This problem solving activity has a number focus.
The twins each have three darts and a dartboard with three rings.
Here is Hana's board. What possible totals can she score if all of her darts land on the board?
Here is Henri's board. Not all of the numbers on his are the same as on Hana’s.
The possible totals that Henri can make are 9, 11, 13, 15, 16, 18, 20, 23, 25, 30.
What numbers are on Henri’s board?
This problem gives students the opportunity to be systematic in the way they combine numbers, be alert to number patterns, and to express patterns in simple expressions and with the use of their own language.. The challenge is to see what scores can be made.
Similar Number problems include: Darts, Level 3; Super darts, Level 3; and More Dartboards, Level 4.
The twins each have three darts and a dartboard with three rings.
Here is Hana's board. What possible totals can she score if all of her darts land on the board?
Here is Henri's board. Not all of the numbers on his are the same as on Hana’s.
The possible totals that Henri can make are 9, 11, 13, 15, 16, 18, 20, 23, 25, 30.
What numbers are on Henri’s board?
Suppose that you had a board something like Hana’s and Henri’s but with not necessarily the same numbers. How many possible totals could you get by using three darts?
Could any two of these totals be the same?
Using a systematic approach on Hana’s board:
The darts might all fall into the same number ring; two might be the same and one different; and all the numbers might be different. So the totals would be:
2 + 2 + 2 = 6 | 5 + 5 + 5 = 15 | 9 + 9 + 9 = 27 |
2 + 2 + 5 = 9 | 2 + 2 + 9 = 13 | |
5 + 5 + 2 = 12 | 5 + 5 + 9 = 19 | |
9 + 9 + 2 = 20 | 9 + 9 + 5 = 23 | |
2 + 5 + 9 = 16 |
This gives ten totals: 6, 9, 12, 13, 15, 16, 19, 20, 23, 27.
The biggest number has to be a multiple of 3 too. Since the biggest total is 30, the biggest number is 10.
How can we get the other number from Henri’s board? What is the next smallest total? It is formed by adding two of the smallest number on the board to the middle number on the board?
11 = 2 x 3 + the middle number.
This problem provides more information than is needed.
Suppose that the three numbers were a, b and c. Then we could list the totals in the same way that we listed the totals on Hana’s board.
This gives 10 possible totals. This is at least consistent with the two examples that we’ve had so far.
Could any of these 10 totals be the same? Let’s assume that a < b < c, otherwise there are going to be lots of totals that are the same.
3a must be different from any other total since a is the smallest number.
Similarly 3c must be the biggest number.
What happens to 3b? Is it clear that 3b isn’t equal to any of 2a + b, 2b + a, 2b + c or 2c + b? So let’s try putting 3b = 2a + c. We’ll do this using an example. Try a = 5 and c = 11. Then b = 7. So what totals do we get with 5, 7 and 11 on the board?
So here there are only 9 totals. These are 15, 17, 19, 21, 23, 25, 27, 29, 33.
Must we always get at least 9 different totals?
Printed from https://nzmaths.co.nz/resource/dartboards at 5:01am on the 22nd May 2024