A Disastrous Day
This is a level 4-4+ activity from the Figure It Out series.
You can use this game as an opportunity to observe the numeracy skills of your class and to assess a student’s ability or level of development. Observing the game could help you to group students into teams of similar numeracy skill or development. If you wish, you can easily write your own sets of cards emphasising different skills.
The game provides an incentive for students to improve their numeracy skills. Each throw of the dice requires them to perform mental calculations using the digits. Encourage them to do all possible calculations and not just settle for the addition option. By doing this, they will gain the maximum advantage in terms of outcome.
After some practice, the students will develop a “feel” for the numbers and start choosing the optimal calculation intuitively. Notice that because of the competitive nature of the game, players will be doing the calculations after every throw to check on their peers!
Over time, the students should notice that subtracting the square of the smaller number from the square of the larger number generally gives the greatest number but that there is an important class of exceptions to this.
More able students may enjoy analysing the results from the different methods more closely.
Algebraically, the approaches can be written as follows (a is the larger of the two numbers if a and b are different):
1. a + b
2. a2 – b2 = (a + b)(a – b)
3. 2(a – b)
When the two numbers are the same, (a – b) = 0 and methods 2 and 3 will both give zero. So the best approach will be to add the two numbers.
When there is a difference of 1 between the numbers, (a – b) = 1, so methods 1 and 2 will both give the same result (which will be 3 or more). Method 3 will give the number 2. When (a – b) is greater than 1, method 2 will give a bigger number than method 1.
Method 3 will never give the largest number. This is because it always gives a number less than or equal to that given by method 2. Both methods have a common factor of (a – b), but (a + b) is always greater than or equal to 2.
The Safe House and Lifesaver cards offer further opportunity and incentive to improve numeracy skills. Because some of the cards have a lot of information to be processed, some students may need to read them for themselves. Have another student hold the card up with the answer covered. A word of warning: most students have acute memories for correct answers when playing games, so adding to the collection of cards by using the same question structures and changing the numbers may be a useful ploy.
Answers to Game
A game involving manipulating numbers.