This study originated from the realization that many community college students have significant difficulties with the concepts related to linear functions and their graphs, even though these topics are included in the mathematics curriculum from about fourth grade onward. It is an attempt to identify possible gaps in the math standards in this area, as well as possible oversights in emphasis in classroom practice at various levels.
Several of these problems overlap with misconceptions about fractions. One of these is the unspoken assumption that the only numbers on the number lines are the integers. Therefore the act of connecting several points on the graph of a line is merely demonstrating that they all lie on the line, implying nothing about function values for non-integer inputs. Another misconception lies in seeing slope exclusively as a ratio of two numbers (rise over run) and not as a unit rate (a single number). These misconceptions can have consequences for future difficulties even with calculus.
In the following document we have quoted standards pulled from several sources, (beginning with) including California State Standards(CSS) the Common Core Standards for Mathematics (CCM), Statement on Competencies in Mathematics Expected of Entering College Students, and the National Common Core Standards for Mathematics. In addition, the Calpass deconstruction documents for Algebra I through Precalculus and the San Diego Department of Education's backward mapping project for algebra supplied the example problems throughout the document. We have also made comments and noted gaps and omissions.
This study originated from the realization that many community college students have significant difficulties with the concepts related to linear functions and their graphs, even though these topics are included in the mathematics curriculum from about fourth grade onward. It is an attempt to identify possible gaps in the math standards in this area, as well as possible oversights in emphasis in classroom practice at various levels.
Several of these problems overlap with misconceptions about fractions. One of these is the unspoken assumption that the only numbers on the number lines are the integers. Therefore the act of connecting several points on the graph of a line is merely demonstrating that they all lie on the line, implying nothing about function values for non-integer inputs. Another misconception lies in seeing slope exclusively as a ratio of two numbers (rise over run) and not as a unit rate (a single number). These misconceptions can have consequences for future difficulties even with calculus.
In the following document we have quoted standards pulled from several sources, (beginning with) including California State Standards(CSS) the Common Core Standards for Mathematics (CCM), Statement on Competencies in Mathematics Expected of Entering College Students, and the National Common Core Standards for Mathematics. In addition, the Calpass deconstruction documents for Algebra I through Precalculus and the San Diego Department of Education's backward mapping project for algebra supplied the example problems throughout the document. We have also made comments and noted gaps and omissions.