UNIT OF STUDY-2mrstephenshillsideclass.weebly.com/uploads/8/5/4/7/... · · 2017-01-04UNIT OF...

UNIT OF STUDY-2GSE ADV. ALGEBRA/ ALGEBRA IIMATHEMATICSTEACHER RESOURCE GUIDE

OPERATIONS WITH POLYNOMIALS:(September 6 – October 6)

This unit develops the structural similarities between the system of polynomials and the systemof integers. Students draw on analogies between polynomial arithmetic and base-tencomputation, focusing on properties of operations, particularly the distributive property.Students connect multiplication of polynomials with multiplication of multi-digit integers, anddivision of polynomials with long division of integers. Students will find inverse functions andverify by composition that one function is the inverse of another function.

APS Literacy

Page | 1Some of the language used in this document is adapted from the GA Frameworks and Common Core Progressions Documents.

Table of Contents

Standards Addressed ................................................................................................................................... 2

Enduring Understandings.............................................................................................................................5

Lesson One Progression ...............................................................................................................................6

Lesson Two Progression .............................................................................................................................27

Lesson Three Progression...........................................................................................................................35

Sample Fluency Strategies for High School ...............................................................................................44


StandardsAddressed

Duration: (maximum of 13 instructional days on an A/B schedule)

Georgia Standards of Excellence - MathematicsContent Standards

(Cluster emphasis is indicated by the following icons. Please note that 70% ofthe time should be focused on the Major Content. ◊ Major Content □Supporting Content)

Perform arithmetic operations on polynomials

MGSE9-12.A.APR.1 Add, subtract, and multiply polynomials; understand thatpolynomials form a system analogous to the integers in that they are closed underthese operations.

Use polynomial identities to solve problems

MGSE9-12.A.APR.5 Know and apply that the Binomial Theorem gives the expansionof (x + y)n in powers of x and y for a positive integer n, where x and y are anynumbers, with coefficients determined using Pascal’s Triangle.

□ Rewrite rational expressions

MGSE9-12.A.APR.6 Rewrite simple rational expressions in different forms usinginspection, long division, or a computer algebra system; write a(x)/b(x) in the formq(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x)less than the degree of b(x).

◊ Build a function that models a relationship between two quantities

MGSE9-12.F.BF.1 Write a function that describes a relationship between twoquantities.

MGSE9-12.F.BF.1b Combine standard function types using arithmetic operations incontextual situations (Adding, subtracting, and multiplying functions of differenttypes).

MGSE9-12.F.BF.1c Compose functions. For example, if T(y) is the temperature in theatmosphere as a function of height, and h(t) is the height of a weather balloon as afunction of time, then T(h(t)) is the temperature at the location of the weatherballoon as a function of time.


Build new functions from existing functions

MGSE9-12.F.BF.4 Find inverse functions.

MGSE9-12.F.BF.4a Solve an equation of the form f(x) = c for a simple function f thathas an inverse and write an expression for the inverse. For example, f(x) =2(x3) orf(x) = (x+1)/(x-1) for x ≠ 1.

MGSE9-12.F.BF.4b Verify by composition that one function is the inverse of another.

MGSE9-12.F.BF.4c Read values of an inverse function from a graph or a table, giventhat the function has an inverse.

Standards for Mathematical Practice

SMP2: Reason Abstractly and Quantitatively.Students are able to represent a situation symbolically using function notationand compose new functions while making meaning of the composition from areal-world situation.

Students: Because Teachers: Use mathematical symbols to

represent situations Take quantities out of context to

work with them (decontextualizing) Put quantities back in context to

see if they make sense(contextualizing)

Consider units when determining ifthe answer makes sense in terms ofthe situation

Provide a variety of problems indifferent contexts that allow studentsto arrive at a solution in different ways

Use think aloud strategies as theymodel problem solving

Attentively listen for strategiesstudents are using to solve problems

SMP 4: Modeling with MathematicsStudents use mathematical representations to describe situations withpolynomial equations, graphs, tables or diagrams. They interpret the results of acontextual situation arising from composing new functions and reflect onwhether to improve or revise the model.

Students: Because Teachers: Use mathematical models (i.e.

formulas, equations, symbols) tosolve problems in the world

Use appropriate tools such asobjects, drawings, and tables to

Provide opportunities for students tosolve problems in real life contexts

Identify problem solving contextsconnected to student interests


create mathematical models Make connections between

different mathematicalrepresentations (concrete, verbal,algebraic, numerical, graphical,pictorial, etc.)

Check to see if an answer makessense within the context of asituation and changing the modelas needed

SMP 5: Use appropriate tools strategicallyStudents incorporate the use of algebra tiles to concretely represent operationswith polynomials. They use the graphing calculator to explore theirunderstanding of function combinations and inverses. They verify functioninverses by looking for rotational symmetry about the origin.

Students: Because Teachers: Use technological tools to explore

and deepen understanding ofconcepts

Decide which tool will best helpsolve the problem. Examples mayinclude:

o Calculatoro Concrete modelso Digital Technologyo Pencil/papero Ruler, compass, protractor

Estimate solutions before using atool

Compare estimates to solutions tosee if the tool was effective

Make a variety of tools readilyaccessible to students and allowingthem to select appropriate tools forthemselves

Help students understand the benefitsand limitations of a variety of mathtools

SMP7: Look for and make use of structure.Students look for patterns and apply general operation rules to polynomials.They understand that polynomials are closed under addition, subtraction, andmultiplication. They see complex polynomials as being composed of multiplelinear binomials.

Students: Because Teachers: Find structure and patterns in

numbers Find structure and patterns in

diagrams and graphs Use patterns to make rules about

math

Provide sense making experiences forall students

Allow students to do the work of usingstructure to find the patterns forthemselves rather than doing thiswork for students


Use these math rules to help themsolve problems

SMP8: Look for and express regularity in repeated reasoning.Students demonstrate repeated reasoning when expanding and multiplying(x + y)n . They understand the broader application of Pascal’s triangle byidentifying the patterns and making connections.

Students: Because Teachers: Looking for patterns when working

with numbers, diagrams, tables,and graphs

Observing when calculations arerepeated

Using observations from repeatedcalculations to take shortcuts

Providing sense making experiencesfor all students

Allowing students to do the work offinding and using their own shortcutsrather than doing this work forstudents

Note: All of the Standards for Mathematical Practice (SMPs) are critical to students fullyand appropriately attending to the content. Not all SMPs will occur in every lesson,however SMPs 1, 3, and 6 should be regularly apparent. All SMPs should be taught intandem with the content standards.

