Proportional and Non-Proportional

Base Ten Materials

As students begin to work with larger numbers, we represent those numbers in increasingly abstract ways. In this MathUP video, Marian describes how we transition from proportional base ten materials to non-proportional materials. She explains the use of 10-frames, base ten blocks, and place value charts, as well as some common challenges students face when using more abstract, non-proportional materials. She also highlights the importance of working with students to prepare them for transitions to increasingly abstract representations of numbers.

Number Strings

What do people mean when they talk about strings in math?

A number string is a set of related questions that build on each other from simpler to more complex and highlight number relationships.

For example, at the K–2 level, a string like the one below highlights the fact that if you add the next consecutive number to itself (which is, of course, doubling it), you add 2 and the only results you get are even.

1 + 1 = 2
2 + 2 = …
3 + 3 = …
4 + 4 = …
…

At the Grade 3–5 level, a string like the one below highlights the fact that if you have four groups and increase the group size by 1, the total increases by 4.

4 × 4 = 16
4 × 5 = …
4 × 6 = …
4 × 7 = …
…

At the Grade 6–8 level, a string like the one below highlights the fact that if you divide a factor by 10, you also divide the quotient by 10. It also highlights the consistency of digits wherever the decimal point might be.

3 × 1428 = 4284
3 × 142.8 = …
3 × 14.28 = …
3 × 1.428 = …
…