1. Thinking about Covariation
Ted Coe, Ph.D.
Director of Mathematics, Achieve
October, 2016

2. Teaching and Learning Mathematics
Ways of doing
Ways of thinking
Habits of thinking

3. The Rules of Engagement
Speak meaningfully — what you say should carry meaning to others;
Exhibit intellectual integrity — base your conjectures on a logical foundation; don’t pretend to understand when you don’t;
Strive to make sense — persist in making sense of problems and your colleagues’ thinking.
Respect the learning process of others — allow them the opportunity to think, reflect and construct. When assisting your peers, pose questions to help them construct meaning rather than show them how to get the answer.
Marilyn Carlson, Arizona State University, Project Pathways

4. Adapted from “Top Rugby Players 100m sprint” https://youtu
Adapted from “Top Rugby Players 100m sprint” CC-BY 187sports

5. Lachlan Turner: 100m, 11.1 seconds
What can we mathematize?

6. A Hands-On Activity

8. Adapted from “My bungee jump” https://youtu
Adapted from “My bungee jump” CC-BY Leonids Stepanovs

10. Adapted from “Austin Coleman Run 1 BMX Vert final X Games Barcelona 2013” CC-BY XGAMESUSA

12. Carlson, M. , Jacobs, S. , Coe, E. , Larsen, S. , & Hsu, E. (2002)
Carlson, M., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002). Applying covariational reasoning while modeling dynamic events: A framework and a study. Journal for Research in Mathematics Education, (p.360)

13. Carlson, M. , Jacobs, S. , Coe, E. , Larsen, S. , & Hsu, E. (2002)
Carlson, M., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002). Applying covariational reasoning while modeling dynamic events: A framework and a study. Journal for Research in Mathematics Education, (p.360)

14. Thompson, P. W. (2016). Researching mathematical meanings for teaching
Thompson, P. W. (2016). Researching mathematical meanings for teaching. In English, L., & Kirshner, D. (Eds.), Handbook of International Research in Mathematics Education (pp ). London: Taylor and Francis.
(p.448)

15. 6.EE.9 Represent and analyze quantitative relationships between
dependent and independent variables.
9. Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.
National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC: Authors

16. 6.EE.9 Represent and analyze quantitative relationships between
dependent and independent variables.
9. Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.
National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC: Authors

17. Grade 8 Introduction
(1) Students use linear equations and systems of linear equations …They understand that the slope (m) of a line is a constant rate of change, so that if the input or x-coordinate changes by an amount A, the output or y-coordinate changes by the amount m·A.
National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC: Authors

18. Grade 8 Introduction
(1) Students use linear equations and systems of linear equations …They understand that the slope (m) of a line is a constant rate of change, so that if the input or x-coordinate changes by an amount A, the output or y-coordinate changes by the amount m·A.
National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC: Authors

19. “From the perspective of a mathematician or a mathematics education researcher, it is easy to see covariation in such statements. However, research tells us that students and teachers typically do not.”
-Thompson and Carlson (in press) referring to the example in CCSS 6.EE.9 and Grade 8 introduction (1).
Thompson, P. W., & Carlson, M. P. (in press). Variation, covariation, and functions: Foundational ways of thinking mathematically. In J. Cai (Ed.), Compendium of research in mathematics education. Reston, VA: National Council of Teachers of Mathematics.

20. 8.F.5
5. Describe qualitatively the functional relationship between two
quantities by analyzing a graph (e.g., where the function is increasing
or decreasing, linear or nonlinear). Sketch a graph that exhibits the
qualitative features of a function that has been described verbally.
National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC: Authors

21. Grade 8 Introduction
(2) Students grasp the concept of a function as a rule that assigns to each input exactly one output. They understand that functions describe situations where one quantity determines another. They can translate among representations and partial representations of functions (noting that tabular and graphical representations may be partial representations), and they describe how aspects of the function are reflected in the different representations.
National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC: Authors

22. F-IF.4
Interpret functions that arise in applications in terms of the context
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.★
National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC: Authors

23. F-LE.1
Construct and compare linear, quadratic, and exponential models
and solve problems
1. Distinguish between situations that can be modeled with linear functions and with exponential functions.
a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC: Authors

24. You have an investment account that grows from $60 to $103
You have an investment account that grows from $60 to $ over three years.
Source:

25. What if we shift from giving the “y-intercept”
What if we shift from giving the “y-intercept”? You have an investment account that was worth $60 after one year and $ at four years…

26. Other things that start to make sense…