From Skip Counting to Linearity: How Do We Get There? Mathematics Teaching Specialists, Milwaukee Public Schools Astrid Fossum, Mary Mooney, The Milwaukee Mathematics Partnership (MMP), an initiative of the Milwaukee Partnership Academy (MPA), is supported with funding from the National Science Foundation.

Distributed Leadership Student Learning ContinuumTeacher Learning Continuum Mathematics Framework

Comprehensive Mathematics Framework

Session Goals:  To deepen our understanding of linearity from early through upper grades.  To explore the interconnectedness of recursive patterning in number and algebraic reasoning.  To illustrate the bridge between algebraic reasoning and symbolic representation in algebra.

Concept Map for Patterns Patterns Functional Relationships Explicit or Functional Strategies Recursive Strategies Using repeated patterns to think functionally Repeating Patterns

Cube Buildings Make a cube building that is five floors tall with three rooms on each floor. If the building has 5 floors, how many rooms are there in the whole building? If the building has 10 floors, how many rooms are there in the whole building? Create a representation for the task.

Questions to Consider and Classroom Implications:  Do students count all of the rooms individually?  Do students count on from three for each floor?  Do students skip count by 3s?  Do students double the number of cubes in five floors?

Shift 1: Students need to make a transition from focusing on only one quantity to realizing that two quantities are important. Lobato, Ellis, Charles, Zbiek, 2010.

Think – Pair - Share What modes of representation did the task allow for? What modes of representation did you use to solve the task?

As children move between and among these representations for concepts, there is a better chance of a concept being formed correctly and understood more deeply. Manipulative models Pictures Real-world situations Oral/Written language Written symbols Modes of representation of a mathematical idea Lesh, Post & Behr (1987)

What could this look like in Middle School?  Floors * 3 is number of rooms  r=3f  y=3x  Start at 0, add 3

Process Standard: Representation A Scaffold for Learning  When learners are able to represent a problem or mathematical situation in a way that is meaningful to them, the problem becomes more accessible.  When students gain access to mathematical representations and the ideas they represent, they have a set of tools to significantly expand their capacity to think and communicate mathematically.

Windows and Towers Make a cube building that is five floors tall with two rooms on each floor. If the building has five floors, how many windows, including skylights, are there in the whole building? If the building has ten floors, how many windows, including skylights, are there in the whole building? Write an arithmetic expression that shows how you figured these out.

Questions to Consider and Classroom Implications:  Do students correctly determine the number of windows and skylights on the towers?  Do students fill in the tables by adding the same amount each time?  Do students use multiplication to show how the number of windows is related to the number of floors?

Shift 2: Students need to make a transition from making an additive comparison to forming a ratio between two quantities. Lobato, Ellis, Charles, Zbiek, 2010.

Questions to Consider and Classroom Implications:  Do students write arithmetic expressions that correctly represent the number of windows?  Do students begin to articulate a general rule for finding the number of windows?  Can students relate the numbers in their rules to features of the tower?

What could this look like to a middle school student?  windows are six times number of floors plus two  w= 6*f +2  y = 6x + 2

Proportional thinking is developed through activities involving comparing and determining the equivalence of ratios and solving proportions in a wide variety of problem based contexts and situations without recourse to rules or formulas. Van de Walle, J., (2004).

19 MPS Students Benchmark 3 – Grade 7 CR Item From a shipment of 500 batteries, a sample of 25 was selected at random and tested. If 2 batteries in the sample were found to be defective, how many defective batteries would be expected in the entire shipment?

21 Proportional Reasoning Proportional reasoning has been referred to as the capstone of the elementary curriculum and the cornerstone of algebra and beyond. It begins with the ability to understand multiplicative relationships, distinguishing them from relationships that are additive. Van de Walle,J. (2004). Elementary and middle school teaching developmentally. Boston, MA: Pearson Education.

 An important objective for all students to achieve is the ability to create, describe, and analyze their own sequential patterns.  Introducing and reinforcing recursive thinking, with sufficient time for discussion and reflection throughout the elementary school curriculum, helps prepare students to reason inductively in the middle grades. Bezuszka & Kenney, 2008

Personal Reflections An idea that squares with my beliefs... A question or concern going around in my head... A point I would like to make...

Session Goals:  To deepen our understanding of linearity from early through upper grades.  To explore the interconnectedness of recursive patterning in number and algebraic reasoning.  To illustrate the bridge between algebraic reasoning and symbolic representation in algebra.

Resources  Assessment Resource Banks,  Bezuszka, S., & Kenney, M., (2008). Algebra and Algebraic Thinking in School Mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.  Lappan, G., Fey, J., et al. (2006). Connected Mathematics 2. East Lansing, Michigan State University: Pearson Education, Inc.  Lobato, J., Ellis, A.B., Charles, R., Zbiek, Rose Mary. (2010). Developing Essential Understanding of Ratios, Proportions & Proportional Reasoning, Grades 6-8. Reston, VA: National Council of Teachers of Mathematics, Inc.  Russell, S.J., & Economopoulos, K. (2008). Investigations in Number, Data and Space. Cambridge, MA: Pearson Education, Inc.  Van de Walle, J. (2004). Elementary and Middle School Mathematics, Teaching Developmentally. Boston: Pearson Education, Inc.

The Milwaukee Mathematics Partnership (MMP), an initiative of the Milwaukee Partnership Academy (MPA), is supported with funding from the National Science Foundation