The point is that once they get one answer they can get another five simply by using the rotations of the equilateral triangle.

Because we can get any one of six from the others here, we might as well say that they are all the same. So what answers exist and how many of them are there? Trial and error will produce four at least. We show them below. First, we need to check that the sum on all three sides is the same to be sure that we have a proper answer. Second, the sum will prove to be fundamental in what follows. The corner circles contain 1, 2, 3; 4, 5, 6; 1, 3, 5; and 2, 4, 6. These are the smallest and largest consecutive numbers, the odd numbers and the even numbers.

If you move the numbers on one answer all round one circle, you get another answer.


Who's in Your Inner Circle?

This causes the same switches as the last item on the list. Look at A. If you replace every number m by 7 — m, you get another answer. So in B if you replace 1 by 6, 2 by 5, and so on, B turns into C. Teaching Sequence Treat this like any of the Problem lessons that can be found elsewhere on this site. Introduce the problem and discuss it to ensure that all of the students understand what the question is asking and how it might be tackled. Then, in groups of 2 to 4, give students the chance to solve the problem. Do this even when they have obtained all four answers.

Also encourage them to think about when two answers are the same or different. This may mean that you will need to suggest some corner numbers or if they have some numbers in correct positions, you may need to tell them that those are OK but they might think about moving others somewhere else. When they think that they have found all of the possible answers, encourage them to prove that the answers that they have found are the only ones that exist.

Have a class discussion on the results of the exploration so far. Get different students to write up one answer on the board until they see when two answers are the same because of the symmetry of the triangle and that there seem to be only four answers. Produce a conjecture on the number of answers. Finally have a think-tank session to generate ideas for the next lesson. What are the key ideas? How might we restrict the problem?

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Session 2 Show that there are only four answers to the Six Circle problem. So there must be only four. Parity — odds and evens. Reduce the sums and produce the sums. A little bit of algebra. Suppose the sum along each side is s. Teaching sequence Recall the problem and discuss the conjecture that was made in the last lesson. How can we prove this conjecture? Note that it would be good to restrict the possible side sums. In groups of 2 or 4, let them work on the ideas that they have generated. Our experience is that, with minimal scaffolding students can come up with the idea behind Method 4.

Any of the quicker groups should be encouraged to try to find another solution. When a few groups have come up with a proof possibly with your help , have a reporting back session. Let one of the students say what their group has come up with. Ask the rest of the class to test out what this student claims. Show the class another way to prove that there are only four answers. In the end they should find a total of six answers.

We list them below. Teaching Sequence Get students to think about extensions or generalisations to the Six Circle Problem. Try to get several ideas from the class some possibilities are to be found in this unit. Lead them to the Eight Circle Problem. Follow the steps of Session 1 in letting them solve the Eight Circle Problem. This is harder than the Six Circle Problem so it may be useful if they divide certain parts of the problem up among themselves.

Get them together when they think that they have most of the answers. Discuss how they might get the rest. Discuss how they might prove that there are only six answers. Session 4 Generalise the Six Circle Problem by finding which sets of six numbers can replace 1, 2, 3, 4, 5, 6 and in balancing the sums on either side of the triangle. We know already for the four answers that we got in Session 1, that The difference between opposite corner and middle numbers is the same. So we get a legitimate set.

Teaching Sequence Having considered an extension of the Six Circle Problem it is time to look at a generalisation. By discussion, lead them to think about what sets of six numbers might work in the six circles. Encourage them to see what these sets have in common. You may need to bring the class back together to think about the sets they have found. Have they found any sets other than linear combinations of the original numbers? Encourage them to prove the conjecture. Give them the opportunity to discuss their proofs in front of the whole class. Let them write up a proof.

Session 5 Discover and prove that in the six Circle problem there are only zero, two or four answers for any set of six numbers.