This problem solving activity has a statistics focus.
Luana is playing a game. She has 6 cards numbered 1 to 6.
She has to place them into the three positions of this grid to make the highest possible total she can.
The right-most position represents the units’ column for the number that she will finally produce.
The middle position is the tens’ column and the left-most position is for the hundreds.
She has to position the cards after rolling a dice.
She rolls a 3 first and decides to place card 3 into the tens’ column.
She then rolls a 5. Where should she place the 5 to get the highest 3-digit number on average?
(She only has 1 of each card, if she rolls another 3, or another 5, she just rolls again.)
Students should understand the the chances involved in this problem.
The problem can be solved empirically, requiring the students to do some experimenting or simulating. It can also be solved theoretically by students applying their logic and reasoning to the situation.
Luana is playing a game. She has 6 cards numbered 1 to 6. She has to place them into the three positions of this grid to make the highest possible total she can.
The right-most position represents the units’ column for the number that she will finally produce. The middle position is the tens’ column and the left-most position is for the hundreds.
She has to position the cards after rolling a dice. She rolls a 3 first and decides to place card 3 into the tens’ column.
She then rolls a 5. Where should she place the 5 to get the highest 3-digit number on average? [She only has 1 of each card, if she rolls another 3, or another 5, she just rolls again.]
Pose the problem situation to the whole class.
Where is the best place to position the 5?
The students should observe that there are only two options, either the 5 goes in the ones’ column, or it goes in the hundreds column.
1. Set up a similar situation with the same game.
For example
The second roll is a 5 - where should it be placed?
Method 1: Empirical. Make a table.
Try the experiment 10 times each way and record the results.
Left | Right |
536 531 532 532 534 531 etc | 135 635 235 235 etc |
How many times does the left side of the table win over the right side?
Method 2: Theoretical
(i)
There are 4 possibilities for the next roll (635, 435, 235, 135). All are equally likely.
(ii)
The 4 possible results are 536, 534, 532, or 531.
If a 6 is thrown, 635 beats 536. However, if any other number is thrown, it is better to have the 5 in the hundreds’ column. ‘Any other number’ is thrown three times (1, 2, or 4). So on average, we would expect that the 5 in the hundreds’ column would win 3 times out of 4 (or with a probability of ¾).
Again, try it in both places to see what will happen. If you try 3 – 1 first, then you get the possibilities 361, 351, 341 and 321. If you then try – 3 1, you’ll get 631, 531, 431 and 231. Having the 3 in the tens’ column wins 3 times out of 4 on average.
With the game that uses all ten digits, we’ll assume that if 0 goes in the hundreds’ column, then it makes the number a 2-digit number. So, if we start out with 5 4 - , we can get 549, 548, 547, 546, 543, 542, 541, 540. On the other hand if we start with – 4 5, then we get 945, 845, 745, 645, 345, 245, 145, or 045. For 9, 8, 7 and 6 it is better to have the 5 in the units’ column but for 3, 2, 1 and 0 it is better to have the 5 in the hundreds’ column. In this case things come out equal.
From here you could explore the game from scratch. That is, no matter what numbers come up, and in what order, where is the best place to put them?
Another variation of this game is to look for the total of 10 numbers produced by rolling a dice. Does this variation need an alternative method of analysis?
Printed from https://meaningfulmaths.nt.edu.au/mmws/nz/resource/make-highest-total at 8:59pm on the 26th February 2024