hrsb, 2008 act ix cohort sohael abidi november 7 th, 2008 day 2

HRSB, 2008HRSB, 2008

ACT IX CohortACT IX Cohort

Sohael AbidiSohael Abidi

November 7November 7thth, 2008, 2008

DAY 2DAY 2


Differentiated InstructionDifferentiated Instruction

WhenWhen, , HowHow & & WhyWhy we we Differentiate within Mathematics Differentiate within Mathematics

Instruction?Instruction?


DifferentiationDifferentiation Mathematics is no longer for the selected Mathematics is no longer for the selected

few. All children few. All children mustmust be expected to be expected to succeed in mathematics.succeed in mathematics.

NCTM NCTM Curriculum StandardsCurriculum Standards

What’s What’s Differentiated InstructionDifferentiated Instruction??

- - It is a chance to offer a variety of learning It is a chance to offer a variety of learning options that address different levels, options that address different levels, interest and learning styles of diverse interest and learning styles of diverse needs of students in mixed ability needs of students in mixed ability classrooms. classrooms.


How will Differentiated Instruction change How will Differentiated Instruction change classroom procedures?classroom procedures?

When students are busy making up their own When students are busy making up their own minds, the role of the teacher shifts. minds, the role of the teacher shifts.

This new focus defines the teacher as one who is:This new focus defines the teacher as one who is:

circulating, redirecting, disciplining, questioning, assessing, circulating, redirecting, disciplining, questioning, assessing, guiding, directing, validating, facilitating, moving, monitoring, guiding, directing, validating, facilitating, moving, monitoring, challenging, motivating, watching, moderating, diagnosing, challenging, motivating, watching, moderating, diagnosing, trouble-shooting, observing, encouraging, suggesting, trouble-shooting, observing, encouraging, suggesting, watching, modeling and clarifying. watching, modeling and clarifying.

http://members.shaw.ca/priscillatheroux/teacherrole.htmlhttp://members.shaw.ca/priscillatheroux/teacherrole.html


The Truth…The Truth…

A student – centered, constructivist approach A student – centered, constructivist approach to teaching & learning can sometimes be:to teaching & learning can sometimes be:

▪ ▪ time consuming time consuming ▪ ▪ messy messy ▪ ▪ inefficientinefficient

The benefits, however, can be…?The benefits, however, can be…?


Benefits For the StudentBenefits For the Student Every student has an opportunity to succeed;Every student has an opportunity to succeed; A single experience with success is enough for a student to A single experience with success is enough for a student to

approach new learning situations with confidence and approach new learning situations with confidence and motivationmotivation

Opportunity is there to discover personal strengths and Opportunity is there to discover personal strengths and show multiple intelligencesshow multiple intelligences

Less frustration due to confusion or boredomLess frustration due to confusion or boredom

Benefits For the TeacherBenefits For the Teacher More sense of control over each student’s learning progressMore sense of control over each student’s learning progress A greater understanding of each students ability to learn A greater understanding of each students ability to learn The reward of having a classroom that allowsThe reward of having a classroom that allows equal equal

opportunity for success for all studentsopportunity for success for all students


Differentiation Builds Upon…Differentiation Builds Upon…

Prior knowledgePrior knowledge Prior experiencesPrior experiences Culturally defined values and normsCulturally defined values and norms Biological differences in cognitive developmentBiological differences in cognitive development Home environmentHome environment Maturity levelMaturity level Self-efficacySelf-efficacy Culturally determined perceptions of school Culturally determined perceptions of school

and learningand learning

http://www.smcm.edu/academics/EdStudy/d7-Proj/Projects/ResearchSites/acbrowning/index.htmhttp://www.smcm.edu/academics/EdStudy/d7-Proj/Projects/ResearchSites/acbrowning/index.htm


