Models for Success Debbie Lane Math Team Assistance Highline, Winlock, Boistfort School Districts darlane32@earthlink.net

Math Practices Make Sense of Problems and Persevere in Solving Them Reason Abstractly and Quantitatively Construct Viable Arguments and Critique Others’ Reasoning Constructively Model with Mathematics Use Appropriate Tools Strategically Attend to Precision Look for and Make Use of Structure Look for and Express Regularity in Repeated Reasoning

What does “Modeling” mean? “Modeling” is mathematizing a situation (which includes structuralizing, idealizing, and making assumptions) or making use of a given or constructed model by interpreting or validating it in relation to the context. Modeling is important because it links classroom mathematics and statistics to everyday life. MP4, “Model with mathematics,” directly addresses this practice. Mydigitalchalkboard.org

Introduction to Modeling It could be the missing link between children being good at ARITHMETIC but not confident enough to solve word problems.

The Program for International Student Assessment (PISA) rubric defines modeling problems on a continuum of four different levels, shown below. Level 0 problems are purely computational or context is unnecessary for solving them. Level 1 problems are directly translatable from a context. An example is a simple word problem from which students can formulate an equation.

The Program for International Student Assessment (PISA) rubric defines modeling problems on a continuum of four different levels, shown below. Level 2 problems use models and modify them to satisfy changed conditions. Such problems allow students to study patterns and relationships between quantities, and represent these patterns and relationships using words, numbers, symbols, and pictures. These can be problems that have multiple solution strategies, but usually have only one correct solution.

The Program for International Student Assessment (PISA) rubric defines modeling problems on a continuum of four different levels, shown below. Level 3 problems have no predetermined solution. Such problems require students to make assumptions about the context, develop strategies to solve them, check their answers, present results, and possibly revise their solution strategies and begin the process over again.

Grade K-2 – Tiling Pool Problem: Tat Ming is designing square swimming pools. Each pool has a square center that is the area of the water. Tat Ming uses blue tiles to represent the water. Around each pool there is a border of white tiles. (modified Ferrini-Mundy, et al., 1997) Bring tub of tiles

Grade K-2 – Tiling Pool Problem: Bring tub of tiles

Prompts… Is the Child expected to be curious Prompts… Is the Child expected to be curious? What level of modeling would this be identified? For each square pool, sort the tiles into blue tiles for the water and white tiles for the border.  Count how many tiles are in each pile. Are there more blue tiles than white tiles?  How many tiles are in the next largest pool? Check your answer by building the square.  Describe your methods for counting the different tiles. What patterns do you see?

How can we increase the engagement of our students? Could we have started with the picture and see if they could expand it? Could we have let the students decide what it would look like - does the picture take away complexity? Could K-2 graders build tables to keep track of how many? Is that appropriate for that age?

Grade 3-5 – Tiling Pool Problem: Tat Ming is designing square swimming pools. Each pool has a square center that is the area of the water. Tat Ming uses blue tiles to represent the water. Around each pool there is a border of white tiles. (modified Ferrini-Mundy, et al., 1997) https://www.mydigitalchalkboard.org/portal/default/Content/Viewer/Content?action=2&scId=306591&sciId=11854

Wondering…. What other content areas could be assessed by observing the students with simply the prompt… no picture? Any Geometry? Measurement standards? Generate opportunities to MP 8 – Analyze our own work and Critique others mathematically?

Prompts… Is the Child expected to be curious Prompts… Is the Child expected to be curious? Are we telling children what to do, or are they expected to think through to generate more questions? Which prompts move from low complexity to higher complexity? Build the first three pools and record the data in a table. Extend the table for the next two pools. How do you know your answers are correct?  If there are 32 white tiles in the border, how many blue tiles are there? Explain how you got your answer. How many tiles are in the next largest pool? Check your answer by building the square. 

Prompts continued Can you make a pool with 49 blue tiles? Explain why our why not. Can you make a pool with 12 blue tiles? Explain why our why not. In each of the first three pools, decide what fraction of the square's area is blue for the water and what fraction is white for the border? What patterns do you see? What fractions will occur in the next two rows of the table? How do you know that your answers are correct? Below is a picture of Salina's backyard. If each tile has a side length of 10 centimeters, what is the largest pool someone could put in Salina's backyard?

Hmmm, do we have enough information? Could we ask some questions?

