Math & Science Collaborative:

Problem Solving Via Functions Tasks

Day 1

• Common Core Content Standards• Standards of Mathematical Practice• What is functional thinking?• How do we develop functional thinking?

Math & Science Collaborative at the Allegheny Intermediate Unit

Overview of Course

Day 1 Agenda

Focus: E.U. 4a- Functions provide a tool for describing how variables change together. Using a function in this way is called modeling, and the function is called a model.• Professional Practice Norms• What is Functional Thinking?• Standards for Mathematical Practice• Learning Progressions• Planning

The Teaching Principal (NCTM 2000)

Effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well

NCTM 2000, pg. 16

• Take a minute and think about what would make this a risk-free learning environment.

Working Together

Math & Science Collaborative at the Allegheny Intermediate Unit

Tiling a Patio

Alfredo Gomez is designing patios. Each patio has a rectangular garden area in the center. Alfredo uses black tiles to represent the soil of the garden. Around each garden, he designs a border of white tiles. The pictures shown below show the three smallest patios that he can design with black tiles for the garden and white tiles for the boarder.

a. Draw patio 4 and patio 5. How many white tiles are in patio 4? Patio 5?b. Make some observations about the patios that could help you describe larger patios.c. Describe a method for finding the total number of white tiles needed for patio 50 (without

constructing it).d. Write a rule that could be used to determine the number of white tiles needed for any patio.

Explain how your rule relates to the visual representation of the patio.e. Write a different rule that could be used to determine the number of white tiles needed for any

patio. Explain how your rule relates to the visual representation of the patio.

Mathematical Task Questions

• Consider your own method for solving the problem? Why did it make sense for you to solve it this way?

• What are some of the ways students might solve it? What misconceptions might they bring?

• What might a teacher need to do to prepare to use this task with students?

• Take a few minutes to jot down some thoughts about what functional thinking is. Fill in the left side of your sheet.

Math & Science Collaborative at the Allegheny Intermediate Unit

What is Functional Thinking?

I think functional thinking is … But now I know it is . . .

Function

• Functions provide a means for describing and understanding relationships between variables. They can have multiple representations- in algebraic symbols (equations), situations, graphs, verbal descriptions, tables, and so on- and they can be classified into different families with similar patterns of change.

NCTM- Developing Essential Understandings of Expressions, Equations, and Functions For Teaching Mathematics in Grades 6-8

Functions- as defined by Intel Math (handout)

• Like a process, a function assigns a unique output to a given input. Each input is assigned to only one output

• Functions are used to describe and predict relationships where one quantity determines another, based on the function rule and the input.

• Functions provide a powerful organizational tool that unifies the notion of a table, pattern, graph, and formula

• Some functions are invertibleIn your binder, complete the exercises “Functions”

Connecting to the Tiling Patio Task

• Can this task be modeled by a function? Explain.

• Write a process/function diagram for the tiling patio task, part d.

• What values define the domain for this function? Range?

• Write the rule for part d as a function

Take time to read the Standards for Mathematical Practice.• What is something that stood out/interested

you? Something you have a question about?• What are the implications for teaching?

Learning?• Which SMPS did you most engage with in the

Tiling Patio Task? Provide evidence.

Math & Science Collaborative at the Allegheny Intermediate Unit

Standards for Mathematical Practice

NCTM- Essential Understandings of Expressions, Equations, & Functions , pages 7-11

Big Idea 4. Representing and Analyzing Functions• Essential Understanding 4a. Functions provide a tool for describing how variables change

together. Using a function in this way is called modeling, and the function is called a model.

• Essential Understanding 4b. Functions can be represented in multiple ways—in algebraic symbols, situations, graphs, verbal descriptions, tables, and so on—and these representations, and the links among them, are useful in analyzing patterns of change.

• Essential Understanding 4c. One important way of describing functions is by identifying the rate at which the variables change together. It is useful to group functions into families with similar patterns of change because these functions, and the situations that they model, share certain general characteristics.

• Essential Understanding 4d. Some representations of a function may be more useful than others, depending on how they are used.

• Essential Understanding 4e. Linear functions have constant rates of change.• Essential Understanding 4f. Quadratic functions are characterized by rates of change that

change at a constant rate.• Essential Understanding 4g. In exponential growth, the rate of change increases over the

domain, but in exponential decay, the rate of change decreases over the domain.

NCTM- Essential Understandings of Functions -grades 9-12,pages 7-11

• Big Idea 2: Covariation and Rate of Change • B.I. 3: Families of Functions and Their Role in Modeling

– Real-World Phenomena – Linear functions– Quadratic functions – Exponential functions – Trigonometric functions

• B.I. 5: Multiple Representations of Functions– Algebraically (equations), geometrically (graphs), Tables,

Word Descriptions

The Learning Principle- NCTM Principles & Standards

• Students must learn mathematics with understanding, actively building new knowledge from experiences and prior knowledge.

NCTM 2000, pg. 16

Functional Thinking Stream: grades 3-7

A learning progression through tasks• Look at tasks from 3rd grade – 7th grade• Match each task with the CCCSS Mathematics

Standard to which the task best aligns. Only one task per standard.

Functional Thinking Stream: grades 3-7

Read the assigned learning progressions for • Operations and Algebraic Thinking (Grades K-5),• Expressions and Equations (6-8)• Ratios and Proportional Relationships (6-7)• With your group (people at your table that have

the same grade level progression), be prepared to share out 2 big ideas that show the essence of the standards at your grade level.

Functional Thinking Stream: grades 3-7

• What new insights do you have about the progression of functional thinking via Operations and Algebraic Thinking to Expressions & Equations to Ratios and Proportional Relationships?– How is understanding being developed and

deepened across the grades?

Reflection

• What is functional thinking? What new insights have you gained? How has your thinking changed?

• Plan task/activity to develop/extend students’ functional thinking. This will be submitted (electronically) for inclusion on the course wikipage

• Focus on one content standard and one standard for math practice

• Resources are found at www.mscsummercourses2013.wikispaces.com

Functional Thinking in Your Classroom

Math & Science Collaborative at the Allegheny Intermediate Unit

• Complete the task/activity you started working on this afternoon.

• Read pages 1-11 and 31-32 in Essential Understanding of E.E.F. Tomorrow morning you will participate in a wiki discussion board and share your thoughts about the assigned reading

• Join wiki- mscsummercourses2013.wikispaces.com.

• Bring a laptop/tablet for days 2 and 3.

Homework

Math & Science Collaborative at the Allegheny Intermediate Unit