Differentiated PlanningDifferentiated Planning Takes effort and practiceTakes effort and practice Begin with the Begin with the “Big Idea,”“Big Idea,” or or “Enduring Questions”“Enduring Questions” Plan learning experiences that aim to enhance Plan learning experiences that aim to enhance

understanding of the understanding of the “Big Idea”“Big Idea” The major concept for every child is the sameThe major concept for every child is the same The teacher plans several ways to approach The teacher plans several ways to approach

learning the same conceptlearning the same concept The “Big Idea” serves as the anchor for the lessonThe “Big Idea” serves as the anchor for the lesson Students travel on different paths of learning but Students travel on different paths of learning but

end up at the same point with an understanding of end up at the same point with an understanding of the same major conceptthe same major concept


A Key to PlanningA Key to Planning The Pre-Assessment The Pre-Assessment (formative assessment) (formative assessment) before the actual before the actual

lesson planninglesson planning Gathering information about what the students already know, Gathering information about what the students already know,

and what they need to learnand what they need to learn The Pre-Assessment paints a picture of the number of students The Pre-Assessment paints a picture of the number of students

who have developed who have developed concept masteryconcept mastery, who show some , who show some understanding, or who show a need for additional focus or understanding, or who show a need for additional focus or instructioninstruction

This information will help determine how many levels of a lesson This information will help determine how many levels of a lesson need to be prepared, or how one could plan a lesson that is need to be prepared, or how one could plan a lesson that is neither above nor below the capabilities of the studentsneither above nor below the capabilities of the students

(see handout: “The Role of (see handout: “The Role of Assessment in a Differentiated Assessment in a Differentiated

Classroom”)Classroom”)http://www.scarsdaleschools.org/mathsymposium/RoleofAssessmentinaDifferentiatedClassroom_Handout.pdfhttp://www.scarsdaleschools.org/mathsymposium/RoleofAssessmentinaDifferentiatedClassroom_Handout.pdf


What we can DifferentiateWhat we can Differentiate

ContentContent refers to the concepts and sub-concepts refers to the concepts and sub-concepts each student learns in a particular lesson.each student learns in a particular lesson.

Process Process refers to the learning experiences that you refers to the learning experiences that you choose to provide for your students to achieve an choose to provide for your students to achieve an understanding of the content in a lesson.understanding of the content in a lesson.

ProductsProducts are the end result of the lesson. Each are the end result of the lesson. Each student applies what she or he has learned in the student applies what she or he has learned in the lesson to create a final product or to show their lesson to create a final product or to show their acquired skillacquired skill


Strategies for DifferentiatingStrategies for Differentiating ContentContent, , ProcessProcess & & ProductProduct

Adjusting QuestionsAdjusting Questions Compacting CurriculumCompacting Curriculum Tiered LessonsTiered Lessons Flexible GroupingFlexible Grouping Others:Others:

Acceleration/Deceleration Acceleration/Deceleration Student Interest Student Interest Peer Teaching Peer Teaching Anchoring Activities Anchoring Activities Learning Profiles/Styles Learning Profiles/Styles Buddy-Studies Buddy-Studies Independent StudyIndependent Study Learning Centers Learning Centers Projects Projects Readiness / Ability Readiness / Ability Learning ContractsLearning Contracts


Strategies to Differentiate:Strategies to Differentiate:Adjusting QuestionsAdjusting Questions

Adjusting the level of complexity of questions while Adjusting the level of complexity of questions while considering:considering:

How does a student understand, define, or explain a task?How does a student understand, define, or explain a task?How does a student organize their approach to the task?How does a student organize their approach to the task?Does the student see relationships?Does the student see relationships?Does the student relate this work to similar problems?Does the student relate this work to similar problems?Does the student vary their approach to different problems?Does the student vary their approach to different problems?Can they describe their strategy?Can they describe their strategy?Does the student show evidence of thinking ahead or Does the student show evidence of thinking ahead or

backwards?backwards?Can the student generalize the process or results?Can the student generalize the process or results?Can the student self-evaluate?Can the student self-evaluate?How does the student work in a group?How does the student work in a group?


THE LOCKER PROBLEMTHE LOCKER PROBLEM A school has 1,000 lockers and 1,000 students. The A school has 1,000 lockers and 1,000 students. The

students decide to have fun one day, so they take turns students decide to have fun one day, so they take turns opening and closing the lockers, according to the following opening and closing the lockers, according to the following plan.plan.

