1. Functions and Everyday Situations
Performance Task
Functions and Everyday Situations

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

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

4. 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?

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

6. 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?

7. 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

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

9. 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.”

10. 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

11. 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

12. 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)

13. 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

14. 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

15. 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

16. 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?

17. 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?

18. 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?

19. Support Understanding Academic Language – Layered Book

20. 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

21. Today’s Lesson

22. 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.

23. 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.

24. 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.

25. 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

26. 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.

27. 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

28. 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.

29. 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?

30. PD Evaluation