Algebra 1 Unit 3: Systems of Equations - …secondarymath.cmswiki.wikispaces.net/file/view/Algebra_1... · Web viewInclude equations arising from linear and quadratic functions, and

Algebra I/Math I Unit 4: Expressions and Equations

BY THE END OF THIS UNIT:

Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. 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.

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. What are the characteristics of a quadratic function? 4. How can you use functions to model real-world situations?

Students will know…

The Quadratic Formula

Rules of exponents – (focus on negative,zero and fractional exponents

Types of Factoring (excluding sum or difference of cubes) Special products

Students will be able to… Operate with polynomials Factor polynomials Solve quadratic equations by factoring Solve quadratic equations by completing the square Solve quadratic equations by using the quadratic formula Extend properties of exponents to rational exponents

Unit ResourcesLearning Task:Concept Byte: Using Models to FactorPerformance Task:Ch 8 of Alg 1 textbook pg 67 – “Pull-it-all-together” tasks #1, 2, and/or 3Ch 9 of Alg 1 textbook pg 151 – “Pull-it-all-together” task #1, 2, Project:Unit Review Game:MathForward:

Culminating performance task: NYCDOE Aussie Fir Tree

Mathematical Practices in Focus:

MP 2 Reason abstractly and quantitatively

MP 7 Look and make use of structure

MP 8 Look for and express regularity in repeated reasoning

Common Core Standards Addressed: A.CED.1, A.CED.2, A.CED.3, A.CED.4, A.SSE.1, A.SSE.2, A.SSE.3, A.APR.1, A.REI.1, A.REI.3, A.REI.4, N.Q.1, N.Q.2, N.Q.3, N.RN.1, N.RN.2

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


CORE CONTENTCluster 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 TEACHERSCritical 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 inequalities

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




Sample Formative Assessment TasksSkill-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 CONTENT



Cluster 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

Write and graph an equation to represent a linear relationship Write and graph an equation to represent an exponential relationship Model a data set using an equation Choose the best form of an equation to model linear and exponential functions


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

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

Sample Formative Assessment TasksStandards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. 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.








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 relationships



Standard A.CED.3: Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional and const constraints on combinations of different foods.Concepts and Skills to Master

Determine whether a point is a solution to an equation or inequality Determine whether a solution has meaning in a real-world context Write and use a system of equations and/or inequalities to solve a real-world problem Recognize that the equations and inequalities represent the constraints of the problem. Use the Objective Equations and

the Corner Principle to determine the solution to the problem. (Linear Programming)


Ability to read and write inequality symbols Ability to graph equations and inequalities on the coordinate plane

Academic VocabularyConstraint, greater than, >, less than, <, greater than or equal to, ≥, less than or equal to, ≤ , inequality, viableSuggested Instructional Strategies

Use a story context such as those found in linear programming problems to write and graph equations with constraints.

Analyze real-world problems connected to student interest

Resources Textbook Correlation: 6-5, 6-6, CC-7 MARS Concept Development Lessons:

Defining Regions using InequalitiesSolving Linear Equations in Two Variables

MARS Problem Solving Lesson: Optimization problems

Sample Formative Assessment Tasks


http://map.mathshell.org/materials/lessons.php?taskid=207&subpage=problem

http://map.mathshell.org/materials/lessons.php?taskid=209&subpage=concept



Skill-based task

Given y ≤ 2x + 1 and y > x – 3 find a point that:

a. Satisfies bothb. Satisfies one, but not the otherc. Satisfies neither

Problem Task

Iced tea costs $1.50 a glass and lemonade costs $2.00. If you have $12, what can you buy? Justify your answer using multiple representation.

CORE CONTENTCluster Title: Create equations that describe numbers or relationshipsStandards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. 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.


Standard A.CED.4: Rearrange formulas to highlight a quantity of interestConcepts 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.


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.

Resources Textbook Correlation: 2-5


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: Interpret the Structure of ExpressionsStandard A.SSE.1: Interpret expressions that represent a quantity in terms of its context.Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. 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.


a. Interpret parts of an expression, such as terms, factors, and coefficientsb. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)n as the product of P

and a factor not depending on P.Concepts and Skills to Master

Given an expression, identify the terms, bases, exponents, coefficients, and factors. Determine the real world context of the variables in an expression. Identify the individual factors of a given term within an expression. Explain the context of different parts of a formula.


Understand 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 Resources

Given a word problem and a formula have students examine the structure and explain the context of different parts of the formula.

Design a game around identifying terms, bases, exponents, coefficients, and factors.

