Magic Figures. We note that in the example magic square the numbers 1 9 are used. All three rows (columns) have equal sum, called the magic number.

Transkript

1 Magic Figures Introduction: This lesson builds on ideas from Magic Squares. Students are introduced to a wider collection of Magic Figures and consider constraints on the Magic Number associated with such figures. Here is an example of a Magic Square: In this case each row, column and diagonal sums to 15. So, the magic number is 15. Group Discussion: We note that in the example magic square the numbers 1 9 are used. What is the sum of the numbers 1 9? All three rows (columns) have equal sum, called the magic number. How is this magic number connected with the sum of the numbers 1 9?

2 Magic V: Now we introduce another Magic Figure, a Magic V. It has two arms which meet at a vertex or corner. We want to place the numbers 1, 2, 3, 4 and 5 so that the sums along each arm are equal. Vertex One solution is:

3 We should agree that when we swap numbers that belong to the same arm and only to that arm, it does not give us a new solution. So, for example is the same solution again. Also, it is clear that swapping the left and right arms does not give a new solution. So, for example is again the same solution.

4 Activity 1: Magic Vs Resources: A copy of Sheet 1 for each student. Tutor instructions: Please ensure that students understand that we are looking for all distinct solutions. (There are only three and one is already given.) Solutions:

5 Adding Odds and Evens: Have you ever noticed that you always get an even number when you add two odd numbers together? We box this fact along with two other facts about adding numbers: odd + odd = even even + even = even odd + even = odd Group Discussion: In Activity 1 we found that in all three solutions we have an odd number at the vertex. By using what we know about summing odds and evens we try to understand why this is the case. Note that three of the numbers to be placed are odd (1,3 and 5) and two are even (2 and 4). Start with an empty V and put the even number 2 at the vertex. Next, place the 4 anywhere you like. Three numbers remain to be placed. All of them are odd. Consider the possible sums along each arm in terms of being odd or even. Is it possible to have a Magic V with 2 at the vertex? Would it be possible with 4? What do you conclude?

6 Activity 2: Magic Letters Resources: A copy of Sheet 2 for each student. Solutions: There are two possible solutions for the Magic A Magic Number: Magic Number: 9 There are six possible solutions for the Magic T: Magic Number: Magic Number: Magic Number: 9

7 Magic Number: Magic Number: Magic Number: 10 The solution for the Magic V is: Magic Number: 33 The solution for the Magic E is: Magic Number: 75 The solution for the Magic N is: Magic Number: 42

8 Activity 3: Magic Triangles Resources: A copy of Sheet 3 for each student. Solutions: Only one distinct solution exists for each Magic Triangle: Magic Number: Magic Number: Magic Number: Magic Number: 12

9 Activity 4: The Treasure Chest Resources: A copy of Sheet 4 for each student. Solutions: Many solutions exist. Each solution must have an odd number at the vertex. Having the 1 there gives the sum 23, the 3 there yields 24, the 5 yields 25, the 7 yields 26 and the 9 at the vertex gives the sum 27.

10 Sheet Magic number: 10 Magic number: Magic number: Magic number: Magic number: Magic number:

11 Sheet 2 For these two letters A and T, fill in the numbers 1 to 5 so that the letter is magic! Magic number: 33 Magic number: Magic number: Magic number: Magic number: 42

12 Sheet 3 On this sheet the numbers 1 to 6 are available. Remember you use each number exactly once! Magic number: 10 Magic number: 9 Magic number: 11 Magic number: 12

13 Sheet 4 The treasure chest has a number lock. To open the lock you must place the number blocks so that the horizontal sum matches the vertical sum a) Can you find a solution? b) How big is your sum? c) How did you work it out? d) Are there other solutions?! The grid on the left shows an example solution. There are more grids to use on the other side when you are looking for your own solutions.!

14