Performance Task Functions and Everyday Situations http://map.mathshell.org/materials/index.php
Dec 21, 2015
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 FunctionTraditional
f(x)x y
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)}
x y 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)}
7
3
0
-2
-5
4
3
2
0
-1
How do these lessons ensure students understand functions?
Function – Vertical Line TestIf any vertical line passes through the graphed function at more than
one point simultaneously, then that relation is not a function. Are these functions?
FUNCTION FUNCTION NOT A FUNCTION
How do these lessons ensure students understand functions?
Sample Lesson on FunctionCommon Core
f(x)x y
Common Core Math Standards
PracticeStandards
ContentStandards
Today’s Common Core Math Practices
MP 1M
M2
MP4
MP 5
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively
Model with Mathematics
Use appropriate tools strategically
MP1
MP2
MP4
MP5
Content Clusters
• Interpret functions that arise in applications in terms of a contextF-IF
• Analyze functions using different representationsF-IF
• Construct and compare linear, quadratic, and exponential models and solve problem
F-LE
Unwrap the Standards
• 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)
F-IF.4
Creating a Pathway to our Bloom’s/DOK Identified Level
13
Revised Bloom’s Taxonomy Webb’s DOK Level 1Recall & Reproduction
Webb’s DOK Level 2Skills & Concepts
Webb’s DOK Level 3Strategic Thinking/ Reasoning
Webb’s DOK Level 4Extended Thinking
RememberRetrieve knowledge from long-term memory, recognize, recall, locate, identify
o Recall, observe, & recognize facts, principles, properties
o Recall/ identify conversions among representations or numbers (e.g., customary and metric measures)
UnderstandConstruct 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
o Evaluate an expressiono Locate points on a grid or number on number lineo Solve a one-step problemo Represent math relationships in words, pictures, or
symbolso Read, write, compare decimals in scientific notation
o Specify and explain relationships (e.g., non-examples/examples; cause-effect)
o Make and record observationso Explain steps followedo Summarize results or conceptso Make basic inferences or logical predictions from
data/observationso Use models /diagrams to represent or explain mathematical
conceptso Make and explain estimates
o Use concepts to solve non-routine problemso Explain, generalize, or connect ideas using supporting
evidenceo Make and justify conjectureso Explain thinking when more than one response is possibleo Explain phenomena in terms of concepts
o Relate mathematical or scientific concepts to other content areas, other domains, or other concepts
o Develop generalizations of the results obtained and the strategies used (from investigation or readings) and apply them to new problem situations
ApplyCarry out or use a procedure in a given situation; carry out (apply to a familiar task), or use (apply) to an unfamiliar task
o Follow simple procedures (recipe-type directions)o Calculate, measure, apply a rule (e.g., rounding)o Apply algorithm or formula (e.g., area, perimeter)o Solve linear equationso Make conversions among representations or numbers,
or within and between customary and metric measures
o Select a procedure according to criteria and perform ito Solve routine problem applying multiple concepts or decision
pointso Retrieve information from a table, graph, or figure and use it
solve a problem requiring multiple stepso Translate between tables, graphs, words, and symbolic
notations (e.g., graph data from a table)o Construct models given criteria
o Design investigation for a specific purpose or research question
o Conduct a designed investigationo Use concepts to solve non-routine problemso Use & show reasoning, planning, and evidenceo Translate between problem & symbolic notation when not a
direct translation
o Select or devise approach among many alternatives to solve a problem
o Conduct a project that specifies a problem, identifies solution paths, solves the problem, and reports results
AnalyzeBreak into constituent parts, determine how parts relate, differentiate between relevant-irrelevant, distinguish, focus, select, organize, outline, find coherence, deconstruct
o Retrieve information from a table or graph to answer a question
o Identify whether specific information is contained in graphic representations (e.g., table, graph, T-chart, diagram)
o Identify a pattern/trend
o Categorize, classify materials, data, figures based on characteristics
o Organize or order datao Compare/ contrast figures or datao Select appropriate graph and organize & display datao Interpret data from a simple grapho Extend a pattern
o Compare information within or across data sets or textso Analyze and draw conclusions from data, citing evidenceo Generalize a patterno Interpret data from complex grapho Analyze similarities/differences between procedures or
solutions
o Analyze multiple sources of evidenceo analyze complex/abstract themeso Gather, analyze, and evaluate information
EvaluateMake judgments based on criteria, check, detect inconsistencies or fallacies, judge, critique
o Cite evidence and develop a logical argument for concepts or solutions
o Describe, compare, and contrast solution methodso Verify reasonableness of results
o Gather, analyze, & evaluate information to draw conclusions
o Apply understanding in a novel way, provide argument or justification for the application
CreateReorganize elements into new patterns/structures, generate, hypothesize, design, plan, construct, produce
o Brainstorm ideas, concepts, or perspectives related to a topic
o Generate conjectures or hypotheses based on observations or prior knowledge and experience
o Synthesize information within one data set, source, or texto Formulate an original problem given a situationo Develop a scientific/mathematical model for a complex
situation
o Synthesize information across multiple sources or texts
o Design a mathematical model to inform and solve a practical or abstract situation
Before the Lesson
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
HANDOUTS # 4
Before the LessonAssessing 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
• 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
Guided Questions
• 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
Number of Workers
Ti
me
• 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?
Number of Workers
Ti
me
Support UnderstandingAcademic Language – Layered Book
Examples Non-ExamplesFunction
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).
CharacteristicsDefinition
-2
-1
0
1
2
0 1 2 3 4
Y
X
0
1
2
0 1 2 3 4
Y
X
Additional Instructional Strategies for Academic Language – Frayer’s Model
Today’s Lesson
100)(
xxf
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.
During the Lesson:Matching Situation to Graph
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
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.
During the Lesson:Matching Situation and Graphs to Formulas
After the Lesson
Student will do the “Another Four Situations”
Check students’ understanding of functions and their types such as continuous or discrete functions
HANDOUT # 5
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• What else?
Reflect on the Lesson based onUsing 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