This is an NRICH activity that I've had on my radar for a while. I even made a pencilcode program to explore the activity in reverse. True to their other activities (check them out!!!) 5 steps to 50 requires very little explanation, is accessible to students with limited background, but has depth and richness.
Our lesson outline
I explained the basic activity and did an example at the board. To get my starting value, I had one student roll for the 10s digit and one for the 1s digit. Then we talked through together as we added 10s and 1s.
I then distributed dice and had the kids try 3 rounds. As they worked, I confirmed several rules:
- the only operations allowed are +1, -1, +10, -10
- we must use exactly five steps (I note that this is ambiguous on the NRICH description, they say "you can then make 5 jumps")
- we are allowed to do the operations in any order
- we can mix addition and subtraction operations
- Which starting numbers can jump to 50?
- Which starting numbers cannot jump to 50?
We helped the kids resolve disagreements and then posed the following:
- What is the smallest number that can jump to 50?
- What is the largest number that can jump to 50?
To engage with the activity, some of the kids just started trying operations without much planning. This quickly reinforced the basic points about addition and place-value and commutativity of addition.
For these kids, it was helpful to ask a couple of prompting questions:
- What do you notice? This is a standard that never gets old!
- If this path doesn't get to 50, does that mean there is no path to 50?
This second question, particularly, raises the interesting observation that it is easy to show when a number can jump to 50 (just show a path) but to show that no path is possible requires a different type of thinking.
Getting more advanced
The next level of sophistication was really about noticing that the key consideration is the distance to 50. In particular, this identified a symmetry, where n could jump to 50 if 2*50 - n can jump to 50. Of course, the kids didn't phrase this relationship in this way....
The next major step is thinking about a way to systematically write down the paths.