Transition to College Algebra Day 2: Content Training

Agenda Developing number sense Functions, equations, and multiple representations Linear Non-linear Exponential Quadratic

Decomposition

Levels of covariational reasoning 1.Coordination 2.Direction 3.Quantitative 4.Average rate 5.Instantaneous Rate Carlson et al., 2002 as cited in Putting Essential Understanding of Functions into Practice

Covariational Reasoning Local perspective Focus on details of the graph Values of points Slopes of individual lines Quantitative Global perspective Focus on relationships Relationships between of the lines Shape of the graph Qualitative Leinhardt, Zaslavsky, and Stein, 1990 as cited in Putting Essential Understandings of Functions into Practice

Flowing Liquid P-6

Flowing Liquid P-7

Liquid flowing out of the top prism 1 P-8

Liquid flowing out of the top prism 2 P-9

Liquid flowing out of the top prism 3 P-10

Liquid flowing out of the top prism 4 P-11

Liquid flowing out of the top prism 5 P-12

Liquid flowing out of the top prism 6 P-13

Graphs P-14

Working Together P-15 1.The graphs represent the flow of a liquid either out of the top prism or into the bottom prism of the container. 2.Take turns to match two cards that represent the movement of water in one container. 3.Place the cards next to each other, not on top, so that everyone can see. 4.When you match two cards, explain how you came to your decision. 5.Your partner should either explain that reasoning again in his or her own words, or challenge the reasons you gave. 6.Some graphs are missing information, such as a scale along an axis. You will need to add this scale. You both need to be able to agree on and explain the match of every card.

Sharing Work P-16 1.If you are staying at your desk, be ready to explain the reasons for your group’s graph matches. 2.If you are visiting another group, copy your matches onto a piece of paper. 3.Go to another group’s desk and check to see which matches are different from your own. If there are differences, ask for an explanation. If you still don’t agree, explain your own thinking. 4.When you return to your own desk, you need to consider as a group whether to make any changes to your own work.

Developing covariation Move between local and global Use multiple representations “You should expect your students to struggle in moving among these representations as they work to make sense of them and the situation that they describe.”

Connection to the curriculum Lesson 24, Part C- meaning of slope Lesson 24, Part D- Meaning of y-intercept, equation of line

Transforming Linear Equations

Large Whiteboard

Covariation Coordination between abstract and concrete representations Don’t spend too much time immersed in a single graphical or tabular representation. “What does this mean in the context of the situation?”

“Clearly a fluid understanding of covariation, one that moves swiftly up Carlson’s hierarchy, does not come cheaply. It requires acts of mental coordination on the part of the students and acts of pedagogocial coordination on the part of the teacher.” p. 52

Connection to the curriculum Lesson 24, Part E- meaning of slope and y-intercept in problem situations Lesson 25, Part A- linear relationships can be expressed in different ways and different forms Lesson 25, Part B- each representation of a line can be used to interpret the relationship, translate between representations

Putting it all together Lesson 26, Part A and B Read through the instructor notes and student pages Which questions are global? Which questions are local? Discuss how you might see different levels of covariational reasoning in student responses.

Multiple Representations Deepens student understanding of concepts and relationships between concepts Look at Misconceptions of functions represented as graphs Ways of connecting representations

Turn and talk “Different representations of functions provide distinct insights into the relationships they model. Analyzing and applying these representations are critical learning strategies for gaining insight into and making sense of models. Although the various representations of a particular function may look very different, students must understand that the covariance relationship that they represent remains constant.” Cooney, Beckmann, and Lloyd, 2010 as cited in Putting Essential Understanding of Functions into Practice, page 93

Sort into function families Quadratic Exponential Square root Rational Stop after question 2

Graphs Key skill for interpreting graphs- screen out attributes not needed to understand the graph Students might have missed instruction on the global perspective

Multiple representations Prepare a whiteboard for the identified question only