• The first student opens every locker.The first student opens every locker.

• The second student closes every second locker.The second student closes every second locker.

• The third student opens every third closed locker.The third student opens every third closed locker.

• The fourth student closes every fourth open locker.The fourth student closes every fourth open locker.

• The students continue in this manner until all 1,000 students have had The students continue in this manner until all 1,000 students have had their turn.their turn.

When all the students are finished, how many lockers When all the students are finished, how many lockers remain open?remain open?


Strategies to Differentiate:Strategies to Differentiate:

Curriculum CompactingCurriculum CompactingWhen a teacher encounters a student who has already When a teacher encounters a student who has already

mastered a concept that other students have not.mastered a concept that other students have not.

Decide on a level of a achievement that indicates “mastery” Decide on a level of a achievement that indicates “mastery” Pre-assess students to decide who has already “mastered” Pre-assess students to decide who has already “mastered”

the materialthe material Plan enrichment activities or adjust content for learners who Plan enrichment activities or adjust content for learners who

attain a mastery levelattain a mastery level Use a record to keep track of the progress of the studentsUse a record to keep track of the progress of the students



Tiered LessonsTiered LessonsContent is presented at varying levels of complexity, but the Content is presented at varying levels of complexity, but the

process is the same for all studentsprocess is the same for all students

Lesson Tier according to:Lesson Tier according to:

- - students’ readiness (ability to understand a particular level of students’ readiness (ability to understand a particular level of content)content)

- learning profiles (style of learning)- learning profiles (style of learning)- interests (student interest in the topics to be studied)interests (student interest in the topics to be studied)

- Students work in teacher-assigned groups according - Students work in teacher-assigned groups according to the chosen tiering strategy to the chosen tiering strategy ((exex. Topic comprehension) . Topic comprehension)


1. Using grid paper, draw a square that has side lengths of two units. Determine the perimeter and area. Try again with squares of different side lengths. Write about the relationship between the side length of a square and its perimeter and its area.

(E4.1 page 5-81)

2. Explore the area and perimeter of squares and rectangles using the website below*. Make two observations about what you notice about changes in sizes of squares or changes in height or width of rectangles. Record one question you have about area or perimeter.

3. A farmer has 100m of fencing to make a pen for his pigs. He decides that a square or a rectangle would be the best shape. What are some possible sizes he could make? What would the area be of each shape? Which pen would you recommend?

(E4.2 page 5-81)

4. Read the book, Spaghetti and Meatballs for All! (Burns, Marilyn. Spaghetti and Meatballs for All! Brainy Day Books, 1995) Retell the story in your own words to a classmate using tiles to illustrate the main ideas.

5. Make a display to show when area and perimeter are used in the real world. You may cut out pictures or draw them yourself or make a graphic organizer using words or pictures.

6. Use string or masking tape to define a rectangle on the floor with an area large enough for four students to sit comfortably during silent reading. Record the perimeter of the rectangle.

Area and Perimeter



Flexible GroupingsFlexible Groupings Groupings can be decided based upon:Groupings can be decided based upon:- student interest; student readiness; student requeststudent interest; student readiness; student request

Allow students to decide to leave a particular group if Allow students to decide to leave a particular group if appropriate. (students may start at a slow pace, but progress appropriate. (students may start at a slow pace, but progress into the group with less guidance.)into the group with less guidance.)

Avoid labeling within a classroomAvoid labeling within a classroom Accommodate differences within an individual; Accommodate differences within an individual; Are always changing; students should not be able to predict Are always changing; students should not be able to predict

what group they will be assigned towhat group they will be assigned to


Differentiating from our TextbookDifferentiating from our TextbookMathematics 9: Focus on UnderstandingMathematics 9: Focus on Understanding

Refer to your differentiating handoutsRefer to your differentiating handouts With a partner, complete each question.With a partner, complete each question. Differentiate each question by creating 3 versions of Differentiate each question by creating 3 versions of

each questioneach question Create one version for students at three different Create one version for students at three different

ability levelsability levels Share with your tableShare with your table Group shareGroup share


Looking at the TR CDLooking at the TR CD Refer to Differentiation From Textbook Refer to Differentiation From Textbook

handoutshandouts With a partner, look at the resource With a partner, look at the resource

handouts printed from the TR CDhandouts printed from the TR CD Match the differentiated handouts to their Match the differentiated handouts to their

corresponding textbook questioncorresponding textbook question Discuss the differentiation evident for each Discuss the differentiation evident for each

question question (type, pros, cons, changes etc.)(type, pros, cons, changes etc.)