EnduringUnderstandings

Viewing an expression as a result of operations on simpler expressions cansometimes clarify its underlying structure.

An important use of functions is modeling real-world phenomenon. The inverse to a simple function has a distinct relationship to the original function

that can be verified graphically.


Lesson One ProgressionDuration 4 days

Focus Standard(s)

Perform arithmetic operations on polynomials

MGSE9-12.A.APR.1 Add, subtract, and multiply polynomials; understand that polynomials form a systemanalogous to the integers in that they are closed under these operations.

Use polynomial identities to solve problems

MGSE9-12.A.APR.5 Know and apply that the Binomial Theorem gives the expansion of(x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficientsdetermined using Pascal’s Triangle.

Rewrite rational expressions

MGSE9-12.A.APR.6 Rewrite simple rational expressions in different forms using inspection, long division, or acomputer algebra system; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) arepolynomials with the degree of r(x) less than the degree of b(x).

Performance-Based ObjectivesAs a result of their engagement with this unit…

MGSE9-12.A.APR.1 – SWBAT identify that the sum, difference, or product of two polynomials will alwaysbe a polynomial IOT define closure and justify how the set of integers and the system of polynomials areanalogous.

MGSE9-12.A.APR.5 – SWBAT expand binomials of the form (x + y)n IOT determine patterns in powers andcoefficients and use these patterns to explain the relationship between polynomial coefficients and termsin Pascal’s Triangle.

MGSE9-12.A.APR.6 – SWBAT determine the quotient and remainder of a rational expression usinginspection, long division, and/or a computer algebra system IOT rewrite simple rational expressions indifferent forms and draw conclusions about the overall structure and patterns inherent in dividingpolynomials.


Building CoherenceAcross Grades:

Within Grades:

Key Terms and Definitions Coefficient: a number multiplied by a variable Degree: the greatest exponent of its variable Leading Coefficient: The coefficient of the term

with highest degree. When a polynomial iswritten in standard form the leading coefficientwill be the coefficient of the first term. If you referto the definition of polynomial na is the leadingcoefficient.

Monomial: a simplified polynomial that hasexactly one term

Pascal’s Triangle: an arrangement of the values ofnCr in a triangular pattern where each rowcorresponds to a value of n

Polynomial: a mathematical expression involving asum of nonnegative integer powers in one or morevariables multiplied by coefficients. A polynomial inone variable with constant coefficients can bewritten in 01

22

11 ... axaxaxaxa n

nn

n

Remainder Theorem: states that the remainder ofa polynomial f(x) divided by a linear divisor (x – c) isequal to f(c)

Roots: solutions to polynomial equations Synthetic Division: Synthetic division is a shortcut

method for dividing a polynomial by a linear factorof the form (x – a). It can be used in place of thestandard long division algorithm

Zero: If f(x) is a polynomial function, then the

Students findquotients andremaiders ofintegers

6thGrade

9thGrade

The exploration of polynomial functions through division sets the stage foran indepth understanding and definition of rational functions in unit 4.



Within Grades:






22

11 ... axaxaxaxa n

nn

n





Students usepolynomialidentites to createfactors of linearbinomials

9thGrade

11thGrade




Within Grades:






22

11 ... axaxaxaxa n

nn

n





Students findquotients andremaiders ofpolynomials anduse Pascals'sTriangle tomultiply binomialfactors

11thGrade



values of x for which f(x) = 0 are called the zeros ofthe function. Graphically, these are the xintercepts.

Guiding Question(s)• How are integer operations and properties of real numbers related to polynomials?• How can two algebraic expressions that appear to be different be equivalent?

Interpretations and Reminders

Note the Common Core State Standards Initiative Appendix A: It is important to ensure that instruction inthis unit goes beyond the investigation of quadratic polynomials as they have explored this specific familyof functions in Algebra 1. In Advanced Algebra/ Algebra II, students should work with a variety ofpolynomials of various degrees.

It might be helpful for students to think of polynomials as a formal number system in a way thatdemonstrates the use of operations between them.

Allow students to use prior knowledge about the ways they have explored operations with integers. Explore connections between the treatment of rational numbers and the treatment of rational expressions

when investigating division of polynomials. Students should be able to see the same structure in operatingwith both.

Have student represent their answers in standard form will allow them to classify the polynomial anddiscuss patterns they may see in the sums, differences, and products.

Misconceptions Some students may see polynomials at equations only. Provide examples of polynomial expressions to

demonstrate that it’s not necessary to have an equations in order to build a polynomial expression. Students may forget to distribute the negative sign when subtracting (i.e. -(8x2 + 15x) = -8x2 – 15x) Students may stop the long division process before the remainder is a polynomial with one degree less

than the original polynomial Operating with exponents sometimes prove problematic for students because they do not distribute

the exponent to all terms in the power of power rule.


Multiplying two binomials often challenges students. They do not see the product as a trinomial. Theyoften just multiply the two end terms.

Learning Progression (Suggested Learning Experiences)Procedural Fluency: (Recommended for 5 - 10 minutes each day: Fluency strategies are useful to activatestudent voice, solicit prior knowledge and develop fluency based on conceptual understandings.) For additionalfluency practice strategies see the table at the end of this document.

Using students prior knowledge about the properties of equality, spend the first 10 minutesallowing for reflection and effortless retrieval of these properties in use.

Expression of the Day – Display the following expression xx 165 on the Active Panel. Allow students toconsider the structure of the expression. Ask students to think of three different ways to rewrite the expressionin order to create equivalent expressions. Students should be able to justify their reasoning in building theirequivalent expressions. Provide students an opportunity to explain their thinking aloud.

Here are some of the equivalent expressions students may create (this list is NOT exhaustive):

)2)(2)(2)(2(

2163

610

)16(

55

5

4

ixixxxx

xxx

xxx

xx

---------------------------------------------------------------------------------------------------------------------------------------------------Graduated Measure (The graduated measure is a quick opportunity to diagnose students’ level of comfort withthe material before you begin the progression.)

Student Directions: Select a question/problem from the table below that you feel best equipped to answersuccessfully.

Beginning Developing ProficientDetermine whether the followingstatement makes sense or not,justify your reasoning.

The following expression hasexactly 5 terms:7x4 – 5x3 + 2x – 21 + 3x2 – x3

Determine whether the followingstatement makes sense or not,justify your reasoning.

