Darts Game
Apply the number operations to single digit numbers.
Use a problem solving strategy to identify all the possible outcomes.
It’s worth pointing out at the start that this is not a probability problem. It doesn’t matter how big the two rings of the dartboard are. We are not trying to see if some scores are more likely than others, we are only interested in seeing what scores might be made.
This problem is really just an exploration of number. It gives students the opportunity to be systematic in producing numbers and inventive in the way they combine numbers.
It’s useful to explore all possibilities of a problem. This sort of skill is useful in playing all sorts of games where there is a chance both to search for good strategies and to think of possible ways to solve problems using a number of options.
The Problem
Jetta has just been given a dart game for her birthday. The board has an outer ring and an inner ring. The outer ring scores 3 points and the inner ring 7. The game has three darts.
- What scores can Jetta get?
- Jetta thought that was a bit boring so she decided that she could add and subtract the 3s and 7s. What possible positive scores could she get now?
- What whole positive numbers can Jetta get if she is allowed to use any of the four operations of addition, subtraction, multiplication and division?
Teaching Sequence
- Introduce the problem with a discussion about the game of darts. Using the picture ask the students to share their understanding of the game.
- Display or draw the game board that is used in the problem.
- Ask the students to identify the highest score that can be obtained from throwing three darts.
- In pairs ask the students to find all the possible scores obtained from throwing three darts.
- Share the possible scores.
- Pose parts (b) and then (c) of the problem for the students to work on. Check that they understand what is required.
- As they work ask questions that focus on their manipulation of numbers using the number operations. Encourage them to justify their solutions.
How did you begin part (b)?
Why did you start there?
How did you work out the scores?(Elaboration of mental strategies)
Do you think that you have found all the possible scores? How do you know? - As the students work, encourage them to record their solutions so that they can be shared with others in the class. Or alternatively you might ask them to write their solution in the form of a letter to a friend who is working on the problem in another school.
- Share solutions.
Other Contexts
The first part of this problem could be set in the context of stamps or coins. The second part might be put in a measuring context.
Extension to the problem
Students could be given more darts; there might be different points for the two rings of the board; more rings could be used.
They might also be asked how many ways there are of getting each number.
Solution
Remember that Jetta can get three 3s; two 3s and a 7; one 3 and two 7s; or three 7s.
- Jetta can only get 9, 13, 17 or 21.
- Jetta can now get 1 = 7 – 3 – 3; 3 = 3 + 3 – 3; 7 = 7 + 3 – 3; 11 = 7 + 7 – 3; in addition to the scores in (a). (It’s interesting to note here that it is not possible to get an even number using three odd numbers. Is it clear why?)
- And finally, Jetta can, as well as the addition and subtraction combination, get:
| 2 = 3x3 – 7 | 2 = (7 + 7) ÷ 7 | 2 = (3 + 3) ÷ 3 | 3 = 3 + 7 - 7 | 3 = 3 x 3 ÷ 3 |
| 3 = 3 x 7 ÷ 7 | 3 = 7 x 3 ÷ 7 | 3 = 3 ÷ 3 x 3 | 3 = 7 x 3 | 4 = 3 + 3/3 |
| 4 = 3 + 3/3 | 4 = 3 ÷3 + 3 | 4 = 3 + 7 ÷ 7 | 4 = 7 ÷7 + 3 | 6 = 3x3 – 3 |
| 7 = 3 ÷ 3 + 7 | 7 =7 x 7 ÷ 7 | 7 = 7 x 3 ÷ 3 | 7 = 3 x 7 ÷ 3 | 7 = 3 ÷3 x 7 |
| 7 = 7 ÷7 x 7 | 8 = 7 + 7/7 | 8 = 7 ÷ 7 + 7 | 10 = 7 + (3 ÷ 3) | 12 = 3 x 3 + 3 |
| 12 = (7 - 3) x 3 | 14 = 3 x7 – 7; | 16 = 3 x3 + 7; | 18 = 3x 7– 3 | 18 = (3 + 3) x 3 |
| 24 = 3 x 7+ 3 | 27 = 3 x 3 x 3 | 28 = (7 – 3)x7 | 28 = 3 x 7 + 7 | 30 = (7 + 3)x3 |
| 42 = (7 + 7)x3 | 42 = 7 x 7 + 7 | 42 = (3 + 3) x 7 | 46 = 7x7– 3 | 46 = 7x7– 3 |
| 52 = 7 x 7 + 3 | 56 = 7 x 7+ 7 | 63 = 3 x 3 x 7 | 70 = (7 + 3) x7 | 98 = (7 + 7)x 7 |
| 147 = 3 x7x7 | 343 = 7 x 7 x7 |
Although we think we have found all the possible scores there may be more.
| Attachment | Size |
|---|---|
| DartsGame.pdf | 38.16 KB |
| DartsGameMaori.pdf | 51.42 KB |
| DartsGamePicture.pdf | 33.13 KB |
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