Group TR CD surf!Group TR CD surf!


Effective Questioning:Effective Questioning:

Raising the Raising the Cognitive Cognitive DemandDemand


What Role does What Role does Questioning play in Questioning play in our daily lessons?our daily lessons?


Effective Questioning:Effective Questioning: Helps control the flow of informationHelps control the flow of information Keeps students focused on important mathematical Keeps students focused on important mathematical

ideasideas Helps students make sense of mathematicsHelps students make sense of mathematics Moves discussions from discrete, unrelated responses Moves discussions from discrete, unrelated responses

to in-depth dialogueto in-depth dialogue Supports and encourages student thinkingSupports and encourages student thinking

* Ultimately, effective questioning * Ultimately, effective questioning helps to raise the ‘helps to raise the ‘cognitive demand’cognitive demand’ placed on our students during our placed on our students during our daily lessonsdaily lessons


Activity: Activity:

“The Ways I Ask for “The Ways I Ask for Information”Information”

Questioning handout 1Questioning handout 1 Write questions and purposes Write questions and purposes

independentlyindependently Complete & share with a partnerComplete & share with a partner Group sharing sessionGroup sharing session


Discussion: What is the Discussion: What is the Purpose?Purpose?

One can categorize questions in many ways, but One can categorize questions in many ways, but an important part of questioning is:an important part of questioning is:

The Teacher’s The Teacher’s PurposePurpose for for asking the questionsasking the questions

That is, what is the teacher trying to accomplish?That is, what is the teacher trying to accomplish? In partners, think about a specific problem where In partners, think about a specific problem where

questions are designed merely to get answers ‘on questions are designed merely to get answers ‘on the table.’the table.’

Jot down the example and some questions that Jot down the example and some questions that may be asked.may be asked.

Now create questions requiring higher level Now create questions requiring higher level thinking for your example…raise the thinking for your example…raise the cognitive cognitive demanddemand of your questions. of your questions.


Classifying Questions based on Classifying Questions based on PurposePurpose

Engaging:Engaging: invite students into a discussion; keep invite students into a discussion; keep them engaged in conversation; them engaged in conversation;

invite invite them to share their work, or get them to share their work, or get answers answers on the table on the table

Refocusing:Refocusing: help students get back on track or help students get back on track or movemove

away from a dead-end strategyaway from a dead-end strategy

Clarifying:Clarifying: help students explain their thinking or help students explain their thinking or

help you understand their thinkinghelp you understand their thinking


DiscussionDiscussion Thoughts? Is there a ‘grey-area?’Thoughts? Is there a ‘grey-area?’ In partners, describe an instructional In partners, describe an instructional

situation that you have experienced situation that you have experienced and give an example of a question from and give an example of a question from each category.each category.

Would an observer in your class know Would an observer in your class know what your purpose is for each of these what your purpose is for each of these questions?questions?


Examples:Examples:

EngagingEngaging““What strategies might we use to solve this problem?”What strategies might we use to solve this problem?”

““If you wanted to graphIf you wanted to graph this function, how would you this function, how would you label the axis?”label the axis?”

RefocusingRefocusing – ( – (if students are if students are stuckstuck working on): working on):

Similar Figures – Similar Figures – ““What does it meanWhat does it mean for two figures for two figures

to be similar?”to be similar?”

Ratio – Ratio – “What quantities are you “What quantities are you comparing?”comparing?”

ClarifyingClarifying ““How did you figure out your answer?”How did you figure out your answer?”

““Why did you start with that number?”Why did you start with that number?”


