Problem String Studio

Grade 8 problem strings

A problem string is a purposeful sequence of related problems, played one at a time, with student thinking recorded on a model. Each string here builds one strategy — and names the algorithm that would let students finish it without building anything.

GRADE 8

1Pick a string

Grouped by the level of reasoning each one builds, innermost last. A string is a strategy, not a topic — earlier problems hand students a way into later ones. Press the same string again for new numbers.

2Your string

Each problem is labelled with the job it does — anchor is the fact the string leans on, target is a problem meant to be solved using an earlier one, break is placed to fail on purpose. Reorder with ↑↓, edit with ✎, remove with ✕.

3Play it

Full screen, one problem at a time. Teaching notes start hidden so nothing on the board gives the strategy away — press N or the notes button when you want them. Type what students say into the line under each problem; the model is how their strategy becomes visible to the rest of the class.

4Save & share

Nothing saves on its own. The whole string — problems, notes, closing question — packs into one code you can paste in a doc or send to a colleague.

The framework. Every string is tagged with the level of reasoning it is meant to build — additive, multiplicative, proportional, functional — following the nested hierarchy in Pamela Weber Harris, Developing Mathematical Reasoning: Avoiding the Trap of Algorithms (Corwin, 2025). Each string also names the trap: the algorithm or rule that would let students produce correct answers using less sophisticated reasoning than the string is after. That is the book’s central warning, and it is the reason the strings are sequenced the way they are.

The routine. Problem strings — purposeful sequences of related problems, played one at a time with student strategies represented on the open number line, the open array, or the ratio table — are developed for this grade band in Harris, Building Powerful Numeracy for Middle and High School Students (Heinemann, 2011) and Lessons and Activities for Building Powerful Numeracy (Heinemann, 2014), and for the elementary grades in the Developing Mathematical Reasoning companions for Grades K–2 and 3–5 (Corwin, 2025). The routine originates with Catherine Twomey Fosnot and colleagues, Young Mathematicians at Work. Strategy names used here — Over, Give and Take, Constant Difference, Smart Partial Products, Doubling/Halving, Using Quarters and Scaling, Flexible Factoring, Smart Partial Quotients, Equivalent Ratios — are Harris’s.

Content sequence. Unit order follows Illustrative Mathematics (IM K–5 Math, IM 6–8 Math / Open Up Resources, IM Algebra 1), openly licensed CC BY 4.0. The problems are original, written to fit those progressions. Further grounding: Bay-Williams & Kling, Math Fact Fluency (ASCD, 2019); Van de Walle et al., Elementary and Middle School Mathematics (10th ed., Pearson, 2019); Small, Understanding the Math We Teach and How to Teach It, K–8 (Stenhouse, 2019).

Problem string