Create formulas based on context

Textbook Correlation: 1-1, 1-2, 1-7, 3-7, 4-5, 5-3, 5-4, 7-7, 8-5, 8-6, 8-7, 8-8, 9-5, CC-2, CC-10

MARS Concept Development Lesson:Sorting Equations and Identities

Sample Formative Assessment TasksSkill-based taskConsider the formula Surface Area = 2B + Ph

a. What are the terms of this formula?b. What are the coefficients?

Problem Task

Interpret the expression: 5 – 3(x – y)2. Explain the output values possible.

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)2 – (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.


Clear and concrete understanding of the distributive property and factoring polynomialsAcademic Vocabulary

Suggested Instructional Strategies Resources Use algeblocks or algebra tiles to explore structure

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

Sample Formative Assessment TasksSkill-based task: Interior DesignA square rug has an area of 49x2 -56x + 16. A second square rug has an area of 16x2 + 24x + 9. What is an expression that represents the difference of the areas of the rugs. Show two different ways to find the solution.

Problem Task:

CORE CONTENTCluster Title: Write expressions in equivalent forms to solve problemsStandard A.SSE.3: Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity



represented by the expression.a. Factor a quadratic expression to reveal the zeros of the function it definesb. Complete the square in a quadratic expression to reveal the maximum or minimum values of the function it defines.c. Use the properties of exponents to transform expressions for exponential functions.

Concepts and Skills to Master Given a quadratic function explain the meaning of the zeros of the function. e.g. if f(x) = (x – c)(x – a), then f(a) = 0 and f(c) = 0 Given a quadratic expression, explain the meaning of the zeros graphically. e.g. for an expression (x – c)(x – a), a and c correspond

to the x-intercepts (if a and c are real numbers). Write the vertex form of a quadratic expression by completing the square. Use the vertex form to find the maximum or minimum of a quadratic function and explain the meaning of the vertex.

SUPPORTS FOR TEACHERSCritical Background KnowledgeStudents should have a good understanding of simple factoring techniques to this point. Students should also understand about the GCF and how to identify or determine the GCF.Academic Vocabulary

Suggested Instructional Strategies: Teachers should be familiar with most all aspects of trinomial

factoring. Teachers should be particularly well versed in factoring using the GCF (greatest common factor). A brief review of factoring with algebra tiles and GCF would greatly help the teacher with this lesson.

Use algeblocks or algebra tiles to explore the meaning of “completing the square” and discover the algorithm

Resources: Textbook Correlation: 7-7, CC-15

Sample Formative Assessment TasksSkill-based task Problem Task

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 Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. 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.


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: Resources

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

Interpreting Algebraic Expressions


CORE CONTENTCluster Title: Reason quantitatively and use units to solve problemsStandard N.Q.1: Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. 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.



Concepts and Skills to Master Select and use appropriate units of measurement for problems with and without context Given a graph draw conclusions and make inferences Choose appropriate scales to create linear and exponential graphs Determine from the labels on a graph what the units of the rate of change are (e.g. gallons per hour)


Know how various attributes are reasonably measuredAcademic VocabularyScale, units of measurementSuggested Instructional Strategies Resources

Explore a variety of examples of measurements used in graphs

Construct graphs using a variety of data sets

Textbook Correlation: 2-5, CB 2-5, 2-6, 2-7, 4-4, 5-7, 12-2, 12-4

Sample Formative Assessment TasksSkill-based taskWhat is the area of strip of wall that is 48 inches by 10 yards?

Problem TaskYour college savings fund has $1800 in it and you plan to spend $30 a week. What would be an appropriate viewing window and scale to see the remaining balance each week until the money is gone? Explain.

CORE CONTENTCluster Title: Reason quantitatively and use units to solve problemsStandard N.Q.2: Define appropriate quantities for the purpose of descriptive modelingConcepts and Skills to Master



Choose appropriate measures and units for problem situations Create a relationship among different units (i.e. feet per second, bacteria per hour, miles per gallon)


Compute unit rates associated with ratios of fractions Recognize and calculate basic conversions (e.g. 3 feet = 1 yard)

Academic VocabularyUnits rates, modeling, quantity, unit conversion, proportion, ratioSuggested Instructional Strategies Resources

Integrate this objective into problem solving throughout the curriculum.

Place an emphasis on relationships between two different units (e.g. dollars per hour, pressure over altitude, calories per gram)

Textbook Correlation: 2-6, 3-3, 4-5, 5-2, 5-5, 12-3, CC-7

Sample Formative Assessment TasksSkill-based taskHow would you measure the rate at which a bathtub fills? Justify your answer.

Problem Task(re-)Create a scenario you have encountered involving two changing quantities and determine appropriate units to describe the relationship between the quantities.

