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