**Session Author: **James Tanton

Good visual models illustrate mathematical concepts with clarity—bridging the concrete and the abstract. Exploding Dots are one such model, supporting interactive modelling and problem solving in arithmetic, representation of integers, and polynomial division. They provide an engaging and dynamic representation that is consistent across a broad range of concepts.

The action of the machines can be viewed as coding and decoding decimal integers. Some interesting mathematical questions to explore are:

For a given machine, how does the coding change as the value being coded increases?

Given a coded value, how can the decimal value be recovered?

What relationship is there, if any, between the results of coding some value using each of a 1 ← 2, 1 ← 4 and a 1 ← 8 machine?

What other machines might exhibit interesting relationships in their coded results?

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**Strategies:**

Try a smaller problem, Work backwards, Problem posing**Topics:**

Algebra / Arithmetic, Number theory, Problem solving**Styles:**

Illustrates a concept**Mathematical Practices:**MP1, MP2, MP4, MP7, MP8