algebra 1 unit 3: systems of equationssecondarymath.cmswiki.wikispaces.net/file/view/unit 6...

Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.

Course Name: MS Algebra I Unit # 6 Unit Title: Polynomials

BY THE END OF THIS UNIT:

Enduring understanding (Big Idea):Students will understand that manipulation of polynomials will enable them to model and analyze real-world non-linear situations.

Essential Questions: 1. How are the properties of real numbers related to polynomials?2. How can two algebraic expressions that appear to be different be the same?3. How can you use functions to model real-world situations?

Students will know…

Vocabulary:Square Root; Perfect Square Trinomials; Factoring Quadratic Equations; Zeros, Roots, Solutions; Polynomial, trinomial, binomial, linear terms, quadratic terms

Students will be able to…A.CED.1:I can write equations in one variable and apply them to the real world.I can write inequalities in one variable and apply them to the real world.A.CED.2:I can write/create an equation with 2 or more variables.I can create a coordinate plane using appropriate labels and scales.I can graph an equation on a coordinate plane with 2 or more variables.I can represent/ interpret/ identify relationships between quantities from equations and graphs.A.CED.4:I can rearrange/rewrite formulas to solve for a given variable, using the same steps to solve equations.A.REI.4a:I can solve a quadratic equation by completing the square.I can derive the quadratic formula by completing the square.I can write a quadratic equation as a binomial square.A.REI.4b:I can solve quadratic equations using square roots.I can solve quadratic equations by completing the square.I can solve quadratic equations by factoring.I can solve quadratic equations by inspection.I can identify which method to use to solve a quadratic equation.N.RN.2:I can write expressions using radical notation.I can write expressions using rational exponent notation.A.SSE.2:I can recognize the patterns of special expressions (i.e. difference of perfect squares or sum of perfect cubes)I can rewrite special expressions. A.APR.1:I can add polynomials.I can subtract polynomials.

Mathematical Practices in Focus1. Make sense of problems and persevere in solving them2. Reason abstractly and quantitatively3. Construct viable arguments and critique the reasoning of others4. Model with mathematics

CCSS-MIncluded:8.EE.1, A.CED.1-2, 4, A.REI.4, N.RN.2, and A.SSE.2, A.APR.1

Suggested Pacing: 25 Days

Released Test Question:3, 5, 8, 13, 16, 20, 33

Algebra I Project Binder:Pages 67-88, 99-108

Unit ResourcesLearning Task:Concept Byte: Using Models to FactorPerformance Task:Ch 8 Alg1 textbook pg 67–“Pull-it-all-together” tasks #1, 2, and/or 3Ch 9 Alg1 textbook pg 151 – “Pull-it-all-together” task #1, 2, Project:Unit Review Game:Culminating performance task: NYCDOE Aussie Fir Tree

http://secondarymath.cmswiki.wikispaces.net/Algebra+1+Unit+4+Resources



CORE CONTENTCluster Title: Expressions and Equations: Work with radicals and integer exponents.Standard 8.EE.1: Know and apply the properties of integer exponents to generate equivalent numerical expressions. For

example: 3² x3−5= 3−3 = 13 ³ =

127

Concepts and Skills to Master: Know the properties of integer exponents. Apply the properties of integer exponents to simplify and evaluate numerical expressions.

SUPPORTS FOR TEACHERSCritical Background Knowledge

Understand exponents as repeated multiplication. (6.EE.1) Compute fluently with integers (add, subtract, and multiply).

Academic Vocabularyexponent, base, power, integerSuggested Instructional Strategies:

Use repeated multiplication and division to informally derive the exponent rules.Have students examine equivalent numerical expressions with exponents.

NCDPI Unpacking:Students understand:

Bases must be the same before exponents can be added, subtracted or multiplied. (Example 1)

Exponents are subtracted when like bases are being divided (Example 2)

A number raised to the zero (0) power is equal to one. (Example 3)

Negative exponents occur when there are more factors in the denominator. These exponents can be expressed as a positive if left in the denominator. (Example 4)

Exponents are added when like bases are being multiplied (Example 5)

Exponents are multiplied when an exponents is raised to an exponent (Example 6)

Several properties may be used to simplify an expression (Example 7)

Example 1:

23

52 =

825

Example 2:

Resources:Textbook CorrelationGrowing, Growing, Growing (CMP2)Investigations 5Common Core Investigation (CMP2)Investigation 1.1 Negative Exponent

MARS AssessmentTask (HS): E06: “Ponzi” Pyramid Schemes

Texas Instrument 8.EE.1 Lessons

CMP2 Resources

http://www.phschool.com/webcodes10/index.cfm?fuseaction=home.gotoWebCode&wcprefix=apk&wcsuffix=0099

http://education.ti.com/calculators/downloads/US/Activities/Search/Standards

http://map.mathshell.org/materials/tasks.php?taskid=278#task278



22

26 =

22 6 =

2 4 =

124 =

116

Example 3:60 = 1 Students understand this relationship from examples such

as

62

62 . This expression could be simplified as

3636 = 1.