7x2y3 + 7x3y2 will give the sameresult as (7x2y3)(7x3y2)


(5x – 2)2 = (5x)2 – (2)2







)2)(2)(2)(2(

2163

610

)16(

55

5

4

ixixxxx

xxx

xxx

xx








(5x – 2)2 = (5x)2 – (2)2







)2)(2)(2)(2(

2163

610

)16(

55

5

4

ixixxxx

xxx

xxx

xx








(5x – 2)2 = (5x)2 – (2)2


Defining Polynomials –While students at this level have worked with polynomials for years, they may not have used the appropriatevocabulary or even realized that linear and quadratic expressions belonged to the polynomial family. Definepolynomials to students as a way of introducing new terminology and reinforcing prior knowledge.

2711254853567 xxxxx

Identifying Polynomials by the Number of Terms Identifying Polynomials by Degree

Monomial 3x4 (One Term)Binomial -2x6 + 8 (Two Terms)Trinomial 5x2 – 7x + 26 (Three Terms)Polynomial 5 + 2x3 – 11 (any number of terms)

Constant Polynomial -8Linear Polynomial 5x + 7Quadratic Polynomial 3x2

Cubic Polynomial 4x3 – 8x2

Quartic Polynomial 11x4 – 7x2 + 8Quintic Polynomial 4x5 + 2

Note: It is important to point out that monomials,binomials, and trinomials are still polynomials. So theterm 18 is a polynomial

Note: Facilitate a discussion around the differencebetween the Binomial example in the first columnand the Linear Polynomial in this column.

Ask students: “Is there another way to name some ofthese terms?”

Closure Property– The set of polynomials is said to be closed under addition, subtraction, and division. Allowstudents to discover this through the following learning experiences. They should note that the sum, differenceand product is always a new polynomial, making the set closed. Have them compare this property inoperations with integers to decide whether it is true to claim that the set of polynomials form a systemanalogous to that of integers.

Polynomials are written in standard form whenterms are in descending order based on the

degree of each term.

Point out that whilethis polynomial is

missing a cubic term itis still in standard

form.

The degree of apolynomial is indicated bythe term with the greatestpower in the polynomial.


Addition and Subtraction:Spend some time demonstrating how to use algebra tiles to model addition and subtractions of integers. Thegoal is to highlight the use of the manipulative and remind students of their prior knowledge with integeroperations. You will be able to address anchored misconceptions specifically around subtraction andmultiplication of integers during this focus lesson. It will also provide a springboard for the investigation ofpolynomials using the manipulatives.

Introduce students to the algebra tiles and what they will model for integer operations:

Let the small squares represent 1 (yellow) and -1 (red)

Ask: “What does combining one red and one yellow square mean?”

Use the language of “zero pairs” to correct students who may say, “They cancel each other out”. The dangerin this language is that students neglect to recognize the additive inverse property in action.

Let the long rectangles represent 10 (green) and -10 (red)

Let the large squares represent 100 (blue) and -100 (red)

Model zero pairs and be explicit by saying that pairing one red with is alternate color is a way of representing zero

Discuss how these are additiveinverses of each other.


Note that subtraction will work in a similar way when the signs are the same. However it is worth ademonstration of subtraction when the signs are different.

Make the connectionhere between

gathering tiles of thesame type and

combining like-terms

Show students howsince zero pairs

model a zero we canremove them

As you move througheach example think aloud

about the integer rulethat is being

demonstrated. Makingcertain students recall

these properties.


Note: If you don’t have what you need you must add enough zero pairs to take away the amount of thesubtrahend.

This is our new representationof 12. By adding in 25 zero

pairs we now have 25negative tiles (red tiles) to

remove.

When students are introduced tosubtraction they conceptualize the idea

of “taking away”. The use of AlgebraTiles builds on this concept as students

only model the minuend (the firstnumber in the sentence) and then take

away the amount of the subtrahend.

After removing -25


Repeat this example if needed to ensure that students understand how to model subtraction with the tiles.Next, redefine the manipulatives to reflect variable expressions and allow students to practice with adding andsubtracting polynomials using the tiles.

The small squares will still represent 1 (yellow) and -1 (red)

Let the long rectangles represent x (green) and -x (red)

Let the large squares represent x2 (blue) and –x2 (red)

The Standard Algorithm

Model the sum and difference of two polynomials to illustrate the connection between place value in integeroperations and like terms in polynomial addition and subtraction.

Using the standard algorithm for addition and subtraction involves lining up digits by their place value andadding in a vertical way from the ones column to the highest place.

Now I can removeany zero pairs thatare left which willreveal my solution

of -13

Discuss how these are still additiveinverses of each other.


Ask students to make conjecture about the connection between the standard algorithm for addition andsubtraction and the vertical sum and difference of polynomials.

MultiplicationMultiplication of polynomials follows nicely from the standard algorithm that students have been introduced toin elementary and middle grades. There are several ways, however, to model this operation and studentsshould be exposed to a variety of methods and given the opportunity to choose their preferred method.

Algebra Tiles or Area ModelAlgebra tiles can also be used to scaffold discussion around the area model of multiplication. This is animportant learning experience as many contextual problems ask students to find area or surface area ofrectangular spaces with algebraic expressions as dimensions.

The rectangular model above can be used to connect several equivalent expressions:ex. 1 (x + x)(x +x +3y)ex. 2 (2x)(2x+3y)

Looking at the specific areas of each individual rectangle would yield:


ex. 3 x2 + x2 +2x(x+3y)

The Standard AlgorithmModel the product of two-digit numbers to remind students of the standard algorithm for multiplication.

Note: The distributive property is at work within this model. You may want to share the distributive propertymethod as well, however if you are multiplying polynomials greater than binomials students may have troubleorganizing all of the terms in a horizontal fashion.


Operations withpolynomials FA.docx

Polynomial Identities and Special ProductsStudents should be familiar with special products from their investigations with equivalent expressions andquadratic equations in Algebra I. Remind them of the special products below by discussing the validity of thetheir expansions:

222

222

22

2)(

2)(

))((

bababa

bababa

bababa

In this unit, experience with polynomial identities will also include acquaintance with the following:

32233

32233

33)(

33)(

babbaaba

babbaaba

Binomial ExpansionOne of the big misconceptions with multiplying binomials arises when the binomial is raised to a power.Ask: What does it mean to expand a binomial?

Students may see this When they should see this(2x + 5)3 = (2x3 + 53) (2x+ 5)3 = (2x+5)(2x+5)(2x+5)

INCORRECT CORRECT

Next ask students to verify the following identity.32233 33)( yxyyxxyx

Have them write the expansion first to ensure that they are multiplying the binomials rather than distributingthe power.

3223223 332 yxyyxxyxyxyxyxyxyxyx

Ask: What will the product be if I raise (x + y)4?Allow time for students to multiply the binomial by the result from the previous identity. For students who


finish quickly invite them to expand (x + y)5

Ask: Are there any patterns that you notice in the expansions?

