# Number string structure and design

How are number strings designed? Typically, people tend to describe number strings as having the following structure

Entry problem

Helper problems

Challenge problem (or clunker)

This post from Math Coach on Demand (which also has a bunch of addition and subtraction number strings) describes the structure like this: Again, the concept of helper problems. But is there just one “formula” for a number string?

Many number strings have this helper problem structure, such as strings that develop jumps of ten (and multiples of ten) in addition and subtraction, like the number string above, as well as number strings that develop the use of the distributive property of multiplication such as:

24 x 9

24 x 10

24 x 20

24 x 29

52 x 18

Again, you can see the series of helper problems that scaffold increasingly difficult problems. But what about this string for early multiplication?

3 x 8

6 x 4

12 x 2

24 x 1

5 x 8

Here, the number string is designed with equivalent problem structure.  The first four problems are all equivalent, allowing students to think through why doubling one factor and halving the other results in an equivalent product. Other number strings with equivalent problem structure are constant difference in subtraction strings, such as:

130 – 65

131 – 66

129 – 64

128 – 63

183-49

Here the first four problems are equivalent to illustrate how constant difference can make a seemingly difficult problem like 131 – 66 simpler to solve, by shifting the difference up the number line to 130 – 65.

There are also number strings that have a compare structure, such as this number string which asks students to consider whether to add on or remove in subtraction

145 – 9

145 – 136

267 – 31

267 – 236

450 – 421

450 – 29

This distinction matters when you are designing strings, and also when you are facilitating them.  When you are facilitating the number string above with a compare structure, you want to make sure to draw out a discussion of which strategy works for which set of numbers, and why. Much like a compare discussion in Elham Kazemi and Allison Hintz’s Intentional Talk, the structure of the mathematics leads to a particular structure for discussion. For a number string with an equivalency structure, you want students to be talking about the equivalency and proving that these problems are truly equivalent. For a number string with a helper problem structure, discussion can focus on which simpler problems helped you solve a more challenging problem. For all the problems, discussion can center around the patterns that they notice in the number string. But for different problem structures, those patterns are different.