Using the laws of exponents this expression could also be written as 62–2 = 60. Combining these gives 60 = 1.Example 4:

3 2

24 =

3 2

x

124

=

132

x

124

=

19

x

116

=

1144

Example 5:

(32) (34) = (32+4) = 36= 729

Example 6:

(43)2 = 43x2 = 46 = 4,096

Example 7:

(32)4

(32)(33)

32x4

323

38

35

38 5

33

=

= = = = 27



Sample Assessment Tasks



Skill-based task

Simplify 57

53 ( ( (2)²)²)² −50 2−2 * 4

Problem Task

Explain why 35 * 32 = 37 and not97.

Write three expressions equivalent to 32 * 92.

CORE CONTENT



Cluster Title:.Create equations that describe numbers or relationshipsStandard A.CED.1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.Concepts and Skills to Master:

Create one-variable linear equations and inequalities from contextual situations (stories). Create one-variable exponential equations and inequalities from contextual situations (stories). Solve and interpret the solution to multi-step linear equations and inequalities in context. Use properties of exponents to solve and interpret the solution to exponential equations and inequalities in

context.SUPPORTS FOR TEACHERS

Critical Background Knowledge Understand and use inverse operations to isolate variables and solve equations. Efficiently use order of operations Understand notation for inequalities Understand and use properties of exponents

Academic VocabularyGreater than, less than, at most, at least, =, < ,>, ≤ ,≥ , no more than, no less thanSuggested Instructional Strategies

Convert contextual information into mathematical notation. Use story contexts to create linear and exponential equations

and inequalitiesNCDPI Unpacking:From contextual situations, write equations and inequalities in one variable and use them to solve problems. Include one-variable equations that arise from functions by the selection of a particular target y-value.For example, in the radioactive decay problem below, 25 would be substituted for y in the

equation 𝑦 = 100 ! !,

which results in the one-variable equation 25 = 100 !

. Note, the resulting equation can be

solved in Level I using a table or graph. See A-REI.11.

Ex. The Tindell household contains three people of different generations. The total of the ages of the three family members is 85.a. Find reasonable ages for the three Tindells.b. Find another reasonable set of ages for them.c. One student, in solving this problem, wrote C + (C+20)+ (C+56) = 85What does C represent in this equation? What do you think the student had in mind when using the numbers 20 and 56? What set of ages do you think the student came up with? Ex. A salesperson earns $700 per month plus 20% of sales. Write an equation to find the minimum amount of sales needed to receive a salary of at least $2500 per month.Ex. A scientist has 100 grams of a radioactive substance. Half of it decays every hour. Write an equation to find how long it takes until 25 grams are left.

Resources

Textbook Correlation: 1-8, 2-1, 2-2, 2-3, 2-4, 2-5, 2-7, 2-8, 3-2, 3-3, 3-4, 3-6, 3-7, 7-1, 7-3, 7-4, 7-5, 9-3, 9-4, 9-5, 9-6, 11-5

MARS Apprentice Tasks:FunctionsMultiplying CellsPrinting Tickets

MARS Expert Tasks: Fearless Frames Skeleton Tower Best Buy Tickets


http://map.mathshell.org/materials/tasks.php?taskid=286&subpage=expert



http://map.mathshell.org/materials/tasks.php?taskid=260&subpage=apprentice





Skill-based task

Juan pays $52.35 a month for his cable bill and an additional $1.99 for each streamed movie. If his last cable bill was $68.27, how many movies did Juan watch?

Problem Task

Juan pays $52.35 a month for his cable bill and an additional $1.99 for each streamed movie. Gail pays $40.32 a month for her cable bill and an additional $2.59 for each streamed movie. Who has the better deal? Justify your choice.

CORE CONTENTCluster Title: Create equations that describe numbers or relationshipsStandard A.CED.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axe with labels and scales.



Concepts and Skills to Master: Solve a system of equations exactly (with algebra) and approximately (with graphs) Test a solution to the system in both original equations (both graphically and algebraically) Analyze a system of equations using slope to predict one, infinitely many or no solutions


Graph points Choose appropriate scales and label a graph Understand slope as a rate of change of one quantity in relation to another quantity

Academic VocabularyVariable, dependent variable, independent variable, domain, range, scaleSuggested Instructional Strategies:

Use story contexts to create linear and exponential graphs.

Use technology to explore a variety of linear and exponential graphs.