543223455

4322344

32233

510105

464

33)(

yxyyxyxyxxyx

yxyyxyxxyx

yxyyxxyx

Students may notice the progression of the descending/ascending nature of the exponents in the chain of terms.They may also notice that the degree of each term is equal to the original power.

Expanding linear binomials can be a tedious process even with a power as small as 3 like in the example above.The Binomial Theorem will assist with expanding linear binomials and can be seen using Pascal’s Triangle.

Pascal’s Triangle

Ask: Is there a way to predict what the next row will look like?

Present students with a copy of Pascal’sTriangle.

Ask: What patterns do you see?

Students should notice many differentpatterns. And should be allowed to exploreeach pattern they see.


Record students’ answers on the board and work with students to generate the 11th row of the triangle basedon their patterns.

Then Ask: What is the connection between Pascal’s Triangle and polynomial identities? Is there a connection atall?

Students should see the coefficients of their polynomial in the rows of Pascal’s Triangle.

DivisionDivision of polynomials follows nicely from the standard algorithm that students have been introduced to inelementary and middle grades. There are several ways, however, to model this operation and students shouldbe exposed to a variety of methods and given the opportunity to choose their preferred method.

A note about Closure – as a contrast to the previous examples students should draw their own conjecturearound quotients in the division of polynomials. They should note that the quotient is not always a polynomialand therefore polynomials are not closed under this operation.

By InspectionStudent experiences with division of polynomials should include:- Division of polynomials by a monomial (Division by inspection)- Division of polynomials by a binomial- Division where the dividend is a polynomial of degree 3 and greater

It is often easy to see the quotient when dividing a polynomial by a monomial. Students should be able tosimply divide each term in the polynomial by the monomial by inspection.

Ex. 65

30

5

5

5

305

t

tt

Ex.2

23222

3

2

4

2

5

2

345 4516103

2

8

2

10

2

32

2

20

2

6

2

81032206

xxxxx

xx

x

x

x

x

x

x

x

x

xxxx

Gradual Release of Responsibility

Remind student how exponents are treatedunder division.

Expand the following using Pascal’sTriangle:

1) (2x – 3)5

2) (n + 2m)7

3) (3x + y)4


Focus LessonAsk: If 6 X 7 = 42, then 42 7 = what? Allow students to discuss the connection between multiplication anddivision then introduce the division algorithm:

Division AlgorithmIf a(x) and b(x) are polynomials, with 0)( xb , and the degree of b(x) is less than or equal to the degree ofa(x), then there exists unique polynomials q(x) and r(x) such that:

Engage students in discourse around the division algorithm that allows them to translate it into studentfriendly language.

Demonstrate the following problem with algebra tiles:

61 xx

x+1

x + 6

The area filled in is represented by1 x2-tile, 7 x-tiles and 6 1’s-tiles:

x2 + 7x + 6

a(x) = b(x) . q(x) + r(x)

QUOTIENT

REMAINDERDIVISOR

DIVIDEND


Guided PracticeProvide students with their own sets of algebra tiles. Guide them through the Modeling Division problem set.Students will identify the quotients based on their models. The last problem on the sheet illustrates a modelwith a remainder. Ask students to make conjectures about the leftover tiles in preparation for a more detailedexploration below.

ModelingDivision.docx

Once you have worked through the Modeling Division problem set. Present students with the problem below:

Now, walk this model backward starting withthe area and one of the original factorswhich now represents the divisor. Askstudents, “What part to this division problemdoes the area represent?”

Students should recognize that the area isthe dividend

Note: The quotient in this example belongsto the set of polynomials, however this is notalways the case and students should trymultiple example to illustrate the lack ofclosure around division of polynomials.


Ex. 12

123 2

x

xx Ask students to model this problem as you circulate the room to check students’ tiles.

This is a difficult model because it requires theaddition of zero pairs to complete it.

Ask student to justify their thinking as they createtheir models. Then place the following model on the

board.

Show students how this models is the dividend, andexplain why you arranged your tiles this way. (You

know the divisor is (x+2) so you expect to placeanother positive green x tile there.) Notice how in

filling in the remaining area 2 red tiles will be left out.

Ask Students:

How did we alter the original dividendto complete the rectangular model?

Does this new model still represent3x2+x-12? How do you know?

Is the divisor of (x+2) represented onthe left of the rectangular area?

What is the quotient that isrepresented along the top of therectangular area?

Way do the two remaining red squaresrepresent?


therefore,2

2532

1223

x

xx

xx

Collaborative PracticeRemind students of the division algorithm. And model an example of dividing polynomials using the standardalgorithm.

1675543

2

23

xxxxxx

x3 – 3x2

7x2 – 5x7x2 – 21x

16x + 516x – 48

53

Divisor(X + 2)

Quotient3x – 5

Remainder-2

This is an appropriate time to remind students that remainders are written as

the leftover part of the original divisor.2

2

x

Dividend3x2 + x - 12

Dividendx3 + 4x2 – 5x + 5

Remainder 53, written as3

53

x

Quotientx2 + 7x + 16

Divisor(X – 3)


Note about long division: Students should make certain that the divisor and dividend are written in standardform and the dividend must include place holders for any missing power families.Ex. 3x3 + 4x – 1 must be written as 3x3 + 0x2 + 4x – 1

Divide students into strength pairs (Strength pairs should include two students with similar strength levelsbased on the guided practice). Arrange the desks in a circle and hand each pair an index card with a divisionproblem. As each pair works the problem using the standard algorithm the teacher should rotate the circle andcheck for accuracy while posing questions to struggling students. Once a pair has completed a problem place astamp or mark on their paper indicating that the problem is correct and hand them a new problem. Continueuntil each pair has successfully completed 10 problems. Some pairs will work more efficiently than others.Provide appropriate scaffolding to pairs in need of additional assistance.

During the collaborative practice allow students to choose between continued use of the manipulatives andthe creation of visual representations and rectangular diagrams. You may want to leave only one set of algebratiles in for each pair so that students can use it to verify their solutions.

Independent PracticeDuring Independent practice students should demonstrate their understanding of polynomial division. Provideboth skill problems as well as open ended questions to assess the level of understanding.

Sample independent practice sheet:Sample

Independent Practice Sheet.docx

Note the absence of Synthetic Division from the writers of the common core:A particularly important application of polynomial division is the case where a polynomial p(x) is divided by a linearfactor of the form x – a, for a real number a. In this case the remainder is the value p(a) of the polynomial at x = a. It is apity to see this topic reduced to “synthetic division,” which reduced the method to a matter of carrying numbersbetween registers, something easily done by a computer, while obscuring the reasoning that makes the result evident.

