1. The Empty Number Line: A Model For Thinking Math Alliance Originally: April 27, 2010

2. Representations: Models for Thinking Strengthening the ability to move between and among representations improves the growth of children’s conceptual understanding. Vandewalle, J. Elementary and Middle School Mathematics Teaching Developmentally. Pearson Education, 2007.. pictures manipulative models Real-world situations written symbols oral language

3. Models for thinking A model for thinking about a mathematical concept refers to any object, picture, or drawing that represents the concept. To see a concept in a model you must have some relationship in your mind to impose on the model. Models give children something to think about, explore with, talk about, and reason with.

4. What is Cosette thinking? The book cart has an assortment of things to read for indoor recess. Out of the 70 books, 37 are comic books. How many are not comic books? Solve it (no standard algorithms allowed). Then turn & share.

5. Cosette’s Model What mathematical understanding is Cosette demonstrating in her thinking? How is she thinking about 70 – 37 = ? How would she solve 80 – 52 = ?

6. Looking At Student Work Pass out the four pieces of student work. Analyze what the student is doing. How are students keeping track of their thinking? What relationships are students demonstrating as they use the empty number line as a model for their thinking? How is the work similar? How is it different? Can you connect the work to any of the strategies from the article?

7. Big ideas behind the Empty Number Line Numbers can be decomposed and the subunits or smaller amounts can be added or subtracted in varying orders, yet still be equivalent. 25 +17 = 25 + 10 + 7 25 + 10 + 5 + 2 25 + 5 + 10 + 2 Place value patterns occur when adding on groups of ten. 27, 37, 47, 57 Unitizing 1 jump of 10 is the same as counting 10 units

8. Number Line Flexibility Write a story problem using content from the grade level you teach that could be solved using an open number line. In your groups: Share your problem. Discuss the student(s) you had in mind who would benefit from using an open number line to solve this problem.

9. The empty number line: A useful tool or just another procedure? Bobis, J. Teaching Children Mathematics. April 2007 What are the benefits for using an open number line for students? What are the benefits for using an open number line for teachers? What needs to be considered when introducing the open number line to students?