Use data sets to generate linear and exponential graphs and equations

NCDPI Unpacking:Given a contextual situation, write equations in two variables that represent the relationship that exists between the quantities. Also graph the equation with appropriate labels and scales. Make sure students are exposed to a variety of equations arising from the functions they have studied.Ex. The height of a ball t seconds after it is kicked vertically depends upon the initial height and velocity of the ball and on the downward pull of gravity. Suppose the ball leaves the kicker’s foot at an initial height of 0.7 m with initial upward velocity of 22m/sec. Write an algebraic equation relating flight time t in seconds and height h in meters for this punt.Ex. In a woman’s professional tennis tournament, the money a player wins depends on her finishing place in the standings. The first-place finisher wins half of $1,500,000 in total prize money. The second-place finisher wins half of what is left; then the third-place finisher wins half of that, and so on. Write a rule to calculate the actual prize money in dollars won by the player finishing in nth place, for any positive integer n. Graph the relationship that exists between the first 10 finishers and the prize money in dollars. What pattern do you notice in the graph? What type of relationship exists between the two variables?

Resources: Textbook Correlation: 1-9, 4-5, 5-2, 5-3, 5-4, 5-5,

7-6, 7-7, 9-1, 9-2, 10-5, 11-6, 11-7, CB 11-7 MARS Apprentice Tasks:

FunctionsMultiplying CellsPrinting Tickets

MARS Expert Tasks: Fearless Frames Skeleton Tower Best Buy Tickets










Skill-based task

Write and graph an equation that models the cost of buying and running an air conditioner with a purchase price of $250 which costs $0.38/hr to run.

Problem Task

Jeanette can invest $2000 at 3% interest compounded annually or she can invest $1500 at 3.2% interest compounded annually. Which is the better investment and why?

CORE CONTENTCluster Title: Create equations that describe numbers or relationshipsStandard A.CED.4: Rearrange formulas to highlight a quantity of interest



Concepts and Skills to Master: Extend the concepts used in solving numerical equations to rearranging multi-variable formulas or literal equations

to solve for a specific variable.SUPPORTS FOR TEACHERS

Critical Background Knowledge Recognize variables as representing quantities in context Solve multi-step equations

Academic VocabularyConstant, variable, formula, literal equationSuggested Instructional Strategies:Use formulas for a variety of disciplines such as physics, chemistry, or sports to explore the advantages of different formats of the same formula

NCDPI Unpacking:A.CED.4: Solve multi-variable formulas or literal equations, for a specific variable. Explicitly connect this to the process of solving equations using inverse operations.

Resources: Textbook Correlation: 2-5




Skill-based task

I = Prt Solve for r.

Problem Task

Paul just arrived in England and heard the temperature in

degrees Celsius. He remembers that . How will Paul find the temperature in Fahrenheit?

CORE CONTENTCluster Title: Solve equations and inequalities in one variableStandard A.REI.4:a. Use the method of completing the square where the leading coefficient of x2 is one (ax2 + bx + c where a=1) to

)32(95

FC



transform any quadratic equation in x into an equation of the form (x - p)2 = q that has the same solutions. Derive the quadratic formula from this form. b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.Concepts and Skills to Master:

Use the method of completing the square where the leading coefficient of x2 is one (ax2 + bx + c where a=1) to transform any quadratic equation in x into an equation of the form (x - p)2 = q that has the same solutions.

Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation.


Square Root; Perfect Square Trinomials; Factoring Quadratic Equations; Zeros, Roots, SolutionsAcademic VocabularyCompleting the Square, Square Roots, factoring, perfect-square trinomial, zeros, solutionsSuggested Instructional Strategies:

Use Algebra Tiles to develop the understanding of what is meant by “completing the square”

NCDPI Unpacking:A.REI.4: Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form.

Resources:

Textbook Correlation: 9-5 Using Algebra tiles to complete the square Video of Completing the Square Completing the square lesson plan

Sample Assessment TasksSkill-based task

Solve the following by completing the squarex2 + 6x – 7 = 0

Problem Task

http://www.acoe.org/acoe/files/EdServices/Completing%20the%20Square%20LessonV7PDF.pdf

http://www.youtube.com/watch?v=GyCuj1hx_zc

http://www.regentsprep.org/Regents/math/algtrig/ATE12/completesq.htm



CORE CONTENTCluster Title:Extend the properties of exponents to rational exponentsStandard: N.RN.1: Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example we define 51/3 to be the cube root of 5 because we want (51/3)3 = 51/3*3 to hold so (51/3)3must equal 5Concepts and Skills to Master:

Understand that the properties of integer exponents extend to rational exponents.SUPPORTS FOR TEACHERS

Critical Background KnowledgeProperties of integer exponents



Academic VocabularyExponent, rational, radicalSuggested Instructional Strategies:

Students can investigate by considering patterns such as

(34) = 31 and ( ) = 31 to arrive at a conjecture.

