First, we wanted to find four consecutive numbers so the class looked at examples of consecutive numbers (1,2,3,4 and 5,6,7,8 etc). We then asked what if we didn't know what the first number was, so we called it x. Seeing how the numbers are built off each other we decided that the four consecutive numbers could be written as x, x+1, x+2, and x+3.
Next, we had to decide how the numbers interacted with each other. How do you create the maximum difference. We decided that in order to create the maximum difference you have to take the largest possible number and subtract it from the smallest. This was found through an investigation using the simple set 2,3,4,5. 5-2 creates the maximum difference. After we explored this we decided that the 4th consecutive number needed to be in the tens place and the 3rd in the units place. This will create the largest possible number. To create the smallest we found that you need to take the 1st number in the tens and the 2nd in the units.
After all of this we had to discover how to write a 2-digit number. We looked at the number 54 and found that it is said as 50,4 or 50 + 4. So we have to multiply the number in the tens place by 10. Here is the work we did:
10(x+3) + (x+2) --> largest number possible
10(x) + (x+1) --> smallest number possible.
Simplify:
10x+30+x+2 --> 11x + 32
10x+x+1 --> 11x + 1
Now subtract:
11x + 32 - (11x + 1) --> 11x + 32 - 11x - 1 --> 31
We did a similar investigation for the smallest difference and found that the 3rd number is in the tens and the 1st in the units for the larger number. For the smaller we found that the 2nd number is in the tens and the 4th number is in the units. Here is our solution:
10(X+2) + x --> 10x + 20 + x --> 11x + 20
10(x+1) + (x+3) --> 10x + 10 + x + 3 --> 11x + 13
Subtracting we get:
11x + 20 - (11x + 13) --> 11x + 20 - 11x - 13 --> 7