Functions and Everyday Situations Performance Task Functions and Everyday Situations http://map.mathshell.org/materials/index.php

Comparing Traditional and Common Core Instruction Function Traditional Definition Table Mapping Vertical Line Test Common Core

Sample Lesson on Function Traditional f(x) x y Sample Lesson on Function Traditional

Function - Definition A function is a relation in which the members of the domain (x-values) DO NOT repeat. So, for every x-value there is only one y-value that corresponds to it. Y-values can be repeated. How do these lessons ensure students understand functions?

Function - Table {(3, 4), (7, 2), (0, -1), (-2, 2), (-5, 0), (3, 3)} How do these lessons ensure students understand functions?

Function - Mapping {(3, 4), (7, 2), (0, -1), (-2, 2), (-5, 0), (3, 3)} -2 -5 4 2 -1 How do these lessons ensure students understand functions?

Function – Vertical Line Test If any vertical line passes through the graphed function at more than one point simultaneously, then that relation is not a function. Are these functions? How do these lessons ensure students understand functions? FUNCTION FUNCTION NOT A FUNCTION

Sample Lesson on Function Common Core f(x) x y Sample Lesson on Function Common Core

Common Core Math Standards Practice Standards Content Standards Instructor Notes: 1 minute 9:05 – 9:10 Here the math CCSS are broken down using a brace map. The brace map helps identify whole and part relationships. It shows the components that make up a whole. The standards for content are grade level specific. They describe what students should know and be able to do at each grade level. The standards for practice are the same through grades K-12, and describe the behaviors that the students should exhibit when working with mathematics. (click) Today we will be exploring the Common Core Standards for Mathematical Practice. When we think about what’s new and different about the CCSS, the practice standards are one of the biggest shifts. To quote the CCSS, “The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important ‘processes and proficiencies’ with longstanding importance in mathematics education.”

Today’s Common Core Math Practices Make sense of problems and persevere in solving them. MP1 MP 1M MP2 Reason abstractly and quantitatively M2 MP4 Model with Mathematics Follow Instructions of slide MP4 MP5 Use appropriate tools strategically MP 5

F-IF F-LE Content Clusters Interpret functions that arise in applications in terms of a context F-IF Analyze functions using different representations Construct and compare linear, quadratic, and exponential models and solve problem F-LE

Unwrap the Standards F-IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in term of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. (Understand, 2), (Apply,2)

Creating a Pathway to our Bloom’s/DOK Identified Level Revised Bloom’s Taxonomy Webb’s DOK Level 1 Recall & Reproduction Webb’s DOK Level 2 Skills & Concepts Webb’s DOK Level 3 Strategic Thinking/ Reasoning Webb’s DOK Level 4 Extended Thinking Remember Retrieve knowledge from long-term memory, recognize, recall, locate, identify Recall, observe, & recognize facts, principles, properties Recall/ identify conversions among representations or numbers (e.g., customary and metric measures)   Understand Construct meaning, clarify, paraphrase, represent, translate, illustrate, give examples, classify, categorize, summarize, generalize, infer a logical conclusion (such as from examples given), predict, compare/contrast, match like ideas, explain, construct models Evaluate an expression Locate points on a grid or number on number line Solve a one-step problem Represent math relationships in words, pictures, or symbols Read, write, compare decimals in scientific notation Specify and explain relationships (e.g., non-examples/examples; cause-effect) Make and record observations Explain steps followed Summarize results or concepts Make basic inferences or logical predictions from data/observations Use models /diagrams to represent or explain mathematical concepts Make and explain estimates Use concepts to solve non-routine problems Explain, generalize, or connect ideas using supporting evidence Make and justify conjectures Explain thinking when more than one response is possible Explain phenomena in terms of concepts Relate mathematical or scientific concepts to other content areas, other domains, or other concepts Develop generalizations of the results obtained and the strategies used (from investigation or readings) and apply them to new problem situations Apply Carry out or use a procedure in a given situation; carry out (apply to a familiar task), or use (apply) to an unfamiliar task Follow simple procedures (recipe-type directions) Calculate, measure, apply a rule (e.g., rounding) Apply algorithm or formula (e.g., area, perimeter) Solve linear equations Make conversions among representations or numbers, or within and between customary and metric measures Select a procedure according to criteria and perform it Solve routine problem applying multiple concepts or decision points Retrieve information from a table, graph, or figure and use it solve a problem requiring multiple steps Translate between tables, graphs, words, and symbolic notations (e.g., graph data from a table) Construct models given criteria Design investigation for a specific purpose or research question Conduct a designed investigation Use & show reasoning, planning, and evidence Translate between problem & symbolic notation when not a direct translation Select or devise approach among many alternatives to solve a problem Conduct a project that specifies a problem, identifies solution paths, solves the problem, and reports results Analyze Break into constituent parts, determine how parts relate, differentiate between relevant-irrelevant, distinguish, focus, select, organize, outline, find coherence, deconstruct Retrieve information from a table or graph to answer a question Identify whether specific information is contained in graphic representations (e.g., table, graph, T-chart, diagram) Identify a pattern/trend Categorize, classify materials, data, figures based on characteristics Organize or order data Compare/ contrast figures or data Select appropriate graph and organize & display data Interpret data from a simple graph Extend a pattern Compare information within or across data sets or texts Analyze and draw conclusions from data, citing evidence Generalize a pattern Interpret data from complex graph Analyze similarities/differences between procedures or solutions Analyze multiple sources of evidence analyze complex/abstract themes Gather, analyze, and evaluate information Evaluate Make judgments based on criteria, check, detect inconsistencies or fallacies, judge, critique Cite evidence and develop a logical argument for concepts or solutions Describe, compare, and contrast solution methods Verify reasonableness of results Gather, analyze, & evaluate information to draw conclusions Apply understanding in a novel way, provide argument or justification for the application Create Reorganize elements into new patterns/structures, generate, hypothesize, design, plan, construct, produce Brainstorm ideas, concepts, or perspectives related to a topic Generate conjectures or hypotheses based on observations or prior knowledge and experience Synthesize information within one data set, source, or text Formulate an original problem given a situation Develop a scientific/mathematical model for a complex situation Synthesize information across multiple sources or texts Design a mathematical model to inform and solve a practical or abstract situation Instructor Notes: 5 -10minutes Now that standard is completed. Circle the levels identified with the standard Then circle in levels to complete a pathway from beginning to end goal Make reference to summative assessment points to end goal Make reference to min. formative assessments as we move through pathway for progress monitoring Touch on the fact that we should always have formative assessments at the level as we progress Make reference to pre-assessments and how they are valuable in saving us time in our planning and instruction

