Transcript Document

Functions and Everyday Situations
Performance Task
http://map.mathshell.org/materials/index.php
Comparing Traditional and Common Core
Instruction
Function
Common
Core
Traditional
Definition
Table
Mapping
Vertical
Line Test
x
f(x)
Sample Lesson on Function
Traditional
y
Function - Definition
A function is a relation in which the members of the domain (xvalues) DO NOT repeat. So, for every x-value there is only one yvalue 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?
x
3
7
0
-2
-5
3
y
4
2
-1
2
0
3
Function - Mapping
{(3, 4), (7, 2), (0, -1), (-2, 2), (-5, 0), (3, 3)}
7
4
3
3
0
2
-2
0
-5
-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
x
f(x)
Sample Lesson on Function
Common Core
y
Common Core Math Standards
Practice
Standards
Content
Standards
Today’s Common Core Math Practices
MP
MP1
Make sense of problems and persevere in solving them.
1M
M2
MP2
MP4
MP4
MP 5
MP5
Reason abstractly and quantitatively
Model with Mathematics
Use appropriate tools strategically
Content Clusters
F-IF
• Interpret functions that arise in
applications in terms of a context
F-IF
• Analyze functions using different
representations
F-LE
• Construct and compare linear,
quadratic, and exponential models
and solve problem
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
Remember
Retrieve knowledge from long-term memory,
recognize, recall, locate, identify
o
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
o
o
o
o
o
o
Webb’s DOK Level 2
Skills & Concepts
Webb’s DOK Level 3
Strategic Thinking/ Reasoning
Webb’s DOK Level 4
Extended Thinking
o
Specify and explain relationships (e.g., nonexamples/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
o
o
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
o
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
o
Design investigation for a specific purpose or research
question
Conduct a designed investigation
Use concepts to solve non-routine problems
Use & show reasoning, planning, and evidence
Translate between problem & symbolic notation when not a
direct translation
o
o
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
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
o
o
o
o
o
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
o
o
o
Analyze multiple sources of evidence
analyze complex/abstract themes
Gather, analyze, and evaluate information
o
Cite evidence and develop a logical argument for concepts or
solutions
Describe, compare, and contrast solution methods
Verify reasonableness of results
o
Gather, analyze, & evaluate information to draw
conclusions
Apply understanding in a novel way, provide
argument or justification for the application
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
o
Recall, observe, & recognize facts, principles,
properties
Recall/ identify conversions among representations or
numbers (e.g., customary and metric measures)
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
o
o
o
o
o
o
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
o
o
o
o
o
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
o
o
o
o
o
Analyze
Break into constituent parts, determine how parts
relate, differentiate between relevant-irrelevant,
distinguish, focus, select, organize, outline, find
coherence, deconstruct
o
o
o
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
o
o
o
o
o
o
Evaluate
Make judgments based on criteria, check, detect
inconsistencies or fallacies, judge, critique
Create
Reorganize elements into new patterns/structures,
generate, hypothesize, design, plan, construct,
produce
o
o
o
o
o
o
o
o
o
o
Brainstorm ideas, concepts, or perspectives related to a
topic
o
Generate conjectures or hypotheses based on observations or
prior knowledge and experience
o
o
o
o
o
o
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
Synthesize information across multiple sources or
texts
Design a mathematical model to inform and solve a
practical or abstract situation
13
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
Create Questions
to Improve
Learning
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
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
Painting the bridge
A group of workers are planning to paint
a bridge.
x = the number of workers
y = the length of time it will take the
workers to paint the bridge
Painting the Bridge
T
i
m
e
• 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
T
i
m
e
Number of Workers
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
2
2
1
Y
Y
1
0
0
1
2
-1
0
0
1
2
X
3
4
-2
X
3
4
Today’s Lesson
x
f ( x) 
100
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
Different
strategies to
match cards
Reflect Student
Work
What was
learned
Whole
Class
Discussion
Explore the
situation in
depth
Focus on
Understanding
Encourage
listening to
other
explanation
During the Lesson:
Matching Situation and Graphs
to Formulas
4
5
6
7
• 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
• Match these cards to the pairs that already have on the table without calculator.
• After matching the function, try to answer the question on the right hand side of the situation
card.
• Allow students to check their answers using calculators.
After the Lesson
HANDOUT # 5
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
• What else?
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