TaskTask

Revisit your “Asking for Information” Revisit your “Asking for Information” sheetsheet

Determine the purpose for asking each Determine the purpose for asking each of your written questionsof your written questions

ex. Engaging, Refocusing, ex. Engaging, Refocusing, ClarifyingClarifying


DiscussionDiscussion VolunteersVolunteers: : Read one of your questions, Read one of your questions,

identify its category, and explain why it is the identify its category, and explain why it is the correct categorycorrect category

Comments??Comments?? Which category was most popular based on Which category was most popular based on

our examples? Least popular? Why might this our examples? Least popular? Why might this be?be?

Could one question fall into more than one Could one question fall into more than one category? Let’s discuss!category? Let’s discuss!


Fraction TracksFraction Tracks

0 1

0

0

0

0

0

0

1

1

1

1

1

1

DEMO:

5/6


Time to Play!Time to Play!

Fraction Tracks HandoutFraction Tracks Handout With a partner, alternating turns With a partner, alternating turns

selecting fraction cards, try and be selecting fraction cards, try and be the first person to move all 7 chips the first person to move all 7 chips from 0 to 1 on their respective from 0 to 1 on their respective number lines.number lines.

GO!GO!


‘‘Post-Game Summary’Post-Game Summary’

Sample Student Card & Comment:

“I drew a card with the fraction 8/8 on it. I don’t know what to do since I can’t use 8/8 on either the fourths or the eighths track.”


Below are a series of questions that you may have Below are a series of questions that you may have asked this student. Read the student answers and asked this student. Read the student answers and decide whether or not the question was effective. decide whether or not the question was effective. Think of another question that could follow up each Think of another question that could follow up each student answer.student answer.

TeacherTeacher:: Why can't you use the entire 8/8 on either of the tracks? Why can't you use the entire 8/8 on either of the tracks?

StudentStudent:: I can't use 8/8 on the fourths track because there's I can't use 8/8 on the fourths track because there's already a already a marker on 1/4, and I can't use 8/8 on the marker on 1/4, and I can't use 8/8 on the eighths track because eighths track because there's already a marker on 2/8. there's already a marker on 2/8.

TeacherTeacher: Can you use part of the 8/8 on either track?: Can you use part of the 8/8 on either track? Student: Student: YesYes. .

TeacherTeacher:: How much could you use on the fourths track? How much could you use on the fourths track?

Student:Student: The marker's at 1/4, so I could use 3/4. The marker's at 1/4, so I could use 3/4.


TeacherTeacher: What is 3/4 equivalent to on the eighths track? : What is 3/4 equivalent to on the eighths track?

Student:Student: It's also 6/8, so I could use 6/8 on the eighths track. It's also 6/8, so I could use 6/8 on the eighths track.


Discussion & Debrief: Discussion & Debrief: Video Video TaskTask

Watch the Watch the Fraction Tracks Fraction Tracks video: video: (1:16-11:16)(1:16-11:16)

http://www.learner.org/resources/series31.htmlhttp://www.learner.org/resources/series31.html

Using the “Case Study” handout:Using the “Case Study” handout:- jot down any questions that are asked by the teacher; - jot down any questions that are asked by the teacher;

- include what you believe was the intended purpose of - include what you believe was the intended purpose of

each question (each question (engagingengaging, , refocusingrefocusing, , clarifying)clarifying)

- after the video, share with a partner- after the video, share with a partner

- group sharegroup share

- DiscussionDiscussion: : volunteers to identify volunteers to identify questions, and how/why questions, and how/why

they classified the purposes they classified the purposes of these questionsof these questions


Post-Video Wrap-upPost-Video Wrap-up Do the students seem to understand the mathematical ideas? Do the students seem to understand the mathematical ideas? Does the teacher ask questions that elicit, extend, and challenge Does the teacher ask questions that elicit, extend, and challenge

the students' thinking? the students' thinking? What other questions could the teacher have asked this student What other questions could the teacher have asked this student

in order to further challenge her mathematical thinking? in order to further challenge her mathematical thinking? Do you think the teacher's questioning of the students will help Do you think the teacher's questioning of the students will help

other students in the class develop their understanding of other students in the class develop their understanding of fractions? fractions?

Do you think the teacher listens, responds, and adapts her Do you think the teacher listens, responds, and adapts her questions effectively? questions effectively?