Unit Assessments –Assessment

Name

Assessment Assessment Type StandardsAddressed

Cognitive Rigor

We’ve Got toOperate

We've Got ToOperate Task.docx

Learning Task MGSE9-12.A.APR.1 2

Divide and Conquer

Divide andConquer.docx

Learning Task MGSE9-12.A.APR.6 2


Powers of 11

Powers of 11.pdf

Partner ExplorationTask

MGSE9-12.A.APR.5 2/3

Differentiated SupportsLearningDifficulty

Students who have learning difficulties may need support with subtracting using theadditive inverse. Here is a lesson from LearnZillion that addresses how to do so.https://learnzillion.com/lesson_plans/5378

Students may also have difficulty with the number of steps involved in long division. Tosupport those students, they can complete the following sequencing activity as a support.The problems should be printed on cardstock, laminated if possible, and then cut. Studentswill sequence not only the problem solution, but the explanation for each step.

Poly Long DivSequencing Activity

High-Achieving

• Challenge high achieving students with the following task:

non-negativepolynomials.pdf

EnglishLanguageLearners

Have students write their understandings in a math journal. You may want to considerallowing EL students to write notes in their first language and annotate by identifying mathspecific words they do not have a translation for.

Allow students to create pictorial flash cards for new math words. Create a graphic organizer for polynomial operations.

Online/Print ResourcesDigitalResources

http://www.math.wfu.edu/Math105/Operations%20on%20Polynomials(revised).pdfThis website provides a step-by-step process on how to add, subtract, multiply and dividepolynomials http://www.youtube.com/watch?v=cy5O1Yu0UFkThis is a video provides a lesson on operations with polynomials.

PrintResources

ManipulatingPolynomials.pdf

ModelingDivision.docx

SampleIndependent Practice Sheet.docx

Manipulatives/Tools

Algebra Tiles TI-84 Handheld Wabbitemu App https://www.desmos.com/calculator


Textbook AlignmentPrecalculus: Mathematics for Calculus. 5th ed.Stewart, Redlin, & Watson

pp. 265-267


Lesson Two ProgressionDuration 4 days

Focus Standard(s)◊ Build a function that models a relationship between two quantities

MGSE9-12.F.BF.1 Write a function that describes a relationship between two quantities.

MGSE9-12.F.BF.1b Combine standard function types using arithmetic operations in contextual situations(Adding, subtracting, and multiplying functions of different types).

MGSE9-12.F.BF.1c Compose functions. For example, if T(y) is the temperature in the atmosphere as a functionof height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature atthe location of the weather balloon as a function of time.

Performance-Based ObjectivesAs a result of their engagement with this unit… MGSE9-12.F.BF.1b – SWBAT decontextualize real-world situations involving arithmetic operations IOT

construct new polynomial equations through combination of standard function types. MGSE9-12.F.BF.1c – SWBAT compose functions and determine the meaning of that composition IOT

describe the relationship between two quantities and make meaning of contextual situations.


Within Grades:

Students explorebuilding functionsfrom contextual

situations

8thGrade

9thGrade

The building of functions provides students with experience decontextualizingreal-world situations which is a reoccuring theme in this course.











Within Grades:

Students buildlinear andquadratic

functions andselect appropriate

models fromcontextualsituations

9thGrade

11thGrade












Within Grades:

Students buildpolynomial

functions andcompose functions

from contextualsituations

11thGrade



Key Terms and DefinitionsDecontextualize: Represent a real-world situationsymbolically by considering the mathematics inisolation from the context.

Functional Relationship: A functional relationshipmodels the connection between two quantities whereone characteristic has a unique changes as a result ofanother. (Example: distance traveled based on speed,height grown based on time.)

Guiding Questions How can real phenomena be modeled by polynomial functions?

How can reasonable conclusions be drawn from modeling with newly composed polynomial functions?

Interpretations and Reminders

Students should first understand that a function operates on a given value to produce another value in anindependent/dependent relationship.

Students should understand function notation and how to substitute values into functions using the orderof operations.

In function composition, order matters and students should work from the inside out in order to eitherevaluate or create new functions.

Function composition allows for the understanding of an operation on a particular function that creates anew function. As students look for and making use of structure (MP.7), “students should also develop thepractice of writing expressions for functions in ways that reveal the key features of the function” (CommonCore Standards Writing Team, 2013, p. 10).

Misconceptions Students can mistake f(x) for “f multiplied by x” rather than “f as a function of x” and likewise ))(( xgf for

multiplication. Explicitly teach the notation for function notation and composition.

Students may compose functions from left to right, rather than from the inside value to the outsidefunction.



Before diving into the algorithmic dialogue around complex numbers remind students about othernumber systems and how those number systems allowed for the solution to real-world problems. For instancethe need for rational numbers and decimals can be seen in scenarios involving construction, pricing, andheight. The idea that many things are not whole can be made clear through examples. Also consider the idea ofintegers through conversations about debt, sea-level, underground height, etc. Solicit examples of equationsthat produce various kinds of numbers.

First ask students to write an equation whose solution is in the group of whole numbers. Next ask students to write an equation whose solution is in the group of integers or rational numbers. Each time ask students to justify their reasoning to their responses.

Graduated Measure (The graduated measure is a quick opportunity to diagnose students’ level of comfort withthe material before you begin the progression.)


Level 1 Level 2 Level 3Given

3

54 2

xxg

xxf

)(

)(

Find )(2f and )( 1g

Given

3

54 2

xxg

xxf

)(

)(

Find ))(( xgf

Given

3

54 2

xxg

xxf

)(

)(

Simplify ))(( xgf

Opening:Remind students that functions can be represented graphically or algebraically. Pass out the graph from theExploration 1. Provide them access to rulers, graph paper, t-charts, and calculators for use if they choose.

Exploration 1: A Sum of FunctionsUsing the graphs below, sketch a graph of the function s(x) = f(x) + g(x).








3

54 2

xxg

xxf

)(

)(

Find )(2f and )( 1g

Given

3

54 2

xxg

xxf

)(

)(

Find ))(( xgf

Given

3

54 2

xxg

xxf

)(

)(

Simplify ))(( xgf










3

54 2

xxg

xxf

)(

)(

Find )(2f and )( 1g

Given

3

54 2

xxg

xxf

)(

)(

Find ))(( xgf

Given

3

54 2

xxg

xxf

)(

)(

Simplify ))(( xgf




Facilitate the task by asking questions about the type of functions being represented by f(x) and g(x) to guidestudents towards combining the function values. Allow students to present their ideas using multiplerepresentations.To assess students’ conceptual understanding, the task can be extended to sketch f(x) – g(x).

