Moving From Additive to Multiplicative Thinking The Road to Proportional Reasoning Core Mathematics Partnership Building Mathematical Knowledge and High-Leverage Instruction for Student Success Thursday Sept. 10, 2015

Root Beer or Cola? During dinner at a local restaurant, the five people sitting at Table A and the ten people sitting at Table B ordered the drinks shown below. Later, the waitress was heard referring to one of the groups as the “root beer drinkers.” To which table was she referring? Table A Table B

Sharing Your Thinking Share your answer and thinking with a neighbor. How are your thoughts alike and how are they different?

Two Perspectives on Thinking Absolute Thinking (additive) Comparing the actual number of root beer bottles from Table A to Table B. How might an additive thinker answer which is the root beer table? How might they justify their reasoning? Relative (multiplicative) Comparing amount of root beers to the total amount of beverages for each table. How might a relative thinker respond to this task? Table A Table B

Learning Intention and Success Criteria We are learning to develop an awareness of proportional situations in every day life. By the end of the session you will be able to…recognize the difference between additive thinking (absolute) and multiplicative thinking (relative) in student work.

Which family has more girls? The Jones Family (GBGBB) The King Family (GBBG)

Thinking about “more” from an absolute and relative perspective After you’ve read turn and talk: How would an additive thinker interpret “more” in this context? How would a relative thinker interpret “more” in this context? In what way will questioning strategies surfacing relative thinking? Read Lamon Reading -

Surfacing relative (multiplicative) thinking… Keeping the relative amount of boys to girls the same, what would happen if… The Jones Family grew to 50? The King family grew to 40?

What happens when… Keeping the ratios of boys to girls the same…. The Jones Family grew to 100? The King family grew to 100?

Which is a better deal? M&M’s were featured in the weekly advertisement from two different stores. Greenwall’s Drug: 2 – 16 oz packages of M & M’s $ 3.00. Drekmeier Pharmacy: 3 – 16 oz packages of M & M’s $ 4.00. Which store offered a better deal?

6.RP.1 Understand ratio concepts and use ratio reasoning to solve problems. Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. Fill instandards Progression sheet

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. (2009). Elementary and middle school teaching developmentally. Boston, MA: Pearson Education.

Proportional Reasoning vs Proportions Proportional reasoning goes well beyond the notion of setting up a proportion to solve a problem—it is a way of reasoning about multiplicative situations. In fact, proportional reasoning, like equivalence is considered a unifying theme in mathematics.

Ratio and Proportion What’s the difference? Ratio and proportion do not develop in isolation. They are part of an individual’s multiplicative conceptual field, which includes other concepts such as multiplication, division, and rational numbers. Lo, J., & Watanabe, T. (1997). Developing ratio and proportional schemes: A story of a fifth grader. Journal for Research in Mathematics Education, 28, 216-236.

What is a ratio? Read 2 paragraphs from the RP Progression. How would you define the word ratio? A pair of non-negative numbers a:b which are not both zero. (CCSSM glossary) An ordered pair of numbers that express a multiplicative (relative) comparison. Uses of ratios Part-to-Part Comparison: number of girls to number of boys Part-to-Whole Comparison: number of girls to number of children in the family Read the progressions p. Ratios, Rates Proportional Relationships 2-3 Make this handout. Read 2 paragraphs starting at bottom of page 2 and complete the top paragraph on page 3.

The ratio of the number of people who own a smart phone to the number of people who own flip phone is 4:3. If 500 more people own a smart phone than a flip phone how many people own each type of phone? We should have student work for this…. Are we looking at it?

Tom and Rob are brothers who like to make bets about the outcomes of different contests that come between them. Before the last bet the ratio of the amount of Tom’s money to the amount of Rob’s money was 4:7. Rob lost the latest competition and the now the ratio of the amount of Tom’s money to the amount of Rob’s money is 8:3. If Rob had $280 before the last competition, how much does Rob have now that he lost the bet? Copy for 7th grade – bring to first class

Looking at Student Work Grade 6 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? Copy for 6th grade Bring to first class

Replace with current student work

Which Tree Grew More? Before tree A was 8’ tall and tree B was 10’ tall. Now, tree A is 14’ tall and tree B is 16’ tall. Which tree grew more? 8’ 10’ 14’ 16’ A B A B

How Do Student’s Make Sense of the Tree Growth? Discuss the student responses to the tree question. Which response is correct? How are they different? Order the responses according to the sophistication of reasoning? Lamon, S. J. (2005). Teaching fractions and ratios for understanding. New York: Routledge

Student Responses Do we need to copy this for everyone?

Learning Intention and Success Criteria We are learning to… develop an awareness of proportional situations in every day life. By the end of the session you will be able to…recognize the difference between additive thinking (absolute) and multiplicative thinking (relative) in student work.

Disclaimer Core Mathematics Partnership Project University of Wisconsin-Milwaukee, 2013-2016   This material was developed for the Core Mathematics Partnership project through the University of Wisconsin-Milwaukee, Center for Mathematics and Science Education Research (CMSER). This material may be used by schools to support learning of teachers and staff provided appropriate attribution and acknowledgement of its source. Other use of this work without prior written permission is prohibited—including reproduction, modification, distribution, or re-publication and use by non-profit organizations and commercial vendors. This project was supported through a grant from the Wisconsin ESEA Title II, Part B, Mathematics and Science Partnerships.