Functions

Overall Themes CNN of OJ = CCLS Concepts spiraled and occur in waves. Multiple representations always present Pictures, tables, formulas, graphs Context! Grade 8 = Sea Level

Grade 8 = Sea Level Students first hear the word “function” in Grade 8. Deep: Input/Output in all representations Shallow: Definition of function “A function is a rule that assigns each input to one output.” Deep: Linear Functions and y = mx + b Shallow: The existence of nonlinear functions Deep: Using scatterplots to illustrate bivariate data Shallow: Linear Regression

Kindergarten… Back to the Future! Pretend you’re back in Eating paste Nap time Skip counting! Patterns!!

3rd Grade Back to the Future! Now, pretend you’re back in Kickball! More Patterns!! Predicting!!!

4th Grade Back to the Future! Pretend you’re back in Pokemon! Patterns!! Measurement!!!

6th Grade Back to the Future! Pretend you’re back in Puberty! Patterns!! Measurement!!! Ratios!!!!

7th Grade Back to the Future! Pretend you’re back in More Puberty! Graphs!! Tables!!! Equations!!!! Direct Variation!!!!!

Back to Sea Level Input/Output in all representations (used for years) First use of the word “function” LinFun and y = mx + b (proportional reasoning and y = cx) First appearance of nonlinear functions Illustrate bivariate data (began previous year) First appearance of Linear Regression

In High School... Build functions from sequences (discrete to continuous) Model & interpret in context Domain & range Compositions Formal notation: f(x)=... Geometric functions (transformations; specifically translations) Recursive vs. explicit functions (recursive functions emphasize rates of change) Inverse functions

Surprise! What’s new, exciting, strange, or potentially difficult to adapt to? Common theme of informal introduction of ideas, then later formalizing them Proportions Multiple representations of linear functions Nonlinear functions Statistics Statistics -- association between two variables, linear/non-linear regression Transformations of functions connected to transformations in geometry 4. NO VERTICAL LINE TEST! Proportions shown informally in 6th grade, formalized in 7th with y=cx & constant of proportionality Multiple representations of linear functions introduced in 7th, then formalized in 8th Nonlinear functions introduced in 8th grade in SOME way (not specified how, but possibly with area of squares), then Algebra shows linear, exponential, and quadratic Stats: Analyzing bivariate data & relationships between two variables when the relationship is linear in Algebra, with some exposure to non-linear associations, then regression with multiple types of functions in Algebra 2 Stats -- How do we help our students, ourselves, our coworkers accept and be comfortable with the emphasis on regression starting as early as 8th grade? Residual plots in Algebra? Nonlinear regression in Alg. 2? What PD is needed? Transformations of functions connected to transformations in geometry Translations especially Slope triangle & dilations NO VERTICAL LINE TEST -- allow for the possibility later of looking at functions in terms of y; what implications does this have for us when we teach? Do we expose them to the convention of assuming y is the output and x is the input unless otherwise stated?