NCDPI Unpacking:Understand that the denominator of the rational exponent is the root index and the numerator is the exponent of the radicand. For example, 51/2 = 5N-RN.1 In order to understand the meaning of rational exponents, students can initially investigate them by considering a pattern such as:

What is the pattern for the exponents? They are reduced by a factor of each time. What is the pattern of thesimplified values? Each successive value is the square root of the previous value. If we continue this pattern, then .Once the meaning of a rational exponent (with a numerator of 1) is established, students can verify that the properties of integer exponents hold for rational exponents as well. For example,since

Ex. Use an example to show why xm/xn = xm-n holds true for expressions involving rational exponents like 1/2or 1/5

Ex: Medical scientists model the amount of active insulin, y, left in the bloodstream after x minutes using the formula (knowing that 10 units of insulin were administered):

What is the amount of active insulin left after 20 seconds?

Ex: Using what you know about properties of exponents, simplify:

Resources:

Textbook Correlation: CC-84 43



Ex: Using the connection between rational exponents and radicals, determine the value of x:




Skill-based task

Using what you know about properties of exponents, simplify:

Using the connection between rational exponents and radicals, determine the value of x:

Problem Task

CORE CONTENTCluster Title: Extend the properties of exponents to rational exponentsStandard N.RN.2: Rewrite expressions involving radicals and rational exponents using the properties of exponents.Concepts and Skills to Master:



Rewrite numbers with rational exponents in radical form. Rewrite numbers in radical form as number with rational exponents.

SUPPORTS FOR TEACHERSCritical Background KnowledgeProperties of integer exponentsAcademic VocabularyExponent, rational, radicalSuggested Instructional Strategies:

NCDPI Unpacking:Students should be able to use the properties of exponents to rewrite expressions involving radicals as expressions using rational exponents. At the Math 1 level, focus on fractional exponents with a numerator of 1.Ex. Simplify the following.Ex. Simplify the following.

,

Resources:

Textbook Correlation: CC-9




Skill-based task

Simplify the followingEx.

Problem Task

Ex. The expression can be written as ( )2. Write these expressions in radical form. How would you confirm that

these forms are equivalent? Considering that = , which form would be easier to simplify without a calculator? Why?

CORE CONTENTCluster Title: Perform Arithmetic Operations on PolynomialsStandard: A.APR.1: Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.Concepts and Skills to Master:



Look for and make use of the structure of addition and subtraction extended to linear and quadratic polynomials Understand the concept of like terms and closure Identify and make use of the structure of special products of linear binomial expressions.


Distributive Property Addition, subtraction and multiplication of integers

Academic VocabularyPolynomial, trinomial, binomial, linear terms, quadratic termsSuggested Instructional Strategies:

Focus on polynomial expressions that simplify to forms that are linear of quadratic in a positive integer power of x.

NCDPI Unpacking:

The Closure Property means that when adding, subtracting or multiplying polynomials, the sum, difference, or product is also a polynomial. Polynomials are not closed under division because in some cases the result is a rational expression.

Resources:

Textbook Correlation: 8-1, 8-2, 8-3, 8-4 MARS Concept Development Lesson: Interpreting

Algebraic Expressions




Skill-based task

A square green rug has a blue square in the center. The side length of the blue square is x inches. The width of the green band that surrounds the square is 6in. What is the area of the green band?

Problem Task

You are given the task of creating a rug to fit into a square room. You must design the rug so that you create a square in the middle. In order for the rug to fit perfectly it must have an area of 49 – 4x2. What is the length of the sides of the outer square and the inner square of the rug?

CORE CONTENTCluster Title: Interpret the Structure of ExpressionsStandard: A.SSE.2: Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2) –(y2)2, thus



recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).Concepts and Skills to Master:

Rewrite an algebraic expression in different forms such as factoring or combining like terms Use factoring techniques such as common factors, grouping, the difference of two squares, the sum or difference

of cubes, or a combination of methods to factor completely. Simplify expressions including combining like terms, using the distributive property and other operations with

polynomials.SUPPORTS FOR TEACHERS

Critical Background KnowledgeUnderstand the meaning of symbols indicating mathematical operations, implied operations (e.g. 2x), the meaning of exponents, and grouping symbols.Academic VocabularyExponents, factors, terms, bases, coefficients, expressionSuggested Instructional Strategies:

Use algeblocks or algebratiles to explore structure of expressions.

NCDPI Unpacking:Students rewrite algebraic expressions by combining like terms or factoring to reveal equivalent forms of the same expression.

Resources:

Textbook Correlation: 5-3, 5-4, 5-5, 8-7, 8-8, CC-12




Skill-based task

Expand the expression 2(x – 1)2 – 4 to show that it is a quadratic expression of the form ax2 + bx + c

Problem Task

algebra 1 unit 3: systems of equationssecondarymath.cmswiki.wikispaces.net/file/view/unit 6...

Documents