What does the teacher do when a student gives an incorrect What does the teacher do when a student gives an incorrect answer? answer?

Do you think the students understand the final answer, or are Do you think the students understand the final answer, or are they just following the teacher's prompting without they just following the teacher's prompting without understanding? understanding?

Do you think this is an effective activity through which students Do you think this is an effective activity through which students learn important mathematical ideas? learn important mathematical ideas?

How could you modify or extend this activity to teach other How could you modify or extend this activity to teach other fraction concepts?fraction concepts?


When Literacy Meets Math…When Literacy Meets Math… Complete the Complete the griney grollersgriney grollers activity activity (see handout)(see handout)

Please Please after question #7.after question #7.STOP


DiscussionDiscussion

What types of questions were found in #1- 7?What types of questions were found in #1- 7? What was their purpose? (if any)What was their purpose? (if any) How meaningful were they in assessing How meaningful were they in assessing

student understanding and/or learning?student understanding and/or learning? Now, continue with #8 thinking about how Now, continue with #8 thinking about how

we can ask questions that increase the we can ask questions that increase the cognitive demand on our students.cognitive demand on our students.

Using this example, what types of questions Using this example, what types of questions could we ask that would promote higher could we ask that would promote higher levels of student thinking and understanding?levels of student thinking and understanding?


HandoutsHandouts I stress the importance of having a I stress the importance of having a “Question “Question

Toolbox”Toolbox” ““Supporting Teachers in Asking Questions Supporting Teachers in Asking Questions

and Choosing Tasks”and Choosing Tasks” ““Developing Mathematical Thinking with Developing Mathematical Thinking with

Effective Questions”Effective Questions” ““Questions that Probe Understanding”Questions that Probe Understanding” ““Motivating Every Student Through Effective Motivating Every Student Through Effective

Questioning”Questioning”

- “Summary of Questioning Techniques- “Summary of Questioning Techniques””

(review with the group)(review with the group)


Higher Level Questioning within Higher Level Questioning within our daily activitiesour daily activities

Refer to the Refer to the “Examples of Good Questioning”“Examples of Good Questioning” handouthandout

Please complete number one, Please complete number one, (top of sheet),(top of sheet), on on your own.your own.

Share strategies with a partner and discuss Share strategies with a partner and discuss the questions that were asked the questions that were asked (i.e. purpose, (i.e. purpose, intension, level of required student thinking etc.)intension, level of required student thinking etc.)

If time, do the same for question two If time, do the same for question two (bottom (bottom of sheet)of sheet)

The Cognitive The Cognitive Demand!Demand!

HRSB, 2008

Scatter plots – Scatter plots – Line of Best FitLine of Best FitRegression!Regression!

A graph of ordered pairs of numeric dataA graph of ordered pairs of numeric data Used to see relationships between two Used to see relationships between two

variables or quantitiesvariables or quantities Helps determine the Helps determine the correlation correlation between between

the Independent & dependent variablesthe Independent & dependent variables CorrelationCorrelation: a measure of how closely the : a measure of how closely the

points on a scatter plot fit a linepoints on a scatter plot fit a line The relationship can be strong, weak, positive The relationship can be strong, weak, positive

or negativeor negative + Correlation – As indep.Var + Correlation – As indep.Var ↑, Dep. Var ↑↑, Dep. Var ↑ - Correlation – As indep. Var ↑, Dep. Var ↓- Correlation – As indep. Var ↑, Dep. Var ↓

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Line of Best FitLine of Best Fit Drawn through as many data points as possibleDrawn through as many data points as possible Aim to have an equal amount of data points Aim to have an equal amount of data points

above and below the lineabove and below the line Does NOT have to go through the originDoes NOT have to go through the origin Allows us to generate an equation that Allows us to generate an equation that

describes the relationship using an equation describes the relationship using an equation form form (ie: y = mx+b)(ie: y = mx+b)

Example 1, Pink Sheet 1Example 1, Pink Sheet 1 – – Discuss (draw LOBF for each)Discuss (draw LOBF for each)

Example 2, Pink Sheet 1, Example 2, Pink Sheet 1, Let’s do together using Let’s do together using thethe

TI-83+TI-83+

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Calculator Applications: Calculator Applications: 10.10.(pg. 38-42 Booklet)(pg. 38-42 Booklet)

Example 2: Line of Best FitExample 2: Line of Best Fit

1.1. 2. 2. 3. 3.