Grade 6-8 – Tiling Pool Problem: Tat Ming is designing square swimming pools. Each pool has a square center that is the area of the water. Tat Ming uses blue tiles to represent the water. Around each pool there is a border of white tiles. (modified Ferrini-Mundy, et al., 1997)

Prompts: Make a table showing the numbers of blue tiles for water and white tiles for the border for the first six pools.  What are the variables in the problem? How are they related? How can you describe this relationship in words?  Make a graph that shows the number of blue tiles in each pool. Make a graph that shows the number of white tiles in each pool. As the number of pool tiles increases, how does the number of white tiles change? How does the number of blue tiles change? How does this relationship show up in the table and in the graph?

Continued: Use your graph to find the number of blue tiles in the seventh pool. Can there ever be a border for a pool with exactly thirty- nine white tiles? Explain why or why not. Below is a picture of Salina's backyard. Design a pool that maximizes the space in the backyard. Use two different colored tiles to create a tile pattern in your pool design. (Note: This question raises the problem to Level 3 modeling.) How?

Resource… https://gfletchy.com/3-act-lessons/ https://gfletchy.com/the-water-boy/ 3.NBT.2 and 5.NBT.7 and 6.RP Have video ready to play… or act out the video…. Get a 2 liter bottle of water……

How did this problem use a model? What increased the engagement?

#1 – Make Sense of Problems and Persevere in Solving Them. Routines for Reasoning, Fostering the Mathematical Practices in All Students. By Kelemanik, Lucenta, Creighton 2016 #1 – Make Sense of Problems and Persevere in Solving Them. All students should participate in this MP#1 daily. All GLAD et al strategies are good For All kids. Create a routine in your classroom with:

Establish a Routine for your lesson Routines for Reasoning, Fostering the Mathematical Practices in All Students. By Kelemanik, Lucenta, Creighton 2016 #1 – Make Sense of Problems and Persevere in Solving Them. Establish a Routine for your lesson Launch Notice Generalize Discuss Reflect

Establish a Routine for students during the lesson Routines for Reasoning, Fostering the Mathematical Practices in All Students. By Kelemanik, Lucenta, Creighton 2016 #1 – Make Sense of Problems and Persevere in Solving Them. Establish a Routine for students during the lesson Individual think time Small group or partner share time to clarify points and gather more information Selected large group shares – gathered by teacher during small group sharing, full group discussion.

Establish strategies that work Routines for Reasoning, Fostering the Mathematical Practices in All Students. By Kelemanik, Lucenta, Creighton 2016 #1 – Make Sense of Problems and Persevere in Solving Them. Establish strategies that work 3-Reads and similar… develop skill by removing numbers and questions to force-focus on the context before working on numbers. Keep your plan simple… Polya, 1945, How to Solve It Understand the problem Devise a plan Carry out the plan Look back

3 Practices are used regularly by students to make sense… Routines for Reasoning, Fostering the Mathematical Practices in All Students. By Kelemanik, Lucenta, Creighton 2016 3 Practices are used regularly by students to make sense… #2 - Reason abstractly and quantitatively or, #7 - Look for and Make use of Structure #8 - Look for and express regularity in repeated reasoning.

How did this problem use math practices? Did you focus on Quantities? MP 2 Did you ask what are the important quantities? I need to see the label; I need to know how many ozs (ml) was in the beginning; are you unable to estimate without some details?

How did this problem use math practices? Did your mind look for something you’ve done before that gives you a structure to follow? MP 7 Did you ask yourself… hmmm, let me watch it a few times… this seems like that problem we did….. Let me step back and look at chunks of the problem then form my probing questions to fit what I know. Can I write it differently so that I can see the answer more clearly without doing a lot of calculating.

How did this problem use math practices? Did your mind immediately try to make similar problems and then work up to the problem at hand? MP 8 Did you start with an easier problem and then calculate based on those imaginary figures? Did you solve an easier problem, then work up into the actual problem? Did you notice a pattern as you worked with your numbers that helped form repetition supporting my rule?

Model with Mathematics. MP#4 What happens no matter which way you think about problems? Expect these with every complex problem – no exceptions. Construct viable Arguments and critique reasoning of others. MP#3 Model with Mathematics. MP#4 Use appropriate tools strategically. MP#5 Attend to Precision. MP#6

Resources: Illustrative Mathematics: https://www.illustrativemathematics.org/content- standards Digital Chalkboard: https://www.mydigitalchalkboard.org/portal/default/Content/Viewer/Cont ent?action=2&scId=306591&sciId=11854 Achieve the Core: https://achievethecore.org/ 3 Act Lessons: https://gfletchy.com/the-water-boy/ Routines for Reasoning, Fostering the Mathematical Practices in all Students by Kelemani, Lucenta, Creighton