CORE CONTENTCluster Title: Reason quantitatively and use units to solve problemsStandard N.Q.3: Choose a level of accuracy appropriate to limitations on measurement when reporting quantitiesConcepts and Skills to Master

Determine whether whole numbers, fractions, or decimals are most appropriateStandards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. 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.


Determine the appropriate power of ten to reasonably measure a quantity Determine the resulting accuracy in calculations Determine what level of rounding should be used in a problem situation


Understand basic units of measurements and their relationship to one another (e.g. a foot is smaller than a yard) Understand how to properly round and estimate

Academic VocabularyPrecision, accuracySuggested Instructional Strategies Resources

Discuss misconceptions in resulting calculations involving measurement, e.g. you cannot increase accuracy through calculation, only through more accurate measurement

Compare the difference between rounding at different places in a calculation and discuss which yields the best result

Discuss how mathematicians maintain precision through the representations that they use.

Textbook Correlation: 2-10, 9-5, 9-6, CC-7

MARS Apprentice Task (N.Q.1 through 3):Leaky Faucet

MARS Apprentice Task (N.Q.1 through 3): Yogurt

Sample Formative Assessment TasksSkill-based task Problem TaskCORE 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 5





Concepts and Skills to Master Understand that the properties of integer exponents extend to rational exponents.


Properties of integer exponentsAcademic VocabularyExponent, rational, radicalSuggested Instructional Strategies

Students can investigate by considering patterns such as (34) = 31 and ( ) = 31 to arrive at a conjecture.

Resources Textbook Correlation: CC-8


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.


Properties of integer exponentsAcademic VocabularyExponent, rational, radicalSuggested Instructional Strategies

Resources

Textbook Correlation: CC-9


CORE CONTENTCluster Title: Understand solving equations as a process of reasoning and explain the reasoningStandard A.REI.1: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.



Concepts and Skills to Master Understand, apply, and explain the results of using inverse operations Justify the steps in solving equations by applying and explaining the properties of equality, inverse and identity.

Justifications may include the associative, commutative, and division properties, combining like terms, multiplication by 1, etc.

Use the names of the properties and common sense explanations to explain the steps in solving an equation


Use order of operations Simplify expressions using properties of algebra

Academic VocabularyConstant, coefficient, properties of operations and properties of equality, like terms, variable, evaluate, justify, viableSuggested Instructional Strategies

Have students share different ways of solving equations that lead to the same answer.

Find and analyze mistakes in student work samples Partner problems: one student solves, the other

writes reasons why steps work. Introduce a two-column proof as a way of

organizing justifications

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


Justify the equation solution by writing the property or reason why

Problem Task

When Sally picks any number between 1 and 20, doubles it, adds Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. 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.


each step works.3x + 7 = 123x + 7 – 7 = 12 – 73x + 0 = 53x = 5(3x)(1/3) = (5)(1/3)1x = 5/3x = 5/3

6, divides by 2 and subtracts 3, she always gets the number she started with. Why? Evaluate and use algebraic evidence to support your conclusion.

CORE CONTENTCluster Title: Solve equations and inequalities in one variableStandard A.REI.3: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.Concepts and Skills to Master

Write equations in equivalent forms to solve problems.Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. 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.


Analyze and solve literal equations for a specified variable. Understand and apply the properties of inequalities. Verify that a given number or variable is a solution to the equation or inequality. Interpret the solution of an inequality in real terms.


Solving linear equationsAcademic VocabularyProperties of inequality as interpreted in table 5 of the CCSS glossary page 90Suggested Instructional Strategies

Solve specified variables, using common formulas used in science, economics, or other disciplines

Examine and prove why dividing or multiplying by a negative reverses the inequality sign.

Use applications from a variety of disciplines to motivate solving linear equations and inequalities

Resources Textbook Correlation: 2-1, 2-2, 2-3, 2-4, 2-5, 2-7, 2-8, 3-2,

3-3, 3-4, 3-5, 3-6


Solve 2(x + 4) – 3 ≥ 4x – 2

Problem Task

The perimeter of a rectangle is given by P = 2W + 2L. Solve for W and restate in words the meaning of this new formula in terms of the meaning of the other variables.



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 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, Solutions

Academic VocabularyCompleting the Square, Square Roots, factoring, perfect-square trinomial, zeros, solutionsSuggested Instructional StrategiesUse Algebra Tiles to develop the understanding of what is meant by “completing the square”

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

Sample Formative Assessment TasksSkill-based task:Solve the following by completing the square

x2 + 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



Documents

Algebra 1 Unit 3: Systems of Equations - …secondarymath.cmswiki.wikispaces.net/file/view/Algebra_1... · Web viewInclude equations arising from linear and quadratic functions, and