1. Mathematical modeling
Ksenija Simic Muller Julia Aguirre
Pacific Lutheran University University of Washington Tacoma

2. Renting to own
You have just moved to a new apartment, and do not have any furniture. There is a rent-to-own business down the street. You decided to visit their website and are now trying to decide if you should rent furniture from them.
How can you use math to make that decision?
I THINK YOU SHOULD DO THE RENT TO OWN PROBLEM

3. Discussion of the problem
Would you recommend renting-to-own?
Why do people rent-to own items?
What other mathematical questions could we ask about this context?

4. Math Modeling Problems
Anhalt (2014)

5. “Choices, assumptions, and approximations are present throughout this cycle.” (CCSS-M, p. 72-73
Modeling Process
KSENIJA: ARE WE KEEPING THE CLIP?

6. Key features of Modeling Tasks
There is a purpose
Comprised of information that is incomplete, ambiguous, or absent
Involves making assumptions and setting conditions, identifying parameters
Requires interpretation representations and data
Involves decision-making and verification
Demands argumentation and justification
Must be sharable and reusable (help in other similar contexts/future)
Includes opportunities to revise and refine the model.
KSENIJA: I PUT THIS BACK. I THOUGHT WE NEEDED TO TALK ABOUT SOME OF THESE ITEMS. MAYBE PROVIDE A LIST, BUT TALK ABOUT AT LEAST A FEW. WE CAN ALSO SKIP THE ONES THAT OVERLAP WITH CYNTHIA’S TABLE.

7. Key features of Modeling Tasks
Helps us interpret our physical and social world (i.e. connects to students’ lives)
Promotes problem posing
Student generated mathematical questions, formulating conjectures based on conditions and evidence
Can include role play (e.g. help a friend, mayor, school board, committee make a decision)
May have multiple solutions
KSENIJA: WHAT DO YOU MEAN BY THE LAST ITEM? It means make the situation better (i.e. get rid of liquor stores; stop school overcrowding; stop racial profiling; make a play park safer and more accessible)

8. Why modeling?
Mathematics can be messy with many answers to a problem (it is not always black and white)
Mathematics is an analytical tool that helps us make sense of and transform/enhance our world (i.e. address issues of equity/justice/civic engagement)
K-8 students can engage in complex mathematical thinking BEFORE knowing basic facts fluently
Because they require a broad range of skills and do not emphasize mere computation, modeling tasks allow a greater variety of students to be successful
KSENIJA: I ADDED BACK SOME ITEMS FROM YOUR EARLIER SLIDE SHOW. I CAN TALK ABOUT MY EXPERIENCES WITH MODELING IN MY CLASS RELATED TO THE THIRD POINT INSTEAD OF ADDING ANOTHER SLIDE. ALSO, THE SECOND BULLET POINT GETS AT THE CONNECTION BETWEEN MODELING AND SOCIAL JUSTICE. I THINK THAT IT’S A NATURAL CONNECTION, AND I AM NOT SURE IF WE NEED TO BELABOR THIS POINT TOO MUCH BEYOND TALKING ABOUT THE BULLET POINT.

9. “[I]n the case of school children, the development of elementary, but powerful mathematical models, should be considered to be among the most important goals of mathematics instruction. So, answers to the question: “What mathematical knowledge and abilities has a given student developed?” should include not only information about “What computations can they do?” but also information about “What kinds of situations can they describe mathematically?” It should include information about “What models have they developed?” as well as information about “What skills have they mastered?”
(Lesh & Lehrer, 2011)
KSENIJA: I LIKE THIS QUOTE BUT WE DO NOT HAVE TO KEEP IT.

10. Creating your own problems
Establish context
Incorporate problem-posing
Represent & interpret
Data
Enhance the context
Listen for community concerns that students and families bring up
Pose, or have students pose, completely open-ended problems that can be investigated in multiple ways using multiple mathematical concepts
Data in tables and spreadsheets provide many opportunities for such modeling
Include a component that takes action or helps make the situation better

11. Working with existing problems
Change the context if necessary
Eliminate all directives and recommendations for how to solve the problem
Eliminate some information and instead let students decide what information they need
Restate the problem to involve making a recommendation, prediction, or drawing a conclusion about the context
HERE IS WHAT I HAVE SO FAR. WILL ADD MORE AS I THINK ABOUT IT MORE THROUGHOUT THE DAY.

12. Context: water and lead
Case 1: Flint
Case 2: Tacoma

13. Corporations Give Water to Flint, MI
Walmart, Coca Cola, Nestlé, PepsiCo said that they will donate bottles of water for school children in Flint, Michigan, to help with the city’s public health crisis over lead contaminated water.
On January 26 the companies said that they are planning to “collectively donate water to meet the daily needs of over 10,000 school children for the balance of the calendar year.” To do so, the companies will send 176 truck loads of bottled water – up to 6.5 million bottles – to Flint.
How do we know how much water will be enough to meet the "daily needs" of Flint school children until December 31, 2016?
Is the companies’ plan a good one?

14. Water prices
“A report released Friday suggests that Flint residents' pricey water bills could double in the next five years due to several factors, even though the state has pledged aid amid the city's ongoing lead- tainted water crisis. The analysis, which was submitted to a governor-created committee working to address the crisis, warns that the average residential bill of nearly $54 a month may rise to $110 — not counting sewer charges — ‘absent any action to increase funding or decrease costs.’ The typical residential water bill has doubled since and is far higher than in other regional cities of Flint's size.” (Chicago Tribune, May 13)

15. PLU: from HR
“PLU is taking proactive steps to test the quality of its drinking water. Environmental, Health, Safety and Emergency Programs personnel will be performing water quality testing in residential, academic and administrative buildings starting on May 9, 2016…. PLU is undertaking this voluntary sampling event to assure the PLU community that additional sources of lead (i.e., old pipes and solder) are not tainting PLU's drinking water above EPA's maximum concentration level of 20 parts per billion… Samples will be collected from representative drinking water coolers, bottle refill stations and kitchen/break room faucets. The standard requires collection of the water sample be taken as a ‘first draw’ after an 8 hour period of inactivity.”  

16. Tacoma Public Schools
TPS testing information

17. Modeling
What kinds of mathematical questions can we ask about these contexts that are directly relevant to us?