Ohio Resource Center
Lessons
Whelk-Come to Mathematics
Discipline
Mathematics
9, 10, 11, 12
Professional Commentary

In this investigation students explore the possible reasons behind the observation that northwestern crows consistently drop a type of mollusk called a whelk from a height of 5 meters to break its shell. Students are given activity sheets and a graphics calculator. Each group of three to four students needs shelled, whole, blanched peanuts and a meter stick. Different kinds of peanuts will produce different ranges of data. Students simulate the dropping of whelks by dropping peanuts to study the relationship between the height of the drop and the number of drops required to crack the shells. Students must reason about the need for horizontal and vertical asymptotes before sketching a graph. Students understand that the minimum number of drops relates to a horizontal asymptote at N = 1. The possible existence of a minimum height and vertical asymptote remains an open issue until the analysis is complete and should be revisited at the end of the activity. After making these conjectures, students typically sketch a graph of a hyperbolic function justifying that a smaller height requires more drops and a larger height requires fewer drops to break open the whelk. (author/sw)

Common Core State Standards for Mathematics
Standards for Mathematical Practice
CCSS.Math.Practice.MP2
Reason abstractly and quantitatively.
CCSS.Math.Practice.MP4
Model with mathematics.
CCSS.Math.Practice.MP6
Attend to precision.
High School - Functions
Interpreting Functions
Interpret functions that arise in applications in terms of the context
HSF-IF.B.4
For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
Analyze functions using different representations
HSF-IF.C.7
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
1. Graph linear and quadratic functions and show intercepts, maxima, and minima.
2. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
3. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
4. (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.
5. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
Building Functions
Build a function that models a relationship between two quantities
HSF-BF.A.1
Write a function that describes a relationship between two quantities.
1. Determine an explicit expression, a recursive process, or steps for calculation from a context.
2. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
3. (+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.
High School - Statistics and Probability
Interpreting Categorical and Quantitative Data
Summarize, represent, and interpret data on two categorical and quantitative variables
HSS-ID.B.6
Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
1. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.
2. Informally assess the fit of a function by plotting and analyzing residuals.
3. Fit a linear function for a scatter plot that suggests a linear association.
Ohio Mathematics Academic Content Standards (2001)
Patterns, Functions and Algebra Standard
Benchmarks (8–10)
C.
Translate information from one representation (words, table, graph or equation) to another representation of a relation or function.
D.
Use algebraic representations, such as tables, graphs, expressions, functions and inequalities, to model and solve problem situations.
I.
Model and solve problem situations involving direct and inverse variation.
Benchmarks (11–12)
A.
Analyze functions by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior.
14.
Differentiate and explain types of changes in mathematical relationships, such as linear vs. nonlinear, continuous vs. noncontinuous, direct variation vs. inverse variation.
16.
Use graphing calculators or computers to analyze change; e.g., interest compounded over time as a nonlinear growth pattern.
12.
Simplify rational expressions by eliminating common factors and applying properties of integer exponents.
3.
Describe and compare the characteristics of the following families of functions: quadratics with complex roots, polynomials of any degree, logarithms, and rational functions; e.g., general shape, number of roots, domain and range, asymptotic behavior.
4.
Identify the maximum and minimum points of polynomial, rational and trigonometric functions graphically and with technology.
Data Analysis and Probability Standard
Benchmarks (11–12)
A.
Create and analyze tabular and graphical displays of data using appropriate tools, including spreadsheets and graphing calculators.
4.
Create a scatterplot of bivariate data, identify trends, and find a function to model the data.
Mathematical Processes Standard
Benchmarks (8–10)
B.
Apply mathematical knowledge and skills routinely in other content areas and practical situations.
E.
Use a variety of mathematical representations flexibly and appropriately to organize, record and communicate mathematical ideas.
F.
Use precise mathematical language and notations to represent problem situations and mathematical ideas.
Benchmarks (11–12)
H.
Use formal mathematical language and notation to represent ideas, to demonstrate relationships within and among representation systems, and to formulate generalizations.
J.
Apply mathematical modeling to workplace and consumer situations, including problem formulation, identification of a mathematical model, interpretation of solution within the model, and validation to original problem situation.
Principles and Standards for School Mathematics
Algebra Standard
Understand patterns, relations, and functions
Expectations (9–12)
understand relations and functions and select, convert flexibly among, and use various representations for them;
analyze functions of one variable by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior;
Represent and analyze mathematical situations and structures using algebraic symbols
Expectations (9–12)
understand relations and functions and select, convert flexibly among, and use various representations for them;
analyze functions of one variable by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior;
use symbolic algebra to represent and explain mathematical relationships;
Use mathematical models to represent and understand quantitative relationships
Expectations (9–12)
understand relations and functions and select, convert flexibly among, and use various representations for them;
analyze functions of one variable by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior;
use symbolic algebra to represent and explain mathematical relationships;
identify essential quantitative relationships in a situation and determine the class or classes of functions that might model the relationships;
use symbolic expressions, including iterative and recursive forms, to represent relationships arising from various contexts;
draw reasonable conclusions about a situation being modeled.
Data Analysis and Probability Standard
Select and use appropriate statistical methods to analyze data
Expectations (9–12)
for bivariate measurement data, be able to display a scatterplot, describe its shape, and determine regression coefficients, regression equations, and correlation coefficients using technological tools;
identify trends in bivariate data and find functions that model the data or transform the data so that they can be modeled.
Connections Standard
Recognize and apply mathematics in contexts outside of mathematics
Representation Standard
Create and use representations to organize, record, and communicate mathematical ideas
Select, apply, and translate among mathematical representations to solve problems
Use representations to model and interpret physical, social, and mathematical phenomena