4.4. 5. 6. 5. 6.

7.7. 8. 9. 8. 9.

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Linear Regression & Correlation Coefficient Linear Regression & Correlation Coefficient (r) (r)

Determining the Equation for the Line of best fit can Determining the Equation for the Line of best fit can be referred to as: be referred to as: Regression AnalysisRegression Analysis

We create a model that can be used to predict values We create a model that can be used to predict values of the Dep. Var. based on values of the Indep. Var.of the Dep. Var. based on values of the Indep. Var.

The ‘r’ value – Correlation CoefficientThe ‘r’ value – Correlation Coefficient

- measures the strength of the association of the 2 - measures the strength of the association of the 2 variables;variables;

(-1 (-1 → +1) – the closer to either, the stronger the → +1) – the closer to either, the stronger the relationshiprelationship

Pink Sheet 3Pink Sheet 3 – – complete in table groups –complete in table groups –

(steps on page 4, 5 pink sheets)(steps on page 4, 5 pink sheets)

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Regression AnalysisRegression AnalysisPg.383, Gr. 9 Text, #13Pg.383, Gr. 9 Text, #13

WindowWindow Scatter plot Scatter plotCorrelationCorrelation

EquationEquation GraphGraph

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Extrapolating data:Extrapolating data: Determining # injured in 2010:Determining # injured in 2010:Change ‘window’ to include this x Change ‘window’ to include this x

parameterparameter(Xmax – 2050) The new graph:(Xmax – 2050) The new graph:Next Key Strokes:Next Key Strokes:22ndnd CALC 1:value CALC 1:value

Type in 2010Type in 2010

Y value when x = 2010, is Y value when x = 2010, is

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Regression Analysis Cont.Regression Analysis Cont.

Example 3, 4: Pink Sheet 3 -Example 3, 4: Pink Sheet 3 - EXTENSIONEXTENSION

- Looking at Parabolic & Exponential Looking at Parabolic & Exponential RelationshipsRelationships

- Complete these problems togetherComplete these problems together

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Algebra Tiles1 -1

x -x

x2-x2

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Algebra TilesAlgebra Tiles

y -y

xy

-y2y2

-xy

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Question 1Question 1

)73()423( 22 yxyx

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)73()423( 22 yxyx

Display first polynomial

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)73()423( 22 yxyx


Add to your display the second

polynomial

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)73()423( 22 yxyx

Simplify by combining like terms

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)73()423( 22 yxyx

Remove “zeros”

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What remains is the answer!

)73()423( 22 yxyx

352 2 yx

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)122()534( yxyx

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)122()534( yxyx

Display first polynomial expression

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)122()534( yxyx

Remove second polynomial expression

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)122()534( yxyxWhat is left?

42 yx

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)33()424( 22 xxxx

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)33()424( 22 xxxx

Display first polynomial expression

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x

)33()424( 22 xxxx

Can we remove the second?

We cannot remove an if is not in the original display…

x

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x

)33()424( 22 xxxxWe have to insert into the display

without changing the value . . .

Add in zero . . . (made of a positive and a negative )

x

x

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)33()424( 22 xxxxNow we can remove the second polynomial

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)33()424( 22 xxxx


132 xx

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)3()122( yxyyxy

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)3()122( yxyyxy


Can you remove the second?

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)3()122( yxyyxy

We cannot remove -3

Add in -3 and +3 (they make zero and will not change the value)

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)3()122( yxyyxyNow we can remove all of

the second polynomial

Your answer is what remains

4 yxy

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)47()513( yy

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)47()513( yy

Display the first expression

Can you remove the second expression?

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)47()513( yy

Yes!

Go ahead and remove the second expression

The answer is what remains!

16 y

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)22()35( 22 yxxx

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)22()35( 22 yxxx


Can we remove the second expression?