The purpose of this task is for students to understand how function addition works graphically, but they mustalso be able to combine functions algebraically.

Explain notation to students: xgxfxgf , xgxfxgf , xgxfxgf , and

)(

/xg

xfxgf

Have students perform each operation on the following functions. 1432 32 xxgxxfNotes:

Students should write terms of their new function in descending order. When dividing, note to students that there are restrictions in the domain. This will be important for them to

understand in Unit 4 with rational functions.

Collaborative Practice:Discussion Question Partner Speaks:

Classify the new functions made from addition, subtraction, and multiplication according to the number of termsand degree. Compare and contrast f(x), g(x), and the new functions. What effect does each operation have on f(x)and g(x)?

Students talk through the problem with their partner and receive feedback before sharing with the class. Pairs then speakfrom the perspective of their partner, re-voicing and paraphrasing their partner’s points. The purpose is to encourageactive listening techniques and speaking for understanding.

Guided Practice

CombiningFunctions Practice Problems(DOK 2)

Exploration 2: Building an Explicit Quadratic Function by Composition (from Illustrative Mathematics)Explain to students that composition is another way to combine two functions to get a new function. Give the

example of12

xxg

xxf

)(

)(. Suppose you want to define h(x) as 11 22 xxfxgfxh )())(()( .


Introduce the task to students. Students should also have access to graphing software, either throughDesmos, Wabbit, or TI-83 calculators. The task provides students the opportunity to develop the conceptsthat, (1) the order of composition is important, (2) the impact of each function on the original, in terms ofshifts and scaling.

Let f be the function defined by f(x) = 2x2 + 4x −16. Let g be the function defined by g(x) = 2(x+1)2−18.a) Verify that f(x)=g(x) for all x.b) In what ways do the equivalent expressions 2x2 + 4x − 16 and 2(x+1)2 − 18 help to understand the function

f?c) Consider the functions h, l, m, and n given by

h(x) = x2, l(x) = x + 1, m(x) = x – 9, n(x) = 2x

Show that f(x) is a composition, in some order, of the functions h, l, m, and n. How do you determine theorder of composition?

d) Explain the impact each of the functions l, m, and n has on the graph of the composition.

Composition Relay

Func CompositionRelay (DOK 2)

In teams of four or five, groups are given one sheet of paper on which to work the problems. To begin therelay, the teacher calls out a number. The student with function number one evaluates his function for thenumber called by the teacher; he must show work (at least substitution of the number into the function) andwrite the answer to his step on the group’s paper. After evaluating the function for the number called out, thatstudent passes the group’s answer sheet to the person with function number two; that student thensubstitutes the value obtained by the first student into function number two. After evaluating function numbertwo using the answer obtained by the first student, the paper is then passed to the student with functionnumber three; that student evaluates function number three using the value obtained by the second student.

gInput: x x2+ 1 fOutput:

12 x


This continues until all five functions have been evaluated using the value obtained by the previous student.

(DOK 2)(Collaborative Practice) Teach-Teach-Trade: Students will work out a given problem on an index card andbecome the “expert” on that problem. They can check their answer with the teacher and possibly get coaching.Once they are ready, students will raise their hands and partner with another student who has a raised hand.They will solve each other’s problem, coaching/teaching and providing support until each student is an expertof a new problem. They will then trade cards and repeat the process.

Additional Unit Assessments –Assessment

Name


Cognitive Rigor

KUTA FunctionOperations

KUTA FunctionOperations

Skills Mastery MGSE9-12.F.BF.1b 2

TemperatureConversions Temp Conversion

Assessment Task MGSE9-12.F.BF.1c 3

FunctionCompositionPractice Problems

FunctionComposition Practice

Skills Practice MGSE9-12.F.BF.1c 2


Differentiated Supports

Learning Difficulty

Students may need extra support with using theorder of operations to simplify expressions. Havethem create a graphic organizer detailing the orderof operations with numerical values first, then usingvariable expressions.

To emphasize the order of composition, providestudents with representational examples of functionoperators, in which they physically place thefunctions on a diagram and evaluate them in order.

CompositionSupport

High-Achieving StudentsChallenge high-achievers with the task found on pages23-26 of the link below. Students may need a refresheron their trigonometric functions before beginning. Theteacher notes follow the task.https://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec3_mod7_mdlfun_te_51914.pdf

English Language Learners

Have students write their understandings in a mathjournal. You may want to consider allowing ELstudents to write notes in their first language andannotate by identifying math specific words they donot have a translation for.

Allow students to create pictorial flash cards for newmath words.

Provide students with a graphing utility to check thevalidity of their work

Online/Print ResourcesDigital Resources

• https://www.khanacademy.org/math/algebra2/manipulating-functions/funciton-composition/v/evaluating-composite-functions-example-1 Video resource showinghow to evaluate a composite function.

• http://www.montville.net/cms/lib3/NJ01001247/Centricity/Domain/564/1-6%20Function%20Operations%20and%20Composition%20of%20Functions.pdf Resource for additional functioncomposition problems with solutions, including problemsin a real-world context.

Print Resources


Manipulatives/Tools

TI-84 Handheld Wabbitemu Apphttps://www.desmos.com/calculator

Textbook AlignmentPrecalculus: Mathematics for Calculus. 5th ed.Stewart, Redlin, & Watson

pp.


Lesson Three ProgressionDuration 5 days

Focus Standard(s)



MGSE9-12.F.BF.4a Solve an equation of the form f(x) = c for a simple function f that has an inverse and write anexpression for the inverse. For example, f(x) =2(x3) or f(x) = (x+1)/(x-1) for x ≠ 1.


MGSE9-12.F.BF.4c Read values of an inverse function from a graph or a table, given that the function has aninverse.Performance-Based ObjectivesAs a result of their engagement with this unit…

MGSE9-12.F.BF.4a – SWBAT solve equations of the form f(x) = c and write an expressions for the inverseIOT describe and compare a function and its inverse graphically and algebraically.

MGSE9-12.F.BF.4b – SWBAT use function composition to verify that one function is the inverse of anotherIOT compare how functions and their inverses compare to one another.

MGSE9-12.F.BF.4c – SWBAT find inverses numerically, algebraically and graphically IOT investigate therelationship between a function and its inverse.


