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Grade 8 - Euler's formula project - Jade Gibson - Model answers

Module by: Wynberg Girls' High School. E-mail the author

Summary: Model answers for Euler's formula project

Grade 8: Mathematics: Euler's formula project – Model answers

Part 1

Table 1
Network A B C D E F G H I J K L M N O P
Can be traveled? y y n y n n n y y n n y y y n y
Number of odd points 2 2 4 2 4 6 4 2 2 4 4 0 0 0 4 0
Number of even points 2 4 2 4 3 2 4 3 0 5 0 1 6 6 1 4

Part 2: Making a conjecture

Conjecture: A network can be traveled whenever...

  • There are only even points.
  • Or when there are two odd points and any number of even points.

It's a good idea to test your solution on some more examples. Let's look at some networks with odd points that we are able to travel (e.g. networks A, B, C, D, H.) Where did you start and where did you end? Did you start/end at an odd or even point? How did you travel them?

Have to start at one of the odd points and end up at the other. And always end up “passing” the odd points twice. (May use other words to describe.)

Part 3: Using your conjecture to solve problems

1) Eighth bridge: either one of the thick lines AC or AB.

Check start and finish, both odd points (D and A/B)

Figure 1
Figure 1 (graphics1.png)

2) The network has 6 odd points so it is impossible. Can be done if he omits two doors: AB and FG or AD and FG.

Check number of points round each point.

Figure 2
Figure 2 (graphics2.png)

3) Shortest distance: 1160m

Route: Start at B; go around the edge and back to B; cross D; back over to C; cross to A then back to B. (Start and finish must be B)

CD and AB crossed twice. (or DE, EC, AB)

4) Yes; it has only two “odd points”, far left and bottom. Must start at one of these odd points and end at the other.

Part 4: Euler's formula

Table 2
Network A B C D E F G H I J K L M N O P
Number of points (P) 4 6 6 6 7 8 8 5 2 9 4 1 6 7 5 4
Number of regions (R) 3 4 4 3 5 5 6 1 1 6 4 2 2 2 5 4
Number of paths or lines (L) 5 8 8 7 10 11 12 4 1 13 6 1 6 7 8 6

Conjecture: If P stands for the number or points in a network, L stands for the number of lines in the network and R stands for the number of regions in the network, then the equation that relates all three for any network is: R – L + P = 2 or R = L – P + 2

Part 5: Use your conjecture to solve each problem:

1) If a network has 26 points and 41 paths, how many regions does the network create?

R = 2 + 41 − 26 = 17 regions R = 2 + 41 − 26 = 17 regions R = 2 + 41 - 26 = 17 regions
(1)

2) If a network has 36 points and 19 regions, how many paths does the network have?

L = 36 + 19 − 2 = 53 paths L = 36 + 19 − 2 = 53 paths L = 36 + 19 - 2 = 53 paths
(2)

3) Draw a network that has 8 points and 10 regions (including the region around the network.) How many paths will you have to draw among the points?

One option is:

Figure 3
Figure 3 (graphics3.png)

The number of paths is 16. P = 8 and R = 10

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