Not the way it is!

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)22()35( 22 yxxx

We need to add in every term from the second expression.Remember to pair these terms with

their opposite so the value of the

expression does not change.

Double Check that you have not changed the value of the original expression . . .

HRSB, 2008

)22()35( 22 yxxx

Now remove the second expression

Your answer is what remains

526 2 yxx

HRSB, 2008


)22()33( 2 yxy

HRSB, 2008

)22()33( 2 yxy


Can we remove the second?

Not the way it is!

HRSB, 2008

)22()33( 2 yxy

Add in the tiles needed.

Make sure the value of the expression has not changed

HRSB, 2008

)22()33( 2 yxy

Remove the second expression.

What remains is the answer.

52 yx

HRSB, 2008


)12()322( yxyyxy

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)12()322( yxyyxy


Add to your display the terms of the

second expression

Combine like terms

HRSB, 2008

)12()322( yxyyxy

Remove “zeros”

What remains is the answer

24 yxy

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)32()233( 222 xyxxy

HRSB, 2008

)32()233( 222 xyxxy

Display first expression

Add the second expression to your

display

HRSB, 2008

)32()233( 222 xyxxy

Combine like terms

Remove any zeros

What remains is the answer

xxy 22 35

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)32()2( xyxyxyxy

HRSB, 2008

)32()2( xyxyxyxy

Display first expression

See if all of the terms from the second expression can be

removed

Not the way it is!

HRSB, 2008

)32()2( xyxyxyxy

Insert zeros as needed

HRSB, 2008

)32()2( xyxyxyxy Remove second expression


xyxy 23

HRSB, 2008

How can we simplify How can we simplify the following expression?the following expression?

)2(3 x

2 of sets 3

means expression This

x

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times three2,x

make totiles-alge use

desk,your On

HRSB, 2008

expression hesimplify t to

termslike combine Now,

simplifiedwhen

63)2(3

thatseecan We

xx

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Can you arrange the tile display to Can you arrange the tile display to create a rectangle?create a rectangle?

What are the dimensions of the rectangle or area model

you created?

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What are the dimensions of the What are the dimensions of the rectangle you created?rectangle you created?

)2(by 3 xThe dimensions are:

3

2x

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Example 2Example 2

4)-2(3x

addition. repeated using

below expression heSimplify t

HRSB, 2008

Remember repeated addition Remember repeated addition means displaying the polynomial as means displaying the polynomial as

many times as indicated by the many times as indicated by the scalarscalar

times2

)43( show

means

)43(2

x

x

HRSB, 2008

times two4,3x

make totiles-alge use

desk,your On

HRSB, 2008

expression hesimplify t to

termslike combine Now

86)43(2

thatseecan We

xx

HRSB, 2008

Can you arrange the tile display to Can you arrange the tile display to form a rectangle?form a rectangle?

Notice that the dimensions are the factors

of the original question

)43(by 2 x

HRSB, 2008

Example 3Example 3

Use repeated addition to simplify Use repeated addition to simplify

the following expressionthe following expression

)12(3 x

HRSB, 2008

)12(3 x

tiles-algeyour with thisShow

12 of sets 3

means hisRemember t

x

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)12(3 xRearrange to combine like terms

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)12(3 xState your answer

36 x

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Can you rearrange your tile display Can you rearrange your tile display to make a rectangle?to make a rectangle?

12 x

3

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Example 4Example 4

Make an area model Make an area model to answer the next questionto answer the next question

)2(4 x

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Example 4Example 4

Use the two factors to create Use the two factors to create the length and width of a the length and width of a

rectangle.rectangle.)2(4 x

)2( islength theand

4 is width The

x

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4

2 xThen fill it in to make a

complete rectangle

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4

2 xWhat is the area?

84 x

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What do you see?What do you see?

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Number 1Number 1

What are the dimensions of the rectangle?

What is the area?

What can you say about the partial products?

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Number 2

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Number 3Number 3

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Number 4Number 4

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Number 5Number 5

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Number 6Number 6

hrsb, 2008 act ix cohort sohael abidi november 7 th, 2008 day 2

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