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?

**Basic level**

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.

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