Studentsdevelop fluency

with inverseoperations an

functionalrelationships

involving inputoutput values

8-9thGrade

11thGrade



Focus Standard(s)










Students findand verigyinverses of

variousfunction types

11thGrade

Students findinverses of

tigonometricfunctions bothalgebraically

and graphically

12thGrade



Focus Standard(s)










Students findinverses of

tigonometricfunctions bothalgebraically

and graphically


Within Grades:

Key Terms and Definitions Composition of Functions: Let f and g be functions

such that the range of g is a subset of the domain off. Then f(g(x)) is called the composite of f with g.

Domain: The set of all first coordinates from theordered pairs of a relation (usually defined as x’s)

Function: A special type of relation in which eachelement of the domain is paired with exactly oneelement of the range

Horizontal Line Test: If ANY horizontal lineintersects your original function in ONLY ONElocation, your function will be a one-to-onefunction and its inverse will also be a function.Note: There are some functions that do not passthe horizontal line test and yet they have an inversefunction on a restricted domain.

Identity Function: h(x)=x Inverse Functions: Two functions f and g are

inverse functions if and only if both of theircompositions are the identity function. That is,

f (g(x)) = x and g(f(x)) = x

Inverse Relations: Two relations are inverserelations if and only if whenever one relationcontains the element (a, b), the other relationcontains the element (b, a)

One-to-oneness: A function is a one-to-onefunction if and only if each second elementcorresponds to one and only one firstelement. (each x and y value is used only once)

Range: The set of all second coordinates from theordered pairs of a relation (usually defined as y’s)

Guiding Questions What is an inverse? How do you prove that two functions are inverses of each other? Is there a function notation for inverses?

Interpretations and Reminders Introduce student to the proper function notation for inverse functions. Make it explicit, however, that

)(

1)(1

xfxf . It is also important to name functions outside of just )(xf so that students understand that

the notation for the inverse can be used for all function names. When students find the inverse of a function it will be necessary for them to understand that the inverse

function does not necessarily have the same domain as the original function.

Understnding the criteria for creating and verifying inverse functions supports theinvestigations in unit 5 which explore exponential and logarithmic relationships


Provide students with example of functions that do not have inverses as well as functions whose inverseshave a restricted domain. For example, 5)( 2 xxf has an inverse 5)(1 xxf provided that thedomain is restricted to x ≥ 5.

We often ensure that inverse functions must pass the horizontal line test, but since inverse “functions” arealso functions students should know that they must pass both the vertical and horizontal line test.

Misconceptions Students may believe that all functions have inverses. Students may mistake the notation for inverses for an exponent. Namely that )(1 xf is like raising the

function to the -1 power. Student may not make the connection that putting the output from the original function into the input of

the inverse results in the original input value.

Learning Progression (Suggested Learning Experiences)Use students’ prior knowledge to begin investigating this cluster of standards. Students have already

Procedural Fluency: (Recommended for 5 - 10 minutes each day: Fluency strategies are useful to activatestudent voice, solicit prior knowledge and develop fluency based on conceptual understandings.) For additionalfluency practice strategies see the table at the end of this document.

Students should develop fluency in seeing structure in expressions.



Beginning Developing ProficientSolve the following equation for h

rhrS 22 2 Solve the following equation for r

5

24 rxrpT

Solve the following equation for a

ba

am

12

Begin by reminding students that they have discussed inverse operations in the middle grades. In order toisolate a variable when solving multi-step equations they “undo” each operation using the inverse operation.














5

24 rxrpT


ba

am

12















5

24 rxrpT


ba

am

12



Provide the following example, think aloud as each step is performed making sure to illustrate how the inverseoperation is used:

Solve: 5x2 + 7 = 27-7 -7

5x2 = 205 5x2 = 4x = 2

Now as you introduce the language of inverse functions extend students’ previous understanding of howinverse operations behave in order to scaffold the new knowledge of inverse functions.

By definition, two functions are inverses if xxfgandxgf ))(())(( for all x in their respective domains. Thismay be a dense definition for some students, consider providing visual representations as an alternativedefinition for students who may struggle with this idea:

For example:

We add in (-7) in line two because it is theadditive inverse of (+7)

We divide 5 because division is the inverse ofmultiplication

We take the square root because squareroots and squaring are inverse operations


Gradual Release of ResponsibilityFocus LessonFinding Inverse Functions

Graphical Exploration: The United States is one of only five countries in the world to use Fahrenheit as thedegree of measure for temperature. Let F be the function that takes Celsius measurements and returns theequivalent Fahrenheit measurement. Let C be the function that takes a Fahrenheit measurement and returnsan equivalent Celsius measurement.

325

9)(

xxF and 32

9

5)( xxC Use a graphing utility to graph both F and C in the same coordinate

plane. Ask students: “How can we verify graphically that these functions are inverses of one another?”Students should use the definition of function inverses to assist with their understanding. Have them drawconjectures from generating a table of input output values in a table for both functions and working thedefinition manually.

x -20 -15 -10 -5 0 5 10 ? 20F(x) -4 5 14 23 32 41 ? 59 ?

x -4 5 14 23 ? 41 50 59 68C(x) -20 -15 -10 -5 0 ? 10 15 20

x -20 -15 -10 -5 0 5 10 15 20F(C(x)) -20 -15 ? ? ? ? ? ? ?C(F(x)) -20 -15 ? ? ? ? ? ? ?

On a graph, if two functions are inverses they should always be symmetric about the identity function y = x


Consider the next pair of functions: Let 5)( 2 xxf and 5)( xxg

When they graph them students should notice a slight difference between the symmetric properties of fagainst g. Engage students in a discussion about why one of the graphs seems to be only half the other.

Ask students:“Why does g(x) only seem to reflect half of the f(x) function across the identity line?”“What is missing from the g(x) function, and why?”“If we create a table of values like the previous example will all of the elements of the domain of f bereflected in the range of g?”

Graphically, functions and their inverses are reflections about the line y = x.


Focus LessonThis task can be done as a class to introduce finding function inverses algebraically. You may have to reviewdeveloping an equation relating two quantities if students struggle with question 1. For the Oprah Winfreyportion of the task, you can use https://www.random.org/lists/ to generate amounts for students. It’s ok ifsome students’ values repeat; they can check values for answers with one another.

Bill Gates Task:You walk into class one day and surprise, Bill Gates is your substitute teacher! He says that he will take the amount ofmoney your pocket, triple it, and then add $100 to that amount.

1. Write an equation that represents how much money Bill Gates is giving to you based on the amount of money inyour pocket. Be sure to define your variables you use in your equation.

_____________________________________________________________________

2. Awww man!!! The principal found out what Bill Gates was up to and doesn’t approve. They ask you to return themoney to Mr. Gates. How will you hand the money back to him? Describe how you will return his money toarrive back to the amount that you have in your pocket.

