In this unit students are asked to investigate mathematical relationships when they select various numbers to be used in a 3 circle configuration and in 6 circle configurations. They will explore both whole number and fractional numbers as they consider how various configurations impact on the relationships between numbers.
The contexts of this unit are configurations of three and six circles but they are used as a basis for the practice of the addition and subtraction of whole numbers and fractions and for investigating mathematical relationships. Students will need to be able to work confidently with basic fractions and have a good knowledge of addition and subtraction facts.
The unit is divided into five sessions: Mathematical Play; Equal Side Sums; Consecutive Side Sums; Six Circles; and Reflecting.
In this section students explore the central idea of the problems. They generate examples and consider the sum-relationships that develop because of their choices. This may include investigating: can the side sums be equal; can two sides be the same; can each side have a different sum; and if they are different what are the characteristics for the numbers within the circles. The circles generate discussions that are used to hone in on what might be especially interesting aspects of the problem and what might lead to interesting conjectures and even proofs (justifications).
Here they start with the simple idea that if the same number is in each circle then it generates equal side sums. But is this the only way to get equal side sums? So is there a unique way to get a particular sum? And can they get all possible sums? This leads to the exploration of using fractions in order to generate odd sums.
Given three consecutive side sums how do you work backward to find the numbers that must have been in the circles? Students will explore problem solving strategies such as Guess and Check. But they should be encourgaed to develop some generalisations such as finding which circle contains the smallest number. If the smallest number was between the sides with the two biggest side sums, then the other two numbers would produce a side sum that was bigger than the two biggest ones. Continuing in this way we can see that the smallest number has to be between the smallest side sums.
Let’s take the example of side sums of 7, 8 and 9 in the diagram. The smallest number has to be at the top. Suppose that we guess that this number is 2, then the bottom left number has to be 5 (to get the left side sum of 7) and the bottom right number has to be 6 (to get a right side sum of 8). But then 5 + 6 = 11. Unfortunately this doesn’t give the other side sum of 9 that we need. So try another guess until you get 3.
However, the problem of the numbers can be solved by algebraic thinking. Suppose the smallest number is blob. Then the bottom left number has to be 7 minus blob and the bottom right number has to be 8 minus blob. If you add these up you get two things. First 15 minus two blobs, and second 9, because the blobby number is the remaining side sum. So
15 minus two blobs = 9.
But then
6 minus two blobs = 0.
Surely then blob is 3.
If you prefer you could, of course, replace blob by b.
The Guess and Check and the algebra are both harder to use with the fractional cases. But since 4 = 0 + 4 = 1 + 3 = 2 + 2, it should soon become clear that with side sums of 4, 5 and 6, the numbers in the circles can’t be whole numbers. So some guessing with fractions has to be tried.
With six circles a much wider range of possibilities for exploration are available. Let’s look at the equal side sums first. Clearly these can be obtained quite easily by putting the same number in each circle. But it is possible to put the numbers 1, 2, 3, 4, 5, 6 into the circles to get side sums of 9 or 10 or 11 or even 12. There is a Level 5 & 6 problem solving unit called Six Circles that you may like to take some ideas from.
What numbers can we put into the circles to get three side sums of 1? The quick answer is 1/3 six times. But can we make a side sum of 1 without using 1/3 six times? Yes. 1 = 1/2 + 1/3 + 1/6, and we can make up two sides using these numbers and the other can be completed using 1/3, 1/3, 1/3. Is this the only way to do this? No. 1 can be made up of many other fractions one of which is 1/3. For instance, 1 = 1/3 + 1/5 + 7/15 and we can make up a Six Circle with side sum 1 using 1/3, 1/3, 1/3, 1/5, 1/5, and 7/15. But can we make up a side sum of 1 using six different fractions? Yes. Recall that we can make up a Six Circle with side sum of 9 using 1, 2, 3, 4, 5, 6. Surely this means that we can make a Six Circle with side sum of 1 using 1/9, 2/9, 3/9 = 1/3, 4/9, 5/9, 6/9 = 2/3?
So it looks as if there are an infinite number of ways to get a side sum of 1 but maybe there is only one way to get a side sum of 1 if we have six different numbers in the circles? Unfortunately this is not the case. Using 1, 2, 3, 4, 5, 7 we can get an arrangement with side sums of 10. So we can get a side sum of 1 using 1/10, 2/10 = 1/5, 3/10, 4/10 = 2/5, 5/10 = 1/2, and 7/10. Maybe there are an infinite number of these that use 6 different numbers?
Reflecting: Here we use a quiz scenario for students to make up their own problems and so increase their fractional fluency.
In the next three teaching periods, the students will investigate the various ideas that surround and extend the Three Circles.
Once again by using student input, recall what has happened in the last four lessons on this topic. In this lesson let your class have time to produce 4 questions each on the ideas around the Three and Six Circle configurations. Then divide the class into groups and let each group choose 10 of the questions that they have made up to form a quiz. Then pairs of groups should interchange their quizzes. You could work on a marking scheme of 2 marks for a correct answer; 1 mark for a small error; 0 for a completely wrong answer.
This week your child can tell you what we have been doing with numbers in circles. This has been a way of refreshing number facts around addition and subtraction. It has also been a way of thinking mathematically about the relationships between numbers. In one lesson we looked at the Three Circles configuration that we show below.
Ask your child to explain how the side-sums are found and what observations they have made so far. Can they show you how to create configurations that have three side sums that:
Printed from https://meaningfulmaths.nt.edu.au/mmws/nz/resource/three-circles-i at 8:32pm on the 26th February 2024