this case study examined the impact of using designed visual representations of simple linear functions on grade 2/3 世界杯2022赛程表淘汰赛 ’ development of algebraic thinking. i also looked for the connections between the use of visual representations of linear functions and young 世界杯2022赛程表淘汰赛 ’ understanding of multiplication. a pre-assessment interview, five-lesson intervention, and post-assessment interview sequence was used over the span of one and a half weeks with a retention task that followed approximately two weeks later. the fivelesson intervention was developed to prominently feature designed visual representations, along with other representations (e.g. table of values, pattern rules, narrative contexts), of simple linear functions to encourage 世界杯2022赛程表淘汰赛 ’ development of explicit reasoning skills. all 世界杯2022赛程表淘汰赛 developed some level of explicit reasoning and were able to generalize about simple linear functions by the end of the study. most 世界杯2022赛程表淘汰赛 moved beyond recursive thinking and were able to generate and apply explicit pattern rules for simple linear functions in order to generalize about any term within that function. 世界杯2022赛程表淘汰赛 ’ development of explicit reasoning in order to work with simple linear functions often sparked a need for a new operation: the invention of repeated addition and, or, multiplication. activities involving explorations of simple linear functions that prominently feature designed visual representations led most of the 世界杯2022赛程表淘汰赛 in the study to develop explicit reasoning skills and an early understanding of multiplication.