_________________________________________________________________________________________________

3. Write an equation describing how you are going to return Mr. Gates’ money based on your description in #2. Usethe same variables you defined in #1.

_____________________________________________________________________

Oprah Winfrey Task:

You come into class the next day and Oprah is there to greet you at the door. Oprah says that you will pick a value out ofa hat with several dollar amounts in it. For whatever amount you choose from the hat, she will add $200 to that amount,square it, and hand it to you.

What value did you pick out of the hat? ___________________

1. How much money you would take home to your family? Be descriptive on you will not receive credit.

_____________________________________________________________________

2. Write an equation for describing the manner in which Oprah is gifting money.

_____________________________________________________________________

You take the jackpot home to your family and everyone is excited! You have a party to celebrate. At the party, UncleRicky asks you what amount you picked from the hat, but you have forgotten.

3. Write an equation that will help you get back to the original amount you picked out of the hat.


_____________________________________________________________________

How can we confirm that the functions for Bill Gates and Oprah Winfrey gifting their money and determininghow much we started with are in fact inverse functions? Prove the functions are inverses graphically.

After students prove the functions are inverses graphically, explicitly teach determining if functions are inversesof one another using function composition.

Collaborative Practice: Find your partner All students are assigned a function. Students must determine who isholding their inverse function. Once students find their partner, they represent their functions graphically,showing that the inverses reflect over the line y = x, and use composition to prove that ( ( )) =and ( ( )) = .

Design a Calculator Exploration:Use a graphing calculator or computer program to compare tabular and graphic representations of exponentialand polynomial functions to show how the y (output) values of the exponential function eventually exceedthose of polynomial functions.

Have students draw the graphs of exponential and other polynomial functions on a graphing calculator orcomputer utility and examine the fact that the exponential curve will eventually get higher than the polynomialfunction’s graph. A simple example would be to compare the graphs (and tables) of the functions y = x 2 and y= 2x to find that the y values are greater for the exponential function when x > 4.Additional Unit Assessments –

Assessment

Name


Cognitive Rigor

KUTA FunctionInverses

KUTA FunctionInverses

Skills Mastery (Canbe used for asproblems for the“Find your partner”

MGSE9-12.F.BF.4bMGSE9-12.F.BF.4c

DOK 3


activity.)Invertible or Not?

Invertible or Not

Performance Task MGSE9-12.F.BF.4bMGSE9-12.F.BF.4c

DOK 3

US Households

US Households.pdf

Student Exploration MGSE9-12.F.BF.4b DOK 3

Differentiated SupportsLearningDifficulty

Use real-world examples of functions and their inverses. For example

Provide students with scaffold notes around the algebraic manipulations needed togenerate a function inverse.

Provide example that compare function inverses graphically and in tables as well asalgebraically. Allow students to make the connection between the differentrepresentations.

High-AchievingStudents

Inverse Functions with Exponential and Logarithmic Functionshttps://www.illustrativemathematics.org/content-standards/HSF/BF/B/4/tasks/615

English LanguageLearners

Provide opportunities for students to explain orally and in writing how to create afunction inverse.

Students should be able to justify each step in verifying whether two functions areinverses using key technical vocabulary in simple sentences.

Provide graphic organizers that illustrate the algebraic manipulation used to generatefunction inverses with annotations around the tier 3 words.

Online/Print ResourcesDigital Resources www.IllustrativeMathematics.org Website offers tasks that demonstrate operations to

prove two functions are inverses, as well as specific types of inverse relationships,including temperature conversions.

https://www.khanacademy.org/math/algebra2/manipulating-functions/introduction-to-inverses-of-functions/v/introduction-to-function-inverses Khan Academy video andpractice questions. Answers what inverse functions are, and how to find an inversealgebraically.

http://www.coolmath.com/algebra/16-inverse-functions

Print Resources

Finding InversesGSE Task

Rainfall.pdf

Textbook Alignment


Precalculus: Mathematics for Calculus. 5th ed.Stewart, Redlin, & Watson

pp.

Sample Fluency Strategies for High SchoolExpression of the Week Choose an expression for the week. Allow students to consider the given expression.

Students will then think of different ways to rewrite that expression in order to createequivalent expressions. As course concepts progress ask students to incorporateadditional concepts in the creation of their expression. Provide students an opportunityto explain their thinking.

Algebraic Talks Extending from the idea of Number Talks. Teachers can help students develop flexibilitythrough Algebraic Talks.

Algebraic Talks (The Norms): No Pencils No calculators Mental Math Only Consider the problem individually during the first few minutes

Place a problem on the board and allow students to consider possible solutionpathways. Next, allow 5 or 6 student to explain their strategy and record the variousmethods on the board for all students to see. Name the strategy (preferably after thestudent who made the conjecture). Allow other students to ask questions about thestrategies while the proposing student is allowed to defend his/her work.

Also see: Jo Boaler Number Talks Videos

Picture This Provide students with a graph, or chart and ask them to come up with a contextualsituation that makes the picture relevant


Find the odd man out

Note 1: This is an open ended question where students can choose any one of thechoices A-D as the odd man as long as they are able to defend their choice. Forexample,A is the odd equation because it is in the only equation with coefficients as 1 or -1B is the odd equation because it is in the only equation with negative slopeC is the odd equation because it is in the only equation whose line passes through theoriginD is the odd equation because it is in the only equation which is not represented in thestandard form.

Note 2: Teacher can provide 4 equations/ 4 triangles / 4 transformations / 4 trig ratioswhere there is more than one way of classifying. Students can choose the odd man bymere inspection (surface level) or by examining the four components with a deeperunderstanding of the concept.

A

x – y = 5

B

2x + 3y = 6

C

2x – 3y = 0

D

3y = 2x – 4


Example 2:

A is the odd equation because it has all positive coefficients or because it has twonegative roots.

B is the odd equation because it has two positive roots.

C is the odd equation because it has one positive and one negative root.

D is the odd equation because the parabola opens downwards

A

y = x2+5x+6

B

y = x2- 5x+6

Cy = x2- x-6

Dy = - x2 + 6

Generate examples andnon-examples Example 1: Draw as many triangles as possible that are similar to

Note: Teachers can ask students to generate examples and non-examples for a varietyof topics to improve fluency. A few more possibilities are listed here:1: Create quadratic equations with imaginary roots2. Create quadratic equations with complex roots3. Create non-examples of complementary angles.4. Create examples of circles whose center is (3, -2)5. Draw triangle ABC where Sin A= 0.3

Locating the error Choose a student’s work to be displayed on board after removing the name and ask theclass to locate the error and also work the solution correctly. This can be done in anytopic where the problem requires multiple steps to solve.

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