Before the Lesson HANDOUTS # 4 Students work individually on this task that is designed to reveal their current understanding and difficulties. Review the solutions and create questions for students to consider to improve their learning

Before the Lesson Assessing Students’ Responses Review Students’ Response Write 1 or 2 Questions on Individual Student Highlight Appropriate Questions from Guided Questions made by Teacher Write a few Questions that will benefit majority of students on the board Create Questions to Improve Learning

Before the Lesson Create Questions to Improve Learning Common Issues Guided Questions Draws continuous lines for all the graphs Cut the axes at inappropriate places Draws a graph that consists of two straight lines of different slopes Unable to interpret and use the formulas correctly Is X a discrete or continuous variable? Why? How many passengers would you need for Y=0? Why does the steepness of your slope change? What does each statement tell you about the value of X and value of Y?

Whole-Class Introduction Give each student a mini-whiteboard, a pen, and eraser. Ask students to sketch a graph that describes this situation. Can you sketch a graph to show how y will depend on x?

Painting the Bridge What does yellow point mean? What does blue point mean? Which graph represents the given situation? Can you suggest a possible algebraic function for each graphs?

Support Understanding Academic Language – Layered Book

Additional Instructional Strategies for Academic Language – Frayer’s Model Definition Characteristics A function is a relation in which each element of the domain is paired with exactly one element of the range. Another way of saying it is that there is one and only one output (y) with each input (x). Function Examples Non-Examples

Today’s Lesson

Matching Situation to Graph During the Lesson: Matching Situation to Graph 1 Organize the class into Groups of two or three students. 2 Students take turns to match situation card to the sketched graphs. Explain their thinking so everyone in the group will agree. Complete the two blank graphs! 3 Arrange pairs side by side so the teacher could check the understanding. Question students to help any misconceptions.

Whole-Group Discussion Strategy Whole Class Discussion Different strategies to match cards What was learned Focus on Understanding Encourage listening to other explanation Explore the situation in depth Reflect Student Work Whole-Group Discussion Strategy What different strategies students have used to match the cards? Ask students from other groups to contribute ideas of alternative matches. Focus on getting students to understand and share their reasoning and explore the different approaches used within the class When discussing a match, encourage students to listen carefully to each other’s explanations and comment on which explanation was easier to follow. Explore the situation in as much depth as possible.

Matching Situation and Graphs to Formulas During the Lesson: Matching Situation and Graphs to Formulas 4 When students have had a chance to match the situations and graphs, give each group the cut up cards: Algebraic Functions, a large sheet of paper, and glue stick for making a poster 5 Match these cards to the pairs that already have on the table without calculator. 6 After matching the function, try to answer the question on the right hand side of the situation card. 7 Allow students to check their answers using calculators.

Student will do the “Another Four Situations” HANDOUT # 5 After the Lesson Student will do the “Another Four Situations” Check students’ understanding of functions and their types such as continuous or discrete functions

Application to the Everyday Situation Write a Function Story Graph the Function Story and provide the rational. Use academic vocabulary that was learned during the lesson. Discuss the work with your elbow partner. In the Learning Log, describe the difference between graphs of functions and non-functions with examples.

What is Common Core Instruction? Before the Lesson Activity: Check Student Learning During the Lesson: Teaching the Concept with Math Practices After the Lesson: Check for Student Understanding

Let’s compare Traditional vs Common Core Practice Topic: Algebra 1 - Interpreting Functions Traditional Skilled Based Learning Drill and Rote Memory Teacher as Lecturer What else? CCSS Concept Based Learning Apply to Everyday Situation (which requires greater understanding) Teacher as Facilitator 1-2 min Using Algebra 1 as an example, compare the California Standards to the CCSS. Note that there was only one California Standard pertaining to Functions, in the CCSS it covers an entire domain with multiple standards.

Reflect on the Lesson based on Using and Citing Evidence What went well with the lesson? Did the lesson go as envisioned? How did the students respond, in their attitudes and their discussion? What will you do differently next time? How might the structure and pedagogy of the common core lesson carry over to other lessons?

PD Evaluation