Across the river
This is a level 4 statistics activity from the Figure It Out series. It is focused on finding all possible outcomes, exploring outcomes in a probability game, finding the probability of the outcome, and designing fair dice for the game. A PDF of the student activity is included.
About this resource
Figure It Out is a series of 80 books published between 1999 and 2009 to support teaching and learning in New Zealand classrooms.
This resource provides the teachers' notes and answers for one activity from the Figure It Out series. A printable PDF of the student activity can be downloaded from the materials that come with this resource.
Specific learning outcomes:
- Find all possible outcomes.
- Explore outcomes in a probability game.
- Find the probability of the outcome.
- Design fair dice for the game.
Across the river
Achievement objectives
S4-3: Investigate situations that involve elements of chance by comparing experimental distributions with expectations from models of the possible outcomes, acknowledging variation and independence.
Required materials
- Figure It Out, Level 4, Statistics, Book One, "Across the river", page 19
- 2 dice
- 24 counters (12 each of 2 different colours)
- a classmate
See Materials that come with this resource to download:
- Across the river activity (.pdf)
Activity
Students can play this game even if they have little understanding of probability. If observant, they will discover that the different totals that can be obtained from throwing two dice are not equally likely. You could discuss why this might be the case. The students can then use either the table suggested in question 2 or a tree diagram as suggested by the curriculum to provide an answer to the question.
Even students who understand some probability can have trouble seeing that there are 36 different ways to get the totals 2 to 12.
Discuss this, along with the idea that there is more than one way to get a total such as 7. You could point out that if you get a 1 on the first roll, you can get the desired total (7) by getting a 6 on the second roll. If you get a 2 on the first roll, it could be followed by a 5 on the second. A 3 could be followed by a 4, or a 4 by a 3. In fact, no matter what you get on the first roll, you have a chance of getting a total of 7 from the two rolls.
Once these facts have been established, you can ask:
- “What is the probability of getting a total of 7?”
The answer can be recorded as a statement (6 out of 36) or as a fraction (6/36 or 1/6 ).
In question 3, the students will first need to realise that they must get each total (1–12) the same number of times. Because there are 36 ways to get a total and there are 12 different totals, each one will have to occur 3 times in a table. Even when they have realised this, they are unlikely to find a solution simply by trial and improvement.
The notes for "Dodgy dice" (page 22 of the students’ book) explain that the probability exemplars give a different view of what students can be expected to understand about probability compared with the curriculum document or the NCEA level 1 achievement standards.
Game
A game for investigating probability.
Activity
1.
a. 1 is impossible. 2 and 12 are very hard to get.
b. 6 and 8 are quite easy. 7 is easiest of all.
2.
|
|
Die One |
|||||
|---|---|---|---|---|---|---|---|
|
1 |
2 |
3 |
4 |
5 |
6 |
|
Die Two |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
|
3 |
4 |
5 |
6 |
7 |
8 |
9 |
|
4 |
5 |
6 |
7 |
8 |
9 |
10 |
|
5 |
6 |
7 |
8 |
9 |
10 |
11 |
|
6 |
7 |
8 |
9 |
10 |
11 |
12 |
|
The probability of obtaining each number is:
- 1: 0/36; 2: 1/36; 3: 2/36; 4: 3/36; 5: 4/36; 6: 5/36; 7: 6/36;
- 8: 5/36; 9: 4/36; 10: 3/36; 11: 2/36; 12: 1/36
3.
The dice should have the numbers 0, 1, 2, 3, 4, 5 and 1, 1, 1, 7, 7, 7.
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